diff --git "a/raw_rss_feeds/https___arxiv_org_rss_math.xml" "b/raw_rss_feeds/https___arxiv_org_rss_math.xml" --- "a/raw_rss_feeds/https___arxiv_org_rss_math.xml" +++ "b/raw_rss_feeds/https___arxiv_org_rss_math.xml" @@ -7,5740 +7,12 @@ http://www.rssboard.org/rss-specification en-us - Fri, 16 Jan 2026 05:00:13 +0000 + Sun, 18 Jan 2026 05:00:03 +0000 rss-help@arxiv.org - Fri, 16 Jan 2026 00:00:00 -0500 + Sun, 18 Jan 2026 00:00:00 -0500 - Saturday Sunday + Saturday - - Graphical C(3)-T(6) implies CAT(0) - https://arxiv.org/abs/2601.09751 - arXiv:2601.09751v1 Announce Type: new -Abstract: Graphical small cancellation extends the classical small cancellation theory and provides a powerful method for constructing groups with interesting features. In the classical setting, C(3)-T(6) small cancellation complexes are known to admit locally CAT(0) metrics. In this paper, we construct locally CAT(0) metrics for graphical C(3)-T(6) complexes. - oai:arXiv.org:2601.09751v1 - math.GR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Huaitao Gui - - - Structure and Decomposition of Deltoids in Abelian Groups - https://arxiv.org/abs/2601.09774 - arXiv:2601.09774v1 Announce Type: new -Abstract: Deltoids provide a natural framework for studying defective (partial) matchings in abelian groups, and we develop both structure and existence results in this setting. Given finite subsets $A$ and $B$ of an abelian group $G$, a matching is a bijection $f:A\to B$ such that $af(a)\notin A$ for all $a\in A$, a definition motivated by the study of canonical forms for symmetric tensors. We provide necessary and sufficient conditions for the existence of a partial matching with any prescribed defect, and then describe the minimal unavoidable defect for a pair $(A,B)$. We also define and examine a defective version of Chowla sets in the matching context. We prove a structure theorem identifying obstructions to the existence of partial matchings with small defect. Finally, within the deltoid setup, we establish max-min results on the partitioning of $A$ and $B$ into left- and right-admissible sets. Our tools mix results from transversal theory with ideas from additive number theory. - oai:arXiv.org:2601.09774v1 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mohsen Aliabadi, Jozsef Losonczy - - - On some Exotic Cylindrical Algebraic Decompositions and Cells - https://arxiv.org/abs/2601.09795 - arXiv:2601.09795v1 Announce Type: new -Abstract: Cylindrical Algebraic Decompositions (CADs) endowed with additional topological properties have found applications beyond their original logical setting, including algorithmic optimizations in CAD construction, robot motion planning, and the algorithmic study of the topology of semi-algebraic sets. In this paper, we construct explicit examples of CADs and CAD cells that refute several conjectures and open questions of J. H. Davenport, A. Locatelli, and G. K. Sankaran concerning these topological assumptions. - oai:arXiv.org:2601.09795v1 - math.AG - cs.SC - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Lucas Michel - - - Witt affine Springer theory - https://arxiv.org/abs/2601.09798 - arXiv:2601.09798v1 Announce Type: new -Abstract: This paper extends the affine Springer theory developed by Bouthier, Kazhdan, and the second author (see [BKV]) to the mixed characteristic case. In particular, we introduce a theory of perfectly placid perfect infinity stacks and establish their dimension theory. Furthermore, we prove that, in the Witt vector setting, the Chevalley morphism between arc spaces is flat. - oai:arXiv.org:2601.09798v1 - math.RT - math.AG - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Noam Nissan, Yakov Varshavsky - - - Spectral projections of an anharmonic oscillator with complex polynomial potential - https://arxiv.org/abs/2601.09800 - arXiv:2601.09800v1 Announce Type: new -Abstract: For a broad class of polynomial potentials $V$, with an important and instructive representative being $V(x) = x^{2a} + i x^b$, $x \in \mathbb R$, $a, b \in \mathbb N$, we show that the system of spectral projections $\{P_n\}_n$ of an anharmonic operator $L = - (\mathrm{d}/ \mathrm{d}x)^2 + V(x)$ does not generate a (Riesz) basis in $L^2(\mathbb R)$ if $a - 1 < b < 2a$. Moreover, for $\sigma = [b - (a - 1)]/(1 + a)$ and $\gamma > 0$ small enough, $\limsup_n \|P_n\|/ \exp(\gamma n^\sigma) = \infty$. Proofs are based on two groups of results which are of great interest on their own: (a) relationship between behavior (growth) of the norms of projections $\|P_n\|$ and of the resolvent $\|(z - L)^{-1}\|$ outside of the spectrum $\sigma(L)$; (b) partial fraction decompositions of special meromorphic functions $1/F$ where $F(w) = \prod_{k=1}^\infty \left( 1 + \frac{w}{a_k} \right)$, $a_{k+1} \geq a_k>0$, $k \in \mathbb N$, and the generalization of the first resolvent identity. - oai:arXiv.org:2601.09800v1 - math.SP - math-ph - math.FA - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Boris Mityagin, Petr Siegl - - - Learning Ecological and Epidemic Processes using Neural ODEs, Kolmogorov-Arnold Network ODEs and SINDy - https://arxiv.org/abs/2601.09811 - arXiv:2601.09811v1 Announce Type: new -Abstract: We consider epidemic and ecological models to investigate their coupled dynamics. Starting with the classical Susceptible-Infected-Recovered (SIR) model for basic epidemic behavior and the predator-prey (Lotka-Volterra, LV) system for ecological interactions, we then combine these frameworks into a coupled Lotka-Volterra-Susceptible-Infected-Susceptible (LVSIS) model. The resulting system consists of four differential equations describing the evolution of susceptible and infected prey and predator populations, incorporating ecological interactions, disease transmission, and spatial dispersal. To learn the underlying dynamics directly from data, we employ several data-driven modeling frameworks: Neural Ordinary Differential Equations (Neural ODEs), Kolmogorov-Arnold Network Ordinary Differential Equations (KANODEs), and Sparse Identification of Nonlinear Dynamics (SINDy). Numerical experiments based on synthetic data are conducted to investigate the learning ability of these models in capturing the epidemic and ecological behavior. We further extend our approach to spatio-temporal models, aiming to uncover hidden local couplings. - oai:arXiv.org:2601.09811v1 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Maria Vasilyeva, Zheng Wei, Kelum Gajamannage, Hyangim Ji, Aleksei Krasnikov, Alexey Sadovski - - - Shallow-KAN Based Solution of Moving Boundary PDEs - https://arxiv.org/abs/2601.09818 - arXiv:2601.09818v1 Announce Type: new -Abstract: Kolmogorov-Arnold Networks (KANs) require significantly smaller architectures compared to multilayer perceptron (MLP)-based approaches, while retaining expressive power through spline-based activations. We propose a shallow KAN framework that directly approximates the temperature distribution T(x,t) and the moving interface $\Gamma(t)$, enforcing the governing PDEs, phase equilibrium, and Stefan condition through physics-informed residuals. To enhance accuracy, we employ interface-focused collocation resampling. Numerical experiments in one and two dimensions show that the framework achieves accurate reconstructions of both temperature fields and interface dynamics, highlighting the potential of KANs as a compact and efficient alternative for moving boundary PDEs. First, we validate the model with semi-infinite analytical solutions. Subsequently, the model is extended to 2D using a level-set based formulation for interface propagation, which is solved within the KAN framework. This work demonstrates that KANs are capable of solving complex moving boundary problems without the need for measurement data. - oai:arXiv.org:2601.09818v1 - math-ph - cs.NA - math.MP - math.NA - physics.comp-ph - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Tarus Pande, V M S K Minnikanti, Shyamprasad Karagadde - - - Kostant cuspidal permutations - https://arxiv.org/abs/2601.09824 - arXiv:2601.09824v1 Announce Type: new -Abstract: In relation to Kostant's problem for simple highest weight modules over the general linear Lie algebra, we prove a persistence result for Kostant negative consecutive patterns. Inspired by it, we introduce the notion of a Kostant cuspidal permutation as a minimal Kostant negative consecutive pattern. It is shown that Kostant cuspidality is an invariant of a Kazhdan-Lusztig left cell. We describe four infinite families of Kostant cuspidal involutions, including a complete classification of Kostant cuspidal fully commutative involutions. In particular, we show that the number of new Kostant cuspidal elements can be arbitrarily large, when the rank grows. This provides some potential explanation why Kostant's problem is hard. - oai:arXiv.org:2601.09824v1 - math.RT - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Samuel Creedon, Volodymyr Mazorchuk - - - When an Approximate Model Suffices for Optimal Control - https://arxiv.org/abs/2601.09826 - arXiv:2601.09826v1 Announce Type: new -Abstract: In this paper, we develop an optimal control framework for dynamical systems when only an approximate model of the underlying plant is available. We consider a setting in which the control strategy is synthesized using a model-based optimal control problem that includes a penalty term capturing deviation from the plant trajectory, while the same control input is applied to both the model and the actual system. For a general class of optimal control problems, we establish conditions under which the control minimizing the model-based Hamiltonian coincides with the plant-optimal control, despite mismatch between the model and the true dynamics. We further specialize these results to problems with quadratic control effort, where explicit and easily verifiable sufficient conditions guarantee equivalence and uniqueness of the resulting optimal control. These results show that accurate control synthesis does not require an exact model of the underlying system, but rather alignment of the optimality conditions that govern control selection. From a learning perspective, this suggests that data-driven efforts can focus on identifying regimes in which model-based and plant-based Hamiltonian minimizers coincide, thereby providing a theoretical basis for robust model-based decision making and the effective use of digital twins under modeling error. We provide examples to illustrate the theoretical findings and demonstrate equivalence of the resulting control trajectories even in the presence of significant model mismatch. - oai:arXiv.org:2601.09826v1 - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Andreas A. Malikopoulos - - - A note on absolutely minimal extensions in finite metric spaces - https://arxiv.org/abs/2601.09840 - arXiv:2601.09840v1 Announce Type: new -Abstract: Absolutely minimal Lipschitz extensions (AMLEs) are known to exist in many infinite metric settings, but the finite case is less settled. In metric spaces with at most four points, every function on a nonempty subset admits an AMLE in the sense that the Lipschitz constant cannot be further reduced on sets that are disjoint from the prescribed domain. We show that in five-point spaces such extensions may fail to exist. - oai:arXiv.org:2601.09840v1 - math.MG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Alberto Dom\'inguez Corella, Tr\'i Minh L\^e - - - High signal-to-noise ratio asymptotics of entropy-constrained Gaussian channel capacity - https://arxiv.org/abs/2601.09864 - arXiv:2601.09864v1 Announce Type: new -Abstract: We study the input-entropy-constrained Gaussian channel capacity problem in the asymptotic high signal-to-noise ratio (SNR) regime. We show that the capacity-achieving distribution as SNR goes to infinity is given by a discrete Gaussian distribution supported on a scaled integer lattice. Further, we show that the gap between the input entropy and the capacity decreases to zero exponentially in SNR, and characterize this exponent. - oai:arXiv.org:2601.09864v1 - cs.IT - math.IT - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Adway Girish, Shlomo Shamai, Emre Telatar - - - Non-commutative Factor theorem for tensor products of lattices in product groups - https://arxiv.org/abs/2601.09875 - arXiv:2601.09875v1 Announce Type: new -Abstract: We establish a non-commutative version of the Intermediate Factor Theorem for crossed products associated with product lattices. Given an irreducible lattice $\Gamma < G= G_1 \times \dots \times G_d$ in higher rank semisimple algebraic groups and a trace-preserving irreducible action $G \curvearrowright (\mathcal{N}, \tau)$, we show that every intermediate von Neumann algebra between $\mathcal{N}\rtimes\Gamma$ and $(L^\infty(G/P,\nu_P)\overline{\otimes}\mathcal{N})\rtimes\Gamma$ is again a crossed product of the form $(L^\infty(G/Q,\nu_Q)\overline{\otimes}\mathcal{N})\rtimes\Gamma$. - oai:arXiv.org:2601.09875v1 - math.OA - math.DS - math.FA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tattwamasi Amrutam, Yongle Jiang, Shuoxing Zhou - - - Asymptotic Stability and Equilibrium Selection in Quasi-Feller Systems with Minimal Moment Conditions - https://arxiv.org/abs/2601.09880 - arXiv:2601.09880v1 Announce Type: new -Abstract: We study equilibrium selection for invariant measures of stochastic dynamical systems with constant step size, under persistent noise and minimal moment assumptions, in a general quasi-Feller framework. Such dynamics arise in projection-based algorithms, learning in games, and systems with discontinuous decision rules, where classical Feller assumptions and small-noise or large-deviation techniques are not applicable. - Under a global Lyapunov condition, we prove that any weak limit of invariant measures must be supported on the set of fixed points of the associated deterministic dynamics. Beyond localization, we establish a sharp exclusion principle for unstable equilibria: strict local maxima and saddle points of the Lyapunov function are shown to carry zero mass in limiting invariant measures under explicit and verifiable non-degeneracy conditions. - Our analysis identifies a local mechanism driven by Lyapunov geometry and persistent variance, showing that equilibrium selection in constant-step dynamics is governed by typical fluctuations rather than rare events. These results provide a probabilistic foundation for stability and equilibrium selection in stochastic systems with persistent noise and weak regularity. - oai:arXiv.org:2601.09880v1 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jean-Gabriel Attali - - - An efficient probabilistic scheme for the exit time probability of $\alpha$-stable L\'evy process - https://arxiv.org/abs/2601.09882 - arXiv:2601.09882v1 Announce Type: new -Abstract: The {\alpha}-stable L\'evy process, commonly used to describe L\'evy flight, is characterized by discontinuous jumps and is widely used to model anomalous transport phenomena. In this study, we investigate the associated exit problem and propose a method to compute the exit time probability, which quantifies the likelihood that a trajectory starting from an initial condition exits a bounded region in phase space within a given time. This estimation plays a key role in understanding anomalous diffusion behavior. The proposed method approximates the {\alpha}-stable process by combining a Brownian motion with a compound Poisson process. The exit time probability is then modeled using a framework based on partial integro-differential equations (PIDEs). The Feynman-Kac formula provides a probabilistic representation of the solution, involving conditional expectations over stochastic differential equations. These expectations are computed via tailored quadrature rules and interpolation techniques. The proposed method achieves first-order convergence in time and offers significant computational advantages over standard Monte Carlo and deterministic approaches. In particular, it avoids assembling and solving large dense linear systems, resulting in improved efficiency. We demonstrate the method's accuracy and performance through two numerical examples, highlighting its applicability to physical transport problems. - oai:arXiv.org:2601.09882v1 - math.NA - cs.NA - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Minglei Yang, Diego del-Castillo-Negrete, Guannan Zhang - - - Multiplicity one for equivariant min-max theory in prescribed homology classes - https://arxiv.org/abs/2601.09884 - arXiv:2601.09884v1 Announce Type: new -Abstract: For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show a generic multiplicity one theorem in equivariant min-max theory, and show in generic sense that there are infinitely many $G$-invariant minimal hypersurfaces in a fixed $G$-homology class. We also establish an equivariant min-max theory for $G$-invariant hypersurfaces of prescribed mean curvature with $G$-index upper bounds. - oai:arXiv.org:2601.09884v1 - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tongrui Wang - - - Graphs of Quasicircles and Quasiconformal Homeomorphisms - https://arxiv.org/abs/2601.09892 - arXiv:2601.09892v1 Announce Type: new -Abstract: We give a combinatorial characterization of the group of quasiconformal homeomorphisms of a closed, oriented surface $S$ of genus at least $2$. In particular, we prove they are exactly the automorphisms of a graph of essential quasicircles on $S$ that respect a canonical coarse ordering induced by quality constants. We also discuss the coarse geometry of this graph. - oai:arXiv.org:2601.09892v1 - math.GT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Katherine Williams Booth, Alexander Nolte, Yvon Verberne - - - Complex Monge-Amp\`ere equation in Orlicz space and Diameter Bound - https://arxiv.org/abs/2601.09893 - arXiv:2601.09893v1 Announce Type: new -Abstract: In this paper, we establish diameter bounds for compact K\"ahler manifolds equipped with K\"ahler metrics $\omega$, assuming the associated measure lies in a specific Orlicz space and satisfies an integrability condition. Firstly, we prove a priori estimates for solutions of the complex Monge-Amp\`ere equation in Orlicz spaces, encompassing $L^{\infty}$ and stability estimates. This is achieved by employing Ko{\l}odziej's approach \cite{Ko98} and the argument of Guo-Phong-Tong-Wang \cite{GuPhToWa21}, respectively. Secondly, building on the work of Guo-Phong-Song-Sturm \cite{GuPhSoSt24-1}, we derive the uniform (local/global) estimates of the Green's function and its gradient for the associated K\"ahler metric $\omega$. - oai:arXiv.org:2601.09893v1 - math.DG - math.CV - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lei Zhang, Zhenlei Zhang - - - One-Cold Poisson Channel: A Simple Continuous-Time Channel with Zero Dispersion - https://arxiv.org/abs/2601.09894 - arXiv:2601.09894v1 Announce Type: new -Abstract: We introduce the one-cold Poisson channel (OCPC), where the transmitter chooses one of several frequency bands to attenuate at a time. In particular, the perfect OCPC, where the number of bands is unlimited, is an extremely simple continuous-time memoryless channel. It has a capacity 1, zero channel dispersion, and an information spectrum being the degenerate distribution at 1. It is the only known nontrivial (discrete or continuous-time) memoryless channel with a closed-form formula for its optimal non-asymptotic error probability, making it the simplest channel in this sense. A potential application is optical communication with a tunable band rejection filter. Due to its simplicity, we may use it as a basic currency of information that is infinitely divisible, as an alternative to bits which are not infinitely divisible. OCPC with perfect feedback gives a generalization of prefix codes. We also study non-asymptotic coding and channel simulation results for the general OCPC. - oai:arXiv.org:2601.09894v1 - cs.IT - eess.SP - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Cheuk Ting Li - - - Lossless Strichartz estimates on the square torus over short time intervals - https://arxiv.org/abs/2601.09895 - arXiv:2601.09895v1 Announce Type: new -Abstract: We prove lossless Strichartz estimates at the critical exponent $q_c = \frac{2(n+1)}{n-1}$ on the square torus for the Schr\"{o}dinger equation with frequency localized initial data on small time windows with length depending on the frequency parameter $\lambda \gg 1$. - oai:arXiv.org:2601.09895v1 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Connor Quinn - - - Birman-Hilden theory for big mapping class groups - https://arxiv.org/abs/2601.09897 - arXiv:2601.09897v1 Announce Type: new -Abstract: Let $S$ and $X$ be two connected topological surfaces without boundary, and assume that $S$ is either of infinite type or has negative Euler characteristic. In this paper, we prove that if $p:S\rightarrow X$ is a fully ramified branched covering map, then $p$ satisfies the Birman-Hilden property. This generalizes a theorem of Winarski, and the known results in the literature, to the context of surfaces of infinite type and branched covering maps of infinite degree. As an application, we show that the mapping class group (respectively, the braid group on $k$-strands) of a non-orientable surface of infinite type can be realized as a subgroup of the mapping class group (respectively, the braid group on $2k$-strands) of its orientable double cover. - oai:arXiv.org:2601.09897v1 - math.GT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Nestor Colin, Ruben Hidalgo, Rita Jim\'enez Rolland, Israel Morales, Sa\'ul Quispe - - - Nonlinear numerical schemes using specular differentiation for initial value problems of first-order ordinary differential equations - https://arxiv.org/abs/2601.09900 - arXiv:2601.09900v1 Announce Type: new -Abstract: This paper proposes specular differentiation in one-dimensional Euclidean space and provides its fundamental analysis, including quasi-Fermat's theorem and the quasi-Mean Value Theorem. As an application, this paper develops several numerical schemes for solving initial value problems for first-order ordinary differential equations. Based on numerical simulations, we select one scheme and prove its first-order consistency and second-order local convergence. - oai:arXiv.org:2601.09900v1 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kiyuob Jung - - - The Morse Local-to-Global Property for Graph Products - https://arxiv.org/abs/2601.09901 - arXiv:2601.09901v1 Announce Type: new -Abstract: The Morse local-to-global property generalizes the local-to-global property for quasi-geodesics in a hyperbolic space. We show that graph products of infinite Morse local-to-global groups have the Morse local-to-global property. To achieve this, we generalize the maximization procedure of Abbott, Behrstock, and Durham for relatively hierarchically hyperbolic groups with clean containers. Under mild conditions satisfied by graph products, we show that stable embeddings into a relatively hierarchically hyperbolic space are exactly those which are quasi-isometrically embedded in the top level hyperbolic space by the orbit map. This shows that graph products of any infinite groups with no isolated vertices are Morse detectable. - oai:arXiv.org:2601.09901v1 - math.GT - math.GR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Joshua Perlmutter - - - Distortion maps for elliptic curves over finite fields - https://arxiv.org/abs/2601.09904 - arXiv:2601.09904v1 Announce Type: new -Abstract: The Weil pairing on elliptic curves has deep links with discrete logarithm problems. In practice, to better suit the functionalities of cryptosystems, one often needs to modify the original Weil pairing via what is called a distortion map. We propose a study on the question of the existence of distortion maps for elliptic curves over finite fields. We revisit results from the literature and provide detailed proofs. We also propose new perspectives at times. - oai:arXiv.org:2601.09904v1 - math.NT - cs.CR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Nikita Andrusov, Sevag B\"uy\"uksimke\c{s}yan, Dimitrios Noulas, Fabien Pazuki, Mustafa Umut Kazanc{\i}o\u{g}lu, Jordi Vil\`a-Casadevall - - - A note on invariants of mixed-state topological order in 2D - https://arxiv.org/abs/2601.09909 - arXiv:2601.09909v1 Announce Type: new -Abstract: The classification of mixed-state topological order requires indices that behave monotonically under finite-depth quantum channels. In two dimensions, a braided $C^*$-tensor category, which corresponds to strong symmetry, arises from a state satisfying approximate Haag duality. In this note, we show that the $S$-matrix and topological twists of the braided $C^*$-tensor category are quantities that are monotone under finite-depth quantum channels. - oai:arXiv.org:2601.09909v1 - math-ph - cond-mat.stat-mech - cond-mat.str-el - math.MP - quant-ph - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Yoshiko Ogata - - - Cylinder type and $p$-divisible sets in $\mathbb{F}_p^3$ - https://arxiv.org/abs/2601.09910 - arXiv:2601.09910v1 Announce Type: new -Abstract: A set of points $S \subseteq \mathbb{F}_p^n$ is called \emph{$p$-divisible} if every affine hyperplane in $\mathbb{F}_p^n$ intersects $S$ in $0 \pmod p$ points. The Strong Cylinder Conjecture of Ball asserts that if - $S$ is a $p$-divisible set of $p^2$ points in $\mathbb{F}_p^3$, then $S$ is a cylinder. In this paper, we show that every $p$-divisible multiset $S$ is both a $\mathbb{F}_p$-linear and $\mathbb{Z}$-linear combination of characteristic functions of cylinders. In addition, the multisets of size $p^2$ are $\Z$-linear combinations of a plane and weighted differences of parallel lines. - oai:arXiv.org:2601.09910v1 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Gergely Kiss, \'Ad\'am Mark\'o, Zolt\'an L\'or\'ant Nagy, G\'abor Somlai - - - Learning-Augmented Perfectly Secure Collaborative Matrix Multiplication - https://arxiv.org/abs/2601.09916 - arXiv:2601.09916v1 Announce Type: new -Abstract: This paper presents a perfectly secure matrix multiplication (PSMM) protocol for multiparty computation (MPC) of $\mathrm{A}^{\top}\mathrm{B}$ over finite fields. The proposed scheme guarantees correctness and information-theoretic privacy against threshold-bounded, semi-honest colluding agents, under explicit local storage constraints. Our scheme encodes submatrices as evaluations of sparse masking polynomials and combines coefficient alignment with Beaver-style randomness to ensure perfect secrecy. We demonstrate that any colluding set of parties below the security threshold observes uniformly random shares, and that the recovery threshold is optimal, matching existing information-theoretic limits. Building on this framework, we introduce a learning-augmented extension that integrates tensor-decomposition-based local block multiplication, capturing both classical and learned low-rank methods. We demonstrate that the proposed learning-based PSMM preserves privacy and recovery guarantees for MPC, while providing scalable computational efficiency gains (up to $80\%$) as the matrix dimensions grow. - oai:arXiv.org:2601.09916v1 - cs.IT - cs.MA - cs.SY - eess.SY - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Zixuan He, Mohammad Reza Deylam Salehi, Derya Malak, Photios A. Stavrou - - - On the Dirichlet boundary value problem on Cartan-Hadamard manifolds - https://arxiv.org/abs/2601.09930 - arXiv:2601.09930v1 Announce Type: new -Abstract: In this paper, we investigate the Dirichlet boundary value problem on Cartan-Hadamard manifolds, focusing on the non-existence of bounded (viscosity) solutions to semi-linear elliptic equations of the form $\Delta u + f(u) = 0$ in domains with prescribed asymptotic boundary, extending previous results by Bonorino and Klaser originally established for hyperbolic spaces. Using a novel comparison technique based on convex hypersurfaces inspired by Choi, G\'alvez, and Lozano, we overcome the absence of totally geodesic foliations, which are instrumental in the hyperbolic space. Our results highlight the interplay between curvature, the spectrum of the Laplacian, and the geometry of the asymptotic boundary. - oai:arXiv.org:2601.09930v1 - math.AP - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Marcos P. Cavalcante, Jos\'e M. Espinar, Diego A. Mar\'in - - - Algebras of distributions suitable for phase-space quantum mechanics. II. Topologies on the Moyal algebra - https://arxiv.org/abs/2601.09934 - arXiv:2601.09934v1 Announce Type: new -Abstract: The topology of the Moyal $*$-algebra may be defined in three ways: the algebra may be regarded as an operator algebra over the space of smooth declining functions either on the configuration space or on the phase space itself; or one may construct the $*$-algebra via a filtration of Hilbert spaces (or other Banach spaces) of distributions. We prove the equivalence of the three topologies thereby obtained. As a consequence, by filtrating the space of tempered distributions by Banach subspaces, we give new sufficient conditions for a phase-space function to correspond to a trace-class operator via the Weyl correspondence rule. - oai:arXiv.org:2601.09934v1 - math.OA - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1063/1.527984 - J. Math. Phys. 29 (1988), 880-887 - Joseph C. V\'arilly, Jos\'e M. Gracia-Bond\'ia - - - Epstein surfaces for $G-$opers - https://arxiv.org/abs/2601.09936 - arXiv:2601.09936v1 Announce Type: new -Abstract: Given a complex semisimple Lie group $G$, we introduce the notion of an Epstein surface associated to a $G$-oper. These surfaces generalize Epstein's classical construction for $G=PGL_2 (\mathbb{C})$. As an application, we provide a criterion that ensures that the holonomy of the oper is $\Delta-$Anosov. Finally, we discuss how the developing map of the oper interacts with domains of discontinuity of the holonomy (whenever Anosov) and the transversality properties it satisfies. Along the way, we provide a quick review of opers that we hope serves as a self-contained introduction. - oai:arXiv.org:2601.09936v1 - math.DG - math.GT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Joaqu\'in Lema - - - On Schur Rings Over Semigroups - https://arxiv.org/abs/2601.09940 - arXiv:2601.09940v1 Announce Type: new -Abstract: We generalize the idea of a Schur ring of a group to the category of semigroups. Fundamental results of Schur rings over groups are shown to be true for Schur rings over semigroups. Examples where Schur rings differ between the two categories are provided. We prove some results for Schur rings over specific families of semigroups. We consider parallels between semigroup extensions and their Schur rings. We fully enumerate the Schur rings for all semigroups of orders 0-7, and some statistical analysis is performed. - oai:arXiv.org:2601.09940v1 - math.GR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Joseph E. Marrow, Andrew Misseldine - - - Reconstructing Reed-Solomon Codes from Multiple Noisy Channel Outputs - https://arxiv.org/abs/2601.09947 - arXiv:2601.09947v1 Announce Type: new -Abstract: The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication setting in which a sender transmits a codeword and the receiver observes K independent noisy versions of this codeword. In this work, we study the problem of efficient reconstruction when each of the $K$ outputs is corrupted by a $q$-ary discrete memoryless symmetric (DMS) substitution channel with substitution probability $p$. Focusing on Reed-Solomon (RS) codes, we adapt the Koetter-Vardy soft-decision decoding algorithm to obtain an efficient reconstruction algorithm. For sufficiently large blocklength and alphabet size, we derive an explicit rate threshold, depending only on $(p, K)$, such that the transmitted codeword can be reconstructed with arbitrarily small probability of error whenever the code rate $R$ lies below this threshold. - oai:arXiv.org:2601.09947v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Shubhransh Singhvi, Han Mao Kiah, Eitan Yaakobi - - - Quantitative Supercritical Bounds for Disconnection in Bernoulli Site Percolation - https://arxiv.org/abs/2601.09950 - arXiv:2601.09950v1 Announce Type: new -Abstract: For any infinite, connected, locally finite graph $G=(V,E)$, any parameter $p>p^{\mathrm{site}}_{c}(G)$, and any (finite or infinite) set of vertices $S\subset V$, we derive explicit exponential-type upper bounds on the disconnection probability $\mathbb{P}_{p}(S\nleftrightarrow\infty)$. The estimates are expressed in terms of a packing profile of $S$, encoded by a $(p,\varepsilon,c)$--packing number, which counts how many well-separated vertices in $S$ exhibit controlled local-to-global connectivity. The proof combines a local functional characterization of $p^{\mathrm{site}}_{c}$ from \cite{ZL24,ZL26} with a packing construction and an amplification-by-independence argument, in the direction of Problem~1.6 in \cite{DC20}. - oai:arXiv.org:2601.09950v1 - math.PR - math-ph - math.CO - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Zhongyang Li - - - Directed strongly regular graphs and divisible design graphs from Tatra association schemes - https://arxiv.org/abs/2601.09955 - arXiv:2601.09955v1 Announce Type: new -Abstract: In this paper, we construct directed strongly regular graphs and divisible design graphs with new parameters merging some basic relations of so-called Tatra associations schemes. We also study the above association schemes, their fusions and isomorphisms. - oai:arXiv.org:2601.09955v1 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mikhail Muzychuk, Grigory Ryabov - - - The Galois Structure of the Spaces of polydifferentials on the Drinfeld Curve - https://arxiv.org/abs/2601.09956 - arXiv:2601.09956v1 Announce Type: new -Abstract: Let $C$ be a smooth projective curve over an algebraically closed field ${\mathbb{F}}$ equipped with the action of a finite group $G$. When $p =\textrm{char}(\mathbb{F})$ divides the order of $G$, the long-standing problem of computing the induced representation of $G$ on the space $H^0(C,\Omega^{\otimes m}_C)$ of globally holomorphic polydifferentials remains unsolved in general. In this paper, we study the case of the group $G = \mathrm{SL}_2(\mathbb{F}_q)$ (where $q$ is a power of~$p$) acting on the Drinfeld curve $C$ which is the projective plane curve given by the equation $XY^q-X^qY-Z^{q+1} = 0$. When $q = p$, we fully decompose $H^0(C,\Omega^{\otimes m}_C)$ as a direct sum of indecomposable $\mathbb{F}[G]$-modules. For arbitrary $q$, we give a partial decomposition in terms of an explicit $\mathbb{F}$-basis of $H^0(C,\Omega^{\otimes m}_C)$. - oai:arXiv.org:2601.09956v1 - math.AG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Denver-James Logan Marchment, Bernhard K\"ock - - - Private Information Retrieval for Graph-based Replication with Minimal Subpacketization - https://arxiv.org/abs/2601.09957 - arXiv:2601.09957v1 Announce Type: new -Abstract: We design new minimal-subpacketization schemes for information-theoretic private information retrieval on graph-based replicated databases. In graph-based replication, the system consists of $K$ files replicated across $N$ servers according to a graph with $N$ vertices and $K$ edges. The client wants to retrieve one desired file, while keeping the index of the desired file private from each server via a query-response protocol. We seek PIR protocols that have (a) high rate, which is the ratio of the file-size to the total download cost, and (b) low subpacketization, which acts as a constraint on the size of the files for executing the protocol. We report two new schemes which have unit-subpacketization (which is minimal): (i) for a special class of graphs known as star graphs, and (ii) for general graphs. Our star-graph scheme has a better rate than previously known schemes with low subpacketization for general star graphs. Our scheme for general graphs uses a decomposition of the graph via independent sets. This scheme achieves a rate lower than prior schemes for the complete graph, however it can achieve higher rates than known for some specific graph classes. An extension of our scheme to the case of multigraphs achieves a higher rate than previous schemes for the complete multi-graph. - oai:arXiv.org:2601.09957v1 - cs.IT - cs.CR - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Vayur Shanbhag, Prasad Krishnan - - - Planar Site Percolation, End Structure, and the Benjamini-Schramm Conjecture - https://arxiv.org/abs/2601.09958 - arXiv:2601.09958v1 Announce Type: new -Abstract: Let $G$ be an infinite, connected, locally finite planar graph and consider i.i.d.\ Bernoulli$(p)$ site percolation. Write $p_c^{\mathrm{site}}(G)$ and $p_u^{\mathrm{site}}(G)$ for the critical and uniqueness thresholds. Using a well--separated Freudenthal embedding $G\hookrightarrow\mathbb S^2$, we introduce a cycle--separation equivalence on ends and associated ``directional'' thresholds $p^{\mathrm{site}}_{c,F}(G)$. - When the set of end--equivalence classes is countable, we show that $p_c^{\mathrm{site}}(G)=\inf_F p^{\mathrm{site}}_{c,F}(G)$ and that for every $p\in\bigl(\tfrac12,\,1-p_c^{\mathrm{site}}(G)\bigr)$ there are almost surely infinitely many infinite open clusters. Combined with the $0/\infty$ theorem of Glazman--Harel--Zelesko for $p\le \tfrac12$, this yields non--uniqueness throughout the full coexistence interval $\bigl(p_c^{\mathrm{site}}(G),\,1-p_c^{\mathrm{site}}(G)\bigr)$, and hence $p_u^{\mathrm{site}}(G)\ge 1-p_c^{\mathrm{site}}(G)$ in this setting. This resolves the extension problem posed by Glazman--Harel--Zelesko for the upper half of the coexistence regime under a natural countability hypothesis. - In contrast, for graphs with uncountably many end--equivalence classes we give criteria guaranteeing infinitely many infinite clusters above criticality, and we construct an explicit locally finite planar graph of minimum degree at least $7$ for which $p_u^{\mathrm{site}}(G)<1-p_c^{\mathrm{site}}(G)$. Consequently, the Benjamini--Schramm conjecture (Conjecture 7 in \cite{bs96}) that planarity together with minimal vertex degree at least 7 forces infinitely many infinite clusters for all $p\in(p_c,1-p_c)$ does not hold in full generality. - Our proofs combine a cutset characterization of $p_c^{\mathrm{site}}$ with a planar alternating--arm exploration organized by an end--adapted boundary decomposition. - oai:arXiv.org:2601.09958v1 - math.PR - math-ph - math.CO - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Zhongyang Li - - - On the Leaky Private Information Retrieval with Side Information - https://arxiv.org/abs/2601.09960 - arXiv:2601.09960v1 Announce Type: new -Abstract: This paper investigates the problem of leaky-private Private Information Retrieval with Side Information (L-PIR-SI), which relaxes the requirement of perfect privacy to achieve improved communication efficiency in the presence of side information. While the capacities of PIR-SI under both $W$-privacy and $(W,S)$-privacy have been partially explored, the impact of controlled information leakage in these settings remains unaddressed. We propose a unified probabilistic framework to construct L-PIR-SI schemes where the privacy leakage is quantified by a parameter $\varepsilon$, consistent with differential privacy standards. We characterize the achievable download costs and show that our results generalize several landmark results in the PIR literature: they recover the capacity of PIR-SI when $\varepsilon \to 0$, and reduce to the known bounds for leaky-PIR when side information is absent. This work provides the first look at the trade-offs between leakage, side information, and retrieval efficiency. - oai:arXiv.org:2601.09960v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yingying Huangfu, Tian Bai - - - Probabilistic heterogeneous Stirling numbers and Bell polynomials - https://arxiv.org/abs/2601.09964 - arXiv:2601.09964v1 Announce Type: new -Abstract: Let Y be a random variable satisfying specific moment conditions. This paper introduces and investigates probabilistic heterogeneous Stirling numbers of the second kind and probabilistic heterogeneous Bell polynomials. These structures unify several classical and probabilistic families, including those of Stirling, Lah, Bell and Lah-Bell. By integrating the heterogeneous framework of Kim and Kim with probabilistic extensions, we derive explicit formulas, Dobi\'nski-like identities, and recurrence relations. We further establish connections to partial Bell polynomials and provide applications for Poisson and Bernoulli distributions. - oai:arXiv.org:2601.09964v1 - math.NT - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Taekyun Kim, Dae San Kim - - - Stochastic Calculus for Rough Fractional Brownian Motion via Operator Factorization - https://arxiv.org/abs/2601.09967 - arXiv:2601.09967v1 Announce Type: new -Abstract: We develop an operator-theoretic framework for stochastic calculus with respect to rough fractional Brownian motion with Hurst parameter H < 1/2. Building on a covariant derivative defined via kernel factorization, we construct a closed unbounded operator on L2(Omega) adapted to the non-semimartingale setting. This approach yields explicit derivative representations for square-integrable functionals and provides a unified analytical framework compatible with rough path techniques. The results extend classical stochastic calculus beyond the semimartingale regime. - oai:arXiv.org:2601.09967v1 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ramiro Fontes - - - Einstein and Yang-Mills implies conformal Yang-Mills - https://arxiv.org/abs/2601.09975 - arXiv:2601.09975v1 Announce Type: new -Abstract: There exist conformally invariant, higher-derivative, variational analogs of the Yang-Mills condition for connections on vector bundles over a conformal manifold of even dimension greater than or equal to six. We give a compact formula for these analogs and prove that they are a strict weakening of the Yang-Mills condition with respect to an Einstein metric. We also show that the conformal Yang-Mills condition for the tractor connection of an even dimensional conformal manifold is equivalent to vanishing of its Fefferman-Graham obstruction tensor. This result uses that the tractor connection on a Poincar\'e-Einstein manifold is itself Yang-Mills. - oai:arXiv.org:2601.09975v1 - math.DG - gr-qc - hep-th - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Samuel Blitz, A. Rod Gover, Jaros{\l}aw Kopi\'nski, Andrew Waldron - - - Stochastic Calculus as Operator Factorization - https://arxiv.org/abs/2601.09976 - arXiv:2601.09976v1 Announce Type: new -Abstract: We present a unified operator-theoretic formulation of stochastic calculus based on two principles: fluctuations factor through differentiation, predictable projection, and integration, and the appropriate stochastic derivative is the Hilbert adjoint of the stochastic integral on the energy space of the driving process. On an isonormal Gaussian space we recover the identity (Id - E)F = delta Pi D F, where D is the Malliavin derivative, Pi is predictable projection, and delta is the divergence operator. Motivated by this factorization, we define for a square-integrable process X admitting a closed stochastic integral an operator-covariant derivative on L2(Omega) via Riesz representation. This yields a canonical Clark-Ocone representation that unifies Malliavin, Volterra-Malliavin, and functional Ito derivatives and clarifies the operator geometry underlying stochastic calculus. - oai:arXiv.org:2601.09976v1 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ramiro Fontes - - - Polynomially effective equidistribution for unipotent orbits in products of $\mathrm{SL}_2$ factors - https://arxiv.org/abs/2601.09983 - arXiv:2601.09983v1 Announce Type: new -Abstract: We sketch the proof of an effective equidistribution theorem for one-parameter unipotent subgroups in $S$-arithmetic quotients arising from $\mathbf K$-forms of $\mathrm{SL}_2^{\mathsf n}$ where $\mathbf K$ is a number field. This gives an effective version of equidistribution results of Ratner and Shah with a polynomial rate. - The key new phenomenon is the existence of many intermediate groups between the $\mathrm{SL}_2$ containing our unipotent and the ambient group, which introduces potential local and global obstruction to equidistribution. - Our approach relies on a Bourgain-type projection theorem in the presence of obstructions, together with a careful analysis of these obstructions. - oai:arXiv.org:2601.09983v1 - math.DS - math.CA - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Elon Lindenstrauss, Amir Mohammadi, Lei Yang - - - Remarks on the convex integration technique applied to singular stochastic partial differential equations - https://arxiv.org/abs/2601.09990 - arXiv:2601.09990v1 Announce Type: new -Abstract: Singular stochastic partial differential equations informally refer to the partial differential equations with rough random force that leads to the products in the nonlinear terms becoming ill-defined. Besides the theories of regularity structures and paracontrolled distributions, the technique of convex integration has emerged as a possible approach to construct a solution to such singular stochastic partial differential equations. We review recent developments in this area, and also demonstrate that an application of the convex integration technique to prove non-uniqueness seems unlikely for a particular singular stochastic partial differential equation, specifically the $\Phi^{4}$ model from quantum field theory. - oai:arXiv.org:2601.09990v1 - math.PR - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hongjie Dong, Kazuo Yamazaki - - - On directional second-order tangent sets of analytic sets and applications in optimization - https://arxiv.org/abs/2601.09991 - arXiv:2601.09991v1 Announce Type: new -Abstract: In this paper we study directional second-order tangent sets of real and complex analytic sets. For an analytic set $X\subseteq\mathbb{K}^n$ and a nonzero tangent direction $u\in T_0X$, we compare the geometric second-order tangent set $T^2_{0,u}X$, defined via second-order expansions of analytic arcs, with the algebraic second-order tangent set $T^{2,a}_{0,u}X$, defined by initial forms of the defining equations. We prove the general inclusion $T^2_{0,u}X\subseteq T^{2,a}_{0,u}X$ and construct explicit real and complex analytic examples showing that the inclusion is strict. - We introduce a second-jet formulation along fixed tangent directions and show that $T^2_{0,u}X=T^{2,a}_{0,u}X$ if and only if the natural second-jet map from analytic arcs in $X$ to jets on the tangent cone $C_0X$ is surjective. This surjectivity is established for smooth analytic germs, homogeneous analytic cones, hypersurfaces with nondegenerate tangent directions, and nondegenerate analytic complete intersections. As an application, we derive second-order necessary and sufficient optimality conditions for $C^2$ optimization problems on analytic sets. - oai:arXiv.org:2601.09991v1 - math.AG - math.CV - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Le Cong Trinh - - - M\"obius-Type Structures in Non-Orientable Singular Semi-Riemannian Manifolds - https://arxiv.org/abs/2601.10009 - arXiv:2601.10009v1 Announce Type: new -Abstract: Our objective is to illuminate the global structure of non-orientable manifolds with signature-changing metrics. Using explicit constructions based on the topology of the M\"{o}bius strip, we produce examples of crosscap manifolds where the gluing junction serves as the locus of signature change. In another set of examples, we convert the M\"{o}bius strip into a singular signature-type changing manifold. For these resulting manifolds, we test whether the metric can be expressed as $\tilde{g}=g+fV^{\flat}\otimes V^{\flat}$, with $g$ a Lorentzian metric and $f$ a smooth interpolation function between the Lorentzian and Riemannian regions, separated by the signature change hypersurface $\mathcal{H}$. Our analysis reveals that the radical of the metric can transition from transverse to tangent at $\mathcal{H}$, pseudo-space orientability is obstructed by the Euler characteristic, and pseudo-time orientability may still hold. These examples illustrate subtle obstructions to applying standard transformation prescriptions for signature change and highlight novel phenomena in compact, non-orientable semi-Riemannian manifolds. - oai:arXiv.org:2601.10009v1 - math.DG - gr-qc - math-ph - math.GT - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Nathalie E. Rieger - - - On the Sasakian Structure of Manifolds with Nonnegative Transverse Bisectional Curvature - https://arxiv.org/abs/2601.10017 - arXiv:2601.10017v1 Announce Type: new -Abstract: In this paper, we concern with the Sasaki analogue of Yau uniformization conjecture in a complete noncompact Sasakian manifold with nonnegative transverse bisectional curvature. As a consequence, we confirm that any $5$-dimensional complete noncompact Sasakian manifold with positive transverse bisectional curvature and the maximal volume growth must be CR-biholomorphic to the standard Heisenberg group $\mathbb{H}_{2}$ which can be stated as the standard contact Euclidean $5$-space $\mathbb{R}^{5}$. - oai:arXiv.org:2601.10017v1 - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Shu-Cheng Chang, Yingbo Han, Chien Lin, Chin-Tung Wu - - - An introduction to weightings along submanifolds - https://arxiv.org/abs/2601.10021 - arXiv:2601.10021v1 Announce Type: new -Abstract: This article is based on a talk given at the Ghent Geometric Analysis Seminar in 2023. We review basic notions from the theory of weightings along submanifolds, with special emphasis on multiplicative weightings for Lie groupoids along subgroupoids. - oai:arXiv.org:2601.10021v1 - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Eckhard Meinrenken - - - A New Overture to Classical Simple Type Theory, Ketonen-type Gentzen and Tableau Systems - https://arxiv.org/abs/2601.10026 - arXiv:2601.10026v1 Announce Type: new -Abstract: In this paper, we introduce a Ketonen-type Gentzen-style classical simple type theory $\bf KCT$. Also the tableau system $\bf KCTT$ corresponding to $\bf KCT$ is introduced. Further inference-preserving Gentzen system $\bf KCT_h$ (equivalent to $\bf KCT$) and tableau system $\bf KCTT_h$ (equivalent to $\bf KCTT$) is introduced. We introduce the notion of Hintikka sequents for $\bf KCTT_h$.The completeness theorem and Takahashi-Prawitz's theorem are proved for $\bf KCTT_h$. - oai:arXiv.org:2601.10026v1 - math.LO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tadayoshi Miwa, Takao Inou\'e - - - Fundamental Limits of Coded Polynomial Aggregation - https://arxiv.org/abs/2601.10028 - arXiv:2601.10028v1 Announce Type: new -Abstract: Coded polynomial aggregation (CPA) enables the master to directly recover a weighted aggregation of polynomial evaluations without individually decoding each term, thereby reducing the number of required worker responses. In this paper, we extend CPA to straggler-aware distributed computing systems and introduce a straggler-aware CPA framework with pre-specified non-straggler patterns, where exact recovery is required only for a given collection of admissible non-straggler sets. Our main result shows that exact recovery of the desired aggregation is achievable with fewer worker responses than required by polynomial coded computing based on individual decoding, and that feasibility is fundamentally characterized by the intersection structure of the non-straggler patterns. In particular, we establish necessary and sufficient conditions for exact recovery in straggler-aware CPA and identify an intersection-size threshold that is sufficient to guarantee exact recovery. We further prove that this threshold becomes both necessary and sufficient when the number of admissible non-straggler sets is sufficiently large. We also provide an explicit construction of feasible CPA schemes whenever the intersection size exceeds the derived threshold. Finally, simulations reveal a sharp feasibility transition at the predicted threshold, providing empirical evidence that the bound is tight in practice. - oai:arXiv.org:2601.10028v1 - cs.IT - cs.DC - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Xi Zhong, J\"org Kliewer, Mingyue Ji - - - Stability and instability of small BGK waves - https://arxiv.org/abs/2601.10030 - arXiv:2601.10030v1 Announce Type: new -Abstract: The aim of this article is to prove that the linear stability or instability of small Bernstein-Green-Kruskal (BGK) waves is determined by the sign of the derivative of their energy distributions at $0$ energy. - oai:arXiv.org:2601.10030v1 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Dongfen Bian, Emmanuel Grenier, Wenrui Huang, Benoit Pausader - - - Convex combination of first and second eigenvalues of trees - https://arxiv.org/abs/2601.10036 - arXiv:2601.10036v1 Announce Type: new -Abstract: For a graph $G$, let $\lambda_1(G)$ and $\lambda_2(G)$ denote the largest and the second largest adjacency eigenvalue of $G$. The sum $\lambda_1(G) + \lambda_2(G)$ is called the \emph{spectral sum} of $G$. We investigate the spectral sum of trees of order $n$ and determine the extremal trees that achieve maximum/minimum. Moreover, for any $\alpha \in [0,1]$, we determine the extremal trees which maximize the convex combination $\alpha \lambda_1 + (1-\alpha)\lambda_2$ in the class of $n$-vertex trees. - oai:arXiv.org:2601.10036v1 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Hitesh Kumar, Bojan Mohar, Shivaramakrishna Pragada, Hanmeng Zhan - - - Recurrence relations for the coefficients of the confluent and Gauss hypergeometric functions in the complex plane - https://arxiv.org/abs/2601.10040 - arXiv:2601.10040v1 Announce Type: new -Abstract: For $a,b,c,z,p, \theta \in \mathbb{C}$, where $\mathbb{C}$ is the complex plane, $-c\notin \mathbb{N\cup }\left\{ 0\right\} $, let \begin{equation*} \mathcal{M}\left( z\right) =\left( 1-\theta z\right) ^{p}M\left(a;c;z\right) =\sum_{n=0}^{\infty }u_{n}z^{n}, \end{equation*} where $|z| <\frac{1}{\theta}$, $|\arg (1-\theta z)| < \pi$, and let \begin{equation*} \mathcal{G}\left( z\right) =(1-\theta z) ^{p}F(a,b;c;z) =\sum_{n=0}^{\infty }v_{n} z^{n}, \end{equation*} where $|z| < 1$, $|\arg (1-\theta z)| < \pi$. In this paper, we prove that the coefficients $u_{n}$ and $v_{n}$ for $n\geq 0$ satisfy a 3-order recurrence relation. These offer a new way to study confluent hypergeometric function $M(a;c;z)$ and Gauss hypergeometric function $F(a,b;c;z)$. And we provide other special functions' recurrence relations of their coefficients, such as error function, Bessel function, incomplete gamma function, complete elliptic integral and Chebyshev polynomials. - oai:arXiv.org:2601.10040v1 - math.CV - math.CA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Zi-Qiao Xu, Zhong-Xuan Mao, Jing-Feng Tian - - - Optimal Proximity Gap for Folded Reed--Solomon Codes via Subspace Designs - https://arxiv.org/abs/2601.10047 - arXiv:2601.10047v1 Announce Type: new -Abstract: A collection of sets satisfies a $(\delta,\varepsilon)$-proximity gap with respect to some property if for every set in the collection, either (i) all members of the set are $\delta$-close to the property in (relative) Hamming distance, or (ii) only a small $\varepsilon$-fraction of members are $\delta$-close to the property. - In a seminal work, Ben-Sasson \textit{et al.}\ showed that the collection of affine subspaces exhibits a $(\delta,\varepsilon)$-proximity gap with respect to the property of being Reed--Solomon (RS) codewords with $\delta$ up to the so-called Johnson bound for list decoding. Their technique relies on the Guruswami--Sudan list decoding algorithm for RS codes, which is guaranteed to work in the Johnson bound regime. - Folded Reed--Solomon (FRS) codes are known to achieve the optimal list decoding radius $\delta$, a regime known as capacity. Moreover, a rich line of list decoding algorithms was developed for FRS codes. It is then natural to ask if FRS codes can be shown to exhibit an analogous $(\delta,\varepsilon)$-proximity gap, but up to the so-called optimal capacity regime. We answer this question in the affirmative (and the framework naturally applies more generally to suitable subspace-design codes). - An additional motivation to understand proximity gaps for FRS codes is the recent results [BCDZ'25] showing that they exhibit properties similar to random linear codes, which were previously shown to be related to properties of RS codes with random evaluation points in [LMS'25], as well as codes over constant-size alphabet based on AEL [JS'25]. - oai:arXiv.org:2601.10047v1 - cs.IT - cs.CC - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Fernando Granha Jeronimo, Lenny Liu, Pranav Rajpal - - - A note on exact approximations - https://arxiv.org/abs/2601.10051 - arXiv:2601.10051v1 Announce Type: new -Abstract: Based on M. Hall's theorem we prove a simple result dealing with real numbers which admit exact approximations by rationals. - oai:arXiv.org:2601.10051v1 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sergei Pitcyn - - - Kov\'acs' conjecture on characterisation of projective space and hyperquadrics - https://arxiv.org/abs/2601.10055 - arXiv:2601.10055v1 Announce Type: new -Abstract: We prove Kov\'acs' conjecture that claims that if the $p^{th}$ exterior power of the tangent bundle of a smooth complex projective variety contains the $p^{th}$ exterior power of an ample vector bundle then the variety is either projective space or the $p$-dimensional quadric hypersurface. This provides a common generalization of Mori, Wahl, Cho-Sato, Andreatta-Wi\'sniewski, Kobayashi-Ochiai, and Araujo-Druel-Kov\'acs type characterizations of such varieties. - oai:arXiv.org:2601.10055v1 - math.AG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Soham Ghosh - - - An Efficient Constant-Coefficient MSAV Scheme for Computing Vesicle Growth and Shrinkage - https://arxiv.org/abs/2601.10057 - arXiv:2601.10057v1 Announce Type: new -Abstract: We present a fast, unconditionally energy-stable numerical scheme for simulating vesicle deformation under osmotic pressure using a phase-field approach. The model couples an Allen-Cahn equation for the biomembrane interface with a variable-mobility Cahn-Hilliard equation governing mass exchange across the membrane. Classical approaches, including nonlinear multigrid and Multiple Scalar Auxiliary Variable (MSAV) methods, require iterative solution of variable-coefficient systems at each time step, resulting in substantial computational cost. We introduce a constant-coefficient MSAV (CC-MSAV) scheme that incorporates stabilization directly into the Cahn-Hilliard evolution equation rather than the chemical potential. This reformulation yields fully decoupled constant-coefficient elliptic problems solvable via fast discrete cosine transform (DCT), eliminating iterative solvers entirely. The method achieves O(N^2 log N) complexity per time step while preserving unconditional energy stability and discrete mass conservation. Numerical experiments verify second-order temporal and spatial accuracy, mass conservation to relative errors below 5 x 10^-11, and close agreement with nonlinear multigrid benchmarks. On grids with N >= 2048, CC-MSAV achieves 6-15x overall speedup compared to classical MSAV with optimized preconditioning, while the dominant Cahn-Hilliard subsystem is accelerated by up to two orders of magnitude. These efficiency gains, achieved without sacrificing accuracy, make CC-MSAV particularly well suited for large-scale simulations of vesicle dynamics. - oai:arXiv.org:2601.10057v1 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zhiwei Zhang, Shuwang Li, John Lowengrub, Steven M. Wise - - - Global convergence of the subgradient method for robust signal recovery - https://arxiv.org/abs/2601.10062 - arXiv:2601.10062v1 Announce Type: new -Abstract: We study the subgradient method for factorized robust signal recovery problems, including robust PCA, robust phase retrieval, and robust matrix sensing. These objectives are nonsmooth and nonconvex, and may have unbounded sublevel sets, so standard arguments for analyzing first-order optimization algorithms based on descent and coercivity do not apply. For locally Lipschitz semialgebraic objectives, we develop a convergence framework under the assumption that continuous-time subgradient trajectories are bounded: for sufficiently small step sizes of order \(1/k\), any subgradient sequence remains bounded and converges to a critical point. We verify this trajectory boundedness assumption for the robust objectives by adapting and extending existing trajectory analyses, requiring only a mild nondegeneracy condition in the matrix sensing case. Finally, for rank-one symmetric robust PCA, we show that the subgradient method avoids spurious critical points for almost every initialization, and therefore converges to a global minimum under the same step-size regime. - oai:arXiv.org:2601.10062v1 - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zesheng Cai, Lexiao Lai, Tiansheng Li - - - Transport equation theory in the Triebel-Lizorkin spaces and its applications to the ideal fluid flows - https://arxiv.org/abs/2601.10071 - arXiv:2601.10071v1 Announce Type: new -Abstract: In this paper, we develop a general theory for the transport equation within the framework of Triebel-Lizorkin spaces. We first derive commutator estimates in these spaces, dispensing with the conventional divergence-free condition, via the Bony paraproduct decomposition and vector-valued maximal function inequalities. Building on these estimates and combining the method of characteristics with a compactness argument, we then obtain the new a priori estimates and prove local well-posedness for the transport equation in Triebel-Lizorkin spaces. The resulting theory is applicable to a wide range of evolution equations, including models for incompressible and compressible ideal fluid flows, shallow water waves, among others. As an illustration, we consider the incompressible ideal magnetohydrodynamics (MHD) system. Employing the general transport theory developed here yields a complete local well-posedness result in the sense of Hadamard, covering both sub-critical and critical regularity regimes, and provides corresponding blow-up criteria for the ideal MHD equations in Triebel-Lizorkin spaces. Our results refine and substantially extend earlier work in this direction. - oai:arXiv.org:2601.10071v1 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Qianyuan Zhang, Kai Yan - - - Simplicial spheres with $g_k=1$ - https://arxiv.org/abs/2601.10072 - arXiv:2601.10072v1 Announce Type: new -Abstract: For $d\geq 4$, Kalai (1987) characterized all simplicial $(d-1)$-spheres with $g_2=0$, and for $k\geq 2$ and $d\geq 2k$, Murai and Nevo (2013) characterized all simplicial $(d-1)$-spheres with $g_k=0$. In addition, for $d\geq 4$, Nevo and Novinsky (2011) characterized all simplicial $(d-1)$-spheres with $g_2=1$. Motivated by these results, we characterize, for any $k\geq 2$ and $d\geq 2k+1$, all simplicial $(d-1)$-spheres with no missing faces of dimension larger than $d-k$ that satisfy $g_k=1$. When $d=2k$, we obtain a characterization of simplicial $(d-1)$-spheres with $g_k=1$ and no missing faces of dimension greater than $k$, under the additional assumption that there exists at least one missing face of dimension $k$. Finally, for $k=3$, we are able to remove this assumption and characterize all simplicial $5$-spheres with no missing faces of dimension larger than $3$ that satisfy $g_3=1$. - oai:arXiv.org:2601.10072v1 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Isabella Novik, Hailun Zheng - - - Sharp propagation of chaos in R\'enyi divergence - https://arxiv.org/abs/2601.10076 - arXiv:2601.10076v1 Announce Type: new -Abstract: We establish sharp rates for propagation of chaos in R\'enyi divergences for interacting diffusion systems at stationarity. Building upon the entropic hierarchy established in Lacker (2023), we show that under strong isoperimetry and weak interaction conditions, one can achieve $\mathsf R_q(\mu^1 \,\lVert\, \pi) = \widetilde O(\frac{d q^2}{N^2})$ bounds on the $q$-R\'enyi divergence. - oai:arXiv.org:2601.10076v1 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Matthew S. Zhang - - - A $p$-adic interpolation of the Cogdell lift - https://arxiv.org/abs/2601.10077 - arXiv:2601.10077v1 Announce Type: new -Abstract: In this paper we obtain several results related to the $p$-adic interpolation of the classical Cogdell lift, mapping special cycles on Picard modular surfaces to elliptic modular forms. The results have a three-fold nature: in the first part of the paper, we $p$-adically interpolate the adjoint Kudla lift, exploiting the previously constructed $\Lambda$-adic Kudla lift. In the second part, we construct higher weight cycles in Kuga-Sato varieties attached to Picard modular surfaces, and show modularity of the generating series of these cycles, thus obtaining a higher weight analogue of the Cogdell lift. Finally, we apply the formalism introduced by Loeffler to construct $p$-adic analytic cohomology classes of special cycles, whose generating series is proved to be a Hida family interpolating the Cogdell lifts in the weight and level variables. - oai:arXiv.org:2601.10077v1 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Francesco Maria Iudica - - - Line-search and Adaptive Step Sizes for Nonconvex-strongly-concave Minimax Optimization - https://arxiv.org/abs/2601.10086 - arXiv:2601.10086v1 Announce Type: new -Abstract: In this paper, we propose a novel reformulation of the smooth nonconvex-strongly-concave (NC-SC) minimax problems that casts the problem as a joint minimization. We show that our reformulation preserves not only first-order stationarity, but also global and local optimality, second-order stationarity, and the Kurdyka-{\L}ojasiewicz (KL) property, of the original NC-SC problem, which is substantially stronger than its nonsmooth counterpart in the literature. With these enhanced structures, we design a versatile parameter-free and nonmonotone line-search framework that does not require evaluating the inner maximization. Under mild conditions, global convergence rates can be obtained, and, with KL property, full sequence convergence with asymptotic rates is also established. In particular, we show our framework is compatible with the gradient descent-ascent (GDA) algorithm. By equipping GDA with Barzilai-Borwein (BB) step sizes and nonmonotone line-search, our method exhibits superior numerical performance against the compared benchmarks. - oai:arXiv.org:2601.10086v1 - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bohao Ma, Nachuan Xiao, Junyu Zhang - - - Admissibility Breakdown in High-Dimensional Sparse Regression with L1 Regularization - https://arxiv.org/abs/2601.10100 - arXiv:2601.10100v1 Announce Type: new -Abstract: The choice of the tuning parameter in the Lasso is central to its statistical performance in high-dimensional linear regression. Classical consistency theory identifies the rate of the Lasso tuning parameter, and numerous studies have established non-asymptotic guarantees. Nevertheless, the question of optimal tuning within a non-asymptotic framework has not yet been fully resolved. We establish tuning criteria above which the Lasso becomes inadmissible under mean squared prediction error. More specifically, we establish thresholds showing that certain classical tuning choices yield Lasso estimators strictly dominated by a simple Lasso-Ridge refinement. We also address how the structure of the design matrix and the noise vector influences the inadmissibility phenomenon. - oai:arXiv.org:2601.10100v1 - math.ST - stat.ME - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Guo Liu (Waseda University) - - - Fano threefolds of genus 12 with large automorphism group in positive and mixed characteristic - https://arxiv.org/abs/2601.10106 - arXiv:2601.10106v1 Announce Type: new -Abstract: We study prime Fano threefolds of genus 12 ($V_{22}$-varieties) with positive-dimensional automorphism groups in positive and mixed characteristic. We classify such varieties over any perfect field. In particular, we prove that $V_{22}$-varieties of Mukai-Umemura type over $k$ exist if and only if $\mathrm{char}\ k \neq 2$, $5$. We also prove the same result for $\mathbb{G}_a$-type. As arithmetic applications, we show that the Shafarevich conjecture holds for $V_{22}$-varieties of Mukai-Umemura type and of $\mathbb{G}_m$-type, while it fails for $V_{22}$-varieties of $\mathbb{G}_a$-type. Moreover, we prove that there exists $V_{22}$-varieties over $\mathbb{Z}$, whereas there do not exist $V_{22}$-varieties over $\mathbb{Z}$ whose generic fiber has a positive-dimensional automorphism group. - oai:arXiv.org:2601.10106v1 - math.AG - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tetsushi Ito, Akihiro Kanemitsu, Teppei Takamatsu, Yuuji Tanaka - - - Shifted bilinear sums of Sali\'e sums and the distribution of modular square roots of shifted primes - https://arxiv.org/abs/2601.10113 - arXiv:2601.10113v1 Announce Type: new -Abstract: We establish various upper bounds on Type-I and Type-II shifted bilinear sums with Sali\'e sums modulo a large prime $q$. We use these bounds to study, for fixed integers $a,b\not \equiv 0 \bmod q$, the distribution ofsolutions to the congruence $x^2 \equiv ap+b \bmod q$, over primes $p\le P$. This is similar to the recently studied case of $b = 0$, however the case $b\not \equiv 0 \bmod q$ exhibits some new difficulties. - oai:arXiv.org:2601.10113v1 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Igor E. Shparlinski, Yixiu Xiao - - - Calabi affine maximal surfaces and centroaffine Bernstein problems - https://arxiv.org/abs/2601.10125 - arXiv:2601.10125v1 Announce Type: new -Abstract: Motivated by Calabi's calculation of the second variation sign for locally strongly convex affine maximal surfaces in equiaffine geometry, we first prove that every Calabi extremal surface is also maximal in the Calabi affine geometry. By employing suitably chosen orthonormal frame fields and analyzing the corresponding Codazzi equations, we then obtain local classifications for certain special classes of Calabi affine maximal surfaces and hyperbolic centroaffine extremal surfaces. These examples inspire the construction of new, complete Calabi affine maximal surfaces and centroaffine extremal hypersurfaces. Notably, the complete centroaffine extremal hypersurfaces we establish answer all five centroaffine Bernstein problems posed by Li- Li-Simon in 2004. - oai:arXiv.org:2601.10125v1 - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yalin Sun, Cheng Xing, Ruiwei Xu - - - Curvature-driven manifold fitting under unbounded isotropic noise - https://arxiv.org/abs/2601.10133 - arXiv:2601.10133v1 Announce Type: new -Abstract: Manifold fitting aims to reconstruct a low-dimensional manifold from high-dimensional data, whose framework is established by Fefferman et al. \cite{fefferman2020reconstruction,fefferman2021reconstruction}. This paper studies the recovery of a compact $C^3$ submanifold $\mathcal{M} \subset \mathbb{R}^D$ with dimension $d<D$ and positive reach $\tau$ from observations $Y = X + \xi$, where $X$ is uniformly distributed on $\mathcal{M}$ and $\xi \sim \mathcal{N}(0, \sigma^2 I_D)$ denotes isotropic Gaussian noise. To project any points $z$ in a tubular neighborhood $\Gamma$ of $\mathcal{M}$ onto $\mathcal{M}$, we construct a sample-based estimator $F:\Gamma\to\mathbb{R}^D$ by a normalized local kernel with the theoretically derived bandwidth $r = c_D\sigma$. Under a sample size of $O(\sigma^{-3d-5})$, we establish with high probability the uniform asymptotic expansion \[ F(z) = \pi(z) + \frac{d}{2} H_{\pi(z)} \sigma^2 + O(\sigma^3), \qquad z \in \Gamma, \] where $\pi(z)$ is the projection of $z$ onto $\mathcal{M}$ and $H_{\pi(z)}$ is the mean curvature vector of $\mathcal{M}$ at $\pi(z)$. The resulting manifold $F(\Gamma)$ has reach bounded below by $c \tau$ for $c>0$ and achieves a state-of-the-art Hausdorff distance of $O(\sigma^2)$ to $\mathcal{M}$. Numerical experiments confirm the quadratic decay of the reconstruction error and demonstrate the computational efficiency of the estimator $F$. Our work provides a curvature-driven framework for denoising and reconstructing manifolds with second-order accuracy. - oai:arXiv.org:2601.10133v1 - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Ruowei Li, Zhigang Yao - - - Function Correcting Codes for Maximally-Unbalanced Boolean Functions - https://arxiv.org/abs/2601.10135 - arXiv:2601.10135v1 Announce Type: new -Abstract: Function-Correcting Codes (FCCs) enable reliable computation of a function of a $k$-bit message over noisy channels without requiring full message recovery. In this work, we study optimal single-error correcting FCCs (SEFCCs) for maximally-unbalanced Boolean functions, where $k$ denotes the message length and $t$ denotes the error-correction capability. We analyze the structure of optimal SEFCC constructions through their associated codeword distance matrices and identify distinct FCC classes based on this structure. We then examine the impact of these structural differences on error performance by evaluating representative FCCs over the additive white Gaussian noise (AWGN) channel using both soft-decision and hard-decision decoding. The results show that FCCs with different distance-matrix structures can exhibit markedly different Data BER and function error behavior, and that the influence of code structure depends strongly on the decoding strategy. - oai:arXiv.org:2601.10135v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rajlaxmi Pandey, Shiven Bajpai, Anjana A Mahesh, B. Sundar Rajan - - - A new contraction principle on the perimeters of triangles and related results - https://arxiv.org/abs/2601.10138 - arXiv:2601.10138v1 Announce Type: new -Abstract: In this article, we introduce a new type of mapping contracting perimeters of triangles in a complete metric space and present related fixed point theorem. We study the metric completeness property of the underlying space in terms of fixed point of our newly introduced mapping. In support of our result, we present several examples. - oai:arXiv.org:2601.10138v1 - math.FA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tanusri Senapati - - - New Second-order Convergent Schemes for Solving decoupled FBSDEs - https://arxiv.org/abs/2601.10149 - arXiv:2601.10149v1 Announce Type: new -Abstract: This paper proposes a new second-order symmetric algorithm for solving decoupled forward-backward stochastic differential equations. Inspired by the alternating direction implicit splitting method for partial differential equations, we split the generator into the sum of two functions. In the computation of the value process Y, explicit and implicit schemes are alternately applied to these two generators, while the algorithms from \citep{ZhaoLi2014} are used for the control process Z. We rigorously prove that the two new schemes have second-order convergence rate. The proposed splitting methods show clear advantages for equations whose generator consists of a linear part plus a nonlinear part, as they reduce the number of iterations required for solving implicit schemes, thereby decreasing computational cost while maintaining second-order convergence. Two numerical examples are provided, including the backward stochastic Riccati equation arising in mean-variance hedging. The numerical results verify the theoretical error analysis and demonstrate the advantage of reduced computational cost compared to the algorithm in \citep{ZhaoLi2014}. - oai:arXiv.org:2601.10149v1 - math.NA - cs.NA - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Wenbo Wang, Guangyan Jia - - - On Quaternionic Fock Spaces: Kernel-induced Integral Operators, Berezin Transforms and Toeplitz Operators - https://arxiv.org/abs/2601.10162 - arXiv:2601.10162v1 Announce Type: new -Abstract: In this paper, we study quaternionic Fock spaces and develop an operator-theoretic framework centered around kernel-induced integral operators, Berezin transforms and Toeplitz operators. More precisely, the following results are obtained: - (i) Global quaternionic Fock structure. We introduce a global Gaussian $L^p$--norm for slice functions on $\mathbb H$ and prove that the resulting global quaternionic Fock space $F_\alpha^p$ coincides with the slice-defined Fock space $\mathfrak F_\alpha^p$, with equivalent norms. In particular, $F_\alpha^2$ becomes a right quaternionic reproducing kernel Hilbert space with an explicit reproducing kernel, yielding a slice-independent Fock projection onto $F_\alpha^2$. - (ii) Kernel-induced integral operators and Fock--Carleson measures. We investigate kernel-induced integral operators and characterize quaternionic Fock--Carleson measures. These embedding theorems provide the measure-theoretic basis that underlies boundedness and compactness criteria for operators on quaternionic Fock spaces. - (iii)Berezin transforms and Toeplitz operators. We define the Berezin transform for slice functions and prove its fundamental properties, including semigroup behavior and fixed-point features. Building on the slice-independent projection and the slice product, we introduce Toeplitz operators with slice-function symbols and with measure symbols, and develop their basic algebraic properties. We then obtain complete boundedness and compactness characterizations for Toeplitz operators with two natural symbol classes: positive measures and slice $\mathrm{BMO}^1$ symbols, expressed in terms of Berezin-type transforms and slice/symmetric averaging quantities. - oai:arXiv.org:2601.10162v1 - math.FA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zhaopeng Lin, Yufeng Lu, Chao Zu - - - Advances on two spectral conjectures regarding booksize of graphs - https://arxiv.org/abs/2601.10163 - arXiv:2601.10163v1 Announce Type: new -Abstract: The \emph{booksize} $ \mathrm{bk}(G) \) of a graph $ G $, introduced by Erd\H{o}s, refers to the maximum integer $ r $ for which $G$ contains the book $ B_r $ as a subgraph. This paper investigates two open problems in spectral graph theory related to the booksize of graphs. - First, we prove that for any positive integer $r$ and any $ B_{r+1} $-free graph $ G $ with $ m \geq (9r)^2 $ edges, the spectral radius satisfies $ \rho(G) \leq \sqrt{m} $. Equality holds if and only if $ G $ is a complete bipartite graph. This result improves the lower bound on the booksize of Nosal graphs (i.e., graphs with $ \rho(G) > \sqrt{m} $) from the previously established $ \mathrm{bk}(G) > \frac{1}{144}\sqrt{m} $ to $ \mathrm{bk}(G) > \frac{1}{9}\sqrt{m} $, presenting a significant advancement in the booksize conjecture proposed Li, Liu, and Zhang. - Second, we show that for any positive integer $r$ and any non-bipartite $ B_{r+1} $-free graph $ G $ with $ m \geq (240r)^2 $ edges, the spectral radius $\rho$ satisfies $\rho^2<m-1+\frac{2}{\rho-1}$, unless $G$ is isomorphic to $S^+_{m,s}$ for some $s\in\{1,\ldots,r\}$. This resolves Liu and Miao's conjecture and further reveals an interesting phenomenon: even with a weaker spectral condition, $\rho^2\geq m-1+\frac2{\rho-1}$, we can still derive the supersaturation of the booksize for non-bipartite graphs. - oai:arXiv.org:2601.10163v1 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mingqing Zhai, Rui Li, Zhenzhen Lou - - - On Existence of Girth-8 QC-LDPC Code with Large Column Weight: Combining Mirror-sequence with Classification Modulo Ten - https://arxiv.org/abs/2601.10170 - arXiv:2601.10170v1 Announce Type: new -Abstract: Quasi-cyclic (QC) LDPC codes with large girths play a crucial role in several research and application fields, including channel coding, compressed sensing and distributed storage systems. A major challenge in respect of the code construction is how to obtain such codes with the shortest possible length (or equivalently, the smallest possible circulant size) using algebraic methods instead of search methods. The greatest-common-divisor (GCD) framework we previously proposed has algebraically constructed QC-LDPC codes with column weights of 5 and 6, very short lengths, and a girth of 8. By introducing the concept of a mirror sequence and adopting a new row-regrouping scheme, QC-LDPC codes with column weights of 7 and 8, very short lengths, and a girth of 8 are proposed for arbitrary row weights in this article via an algebraic manner under the GCD framework. Thanks to these novel algebraic methods, the lower bounds (for column weights 7 and 8) on consecutive circulant sizes are both improved by asymptotically about 20%, compared with the existing benchmarks. Furthermore, these new constructions can also offer circulant sizes asymptotically about 25% smaller than the novel bounds. - oai:arXiv.org:2601.10170v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Guohua Zhang, Xiangya Liu, Jianhua Zhang, Yi Fang - - - On 3-Connected Planar Graphs with Unique Orientable Circuit Double Covers - https://arxiv.org/abs/2601.10171 - arXiv:2601.10171v1 Announce Type: new -Abstract: A circuit double cover of a bridgeless graph is a collection of even subgraphs such that every edge is contained in exactly two subgraphs of the given collection. Such a circuit double cover describes an embedding of the corresponding graph onto a surface. In this paper, we investigate the well-known Orientable Strong Embedding Conjecture. This conjecture proposes that every bridgeless graph has a circuit double cover describing an embedding on an orientable surface. In a recent paper, we have proved that a 3-connected cubic planar graph G has exactly one orientable circuit double cover if and only if G is the dual graph of an Apollonian network. In this paper, we extend this result by demonstrating that this characterisation applies to any 3-connected planar graph, regardless of whether it is cubic. - oai:arXiv.org:2601.10171v1 - math.CO - cs.DM - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Meike Wei{\ss}, Reymond Akpanya, Alice C. Niemeyer - - - A Low-Complexity Architecture for Multi-access Coded Caching Systems with Arbitrary User-cache Access Topology - https://arxiv.org/abs/2601.10175 - arXiv:2601.10175v1 Announce Type: new -Abstract: This paper studies the multi-access coded caching (MACC) problem under arbitrary user-cache access topologies, extending existing models that rely on highly structured and combinatorially designed connectivity. We consider a MACC system consisting of a single server, multiple cache nodes, and multiple user nodes. Each user can access an arbitrary subset of cache nodes to retrieve cached content. The objective is to design a general and low-complexity delivery scheme under fixed cache placement for arbitrary access topologies. We propose a universal graph-based framework for modeling the MACC delivery problem, where decoding conflicts among requested packets are captured by a conflict graph and the delivery design is reduced to a graph coloring problem. In this formulation, a lower transmission load corresponds to using fewer colors. The classical greedy coloring algorithm DSatur achieves a transmission load close to the index-coding converse bound, providing a tight benchmark, but its computational complexity becomes prohibitive for large-scale graphs. To overcome this limitation, we develop a learning-based framework using graph neural networks that efficiently constructs near-optimal coded multicast transmissions and generalizes across diverse access topologies and varying numbers of users. In addition, we extend the index-coding converse bound for uncoded cache placement to arbitrary access topologies and propose a low-complexity greedy approximation. Numerical results demonstrate that the proposed learning-based scheme achieves transmission loads close to those of DSatur and the converse bound while significantly reducing computational time. - oai:arXiv.org:2601.10175v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ting Yang, Minquan Cheng, Xinping Yi, Robert Caiming Qiu, Giuseppe Caire - - - Characteristics of drift effects in the quasi-geostrophic equation arising from nonlinear symmetry - https://arxiv.org/abs/2601.10185 - arXiv:2601.10185v1 Announce Type: new -Abstract: This paper compares two similar diffusion equations that appear in meteorology. One is the quasi-geostrophic equation, and the other is the convection-diffusion equation. Both are two-dimensional bilinear equations, and the order of differentiation is the same. Naturally, their scales also coincide. However, the direction in which the nonlinear effects act differs: one acts along the isothermal surface, while the other acts along the temperature gradient in a specified direction. The main assertion quantifies this difference through the large-time behavior of their solutions. In particular, the nonlinear distortions in the asymptotic profiles of both equations are compared. In this context, the spatial symmetry of the first approximation plays a crucial role, but the solutions require no symmetry. - oai:arXiv.org:2601.10185v1 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Masakazu Yamamoto - - - Outlier eigenvalues and eigenvectors of generalized Wigner matrices with finite-rank perturbations - https://arxiv.org/abs/2601.10204 - arXiv:2601.10204v1 Announce Type: new -Abstract: A generalized Wigner matrix perturbed by a finite-rank deterministic matrix is considered. The fluctuations of the largest eigenvalues, which emerge outside the bulk of the spectrum, and the corresponding eigenvectors, are studied. Under certain assumptions on the perturbation and the matrix structure, we derive the first-order behavior of these eigenvalues and show that they are well separated from the bulk. The fluctuations of these eigenvalues are shown to follow a multivariate Gaussian distribution, and the asymptotic behavior of the associated eigenvectors is also studied. We prove central limit theorems that describe the asymptotic alignment of these eigenvectors with the perturbation's eigenvectors, as well as their Gaussian fluctuations around the origin for non-aligned components. Furthermore, we discuss the convergence of the eigenvector process in a Sobolev space framework. - oai:arXiv.org:2601.10204v1 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Bishakh Bhattacharya, Arijit Chakrabarty, Rajat Subhra Hazra - - - Biharmonic and Interpolating Sesqui-Harmonic Vector Fields with Respect to the varphi-Sasakian Metric - https://arxiv.org/abs/2601.10216 - arXiv:2601.10216v1 Announce Type: new -Abstract: This work investigates biharmonic and interpolating sesqui-harmonic vector fields on the tangent bundle of a para-K\"ahler--Norden manifold (M, varphi, g) endowed with the varphi-Sasaki metric. We derive the first variation of the bienergy and interpolating sesqui-energy functionals, restricted to the space of vector fields. Explicit characterizations are established for vector fields satisfying the corresponding variational conditions-namely, biharmonicity and interpolating sesqui-harmonicity. Furthermore, several examples are presented to illustrate the general theory and to elucidate the distinctions between harmonic, biharmonic, and interpolating sesqui-harmonic behaviors. These results extend and complement existing research on higher-order harmonicity in pseudo-Riemannian geometry. - oai:arXiv.org:2601.10216v1 - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Abderrahim Zagane, Kheireddine Biroud, Medjahed Djilali - - - Nuclear Toeplitz operators between Fock spaces - https://arxiv.org/abs/2601.10217 - arXiv:2601.10217v1 Announce Type: new -Abstract: We study Toeplitz operators with measure-valued symbols acting between Fock spaces. Given $1\le p,q\le\infty$ and a Borel measure $\mu$ on $\mathbb C$, we investigate when the associated Toeplitz operator \[ T_\mu : F^p_\alpha \to F^q_\alpha \] belongs to the nuclear class. For positive measures $\mu$ and in the range $1\le q\le p\le\infty$, we obtain necessary and sufficient conditions for the nuclearity of $T_\mu$ in terms of the Berezin transform of $\mu$. As a consequence, nuclearity in this setting exhibits a rigidity property: if $T_\mu$ is nuclear from $F^p_\alpha$ to $F^q_\alpha$ for some $q\le p$, then it is nuclear for all such $q$. In the case $p<q$, we show that the situation is more delicate. We provide separate necessary and sufficient conditions for nuclearity, indicating that the Berezin transform alone does not yield a complete characterization. The proofs rely on tools from Banach space operator theory combined with kernel estimates on Fock spaces. Our results extend naturally to Fock spaces on $\mathbb C^n$. - oai:arXiv.org:2601.10217v1 - math.FA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tengfei Ma, Yufeng Lu, Chao Zu - - - Introduction to optimization methods for training SciML models - https://arxiv.org/abs/2601.10222 - arXiv:2601.10222v1 Announce Type: new -Abstract: Optimization is central to both modern machine learning (ML) and scientific machine learning (SciML), yet the structure of the underlying optimization problems differs substantially across these domains. Classical ML typically relies on stochastic, sample-separable objectives that favor first-order and adaptive gradient methods. In contrast, SciML often involves physics-informed or operator-constrained formulations in which differential operators induce global coupling, stiffness, and strong anisotropy in the loss landscape. As a result, optimization behavior in SciML is governed by the spectral properties of the underlying physical models rather than by data statistics, frequently limiting the effectiveness of standard stochastic methods and motivating deterministic or curvature-aware approaches. This document provides a unified introduction to optimization methods in ML and SciML, emphasizing how problem structure shapes algorithmic choices. We review first- and second-order optimization techniques in both deterministic and stochastic settings, discuss their adaptation to physics-constrained and data-driven SciML models, and illustrate practical strategies through tutorial examples, while highlighting open research directions at the interface of scientific computing and scientific machine learning. - oai:arXiv.org:2601.10222v1 - math.NA - cs.AI - cs.NA - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Alena Kopani\v{c}\'akov\'a, Elisa Riccietti - - - Unrefinable Partitions into Distinct Parts and Numerical Semigroups - https://arxiv.org/abs/2601.10227 - arXiv:2601.10227v1 Announce Type: new -Abstract: This article investigates structural connections between unrefinable partitions into distinct parts and numerical semigroups. By analysing the hooksets of Young diagrams associated with numerical sets, new criteria for recognising unrefinable partitions are established. - A correspondence between missing parts and the gaps of numerical semigroups is developed, extending previous classifications and enabling the characterisation of partitions with maximal numbers of missing parts. In particular, the results show that certain families of unrefinable partitions correspond precisely to symmetric numerical semigroups when the maximal part is prime. Further structural consequences, examples, and a decomposition of unrefinable partitions by minimal excludant are discussed, together with implications for the study of maximal unrefinable partitions. - oai:arXiv.org:2601.10227v1 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Lorenzo Campioni - - - Inconsistency of Reinhardt cardinals with $\mathsf{ZF}$ - https://arxiv.org/abs/2601.10231 - arXiv:2601.10231v1 Announce Type: new -Abstract: A proof will be presented that the existence of a non-trivial $\Sigma_1$-elementary embedding $j: V_{\lambda+3} \prec V_{\lambda+3}$ is inconsistent with $\textsf{ZF}$. Sections 1 and 2 shall review various important contributions from the literature, notably including \cite{Goldberg2020}, \cite{Schlutzenberg2020}, and \cite{Woodin2010}, the latter reference being where the crucial forcing construction is presented. Section 3 shall introduce some new large cardinal properties, of consistency strength intermediate between $\mathsf{I_3}$ and $\mathsf{I_2}$, and greater than $\mathsf{I_1}$, respectively. The proof of the inconsistency with $\mathsf{ZF}$ of the existence of a non-trivial $\Sigma_1$-elementary embedding $j:V_{\lambda+3} \prec V_{\lambda+3}$ shall be given in Section 4. The claims of Sections 2 and 4 are provable in $\textsf{ZF}$; those of Section 3, with the exception of the last two theorems, in $\textsf{ZFC}$. - oai:arXiv.org:2601.10231v1 - math.LO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Rupert McCallum - - - Synchronization and Hopf Bifurcation in Stuart--Landau Networks - https://arxiv.org/abs/2601.10234 - arXiv:2601.10234v1 Announce Type: new -Abstract: The Kuramoto model has shaped our understanding of synchronization in complex systems, yet its phase-only formulation neglects amplitude dynamics that are intrinsic to many oscillatory networks. In this work, we revisit Kuramoto-type synchronization through networks of Stuart--Landau oscillators, which arise as the universal normal form near a Hopf bifurcation. For identical natural frequencies, we analyze synchronization in two complementary regimes. Away from criticality, we establish topology-robust complete synchronization for general connected networks under explicit sufficient conditions that preclude amplitude death. At criticality, we exploit network symmetries to analyze the onset of collective oscillations via Hopf bifurcation theory, demonstrating the emergence of synchronized periodic states in ring-symmetric networks. Our results clarify how amplitude dynamics enrich the structure of synchronized states and provide a bridge between classical Kuramoto synchronization and amplitude-inclusive models in complex networks. - oai:arXiv.org:2601.10234v1 - math.DS - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kuan-Wei Chen, Ting-Yang Hsiao - - - A flower theorem in $\mathbb{C}^n$ - https://arxiv.org/abs/2601.10235 - arXiv:2601.10235v1 Announce Type: new -Abstract: We prove an analog of the flower theorem for non-degenerate reduced tangent to the identity germs that fix the coordinate hyperspaces in any dimension. - oai:arXiv.org:2601.10235v1 - math.DS - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - K\'emo Morvan (UPCit\'e, IMJ-PRG) - - - Ramsey number of a cycle versus a graph of a given size - https://arxiv.org/abs/2601.10238 - arXiv:2601.10238v1 Announce Type: new -Abstract: In this paper, we prove that for every $k$ and every graph $H$ with $m$ edges and no isolated vertices, the Ramsey number $R(C_k,H)$ is at most $2m+\lfloor \frac{k-1}{2} \rfloor$, provided $m$ is sufficiently large with respect to $k$. This settles a problem of Erd\H{o}s, Faudree, Rousseau and Schelp. - oai:arXiv.org:2601.10238v1 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Stijn Cambie, Andrea Freschi, Patryk Morawski, Kalina Petrova, Alexey Pokrovskiy - - - Model order reduction of piecewise linear mechanical systems using invariant cones - https://arxiv.org/abs/2601.10241 - arXiv:2601.10241v1 Announce Type: new -Abstract: We present a methodology that extends invariant manifold theory to a class of autonomous piecewise linear systems with nonsmoothness at the equilibrium, providing a framework for model order reduction in mechanical structures with compliant contact laws. The key idea is to make the absence of a local linearization around the equilibrium tractable by leveraging the positive homogeneity property. This property simplifies the invariance equations defining the geometry of the invariant cones, from a set of partial differential equations to a system of ordinary differential equations, enabling their effective solution. We introduce two techniques to compute these invariant cones. First, an intuitive graph-style parametrization is proposed that utilizes Fourier expansions and Chebyshev polynomials to derive explicit reduced-order models in closed form. Second, an arc-length parametrization is introduced to robustly compute invariant cones with complex folding geometries, which are intractable with a standard graph-style technique. The approach is demonstrated on mechanical oscillators with unilateral visco-elastic supports, showcasing its applicability for systems with both continuous (unilateral elastic) and discontinuous (unilateral visco-elastic) unilateral force laws. - oai:arXiv.org:2601.10241v1 - math.DS - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - A. Yassine Karoui, Remco I. Leine - - - Restoring similarity in randomized Krylov methods with applications to eigenvalue problems and matrix functions - https://arxiv.org/abs/2601.10248 - arXiv:2601.10248v1 Announce Type: new -Abstract: The randomized Arnoldi process has been used in large-scale scientific computing because it produces a well-conditioned basis for the Krylov subspace more quickly than the standard Arnoldi process. However, the resulting Hessenberg matrix is generally not similar to the one produced by the standard Arnoldi process, which can lead to delays or spike-like irregularities in convergence. In this paper, we introduce a modification of the randomized Arnoldi process that restores similarity with the Hessenberg matrix generated by the standard Arnoldi process. This is accomplished by enforcing orthogonality between the last Arnoldi vector and the previously generated subspace, which requires solving only one additional least-squares problem. When applied to eigenvalue problems and matrix function evaluations, the modified randomized Arnoldi process produces approximations that are identical to those obtained with the standard Arnoldi process. Numerical experiments demonstrate that our approach is as fast as the randomized Arnoldi process and as robust as the standard Arnoldi process. - oai:arXiv.org:2601.10248v1 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Laura Grigori, Daniel Kressner, Nian Shao, Igor Simunec - - - Critical time of the almost 2-regular random degree constrained process - https://arxiv.org/abs/2601.10249 - arXiv:2601.10249v1 Announce Type: new -Abstract: We study the phase transition of the random degree constrained process (RDCP), a time-evolving random graph model introduced by Ruci\'nski and Wormald that generalizes the random $d$-process to the non-regular setting: each vertex of the complete graph $K_n$ has its pre-assigned degree constraint (i.e., a number from the set $\{2,\dots,\Delta \}$), we attempt to add the edges one-by-one in a uniform random order, but a new edge is added only if it does not violate the degree constraints at its end-vertices. Warnke and Wormald identified the critical time of the RDCP when the giant component emerges as $n \to \infty$. R\'ath, Sz\H{o}ke and Warnke identified the local weak limit of the RDCP and gave an alternative characterization of the critical time in terms of the principal eigenvalue of the branching operator of the multi-type branching process that arises as the local limit object. - In the current paper we use this spectral characterization to study the critical time of the RDCP in the almost 2-regular case, i.e., when the degree constraint of most of the vertices is equal to 2. In this case the giant component emerges quite late, and our main result provides the precise asymptotics of the critical time as the model approaches 2-regularity. Interestingly, our formula asymptotically matches the well-known Molloy-Reed formula, despite the fact that Molloy, Surya and Warnke proved that the final graph of the RDCP is not contiguous to the configuration model with the same degree sequence. - oai:arXiv.org:2601.10249v1 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bal\'azs R\'ath, M\'arton Sz\H{o}ke - - - Error-Correcting Codes for the Sum Channel - https://arxiv.org/abs/2601.10256 - arXiv:2601.10256v1 Announce Type: new -Abstract: We introduce the sum channel, a new channel model motivated by applications in distributed storage and DNA data storage. In the error-free case, it takes as input an $\ell$-row binary matrix and outputs an $(\ell+1)$-row matrix whose first $\ell$ rows equal the input and whose last row is their parity (sum) row. We construct a two-deletion-correcting code with redundancy $2\lceil\log_2\log_2 n\rceil + O(\ell^2)$ for $\ell$-row inputs. When $\ell=2$, we establish an upper bound of $\lceil\log_2\log_2 n\rceil + O(1)$, implying that our redundancy is optimal up to a factor of 2. We also present a code correcting a single substitution with $\lceil \log_2(\ell+1)\rceil$ redundant bits and prove that it is within one bit of optimality. - oai:arXiv.org:2601.10256v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lyan Abboud, Eitan Yaakobi - - - Transmission Mask Analysis for Range-Doppler Sensing in Half-Duplex ISAC - https://arxiv.org/abs/2601.10259 - arXiv:2601.10259v1 Announce Type: new -Abstract: In this paper, we analyze the periodic transmission masks for MASked Modulation (MASM) in half-duplex integrated sensing and communication (ISAC), and derive their closed-form expected range-Doppler response $\mathbb{E}\{r(k,l,\nu)\}$. We show that range sidelobes ($k\neq l$) are Doppler-invariant, extending the range-sidelobe optimality to the 2-D setting. For the range mainlobe ($k=l$), periodic masking yields sparse Doppler sidelobes: Cyclic difference sets (CDSs) (in particular Singer CDSs) are minimax-optimal in a moderately dynamic regime, while in a highly dynamic regime the Doppler-sidelobe energy is a concave function of the mask autocorrelation, revealing an inevitable tradeoff with mainlobe fluctuation. - oai:arXiv.org:2601.10259v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Dikai Liu, Yifeng Xiong, Marco Lops, Fan Liu, Jianhua Zhang - - - Controllability score for linear time-invariant systems on an infinite time horizon - https://arxiv.org/abs/2601.10260 - arXiv:2601.10260v1 Announce Type: new -Abstract: We introduce a scaled controllability Gramian that can be computed reliably even for unstable systems. Using this scaled Gramian, we reformulate the controllability scoring problems into equivalent but numerically stable optimization problems. Their optimal solutions define dynamics-aware network centrality measures, referred to as the volumetric controllability score (VCS) and the average energy controllability score (AECS). We then formulate controllability scoring problems on an infinite time horizon. Under suitable assumptions, we prove that the resulting VCS and AECS are unique and that the finite-horizon scores converge to them. We further show that VCS and AECS can differ markedly in this limit, because VCS enforces controllability of the full system, whereas AECS accounts only for the stable modes. Finally, using Laplacian dynamics as a representative example, we present numerical experiments that illustrate this convergence. - oai:arXiv.org:2601.10260v1 - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kota Umezu, Kazuhiro Sato - - - Algebraic Properties of PAC Codes - https://arxiv.org/abs/2601.10262 - arXiv:2601.10262v1 Announce Type: new -Abstract: We analyze polarization-adjusted convolutional codes using the algebraic representation of polar and Reed-Muller codes. We define a large class of codes, called generalized polynomial polar codes which include PAC codes and Reverse PAC codes. We derive structural properties of generalized polynomial polar codes, such as duality, minimum distance. We also deduce some structural limits in terms of number of minimum weight codewords, and dimension of monomial sub-code. - oai:arXiv.org:2601.10262v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Vlad-Florin Dragoi, Mohammad Rowshan - - - Three-dimensional compact Heterotic solitons with parallel torsion - https://arxiv.org/abs/2601.10270 - arXiv:2601.10270v1 Announce Type: new -Abstract: We obtain a rigidity result for compact three-dimensional Heterotic solitons with parallel non-trivial torsion. We show that they are either hyperbolic three-manifolds or compact quotients of the Heisenberg group equipped with a left-invariant metric. In particular, the latter arise both as solitons with completely skew-symmetric torsion as well as with non-vanishing twistorial component. As a corollary, we obtain the universal bound $-24$ for the scalar curvature of Heterotic solitons with parallel skew-symmetric torsion, which prevents it from being arbitrarily large. - oai:arXiv.org:2601.10270v1 - math.DG - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Andrei Moroianu, Miguel Pino Carmona, C. S. Shahbazi - - - On a general identity and a resulting class of umbral operators - https://arxiv.org/abs/2601.10275 - arXiv:2601.10275v1 Announce Type: new -Abstract: We prove a new universal identity for umbral operators. This motivates the definition of a subclass obeying a simplified identity, which we then fully characterize. The results are illustrated with common examples of the theory of umbral calculus. - oai:arXiv.org:2601.10275v1 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Kei Beauduin - - - New Upper Bounds on the Ribbonlength of Alternating Links with Bipartite Dual Graphs - https://arxiv.org/abs/2601.10278 - arXiv:2601.10278v1 Announce Type: new -Abstract: The ribbonlength of a link is a geometric invariant defined as the infimum of the ratio of the length to the width of a folded ribbon realization of the link. In this paper, we prove that if an alternating link admits an alternating diagram with a bipartite dual graph, then its ribbonlength satisfies $$ \mathrm{Rib}(L) \le \sqrt{3} \, c(L). $$ Using this result, we present improved upper bounds on the ribbonlength for several knots and links with small crossing numbers, and determines the exact ribbonlength of the Hopf link to be $2\sqrt{3}$. - oai:arXiv.org:2601.10278v1 - math.GT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Hyungkee Yoo - - - Optimisation of the lowest Robin eigenvalue in exterior domains of the hyperbolic plane - https://arxiv.org/abs/2601.10280 - arXiv:2601.10280v1 Announce Type: new -Abstract: We consider the Robin Laplacian in the exterior of a bounded simply-connected Lipschitz domain in the hyperbolic plane. We show that the essential spectrum of this operator is $[\frac14,\infty)$ and that, under convexity assumption on the domain, there exist discrete eigenvalues below $\frac14$ if, and only if, the Robin parameter is below a non-positive critical constant, which depends on the shape of the domain. As the main result, we prove that the lowest Robin eigenvalue for the exterior of a bounded geodesically convex domain $\Omega$ in the hyperbolic plane does not exceed such an eigenvalue for the exterior of the geodesic disk, whose geodesic curvature of the boundary is not smaller than the averaged geodesic curvature of the boundary of $\Omega$. This result implies as a consequence that under fixed area or fixed perimeter constraints the exterior of the geodesic disk maximises the lowest Robin eigenvalue among exteriors of bounded geodesically convex domains. Moreover, we obtain under the same geometric constraints a reverse inequality between the critical constants. - oai:arXiv.org:2601.10280v1 - math.AP - math.OC - math.SP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Antonio Celentano, David Krejcirik, Vladimir Lotoreichik - - - On holonomy groups of K-contact sub-pseudo-Riemannian manifolds - https://arxiv.org/abs/2601.10286 - arXiv:2601.10286v1 Announce Type: new -Abstract: This article investigates the holonomy groups of K-contact sub-pseudo-Riemannian manifolds. The primary result is a proof that the horizontal holonomy group either coincides with the adapted holonomy group or acts as its normal subgroup of codimension one. The theory is adapted for metrics of indefinite signature, bypassing the problem of subspace degeneracy that previously prevented the use of established orthogonal decomposition methods. It is established that, in the sub-Lorentzian case, the adapted holonomy group corresponds to the holonomy group of a certain Lorentzian manifold. This work also provides a complete classification of codimension-one ideals for Lorentzian holonomy algebras and presents specific examples of structures based on Cahen-Wallach spaces and K\"ahler manifolds. - oai:arXiv.org:2601.10286v1 - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - E. A. Kokin - - - High-Contrast Transmission Resonances for the Lam\'e System - https://arxiv.org/abs/2601.10290 - arXiv:2601.10290v1 Announce Type: new -Abstract: We consider the Lam\'e transmission problem in $\mathbb{R}^3$ with a bounded isotropic elastic inclusion in a high-contrast setting, where the interior-to-exterior Lam\'e moduli and densities scale like $1/\tau$ as $\tau\to0$. We study the scattering resonances of the associated self-adjoint Hamiltonian, defined as the poles of the meromorphic continuation of its resolvent. - We obtain a sharp asymptotic description of resonances near the real axis as $\tau\to0$. Near each nonzero Neumann eigenvalue of the interior Lam\'e operator there is a cluster of resonances lying just below it in the complex plane; in this wavelength-scale regime the imaginary parts are of order $\tau$ with non-vanishing leading coefficients. In addition, near zero (a subwavelength regime), we identify resonances with real parts of order $\sqrt{\tau}$ and prove a lifetime dichotomy: their imaginary parts are of order $\tau$ generically, but of order $\tau^2$ for an explicit admissible set $\mathcal E$. This yields a classification of long-lived elastic resonances in the high-contrast limit. - We also establish resolvent asymptotics for both fixed-size resonators and microresonators. We derive explicit expansions with a finite-rank leading term and quantitative remainder bounds, valid near both wavelength-scale and subwavelength resonances. For microresonators, at the wavelength scale the dominant contribution is an anisotropic elastic point scatterer. Near the zero eigenvalue, the leading-order behaviour is of monopole or dipole type, and we give a rigorous criterion distinguishing the two cases. - oai:arXiv.org:2601.10290v1 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Long Li, Mourad Sini - - - Two dimensional covering systems and possible prime producing $a^m-b^n$ - https://arxiv.org/abs/2601.10296 - arXiv:2601.10296v1 Announce Type: new -Abstract: We exhibit a new application of two dimensional covering systems, examples of integer pairs $a,b$ for which $a^m-b^n$ has a prime divisor from some given finite set of primes, for every pair of integers $m,n\geq 0$. This leads us to conjecture what are the only possible obstructions to $|a^m-b^n|$ taking on infinitely many distinct prime values. - oai:arXiv.org:2601.10296v1 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Andrew Granville, Francesco Pappalardi - - - Global minimizers for a two-sided biharmonic Alt-Caffarelli problem - https://arxiv.org/abs/2601.10297 - arXiv:2601.10297v1 Announce Type: new -Abstract: We study global minimizers of biharmonic analogues of the Alt-Caffarelli functional. It turns out that half-space solutions are global minimizers for the two-sided Alt-Caffarelli functional, but not in the one-sided case. In addition, we identify a further class of global minimizers, all of which have constant Laplacian. Recent work by J. Lamboley and M. Nahon reduces potential global minimizers in dimension two to four possible categories. Our work shows that three of these categories persist in any dimension and are in fact global minimizers. - Moreover, we show that minimizers of the two-sided biharmonic Alt-Caffarelli problem do in general not satisfy a partial differential equation, not even with a signed measure as right-hand-side. This is in sharp contrast to the corresponding one-sided problem. - oai:arXiv.org:2601.10297v1 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Hans-Christoph Grunau, Marius M\"uller - - - Kummer-faithful fields with finitely generated absolute Galois group - https://arxiv.org/abs/2601.10298 - arXiv:2601.10298v1 Announce Type: new -Abstract: This paper studies the structure of the Mordell--Weil groups of semiabelian varieties over algebraic extensions of number fields whose absolute Galois group is finitely generated, with particular emphasis on that generated by a single element. A probabilistic argument using the Haar measure on the absolute Galois group of a number field shows that almost all such fields are Kummer-faithful, i.e., the Mordell--Weil group of any semiabelian variety over any finite extension of such a field has trivial divisible part. This result implies that there exists a Kummer-faithful field algebraic over a number field whose absolute Galois group is abelian. - oai:arXiv.org:2601.10298v1 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Takuya Asayama - - - On refinements of two-term Machin-like formulas - https://arxiv.org/abs/2601.10300 - arXiv:2601.10300v1 Announce Type: new -Abstract: We develop a refinement process for two-term Machin-like formulas: $a_0 \arctan{u_0} + a_1 \arctan{u_1} = \frac{\pi}{4}$ (where $a_0 , a_1 \in \mathbb{Z}$, $u_0 , u_1 \in \mathbb{Q}_+^*$, $u_0 > u_1$) by exploiting the continued fraction expansion of the ratio $\alpha := \frac{\arctan{u_0}}{\arctan{u_1}}$. This construction yields a sequence of derived two-term Machin-like formulas: $a_{- n} \arctan{u_n} + a_{- n + 1} \arctan{u_{n + 1}} = \frac{\pi}{4}$ ($n \in \mathbb{N}$) with positive rational arguments $u_n$ decreasing to zero and corresponding integer coefficients $a_{- n}$. We derive closed forms and estimates for $a_{-n}$ and $u_n$ in terms of the convergents of $\alpha$ and prove that the associated rational sequence $(a_{- n} u_n + a_{- n + 1} u_{n + 1})_n$ converges to $\pi/4$ with geometric decay. The method is illustrated using Euler's two-term Machin-like formula : $\arctan(1/2) + \arctan(1/3) = \pi/4$. - oai:arXiv.org:2601.10300v1 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Bakir Farhi - - - On Force Interactions for Electrodynamics-Like Theories - https://arxiv.org/abs/2601.10308 - arXiv:2601.10308v1 Announce Type: new -Abstract: A framework for premetric p-form electrodynamics is proposed. Independently of particular constitutive relations, the corresponding Maxwell equations are derived as a special case of stress theory in geometric continuum mechanics. Expressions for the potential energy of a charged region in spacetime, as well as expressions for the force and stress interactions on the region, are presented. The expression for the force distribution is obtained by computing the rate of change of the proposed potential energy under a virtual motion of the region. These expressions differ from those appearing in the standard references. The cases of electrostatics and magnetostatics in R^3 are presented as examples. - oai:arXiv.org:2601.10308v1 - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Vladimir Gol'dshtein, Reuven Segev - - - Deformations of Chow groups via cyclic homology - https://arxiv.org/abs/2601.10309 - arXiv:2601.10309v1 Announce Type: new -Abstract: Let $X$ be a smooth projective variety over an arbitrary field $k$ of characteristic zero. We explore infinitesimal deformations of the Chow group $CH^{p}(X)$ via its formal completion $\widehat{CH}^{p}$, a functor defined on the category of local augmented Artinian $k$-algebras. Under a natural vanishing condition on Hodge cohomology groups, for certain augmented graded Artinian $k$-algebras $A$ with the maximal ideal $m_{A}$, we prove that \[ \widehat{CH}^{p}(A) \cong H^{p}(X, \Omega^{p-1}_{X/ k})\otimes_{k}m_{A}. \]This extends earlier results of Bloch and others from the case where $k$ is algebraic over $\mathbb{Q}$ to arbitrary fields of characteristic zero,and gives a partial affirmative answer to a general question linking the pro-representability of Chow groups to a specific set of Hodge-theoretic vanishing conditions. - oai:arXiv.org:2601.10309v1 - math.AG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Sen Yang - - - Conjugate Gradient Methods are Not Efficient: Experimental Study of the Locality Limitation - https://arxiv.org/abs/2601.10322 - arXiv:2601.10322v1 Announce Type: new -Abstract: The convergence of the Conjugate Gradient method is subject to a locality limitation which imposes a lower bound on the number of iterations required before a qualitatively accurate approximation can be obtained. This limitation originates from the restricted transport of information in the graph induced by the sparsity pattern of the system matrix. In each iteration, information from the right-hand side can propagate only across directly connected graph nodes. The diameter of this graph therefore determines a minimum number of iterations that is necessary to achieve an acceptable level of accuracy. - oai:arXiv.org:2601.10322v1 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ulrich R\"ude - - - On gradient stability in nonlinear PDE models and inference in interacting particle systems - https://arxiv.org/abs/2601.10326 - arXiv:2601.10326v1 Announce Type: new -Abstract: We consider general parameter to solution maps $\theta \mapsto \mathcal G(\theta)$ of non-linear partial differential equations and describe an approach based on a Banach space version of the implicit function theorem to verify the gradient stability condition of Nickl&Wang (JEMS 2024) for the underlying non-linear inverse problem, providing also injectivity estimates and corresponding statistical identifiability results. We illustrate our methods in two examples involving a non-linear reaction diffusion system as well as a McKean--Vlasov interacting particle model, both with periodic boundary conditions. We apply our results to prove the polynomial time convergence of a Langevin-type algorithm sampling the posterior measure of the interaction potential arising from a discrete aggregate measurement of the interacting particle system. - oai:arXiv.org:2601.10326v1 - math.ST - cs.NA - math.AP - math.NA - math.PR - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Aur\'elien Castre, Richard Nickl - - - On the Capacity of Noisy Frequency-based Channels - https://arxiv.org/abs/2601.10329 - arXiv:2601.10329v1 Announce Type: new -Abstract: We investigate the capacity of noisy frequency-based channels, motivated by DNA data storage in the short-molecule regime, where information is encoded in the frequency of items types rather than their order. The channel output is a histogram formed by random sampling of items, followed by noisy item identification. While the capacity of the noiseless frequency-based channel has been previously addressed, the effect of identification noise has not been fully characterized. We present a converse bound on the channel capacity that follows from stochastic degradation and the data processing inequality. We then establish an achievable bound, which is based on a Poissonization of the multinomial sampling process, and an analysis of the resulting vector Poisson channel with inter-symbol interference. This analysis refines concentration inequalities for the information density used in Feinstein bound, and explicitly characterizes an additive loss in the mutual information due to identification noise. We apply our results to a DNA storage channel in the short-molecule regime, and quantify the resulting loss in the scaling of the total number of reliably stored bits. - oai:arXiv.org:2601.10329v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yuval Gerzon, Ilan Shomorony, Nir Weinberger - - - On the characterization of geometric distance-regular graphs - https://arxiv.org/abs/2601.10330 - arXiv:2601.10330v1 Announce Type: new -Abstract: In 2010, Koolen and Bang proposed the following conjecture: For a fixed integer $m \geq 2$, any geometric distance-regular graph with smallest eigenvalue $-m$, diameter $D \geq 3$ and $c_2 \geq 2$ is either a Johnson graph, a Grassmann graph, a Hamming graph, a bilinear forms graph, or the number of vertices is bounded above by a function of $m$. In this paper, we obtain some partial results towards this conjecture. - oai:arXiv.org:2601.10330v1 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chenhui Lv, Jack H. Koolen - - - Polymultiplicative maps associated with the algebra of Iterated Laurent series and the higher-dimensional Contou-Carrere Symbol - https://arxiv.org/abs/2601.10335 - arXiv:2601.10335v1 Announce Type: new -Abstract: We study functorial polymultiplicative maps from the multiplicative group of the algebra of $n$-times iterated Laurent series over a commutative ring in $n+1$ variables into the multiplicative group of the ring. It is proven that if such a map is invariant under continuous automorphisms of this algebra, then it coincides, up to a sign, with an integer power of the $n$-dimensional Contou-Carr\`ere symbol. - oai:arXiv.org:2601.10335v1 - math.AG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Levashev Vladislav - - - Triggered urn models for frequently asked questions (FAQ) - https://arxiv.org/abs/2601.10337 - arXiv:2601.10337v1 Announce Type: new -Abstract: We investigate a nonclassic urn model with triggers that increase the number of colors. The scheme has emerged as a model for web services that set up frequently asked questions (FAQ). We present a thorough asymptotic analysis of the FAQ urn scheme in generality that covers a large number of special cases, such as Simon urn. For instance, we consider time dependent triggering probabilities. We identify regularity conditions on these probabilities that classify the schemes into those where the number of colors in the urn remains almost surely finite or increases to infinity and conditions that tell us whether all the existing colors are observed infinitely often or not. We determine the rank curve, too. In view of the broad generality of the trigger probabilities, a spectrum of limit distributions appears, from central limit theorems to Poisson approximation, to power-laws, revealing connections to Heap's exponent and Zipf's law. A combinatorial approach to the Simon urn is presented to indicate the possibility of such exact analysis, which is important for short-term predictions. Extensive simulations on real datasets (from Amazon sales) as well as computer-generated data clearly indicate that the asymptotic and exact theory developed agrees with practice. - oai:arXiv.org:2601.10337v1 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Irene Crimaldi, Andrea Ghiglietti, Leen Hatem, Hosam Mahmoud - - - Convertible Codes for Data and Device Heterogeneity - https://arxiv.org/abs/2601.10341 - arXiv:2601.10341v1 Announce Type: new -Abstract: Distributed storage systems must handle both data heterogeneity, arising from non-uniform access demands, and device heterogeneity, caused by time-varying node reliability. In this paper, we study convertible codes, which enable the transformation of one code into another with minimum cost in the merge regime, addressing the latter. We derive general lower bounds on the read and write costs of linear code conversion, applicable to arbitrary linear codes. We then focus on Reed-Muller codes, which efficiently handle data heterogeneity, addressing the former issue, and construct explicit conversion procedures that, for the first time, combine both forms of heterogeneity for distributed data storage. - oai:arXiv.org:2601.10341v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Anina Gruica, Benjamin Jany, Stanislav Kruglik - - - Characteristic free Galois rings and generalized Weyl algebras - https://arxiv.org/abs/2601.10346 - arXiv:2601.10346v1 Announce Type: new -Abstract: This paper develops from scratch a theory of Galois rings and orders over arbitrary fields. Our approach is different from others in the literature in that there is no non-modularity assumption. We prove, when the field is algebraically closed, the analogue of the Main Theorem of the representation theory of Galois orders by V. Futorny and S. Ovsienko. Then we develop a theory of infinite rank generalized Weyl algebras, which was never explicitly introduced in the literature before, and prove its basic properties. We expect their representation theory to be of interest for future works. Finally we show that under very mild assumptions, the invariants of generalized Weyl algebras under the action of non-exceptional irreducible complex reflection groups are a principal Galois orders, greatly generalizing, in an elementary fashion, results obtained previously for the Weyl algebras. - oai:arXiv.org:2601.10346v1 - math.RT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Joao Schwarz - - - Waring's problem for pseudo-polynomials - https://arxiv.org/abs/2601.10351 - arXiv:2601.10351v1 Announce Type: new -Abstract: Waring's problem has a long history in additive number theory. In its original form it deals with the representability of every positive integer as sum of $k$-th powers with integer $k$. Instead of these powers we deal with pseudo-polynomials in this paper. A pseudo-polynomial is a ``polynomial'' with at least one exponent not being an integer. - Our work extends earlier results on the related problem of Waring for arbitrary real powers $k>12$ by Deshouillers and Arkhipov and Zhitkov. - oai:arXiv.org:2601.10351v1 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Manfred G. Madritsch - - - A New Construction Structure on MISO Coded Caching with Linear Subpacketization: Half-Sum Disjoint Packing - https://arxiv.org/abs/2601.10353 - arXiv:2601.10353v1 Announce Type: new -Abstract: In the $(L,K,M,N)$ cache-aided multiple-input single-output (MISO) broadcast channel (BC) system, the server is equipped with $L$ antennas and communicates with $K$ single-antenna users through a wireless broadcast channel where the server has a library containing $N$ files, and each user is equipped with a cache of size $M$ files. Under the constraints of uncoded placement and one-shot linear delivery strategies, many schemes achieve the maximum sum Degree-of-Freedom (sum-DoF). However, for general parameters $L$, $M$, and $N$, their subpacketizations increase exponentially with the number of users. We aim to design a MISO coded caching scheme that achieves a large sum-DoF with low subpacketization $F$. An interesting combinatorial structure, called the multiple-antenna placement delivery array (MAPDA), can be used to generate MISO coded caching schemes under these two strategies; moreover, all existing schemes with these strategies can be represented by the corresponding MAPDAs. In this paper, we study the case with $F=K$ (i.e., $F$ grows linearly with $K$) by investigating MAPDAs. Specifically, based on the framework of Latin squares, we transform the design of MAPDA with $F=K$ into the construction of a combinatorial structure called the $L$-half-sum disjoint packing (HSDP). It is worth noting that a $1$-HSDP is exactly the concept of NHSDP, which is used to generate the shared-link coded caching scheme with $F=K$. By constructing $L$-HSDPs, we obtain a class of new schemes with $F=K$. Finally, theoretical and numerical analyses show that our $L$-HSDP schemes significantly reduce subpacketization compared to existing schemes with exponential subpacketization, while only slightly sacrificing sum-DoF, and achieve both a higher sum-DoF and lower subpacketization than the existing schemes with linear subpacketization. - oai:arXiv.org:2601.10353v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Bowen Zheng, Minquan Cheng, Kai Wan, Giuseppe Caire - - - An It\^o Formula via Predictable Projection for Non-Semimartingale Processes - https://arxiv.org/abs/2601.10359 - arXiv:2601.10359v1 Announce Type: new -Abstract: We derive an It\^o-type change-of-variables formula for a class of adapted stochastic processes that do not necessarily admit semimartingale structure. The formulation is based on an intrinsic Hilbert-space derivative together with a predictable projection operator, allowing stochastic integrals to be expressed without reliance on quadratic variation or anticipative calculus. - The resulting formula replaces the classical quadratic variation term with a computable second-order contribution expressed as a norm of the projected derivative. In the semimartingale case, the formula reduces to the classical It\^o formula. The approach applies naturally to processes with memory and non-Markovian dependence, providing a unified and intrinsic framework for stochastic calculus beyond the semimartingale setting. - oai:arXiv.org:2601.10359v1 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ramiro Fontes - - - On UC-multipliers for multiple trigonometric systems - https://arxiv.org/abs/2601.10360 - arXiv:2601.10360v1 Announce Type: new -Abstract: We investigate the class of sequences $w(n)$ that can serve as almost-everywhere convergence Weyl multipliers for all rearrangements of multiple trigonometric systems. We show that any such sequence must satisfy the bounds $\log n\lesssim w(n)\lesssim\log^2 n$. Our main result establishes a general equivalence principle between one-dimensional and multidimensional trigonometric systems, which allows one to extend certain estimates known for the one-dimensional case to higher dimensions. - oai:arXiv.org:2601.10360v1 - math.CA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Grigori A. Karagulyan - - - Generalized Weight Structure of Polar Codes: Selected Template Polynomials - https://arxiv.org/abs/2601.10362 - arXiv:2601.10362v1 Announce Type: new -Abstract: Polar codes can be viewed as decreasing monomial codes, revealing a rich algebraic structure governed by the lower-triangular affine (LTA) group. We develop a general framework to compute the Hamming weight of codewords generated by sums of monomials, express these weights in a canonical dyadic form, and derive closed expressions for key structural templates (disjoint sums, nested blocks, complementary flips) that generate the low and intermediate weight spectrum. Combining these templates with the LTA group action, we obtain explicit multiplicity formulas, yielding a unified algebraic method to characterize and enumerate codewords. - oai:arXiv.org:2601.10362v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Mohammad Rowshan, Vlad-Florin Dragoi - - - A two-step inertial method with a new step-size rule for variational inequalities in hilbert spaces - https://arxiv.org/abs/2601.10370 - arXiv:2601.10370v1 Announce Type: new -Abstract: In this paper, a two-step inertial Tseng extragradient method involving self-adaptive and Armijo-like step sizes is introduced for solving variational inequalities with a quasimonotone cost function in the setting of a real Hilbert space. Weak convergence of the sequence generated by the proposed algorithm is proved without assuming the Lipschitz condition. An interesting feature of the proposed algorithm is its ability to select the better step size between the self-adaptive and Armijo-like options at each iteration step. Moreover, removing the requirement for the Lipschitz condition on the cost function broadens the applicability of the proposed method. Finally, the algorithm accelerates and complements several existing iterative algorithms for solving variational inequalities in Hilbert spaces. - oai:arXiv.org:2601.10370v1 - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Jian-Wen Peng, Jun-Jie Luo, Abubakar Adamu - - - A Hybrid Reliability--Weight Framework for Construction of Polar Codes - https://arxiv.org/abs/2601.10376 - arXiv:2601.10376v1 Announce Type: new -Abstract: Polar codes are usually constructed by ranking synthetic bit-channels according to reliability, which guarantees capacity-achieving behavior but can yield poor low-weight spectra at short and moderate lengths. Recent algebraic results express the contribution of individual bit-channels to the multiplicities of minimum and near-minimum weight codewords in closed form. In this work we combine these insights into a mixed (reliability--weight) bit-channel ordering. We define a per-bit cost whose distance term is derived from orbit enumeration of minimum-weight codewords and scaled by a Bhattacharyya-type factor, and show that the resulting mixed construction minimises a truncated SC/ML union-bound surrogate within a class of decreasing monomial codes. We relate the mixed metric to error events in SCL decoding via a pruning/ML decomposition, and prove that mixed designs act as local perturbations of reliability-based constructions whose asymptotic impact vanishes as code-length approaches infinity. Numerical results for short and moderate lengths on BPSK-AWGN, implemented via Gaussian approximation and closed-form weight contributions, illustrate the trade-off between pure reliability-based and mixed constructions in terms of minimum distance, multiplicity, and union-bound approximations. All proofs are deferred to the appendices. - oai:arXiv.org:2601.10376v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Mohammad Rowshan, Vlad-Florin Dragoi - - - On surgeries from lens space $L(p,1)$ to $L(q,2)$ - https://arxiv.org/abs/2601.10377 - arXiv:2601.10377v1 Announce Type: new -Abstract: We mainly use the d-invariant surgery formula established by Wu and Yang \cite{wu2025surgerieslensspacestype} to study the distance one surgeries along a homologically essential knot between lens spaces of the form $L(p,1)$ and $L(q,2)$ where $p,q$ are odd integers. - oai:arXiv.org:2601.10377v1 - math.GT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Boning Wang - - - Phase Space structure on Clifford Algebras - https://arxiv.org/abs/2601.10381 - arXiv:2601.10381v1 Announce Type: new -Abstract: I argue that the Hodge structure on a Euclidean Clifford algebra $Cl(n)$ provides a way to generalise K\"ahler structure to higher dimensions, in the sense that the paired variables are now associated with $k-$ and $(n-k)-$ dimensional subspaces rather than with vectors. This puts a phase space structure on Clifford algebras, and so allows us to construct a Hamiltonian dynamics on these multilinear variables. This construction shows that alternating pairs of subspaces obey commuting and anticommuting dynamics, hinting that this construction is indeed a natural one, with interesting new behaviour. - oai:arXiv.org:2601.10381v1 - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - C. Robson - - - Regularization of linear inverse problems by rational Krylov methods - https://arxiv.org/abs/2601.10389 - arXiv:2601.10389v1 Announce Type: new -Abstract: For approximately solving linear ill-posed problems in Hilbert spaces, we investigate the regularization properties of the aggregation method and the RatCG method. These recent algorithms use previously calculated solutions of Tikhonov regularization (respectively, Landweber iterations) to set up a new search space on which the least-squares functional is minimized. We outline how these methods can be understood as rational Krylov space methods, i.e., based on the space of rational functions of the forward operator. The main result is that these methods form an optimal-order regularization schemes when combined with the discrepancy principle as stopping rule and when the underlying regularization parameters are sufficiently large. - oai:arXiv.org:2601.10389v1 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Stefan Kindermann - - - Algebraic Farkas Lemma and Strong Duality for Perturbed Conic Linear Programming - https://arxiv.org/abs/2601.10390 - arXiv:2601.10390v1 Announce Type: new -Abstract: This paper addresses the study of algebraic versions of Farkas lemma and strong duality results in the very broad setting of infinite-dimensional conic linear programming in dual pairs of vector spaces. To this end, purely algebraic properties of perturbed optimal value functions of both primal and dual problems and their corresponding hypergraph/epigraph are investigated. The newly developed hypergraphical/epigraphical sets, inspired by Kretschmer's closedness conditions \cite{Kretschmer61}, together with their novel convex separation-type characterizations, give rise to various perturbed Farkas-type lemmas which allow us to derive complete characterizations of ``zero duality gap''. Principally, when certain structures of algebraic or topological duals are imposed, illuminating implications of the developed condition are also explored. - oai:arXiv.org:2601.10390v1 - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/publicdomain/zero/1.0/ - P. D. Khanh, V. V. H. Khoa, T. H. Mo - - - Codebook Design for Limited Feedback in Near-Field XL-MIMO Systems - https://arxiv.org/abs/2601.10391 - arXiv:2601.10391v1 Announce Type: new -Abstract: In this paper, we study efficient codebook design for limited feedback in extremely large-scale multiple-input-multiple-output (XL-MIMO) frequency division duplexing (FDD) systems. It is worth noting that existing codebook designs for XL-MIMO, such as polar-domain codebook, have not well taken into account user (location) distribution in practice, thereby incurring excessive feedback overhead. To address this issue, we propose in this paper a novel and efficient feedback codebook tailored to user distribution. To this end, we first consider a typical scenario where users are uniformly distributed within a specific polar-region, based on which a sum-rate maximization problem is formulated to jointly optimize angle-range samples and bit allocation among angle/range feedback. This problem is challenging to solve due to the lack of a closed-form expression for the received power in terms of angle and range samples. By leveraging a Voronoi partitioning approach, we show that uniform angle sampling is optimal for received power maximization. For more challenging range sampling design, we obtain a tight lower-bound on the received power and show that geometric sampling, where the ratio between adjacent samples is constant, can maximize the lower bound and thus serves as a high-quality suboptimal solution. We then extend the proposed framework to accommodate more general non-uniform user distribution via an alternating sampling method. Furthermore, theoretical analysis reveals that as the array size increases, the optimal allocation of feedback bits increasingly favors range samples at the expense of angle samples. Finally, numerical results validate the superior rate performance and robustness of the proposed codebook design under various system setups, achieving significant gains over benchmark schemes, including the widely used polar-domain codebook. - oai:arXiv.org:2601.10391v1 - cs.IT - eess.SP - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Liujia Yao, Changsheng You, Zixuan Huang, Chao Zhou, Zhaohui Yang, Xiaoyang Li - - - Multiaccess Coded Caching with Heterogeneous Retrieval Costs - https://arxiv.org/abs/2601.10394 - arXiv:2601.10394v1 Announce Type: new -Abstract: The multiaccess coded caching (MACC) system, as formulated by Hachem {\it et al.}, consists of a central server with a library of $N$ files, connected to $K$ cache-less users via an error-free shared link, and $K$ cache nodes, each equipped with cache memory of size $M$ files. Each user can access $L$ neighboring cache nodes under a cyclic wrap-around topology. Most existing studies operate under the strong assumption that users can retrieve content from their connected cache nodes at no communication cost. In practice, each user retrieves content from its $L$ different connected cache nodes at varying costs. Additionally, the server also incurs certain costs to transmit the content to the users. In this paper, we focus on a cost-aware MACC system and aim to minimize the total system cost, which includes cache-access costs and broadcast costs. Firstly, we propose a novel coded caching framework based on superposition coding, where the MACC schemes of Cheng \textit{et al.} are layered. Then, a cost-aware optimization problem is derived that optimizes cache placement and minimizes system cost. By identifying a sparsity property of the optimal solution, we propose a structure-aware algorithm with reduced complexity. Simulation results demonstrate that our proposed scheme consistently outperforms the scheme of Cheng {\it et al.} in scenarios with heterogeneous retrieval costs. - oai:arXiv.org:2601.10394v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Wenbo Huang, Minquan Cheng, Kai Wan, Xiaojun Li, Robert Caiming Qiu, Giuseppe Caire - - - A Geometric Multigrid Preconditioner for Shifted Boundary Method - https://arxiv.org/abs/2601.10399 - arXiv:2601.10399v1 Announce Type: new -Abstract: The Shifted Boundary Method (SBM) trades some part of the burden of body-fitted meshing for increased algebraic complexity. While the resulting linear systems retain the standard $\mathcal{O}(h^{-2})$ conditioning of second-order operators, the non-symmetry and non-local boundary coupling render them resistant to standard Algebraic Multigrid (AMG) and simple smoothers for high-order discretisations. We present a geometric multigrid preconditioner that effectively tames these systems. At its core lies the \emph{Full-Residual Shy Patch} smoother: a subspace correction strategy that filters out some patches while capturing the full physics of the shifted boundary. Unlike previous cell-wise approaches that falter at high polynomial degrees, our method delivers convergence with low mesh dependence. We demonstrate performance for Continuous Galerkin approximations, maintaining low and stable iteration counts up to polynomial degree $p=3$ in 3D, proving that SBM can be both geometrically flexible and algebraically efficient. - oai:arXiv.org:2601.10399v1 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Micha{\l} Wichrowski, Ajay Ajith - - - A proof of Alexander's conjecture on an inequality of Cassels - https://arxiv.org/abs/2601.10411 - arXiv:2601.10411v1 Announce Type: new -Abstract: Let $z_1,\dots,z_n$ be complex numbers with $|z_j|\le \rho$, where $\rho>1$. Cassels proved that, under an additional restriction on $\rho$, the inequality \[ \prod_{j\ne k}\bigl|1-\overline{z_j}z_k\bigr| \le \left(\frac{\rho^{2n}-1}{\rho^2-1}\right)^{\!n} \] holds. In a subsequent note, Alexander conjectured that this inequality is in fact valid without any restriction on $\rho$. In this paper, we confirm Alexander's conjecture. - oai:arXiv.org:2601.10411v1 - math.CV - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Myriam Ouna\"ies - - - Optimality in nonlocal time-dependent obstacle problems - https://arxiv.org/abs/2601.10417 - arXiv:2601.10417v1 Announce Type: new -Abstract: This paper showcases the effectiveness of the quasiconvexity property in addressing the optimal regularity of the temporal derivative and establishes conditions for its continuity in nonlocal time-dependent obstacle problems. - oai:arXiv.org:2601.10417v1 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ioannis Athanasopoulos, Luis Caffarelli, Emmanouil Milakis - - - On the Canonical Construction of Simple Lie Superalgebras - https://arxiv.org/abs/2601.10419 - arXiv:2601.10419v1 Announce Type: new -Abstract: Axioms for the generalization of root systems were defined and classified (irreducible) by V. Serganova, which precisely correspond to the root systems of basic classical Lie Superalgebras. Here, we present a unified method for constructing simple Lie Superalgebras from the abstract root system, with the choice of base having the minimal number of isotropic roots. - oai:arXiv.org:2601.10419v1 - math.RA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - J. Dhamothiran, Saudamini Nayak - - - Placement Delivery Array for Cache-Aided MIMO Systems - https://arxiv.org/abs/2601.10422 - arXiv:2601.10422v1 Announce Type: new -Abstract: We consider a $(G,L,K,M,N)$ cache-aided multiple-input multiple-output (MIMO) network, where a server equipped with $L$ antennas and a library of $N$ equal-size files communicates with $K$ users, each equipped with $G$ antennas and a cache of size $M$ files, over a wireless interference channel. Each user requests an arbitrary file from the library. The goal is to design coded caching schemes that simultaneously achieve the maximum sum degrees of freedom (sum-DoF) and low subpacketization. In this paper, we first introduce a unified combinatorial structure, termed the MIMO placement delivery array (MIMO-PDA), which characterizes uncoded placement and one-shot zero-forcing delivery. By analyzing the combinatorial properties of MIMO-PDAs, we derive a sum-DoF upper bound of $\min\{KG, Gt+G\lceil L/G \rceil\}$, where $t=KM/N$, which coincides with the optimal DoF characterization in prior work by Tehrani \emph{et al.}. Based on this upper bound, we present two novel constructions of MIMO-PDAs that achieve the maximum sum-DoF. The first construction achieves linear subpacketization under stringent parameter constraints, while the second achieves ordered exponential subpacketization under substantially milder constraints. Theoretical analysis and numerical comparisons demonstrate that the second construction exponentially reduces subpacketization compared to existing schemes while preserving the maximum sum-DoF. - oai:arXiv.org:2601.10422v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Yifei Huang, Kai Wan, Minquan Cheng, Jinyan Wang, Giuseppe Caire - - - Positivity of Schur forms for Griffiths positive vector bundles of rank three over complex threefolds - https://arxiv.org/abs/2601.10424 - arXiv:2601.10424v1 Announce Type: new -Abstract: In this paper, we prove the positivity of the double mixed discriminant associated with a positive linear map between spaces of \(3\times 3\) complex matrices, thereby settling the three-dimensional case of Finski's open problem. As an application, we show that all Schur forms are weakly positive for Griffiths positive Hermitian holomorphic vector bundles of rank three over complex threefolds. This yields a complete affirmative answer, in the case where both the rank and the dimension are three, to the question posed by Griffiths in 1969. - oai:arXiv.org:2601.10424v1 - math.AG - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xueyuan Wan - - - Algebraic functional equation for big Galois representations over multiple $\mathbb{Z}_p$-extensions - https://arxiv.org/abs/2601.10426 - arXiv:2601.10426v1 Announce Type: new -Abstract: We present a general approach to establish algebraic functional equations for big Galois representations over multiple $\mathbb{Z}_p$-extensions. Our result is formulated in both Selmer group and Selmer complex settings, and encompasses a broad range of Iwasawa-theoretic scenarios. In particular, our result applies to the triple product of Hida families in both balanced and unbalanced cases, as well as the half-ordinary Rankin-Selberg universal deformations recently studied by the first named author and Loeffler. Our result also significantly generalizes many previously known cases of algebraic functional equations and answers a question of Greenberg. - oai:arXiv.org:2601.10426v1 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Zeping Hao, Meng Fai Lim - - - Geometric characterization of frictional impacts by means of breakable kinetic constraints - https://arxiv.org/abs/2601.10432 - arXiv:2601.10432v1 Announce Type: new -Abstract: In the context of geometric Impulsive Mechanics of systems with a finite number of degrees of freedom, we model the roughness of a unilateral constraint ${\mathcal S\/}$ by introducing a suitable instantaneous kinetic constraint ${\mathcal B\/}\subset {\mathcal S\/}$. A constitutive characterization of ${\mathcal B\/}$ based only on the geometric properties of the setup and on the dry friction laws can then be introduced to model the frictional behavior of ${\mathcal S\/}$ in an impact of the system. Such a model restores determinism and avoids the analysis of frictional forces in the contact point, with all its associated theoretical problems of causality. Three examples of increasing complexity, showing a natural stick--slip behavior of the impact, are presented. - oai:arXiv.org:2601.10432v1 - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Stefano Pasquero - - - Linear identities for partition pairs with $4$-cores - https://arxiv.org/abs/2601.10438 - arXiv:2601.10438v1 Announce Type: new -Abstract: We determine an infinite family of linear identities for the number $A_4(n)$ of partition pairs of $n$ with $4$-cores by employing elementary $q$-series techniques and certain $3$-dissection formulas. We then discover an infinite family of congruences for $A_4(n)$ as a consequence of these linear identities. - oai:arXiv.org:2601.10438v1 - math.NT - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Russelle Guadalupe - - - Non-Intrusive Hyperreduction by a Physics-Augmented Neural Network with Second-Order Sobolev Training - https://arxiv.org/abs/2601.10442 - arXiv:2601.10442v1 Announce Type: new -Abstract: The finite element method is an indispensable tool in engineering, but its computational complexity prevents applications for control or at system-level. Model order reduction bridges this gap, creating highly efficient yet accurate surrogate models. Reducing nonlinear setups additionally requires hyperreduction. Compatibility with commercial finite element software requires non-intrusive methods based on data. Methods include the trajectory piecewise linear approach, or regression, typically via neural networks. Important aspects for these methods are accuracy, efficiency, generalization, including desired physical and mathematical properties, and extrapolation. Especially the last two aspects are problematic for neural networks. Therefore, several studies investigated how to incorporate physical knowledge or desirable properties. A promising approach from constitutive modeling is physics augmented neural networks. This concept has been elegantly transferred to hyperreduction by Fleres et al. in 2025 and guarantees several desired properties, incorporates physics, can include parameters, and results in smaller architectures. We augment this reference work by second-order Sobolev training, i.e., using a function and its first two derivatives. These are conveniently accessible and promise improved performance. Further modifications are proposed and studied. While Sobolev training does not meet expectations, several minor changes improve accuracy by up to an order of magnitude. Eventually, our best model is compared to reference work and the trajectory piecewise linear approach. The comparison relies on the same numerical case study as the reference work and additionally emphasizes extrapolation due to its critical role in typical applications. Our results indicate quick divergence of physics-augmented neural networks for extrapolation, preventing its deployment. - oai:arXiv.org:2601.10442v1 - math.DS - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Arwed Sch\"utz, Lars Nolle, Tamara Bechtold - - - Umbral theory and the algebra of formal power series - https://arxiv.org/abs/2601.10443 - arXiv:2601.10443v1 Announce Type: new -Abstract: Umbral theory, formulated in its modern version by S. Roman and G.~C. Rota, has been reconsidered in more recent times by G. Dattoli and collaborators with the aim of devising a working computational tool in the framework of special function theory. Concepts like umbral image and umbral vacuum have been introduced as pivotal elements of the discussion, which, albeit effective, lacks of generality. - This article is directed towards endowing the formalism with a rigorous formulation within the context of the formal power series with complex coefficients $(\mathbb{C}[[ t ]], \partial)$. The new formulation is founded on the definition of the umbral operator $\operatorname{\mathfrak{u}}$ as a functional in the "umbral ground state" subalgebra of analytically convergent formal series $\varphi \in \mathbb{C}\{t\}$. - We consider in detail some specific classes of umbral ground states $\varphi$ and analyse the conditions for analytic convergence of the corresponding umbral identities, defined as formal series resulting from the action on $\varphi$ of operators of the form $f(\zeta \operatorname{\mathfrak{u}}^\mu)$ with $f \in \mathbb{C}\{t\}$ and $\mu, \zeta \in \mathbb{C}$. For these umbral states, we exploit the Gevrey classification of formal power series to establish a connection with the theory of Borel-Laplace resummation, enabling to make rigorous sense of a large class of -- even divergent -- umbral identities. - As an application of the proposed theoretical framework, we introduce and investigate the properties of new umbral images for the Gaussian trigonometric functions, which emphasise the trigonometric-like nature of these functions and enable to define the concept of "Gaussian Fourier transform", a potentially powerful tool for applications. - oai:arXiv.org:2601.10443v1 - math.CA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Roberto Ricci (ENEA, Nuclear Department NUC-DTT, Frascati Research Center, Via E. Fermi 45, 00044 Frascati RM Italy) - - - Energy-Efficient Probabilistic Semantic Communication Over Visible Light Networks With Rate Splitting - https://arxiv.org/abs/2601.10452 - arXiv:2601.10452v1 Announce Type: new -Abstract: Visible light communication (VLC) is emerging as a key technology for future wireless communication systems due to its unique physical-layer advantages over traditional radio-frequency (RF)-based systems. However, its integration with higher-layer techniques, such as semantic communication, remains underexplored. This paper investigates the energy efficiency maximization problem in a resource-constrained VLC-based probabilistic semantic communication (PSCom) system. In the considered model, light-emitting diode (LED) transmitters perform semantic compression to reduce data size, which incurs additional computation overhead. The compressed semantic information is transmitted to the users for semantic inference using a shared knowledge base that requires periodic updates to ensure synchronization. In the PSCom system, the knowledge base is represented by probabilistic graphs. To enable simultaneous transmission of both knowledge and information data, rate splitting multiple access (RSMA) is employed. The optimization problem focuses on maximizing energy efficiency by jointly optimizing transmit beamforming, direct current (DC) bias, common rate allocation, and semantic compression ratio, while accounting for both communication and computation costs. To solve this problem, an alternating optimization algorithm based on successive convex approximation (SCA) and Dinkelbach method is developed. Simulation results demonstrate the effectiveness of the proposed approach. - oai:arXiv.org:2601.10452v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zhouxiang Zhao, Zhaohui Yang, Mingzhe Chen, Chen Zhu, Xin Tong, Zhaoyang Zhang - - - Symmetric spaces, non-formal star products and Drinfel'd twists - https://arxiv.org/abs/2601.10456 - arXiv:2601.10456v1 Announce Type: new -Abstract: These notes refer to a minicourse I gave at the occasion of the conference meeting ``Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Space-Time'' to be held from 7 April to 11 April 2025 at the Centre International de Rencontres Math\'ematiques in Luminy. They consist in a review of a long standing work of mine and collaborators (see references therein) in the field of non-formal deformation quantization admitting a large group of symmetries. But they also contain new material and results. More precisely, in a first part, I present a method (called the Retract Method) to define quantizations/symbolic calculi and associated operator symbol composition formulae (non-formal deformations/star products) of symplectic symmetric spaces such as the hyperbolic plane (Kahler) or symmetric co-adjoint orbits of the Poincar\'e group (non-metric). In a second part, I explain how to derive non-formal Drinfel'd twists for actions of non-Abelian solvable Lie groups (non-Abelian Universal Deformation Formulae) on or Fr \'echet algebras from the non-formal noncommutative symmetric spaces defined in the first part. - oai:arXiv.org:2601.10456v1 - math.QA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pierre Bieliavsky - - - The Wiener Wintner and Return Times Theorem Along the Primes - https://arxiv.org/abs/2601.10459 - arXiv:2601.10459v1 Announce Type: new -Abstract: We prove the following Return Times Theorem along the sequence of prime times, the first extension of the Return Times Theorem to arithmetic sequences: - For every probability space, $(\Omega,\nu)$, equipped with a measure-preserving transformation, $T \colon \Omega \to \Omega$, and every $f \in L^\infty(\Omega)$, there exists a set of full probability, $\Omega_f \subset \Omega$ with $\nu(\Omega_f) =1$, so that for all $\omega \in \Omega_f$, for any other probability space $(X,\mu)$, equipped with a measure-preserving transformation $S : X \to X$, for any $g \in L^{\infty}(X)$, - \begin{align} - \frac{1}{N} \sum_{n \leq N} f(T^{p_n} \omega) g(S^{p_n} \cdot) - \end{align} - converges $\mu$-almost surely; above, $\{ 2=p_1 < p_2 < \dots \}$ are an enumeration of the primes. The Wiener-Wintner theorem along the primes - is an immediate corollary. - Our proof lives at the interface of classical Fourier analysis, combinatorial number theory, higher order Fourier analysis, and pointwise ergodic theory, with $U^3$ theory playing an important role; our $U^3$-estimates for \emph{Heath-Brown} models of the von Mangoldt function may be of independent interest. - oai:arXiv.org:2601.10459v1 - math.DS - math.CA - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Jan Fornal, Anastasios Fragkos, Ben Krause, Michael Lacey, Hamed Mousavi, Yu-Chen Sun - - - On the projective dimension of some deformations of Weyl arrangements - https://arxiv.org/abs/2601.10466 - arXiv:2601.10466v1 Announce Type: new -Abstract: We show that the logarithmic derivation module of (the cone of) the deformation A of a Weyl arrangement associated with a root system of simply laced type has projective dimension one if the deforming parameter ranges from -j to j+2. In addition, we give an explicit minimal free resolution when the root system is of type A3 and B2. Moreover, in the second case, we determine the jumping lines of maximal jumping order of the associated vector bundle. When the deforming parameter of A (respectively A') ranges from -k to k+j (respectively, from -k' to k'+j), with k different from k' and j at least 3, this allows to distinguish D0(A) from D0(A') shifted by 4(k'-k), even though these modules have the same graded Betti numbers. - oai:arXiv.org:2601.10466v1 - math.AG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Takuro Abe (UBE, IMB), Daniele Faenzi (UBE, IMB) - - - Joint Source-Channel Coding for ISAC: Distortion Tradeoffs and Separation Theorems - https://arxiv.org/abs/2601.10470 - arXiv:2601.10470v1 Announce Type: new -Abstract: Integrated Sensing and Communication (ISAC) systems have garnered significant attention due to their capability to simultaneously achieve efficient communication and environmental sensing. A core objective in this field is characterizing the performance tradeoff between sensing and communication. In this paper, we consider a joint source-channel coding (JSCC) framework for the ISAC system that consists of a transmitter with a channel state estimator and a joint source-channel encoder, a state-dependent memoryless channel, and a receiver with a joint source-channel decoder. From an information-theoretic perspective, we establish the tradeoff relationships among channel capacity, distortions in both communication and sensing processes, and the estimation cost. We prove that the separate source and channel coding can achieve joint optimality in this setting. An illustrative example of a binary setting is also provided to validate our theoretical results. - oai:arXiv.org:2601.10470v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gefei Peng, Youlong Wu - - - Optimal error estimates for a discontinuous Galerkin method on curved boundaries with polygonal meshes - https://arxiv.org/abs/2601.10474 - arXiv:2601.10474v1 Announce Type: new -Abstract: We consider a discontinuous Galerkin method for the numerical solution of boundary value problems in two-dimensional domains with curved boundaries. A key challenge in this setting is the potential loss of convergence order due to approximating the physical domain by a polygonal mesh. Unless boundary conditions can be accurately transferred from the true boundary to the computational one, such geometric approximation errors generally lead to suboptimal convergence. To overcome this limitation, a higher-order strategy based on polynomial reconstruction of boundary data was introduced for classical finite element methods in [28, 29] and in the finite volume context in [7, 11]. More recently, this approach was extended to discontinuous Galerkin methods in [32], leading to the DG-ROD method, which restores optimal convergence rates on polygonal approximations of domains with curved boundaries. In this work, we provide a rigorous theoretical analysis of the DG-ROD method, establishing existence and uniqueness of the discrete solution and deriving error estimates for a two-dimensional linear advection-diffusion-reaction problem with homogeneous Dirichlet boundary conditions on both convex and non-convex domains. Following and extending techniques from classical finite element methods [29], we prove that, under suitable regularity assumptions on the exact solution, the DG-ROD method achieves optimal convergence despite polygonal approximations. Finally, we illustrate and confirm the theoretical results with a numerical benchmark. - oai:arXiv.org:2601.10474v1 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Ad\'erito Ara\'ujo, Milene Santos - - - Positive Damping Region: A Graphic Tool for Passivization Analysis with Passivity Index - https://arxiv.org/abs/2601.10475 - arXiv:2601.10475v1 Announce Type: new -Abstract: This paper presents a geometric framework for analyzing output-feedback and input-feedforward passivization of linear time-invariant systems. We reveal that a system is passivizable with a given passivity index when the Nyquist plot for SISO systems or the Rayleigh quotient of the transfer function for MIMO systems lies within a specific, index-dependent region in the complex plane, termed the positive damping region. The criteria enable a convenient graphic tool for analyzing the passivization, the associated frequency bands, the maximum achievable passivity index, and the waterbed effect between them. Additionally, the tool can be encoded into classical tools such as the Nyquist plot, the Nichols plot, and the generalized KYP lemma to aid control design. Finally, we demonstrate its application in passivity-based power system stability analysis and discuss its implications for electrical engineers regarding device controller design trade-offs. - oai:arXiv.org:2601.10475v1 - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Xiaoyu Peng, Xi Ru, Zhongze Li, Jianxin Zhang, Xinghua Chen, Feng Liu - - - On Generalized Strong and Norm Resolvent Convergence - https://arxiv.org/abs/2601.10476 - arXiv:2601.10476v1 Announce Type: new -Abstract: We present a streamlined approach for generalized strong and norm convergence of self-adjoint operators in different Hilbert spaces. In particular, we establish convergence of associated (semi-)groups, (essential) spectra and spectral projections. In addition, we give some applications to Sturm-Liouville operators. - oai:arXiv.org:2601.10476v1 - math.SP - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gerald Teschl, Yifei Wang, Bing Xie, Zhe Zhou - - - A Riemannian Autocorrelation Function and its Application to Non-Local Isoperimetric Energies - https://arxiv.org/abs/2601.10481 - arXiv:2601.10481v1 Announce Type: new -Abstract: We study a family of non-local isoperimetric energies $E_{\gamma,\varepsilon}$ on the round sphere $M = S^n$, where the non-local interaction kernel $K_\varepsilon$ is the fundamental solution of the Helmholtz operator $1 - \varepsilon^2 \Delta$. To analyse these energies, we introduce a Riemannian autocorrelation function $c_\Omega$ associated to a measurable set $\Omega\subset M$, defined on any compact, connected, oriented Riemannian manifold without boundary $(M^n,g)$ of dimension $n\ge2$. This function is intimately linked to Matheron's set covariogram from convex geometry. By establishing a characterisation of functions of bounded variation $BV(M)$ in terms of geodesic difference quotients, we show that $\Omega$ has finite perimeter if and only if $c_\Omega$ is Lipschitz, and we relate the Lipschitz constant to the perimeter of $\Omega$. We show that on the round sphere $E_{\gamma,\varepsilon}$ admits a reformulation in terms of $c_\Omega$, which allows us to compute the limit as $\varepsilon \to 0$ in a variational sense, that is, in the framework of $\Gamma$-convergence. - oai:arXiv.org:2601.10481v1 - math.AP - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Michael Bleher, Denis Brazke, Sebastian Nill - - - High-Dimensional Analysis of Gradient Flow for Extensive-Width Quadratic Neural Networks - https://arxiv.org/abs/2601.10483 - arXiv:2601.10483v1 Announce Type: new -Abstract: We study the high-dimensional training dynamics of a shallow neural network with quadratic activation in a teacher-student setup. We focus on the extensive-width regime, where the teacher and student network widths scale proportionally with the input dimension, and the sample size grows quadratically. This scaling aims to describe overparameterized neural networks in which feature learning still plays a central role. In the high-dimensional limit, we derive a dynamical characterization of the gradient flow, in the spirit of dynamical mean-field theory (DMFT). Under l2-regularization, we analyze these equations at long times and characterize the performance and spectral properties of the resulting estimator. This result provides a quantitative understanding of the effect of overparameterization on learning and generalization, and reveals a double descent phenomenon in the presence of label noise, where generalization improves beyond interpolation. In the small regularization limit, we obtain an exact expression for the perfect recovery threshold as a function of the network widths, providing a precise characterization of how overparameterization influences recovery. - oai:arXiv.org:2601.10483v1 - math.OC - cond-mat.dis-nn - stat.ML - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Simon Martin (DI-ENS, LPENS, SIERRA), Giulio Biroli (LPENS), Francis Bach (DI-ENS, SIERRA) - - - A Construction Framework of Coded Caching Scheme for Multi-Access MIMO Systems via Knapsack Problem - https://arxiv.org/abs/2601.10484 - arXiv:2601.10484v1 Announce Type: new -Abstract: This paper investigates the coded caching problem in a multi-access multiple-input single-output (MAMISO) network with the combinatorial topology. The considered system consists of a server containing $N$ files, $\Lambda$ cache nodes, and $K$ cache-less users, where each user can access a unique subset of $r$ cache nodes. The server is equipped with $L$ transmit antennas. Our objective is to design a caching scheme that simultaneously achieves a high sum Degree of Freedom (sum-DoF) and low subpacketization complexity. To address this challenge, we formulate the design of multi-antenna placement delivery arrays (MAPDA) as a $0$--$1$ knapsack problem to maximize the achievable DoF, thereby transforming the complex combinatorial caching structure into a tractable optimization framework that yields efficient cache placement and flexible delivery strategies. Theoretical and numerical analyses demonstrate that: for networks with combinatorial topologies, the proposed scheme achieves a higher sum-DoF than existing schemes. Under identical cache size constraints, the subpacketization level remains comparable to existing linear subpacketization schemes. Moreover, under specific system conditions, the proposed scheme attains the theoretical maximum sum-DoF of $\min\{L+KM/N, K\}$ while achieving further reductions subpacketization. For particular combinatorial structures, we further derive optimized constructions that achieve even higher sum-DoF with lower subpacketization. ``` - oai:arXiv.org:2601.10484v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Siying Luo, Youlong Wu, Mingming Zhang, Minquan Cheng, Dianhua Wu - - - Finite lattice kinetic equations for bosons, fermions, and discrete NLS - https://arxiv.org/abs/2601.10486 - arXiv:2601.10486v1 Announce Type: new -Abstract: We introduce and study finite lattice kinetic equations for bosons, fermions, and discrete NLS. For each model this closed evolution equation provides an approximate description for the evolution of the appropriate covariance function in the system. It is obtained by truncating the cumulant hierarchy and dropping the higher order cumulants in the usual manner. To have such a reference solution should simplify controlling the full hierarchy and thus allow estimating the error from the truncation. The harmonic part is given by nearest neighbour hopping, with arbitrary symmetric interaction potential of coupling strength $\lambda>0$. We consider the well-posedness of the resulting evolution equation up to finite kinetic times on a finite but large enough lattice. We obtain decay of the solutions and upper bounds that are independent of $\lambda$ and depend on the lattice size only via some Sobolev type norms of the interaction potential and initial data. We prove that the solutions are not sensitive to how the energy conservation delta function is approximated. - oai:arXiv.org:2601.10486v1 - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jani Lukkarinen, Sakari Pirnes, Aleksis Vuoksenmaa - - - A proof of the soliton resolution conjecture for the Benjamin--Ono equation - https://arxiv.org/abs/2601.10488 - arXiv:2601.10488v1 Announce Type: new -Abstract: We give a proof of the soliton resolution conjecture for the Benjamin--Ono equation, namely every solution with sufficiently regular and decaying initial data can be written as a finite sum of soliton solutions with different velocities up to a radiative remainder term in the long--time asymptotics. We provide a detailed correspondence between the spectral theory of the Lax operator associated to the initial data and the different terms of the soliton resolution expansion. The proof is based on a new use of a representation formula of the solution due to the second author, and on a detailed analysis of the distorted Fourier transform associated to the Lax operator. - oai:arXiv.org:2601.10488v1 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Louise Gassot, Patrick G\'erard, Peter D. Miller - - - Malliavin Calculus for the stochastic Cahn-Hilliard equation driven by fractional noise - https://arxiv.org/abs/2601.10490 - arXiv:2601.10490v1 Announce Type: new -Abstract: The stochastic partial differential equation analyzed in this work is the Cahn-Hilliard equation perturbed by an additive fractional white noise (fractional in time and white in space). We work in the case of one spatial dimension and apply Malliavin calculus to investigate the existence of a density for the stochastic solution $u$. In particular, we show that $u$ admits continuous paths almost surely and construct a localizing sequence through which we prove that its Malliavin derivative exists locally, and that its law is absolutely continuous with respect to the Lebesgue measure on $\bf R$, establishing thus that a density exists. A key contribution of this work is the analysis of the stochastic integral appearing in the mild formulation: we derive sharp estimates for the expectation of the $p$-th power ($p \geq 2$) of the $L^{\infty}(D)$-norm of this stochastic integral as well as for the integral involving the $L^{\infty}(D)$-norm of the operator associated with the kernel appearing in the integral representation of the fractional noise, all of which are essential for this study. - oai:arXiv.org:2601.10490v1 - math.PR - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Dimitrios Dimitriou, Dimitris Farazakis, Georgia Karali - - - Coded Caching for Combinatorial Multi-Access Hotplug Networks from $t$-Designs - https://arxiv.org/abs/2601.10503 - arXiv:2601.10503v1 Announce Type: new -Abstract: We study hotplug coded caching in combinatorial multi-access networks, which generalizes existing hotplug coded caching models by allowing users to access multiple caches, while only a subset of caches is online during the delivery phase. We first generalize the Hotplug Placement Delivery Array (HpPDA) framework to the combinatorial multi-access setting. Based on this generalized framework, we propose a t-design-based coded caching scheme for combinatorial multi-access networks. We characterize a class of design parameters under which every active user has access to a sufficient number of coded subfiles to decode its requested file, and show that appropriate parameter choices allow for the elimination of redundant multicast transmissions. As a result, the proposed scheme achieves a family of rate memory trade offs with flexible subpacketization. We present numerical comparisons illustrating that the proposed t-scheme outperforms existing hotplug coded caching schemes in certain memory regimes. - oai:arXiv.org:2601.10503v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Dhruv Pratap Singh, Anjana A. Mahesh, B. Sundar Rajan - - - A New Construction Structure on Coded Caching with Linear Subpacketization: Non-Half-Sum Latin Rectangle - https://arxiv.org/abs/2601.10505 - arXiv:2601.10505v1 Announce Type: new -Abstract: Coded caching is recognized as an effective method for alleviating network congestion during peak periods by leveraging local caching and coded multicasting gains. The key challenge in designing coded caching schemes lies in simultaneously achieving low subpacketization and low transmission load. Most existing schemes require exponential or polynomial subpacketization levels, while some linear subpacketization schemes often result in excessive transmission load. Recently, Cheng et al. proposed a construction framework for linear coded caching schemes called Non-Half-Sum Disjoint Packing (NHSDP), where the subpacketization equals the number of users $K$. This paper introduces a novel combinatorial structure, termed the Non-Half-Sum Latin Rectangle (NHSLR), which extends the framework of linear coded caching schemes from $F=K$ (i.e., the construction via NHSDP) to a broader scenario with $F=\mathcal{O}(K)$. By constructing NHSLR, we have obtained a new class of coded caching schemes that achieves linearly scalable subpacketization, while further reducing the transmission load compared with the NHSDP scheme. Theoretical and numerical analyses demonstrate that the proposed schemes not only achieves lower transmission load than existing linear subpacketization schemes but also approaches the performance of certain exponential subpacketization schemes. - oai:arXiv.org:2601.10505v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yongcheng Yang, Minquan Cheng, Kai Wan, Giuseppe Caire - - - A New Construction Structure on Multi-access Coded Caching with Linear Subpacketization: Cyclic Multi-Access Non-Half-Sum Disjoint Packing - https://arxiv.org/abs/2601.10510 - arXiv:2601.10510v1 Announce Type: new -Abstract: We consider the $(K,L,M,N)$ multi-access coded caching system introduced by Hachem et al., which consists of a central server with $N$ files and $K$ cache nodes, each of memory size $M$, where each user can access $L$ cache nodes in a cyclic wrap-around fashion. At present, several existing schemes achieve competitive transmission performance, but their subpacketization levels grow exponentially with the number of users. In contrast, schemes with linear or polynomial subpacketization always incur higher transmission loads. We aim to design a multi-access coded caching scheme with linear subpacketization $F$ while maintaining low transmission load. Recently, Cheng et al. proposed a construction framework for coded caching schemes with linear subpacketization (i.e., $F=K$) called non-half-sum disjoint packing (NHSDP). Inspired by this structure, we introduce a novel combinatorial structure named cyclic multi-access non-half-sum disjoint packing (CMA-NHSDP) by extending NHSDP to MACC system. By constructing CMA-NHSDP, we obtain a new class of multi-access coded caching schemes. Theoretical and numerical analyses show that our scheme achieves lower transmission loads than some existing schemes with linear subpacketization. Moreover, the proposed schemes achieves lower transmission load compared to existing schemes with exponential subpacketization in some case. - oai:arXiv.org:2601.10510v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mengyuan Li, Minquan Cheng, Kai Wan, Giuseppe Caire - - - Some Eigenvalue Inequalities for the Schr\"odinger Operator on Integer Lattices - https://arxiv.org/abs/2601.10523 - arXiv:2601.10523v1 Announce Type: new -Abstract: In this paper, we establish analogues of the Payne-P\'olya-Weinberger, Hile-Protter, and Yang eigenvalue inequalities for the Schr\"odinger operator on arbitrary finite subsets of the integer lattice $\mathbb{Z}^n$. The results extend known inequalities for the discrete Laplacian to a more general class of Schr\"odinger operators with nonnegative potentials and weighted eigenvalue problems. - oai:arXiv.org:2601.10523v1 - math.SP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Wentao Liu - - - On the suboptimality of linear codes for binary distributed hypothesis testing - https://arxiv.org/abs/2601.10526 - arXiv:2601.10526v1 Announce Type: new -Abstract: We study a binary distributed hypothesis testing problem where two agents observe correlated binary vectors and communicate compressed information at the same rate to a central decision maker. In particular, we study linear compression schemes and show that simple truncation is the best linear scheme in two cases: (1) testing opposite signs of the same magnitude of correlation, and (2) testing for or against independence. We conjecture, supported by numerical evidence, that truncation is the best linear code for testing any correlations of opposite signs. Further, for testing against independence, we also compute classical random coding exponents and show that truncation, and consequently any linear code, is strictly suboptimal. - oai:arXiv.org:2601.10526v1 - cs.IT - math.IT - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Adway Girish, Robinson D. H. Cung, Emre Telatar - - - Three realization problems about univariate polynomials - https://arxiv.org/abs/2601.10529 - arXiv:2601.10529v1 Announce Type: new -Abstract: We consider three realization problems about monic real univariate polynomials without vanishing coefficients. Such a polynomial $P:=\sum_{j=0}^db_jx^j$ defines the sign pattern $\sigma (P):=({\rm sgn}(b_d)$, $\ldots$, ${\rm sgn}(b_0))$. The numbers $p_d$ and $n_d$ of positive and negative roots of $P$ (counted with multiplicity) satisfy the Descartes' rule of signs. Problem~1 asks for which couples $C$ of the form (sign pattern $\sigma$, pair $(p_d,n_d)$ compatible with $\sigma$ in the sense of Descartes' rule of signs), there exist polynomials $P$ defining these couples. Problem~2 asks for which $d$-tuples of pairs $T:=((p_d,n_d)$, $\ldots$, $(p_1,n_1))$, there exist polynomials $P$ such that $P^{(d-j)}$ has $p_j$ positive and $n_j$ negative roots. A $d$-tuple $T$ determines the sign pattern $\sigma (P)$, but the inverse is false. We show by an example that $6$ is the smallest value of $d$ for which there exist non-realizable tuples $T$ for which the corresponding couples $C$ are realizable. The third problem concerns polynomials with all roots real. We give a geometric interpretation of the three problems in the context of degree $4$ polynomials. - oai:arXiv.org:2601.10529v1 - math.CA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/publicdomain/zero/1.0/ - Vladimir Petrov Kostov - - - Network Integrated Sensing and Communication - https://arxiv.org/abs/2601.10538 - arXiv:2601.10538v1 Announce Type: new -Abstract: Integrated sensing and communication (ISAC) is a cornerstone technology for 6G networks, offering unified support for high-rate communication and high-accuracy sensing. While existing literature extensively covers link-level designs, the transition toward large-scale deployment necessitates a fundamental understanding of network-level performance. This paper investigates a network ISAC model where a source node communicates with a destination via a relay network, while intermediate nodes concurrently perform cooperative sensing over specific spatial regions. We formulate a novel optimization framework that captures the interplay between multi-node routing and sensing coverage. For a one-dimensional path network, we provide an analytical characterization of the complete sensing-throughput region. Extending this to general network topologies, we establish that the sensing-throughput Pareto boundary is piecewise linear and provide physical interpretations for each segment. Our results reveal the fundamental trade-offs between sensing coverage and communication routing, offering key insights for the design of future 6G heterogeneous networks. - oai:arXiv.org:2601.10538v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Edward Andrews, Lawrence Ong, Duy T. Ngo, Yao Liu, Min Li - - - Smoothness of martingale observables and generalized Feynman-Kac formulas - https://arxiv.org/abs/2601.10539 - arXiv:2601.10539v1 Announce Type: new -Abstract: We prove that, under the H\"ormander criterion on an It\^{o} process, all its martingale observables are smooth. As a consequence, we also obtain a generalized Feynman-Kac formula providing smooth solutions to certain PDE boundary-value problems, while allowing for degenerate diffusions as well as boundary stopping (under very mild boundary regularity assumptions). We also highlight an application to a question posed on Schramm-Loewner evolutions, by making certain Girsanov transform martingales accessible via It\^{o} calculus. - oai:arXiv.org:2601.10539v1 - math.PR - math-ph - math.AP - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alex Karrila, Lauri Viitasaari - - - Error-Correcting Codes for Two Bursts of t1-Deletion-t2-Insertion with Low Computational Complexity - https://arxiv.org/abs/2601.10540 - arXiv:2601.10540v1 Announce Type: new -Abstract: Burst errors involving simultaneous insertions, deletions, and substitutions occur in practical scenarios, including DNA data storage and document synchronization, motivating developments of channel codes that can correct such errors. In this paper, we address the problem of constructing error-correcting codes (ECCs) capable of handling multiple bursts of $t_1$-deletion-$t_2$-insertion ($(t_1,t_2)$-DI) errors, where each burst consists of $t_1$ deletions followed by $t_2$ insertions in a binary sequence. We make three key contributions: Firstly, we establish the fundamental equivalence of (1) two bursts of $(t_1,t_2)$-DI ECCs, (2) two bursts of $(t_2,t_1)$-DI ECCs, and (3) one burst each of $(t_1,t_2)$-DI and $(t_2,t_1)$-DI ECCs. Then, we derive lower and upper bounds on the code size of two bursts of $(t_1,t_2)$-DI ECCs, which can naturally be extended to the case of multiple bursts. Finally, we present constructions of two bursts of $(t_1,t_2)$-DI ECCs. Compared to the codes obtained by the syndrome compression technique, the resulting codes achieve significantly lower computational complexity. - oai:arXiv.org:2601.10540v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Yajuan Liu, Tolga M. Duman - - - Linear independence properties of the signature components of time-augmented stochastic processes - https://arxiv.org/abs/2601.10545 - arXiv:2601.10545v1 Announce Type: new -Abstract: The addition of the running time as a component of a path before computing its signature is a widespread approach to ensure the one-to-one property between them and leads to universal approximation theorems (Cuchiero, Primavera and Svaluto-Ferro, 2023). However, this also leads to the linear dependence of the components of the terminal value of the signature of the time-augmented path. More precisely, for a given natural number $N$, the signature components associated with words of length $N$ have the same linear span as the signature components associated with words of length not greater than $N$. We generalize this result by exhibiting other subfamilies of signature components with the same spanning properties. In particular we recover the result of Dupire and Tissot-Daguette which states that the spanning of the iterated integrals with the last integrator different from the time variable is the same as the spanning of all iterated integrals. We check that this choice leads to the minimal computation time when the terms of the signature are calculated using Chen's relation in a backward way. The same optimal computation time is symmetrically achieved in a forward way for the iterated integrals with the first integrator different from the time variable. Building on these results, we derive several results regarding the linear independence of the signature components of a time-augmented stochastic process. We show that if the stochastic process we consider is solution to some SDE with additive Brownian noise then any subfamily of components proposed previously is linearly independent. We also prove that the linear independence of these subfamilies of components is still true when we consider the discretization of the sample paths of this stochastic process on a grid with a sufficiently small discretization time step. - oai:arXiv.org:2601.10545v1 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Arthur Bourdon, Benjamin Jourdain, Herv\'e Andr\`es - - - The inducibility of Tur\'an graphs - https://arxiv.org/abs/2601.10548 - arXiv:2601.10548v1 Announce Type: new -Abstract: Let $I(F,n)$ denote the maximum number of induced copies of a graph $F$ in an $n$-vertex graph. The inducibility of $F$, defined as $i(F)=\lim_{n\to \infty} I(F,n)/\binom{n}{v(F)}$, is a central problem in extremal graph theory. In this work, we investigate the inducibility of Tur\'an graphs $F$. This topic has been extensively studied in the literature, including works of Pippenger--Golumbic, Brown--Sidorenko, Bollob\'as--Egawa--Harris--Jin, Mubayi, Reiher, and the first author, and Yuster. Broadly speaking, these results resolve or asymptotically resolve the problem when the part sizes of $F$ are either sufficiently large or sufficiently small (at most four). - We complete this picture by proving that for every Tur\'an graph $F$ and sufficiently large $n$, the value $I(F,n)$ is attained uniquely by the $m$-partite Tur\'an graph on $n$ vertices, where $m$ is given explicitly in terms of the number of parts and vertices of $F$. This confirms a conjecture of Bollob\'as--Egawa--Harris--Jin from 1995, and we also establish the corresponding stability theorem. Moreover, we prove an asymptotic analogue for $I_{k+1}(F,n)$, the maximum number of induced copies of $F$ in an $n$-vertex $K_{k+1}$-free graph, thereby completely resolving a recent problem of Yuster. Finally, our results extend to a broader class of complete multipartite graphs in which the largest and smallest part sizes differ by at most on the order of the square root of the smallest part size. - oai:arXiv.org:2601.10548v1 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Xizhi Liu, Jie Ma, Tianming Zhu - - - Chebyshev Accelerated Subspsace Eigensolver for Pseudo-hermitian Hamiltonians - https://arxiv.org/abs/2601.10557 - arXiv:2601.10557v1 Announce Type: new -Abstract: Studying the optoelectronic structure of materials can require the computation of up to several thousands of the smallest eigenpairs of a pseudo-hermitian Hamiltonian. Iterative eigensolvers may be preferred over direct methods for this task since their complexity is a function of the desired fraction of the spectrum. In addition, they generally rely on highly optimized and scalable kernels such as matrix-vector multiplications that leverage the massive parallelism and the computational power of modern exascale systems. \textit{Chebyshev Accelerated Subspace iteration Eigensolver} (ChASE) is able to compute several thousands of the most extreme eigenpairs of dense hermitian matrices with proven scalability over massive parallel accelerated clusters. This work presents an extension of ChASE to solve for a portion of the spectrum of pseudo-hermitian Hamiltonians as they appear in the treatment of excitonic materials. The new pseudo-hermitian solver achieves similar convergence and performance as the hermitian one. By exploiting the numerical structure and spectral properties of the Hamiltonian matrix, we propose an oblique variant of Rayleigh-Ritz projection featuring quadratic convergence of the Ritz-values with no explicit construction of the dual basis set. Additionally, we introduce a parallel implementation of the recursive matrix-product operation appearing in the Chebyshev filter with limited amount of global communications. Our development is supported by a full numerical analysis and experimental tests. - oai:arXiv.org:2601.10557v1 - math.NA - cs.CE - cs.DC - cs.NA - physics.comp-ph - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Edoardo Di Napoli (J\"ulich Supercomputing Centre, Forschungszentrum J\"ulich, Germany), Cl\'ement Richefort (J\"ulich Supercomputing Centre, Forschungszentrum J\"ulich, Germany), Xinzhe Wu (J\"ulich Supercomputing Centre, Forschungszentrum J\"ulich, Germany) - - - (a,b)-Fibonacci-Legendre Cordial Graphs and k-Pisano-Legendre Primes - https://arxiv.org/abs/2601.10561 - arXiv:2601.10561v1 Announce Type: new -Abstract: Let $p$ be an odd prime and let $F_i$ be the $i$th $(a,b)$-Fibonacci number with initial values $F_0=a$ and $F_1=b$. For a simple connected graph $G=(V,E)$, define a bijective function $f:V(G)\to \{0,1,\ldots,|V|-1\}$. If the induced function $f_p^*:E(G)\to \{0,1\}$, defined by $f_p^*(uv)=\frac{1+([F_{f(u)}+F_{f(v)}]/p)}{2}$ whenever $F_{f(u)}+F_{f(v)}\not\equiv 0\pmod{p}$ and $f_p^*(uv)=0$ whenever $F_{f(u)}+F_{f(v)}\equiv 0\pmod{p}$, satisfies the condition $|e_{f_p^*}(0)-e_{f_p^*}(1)|\leq 1$ where $e_{f_p^*}(i)$ is the number of edges labeled $i$ ($i=0,1$), then $f$ is called $(a,b)$-Fibonacci-Legendre cordial labeling modulo $p$. In this paper, the $(a,b)$-Fibonacci-Legendre cordial labeling of path graphs, star graphs, wheel graphs, and graphs under the operations join, corona, lexicographic product, cartesian product, tensor product, and strong product is explored in relation to $k$-Pisano-Legendre primes relative to $(a,b)$. We also present some properties of $k$-Pisano-Legendre primes relative to $(a,b)$ and numerical observations on its distribution, leading to several conjectures concerning their density and growth behavior. - oai:arXiv.org:2601.10561v1 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - J. D. Andoyo - - - Hydrodynamic Limit with a Weierstrass-type result - https://arxiv.org/abs/2601.10568 - arXiv:2601.10568v1 Announce Type: new -Abstract: We show that any positive, continuous, and bounded function can be realised as the diffusion coefficient of an evolution equation associated with a gradient interacting particle system. The proof relies on the construction of an appropriate model and on the entropy method. - oai:arXiv.org:2601.10568v1 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Gabriel S. Nahum - - - Sparse Signal Recovery from Random Measurements - https://arxiv.org/abs/2601.10569 - arXiv:2601.10569v1 Announce Type: new -Abstract: Given the compressed sensing measurements of an unknown vector $z \in \mathbb{R}^n$ using random matrices, we present a simple method to determine $z$ without solving any optimization problem or linear system. Our method uses $\Theta(\log n)$ random sensing matrices in $\mathbb{R}^{k \times n}$ and runs in $O(kn\log n)$ time, where $k = \Theta(s\log n)$ and $s$ is the number of nonzero coordinates in $z$. We adapt our method to determine the support set of $z$ and experimentally compare with some optimization-based methods on binary signals. - oai:arXiv.org:2601.10569v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Siu-Wing Cheng, Man Ting Wong - - - Mind the gap: A real-valued distance on combinatorial games - https://arxiv.org/abs/2601.10574 - arXiv:2601.10574v1 Announce Type: new -Abstract: We define a real-valued distance metric $wd$ on the space $\mathcal{C}$ of short combinatorial games in canonical form. We demonstrate the existence of Cauchy sequences informed by sidling sequences, find limit points, and investigate the closure $\overline{\mathcal{C}}$, which is shown to partition the set of loopy games in a non-trivial way. Stoppers, enders, and non-stopper-sided loopy games are explored, as well as the topological properties of $(\mathcal{C},wd)$. - oai:arXiv.org:2601.10574v1 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kyle Burke, Michael Fisher, Craig Tennenhouse - - - On Zalcman's and Bieberbach conjectures - https://arxiv.org/abs/2601.10584 - arXiv:2601.10584v1 Announce Type: new -Abstract: The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions $f(z) = z + \sum\limits_2^{\infty} a_n z^n$ on the unit disk satisfy $|a_n^2 - a_{2n-1}| \le (n-1)^2$ for all $n > 2$, with equality only for the Koebe function and its rotations. The conjecture was proved by the author for $n \le 6$ (using geometric arguments related to the Ahlfors-Schwarz lemma) and remains open for $n \ge 7$. - The main theorem of this paper states that these conjectures are equivalent and provides their simultaneous proof for all $n \ge 3$ combining the indicated geometric arguments with a new author's approach to extremal problems for holomorphic functions based on lifting the rotationally homogeneous coefficient functionals to the Bers fiber space over universal Teichmuller space. - oai:arXiv.org:2601.10584v1 - math.CV - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Samuel L. Krushkal - - - Comparison of viscosity solutions for a class of non-linear PDEs on the space of finite nonnegative measures - https://arxiv.org/abs/2601.10586 - arXiv:2601.10586v1 Announce Type: new -Abstract: We establish a comparison principle for viscosity solutions of a class of nonlinear partial differential equations posed on the space of nonnegative finite measures, thereby extending recent results for PDEs defined on the Wasserstein space of probability measures. As an application, we study a controlled branching McKean-Vlasov diffusion and characterize the associated value function as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. This yields a PDE-based approach to the optimal control of branching processes. - oai:arXiv.org:2601.10586v1 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ibrahim Ekren, Xihao He, Tianxu Lan, Xiaolu Tan - - - Schur--Horn type inequalities for hyperbolic polynomials - https://arxiv.org/abs/2601.10602 - arXiv:2601.10602v1 Announce Type: new -Abstract: We establish a Schur--Horn type inequality for symmetric hyperbolic polynomials. As an immediate consequence, we resolve a conjecture of Nam Q. Le on Hadamard-type inequalities for hyperbolic polynomials. Our argument is based on the Schur--Horn theorem, the Birkhoff theorem, and G{\aa}rding's concavity theorem for hyperbolicity cones. Beyond the eigenvalue level, we develop a symmetrization principle on hyperbolicity cones: if a hyperbolic polynomial is invariant under a finite group action, then its value increases under the associated Reynolds operator (group averaging). Applied to the sign-flip symmetries of linear principal minor polynomials introduced by Blekherman et al., this yields a short proof of the hyperbolic Fischer--Hadamard inequalities for PSD-stable lpm polynomials. - oai:arXiv.org:2601.10602v1 - math.FA - math.RT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Teng Zhang - - - Fundamental Limits of Multi-User Distributed Computing of Linearly Separable Functions - https://arxiv.org/abs/2601.10603 - arXiv:2601.10603v1 Announce Type: new -Abstract: This work establishes the fundamental limits of the classical problem of multi-user distributed computing of linearly separable functions. In particular, we consider a distributed computing setting involving $L$ users, each requesting a linearly separable function over $K$ basis subfunctions from a master node, who is assisted by $N$ distributed servers. At the core of this problem lies a fundamental tradeoff between communication and computation: each server can compute up to $M$ subfunctions, and each server can communicate linear combinations of their locally computed subfunctions outputs to at most $\Delta$ users. The objective is to design a distributed computing scheme that reduces the communication cost (total amount of data from servers to users), and towards this, for any given $K$, $L$, $M$, and $\Delta$, we propose a distributed computing scheme that jointly designs the task assignment and transmissions, and shows that the scheme achieves optimal performance in the real field under various conditions using a novel converse. We also characterize the performance of the scheme in the finite field using another converse based on counting arguments. - oai:arXiv.org:2601.10603v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - K. K. Krishnan Namboodiri, Elizabath Peter, Derya Malak, Petros Elia - - - Local times and excursions for self-similar Markov trees - https://arxiv.org/abs/2601.10610 - arXiv:2601.10610v1 Announce Type: new -Abstract: This work builds upon the recent monograph [5] on self-similar Markov trees. A self-similar Markov tree is a random real tree equipped with a function from the tree to $[0,\infty)$ that we call the decoration. Here, we construct local time measures $L(x,dt)$ at every level $x>0$ of the decoration for a large class of self-similar Markov trees. This enables us to mark at random a typical point in the tree at which the decoration is $x$. We identify the law of the decoration along the branch from the root to this tagged point in terms of a remarkable (positive) self-similar Markov process. We also show that after a proper normalization, $L(x,dt)$ converges as $x\to 0+$ to the harmonic measure $\mu$ on the tree. Finally, we point out that using a local time measure instead of the usual length measure $\lambda$ to compute distances on the tree turn the latter into a continuous branching tree. This is relevant to analyze the excusions of the decoration away from a given level. Many results of the present work shall be compared with the recent ones in [22,23] about local times and excursions of a Markov process indexed by L\'evy tree. - oai:arXiv.org:2601.10610v1 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Jean Bertoin, Armand Riera, Alejandro Rosales-Ortiz - - - Malcev classification for the variety of left-symmetric algebras - https://arxiv.org/abs/2601.10613 - arXiv:2601.10613v1 Announce Type: new -Abstract: In this paper, we study three classes of subvarieties inside the variety of left-symmetric algebras. We show that these subvarieties are naturally related to some well-known varieties, such as alternative, assosymmetric and Zinbiel algebras. For certain subvarieties of the varieties of alternative and assosymmetric algebras, we explicitly construct bases of the corresponding free algebras. We then define the commutator and anti-commutator operations on these algebras and derive a number of identities satisfied by these operations in all degrees up to $4$. - oai:arXiv.org:2601.10613v1 - math.RA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - A. Ryskeldin, B. Sartayev - - - The directedness of the Rudin-Keisler order at measurable cardinals - https://arxiv.org/abs/2601.10614 - arXiv:2601.10614v1 Announce Type: new -Abstract: The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals $\lambda\leq \kappa$, the following theories are equiconsistent modulo ZFC: (1) $\kappa$ is a measurable cardinal with $o(\kappa)=\lambda^+$ (resp. $o(\kappa)=\kappa$). (2) The Rudin-Keisler order restricted to the set of $\kappa$-complete (non-principal) ultrafilters on $\kappa$ is $\lambda^+$-directed (resp. $\kappa^+$-directed). The theorem reported here is proved after bridging the directedness of the RK-order with the $\lambda$-Gluing Property introduced by the authors in \cite{HP}. Our result provides what seems to be the first example of a compactness-type property at the level of measurable cardinals whose consistency strength is much lower than the existence of a strong cardinal. As part of our analysis we also answer a question of Gitik by showing that the $\aleph_0$-Gluing Property fails in his classical model from ''Changing cofinalities and the nonstationary ideal". As a consequence of this, in Gitik's model the Rudin-Keisler order fails to be $\aleph_1$-directed. - oai:arXiv.org:2601.10614v1 - math.LO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Yair Hayut, Alejandro Poveda - - - Basis-Spline Assisted Coded Computing: Strategies and Error Bounds - https://arxiv.org/abs/2601.10616 - arXiv:2601.10616v1 Announce Type: new -Abstract: Coded computing has become a key framework for reliable distributed computation over decentralized networks, effectively mitigating the impact of stragglers. Although there exists a wide range of coded computing methods to handle both polynomial and non-polynomial functions, computing methods for the latter class have received traction due its inherent challenges in reconstructing non-polynomial functions using a finite number of evaluations. Among them, the state-of-the-art method is Berrut Approximated coded computing, wherein Berrut interpolants, are used for approximating the non-polynomial function. However, since Berrut interpolants have global support characteristics, such methods are known to offer degraded accuracy when the number of stragglers is large. To address this challenge, we propose a coded computing framework based on cubic B-spline interpolation. In our approach, server-side function evaluations are reconstructed at the master node using B-splines, exploiting their local support and smoothness properties to enhance stability and accuracy. We provide a systematic methodology for integrating B-spline interpolation into coded computing and derive theoretical bounds on approximation error in terms of the number of servers and stragglers. Comparative analysis demonstrates that our framework significantly outperforms Berrut-based methods for various non-polynomial functions. - oai:arXiv.org:2601.10616v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rimpi Borah, J. Harshan, V. Lalitha - - - Quantitative surgery and total mean curvature - https://arxiv.org/abs/2601.10617 - arXiv:2601.10617v1 Announce Type: new -Abstract: We develop quantitative surgery, which extends the classical constructions of Gromov--Lawson and Lawson--Michelsohn. As an application, we prove a conjecture of Gromov on the total mean curvature of fill-ins. - oai:arXiv.org:2601.10617v1 - math.DG - gr-qc - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Georg Frenck, Bernhard Hanke, Sven Hirsch - - - A universal Bochner formula for scalar curvature - https://arxiv.org/abs/2601.10618 - arXiv:2601.10618v1 Announce Type: new -Abstract: We introduce a universal Bochner formula for scalar curvature that contains, as special cases, the stability inequality for minimal slicings, a Schr\"odinger-Lichnerowicz-type formula, and a higher-dimensional version of Stern's level-set identity. - oai:arXiv.org:2601.10618v1 - math.DG - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sven Hirsch - - - Source localisation in simple random walks - https://arxiv.org/abs/2601.10624 - arXiv:2601.10624v1 Announce Type: new -Abstract: We consider the problem of locating the source (starting vertex) of a simple random walk, given a snapshot of the set of edges (or vertices) visited in the first $n$ steps. Considering lattices $\mathbb{Z}^d$, in dimensions $d \geq 5$, we show that the source can be identified (a) with probability bounded away from $0$ using one guess, and (b) with probability arbitrarily close to $1$ using a constant number of guesses. On the other hand, for dimensions $d \leq 2$, we show that one cannot locate the source with positive constant probability. Our arguments apply more generally to strongly transient and recurrent simple random walks on vertex-transitive graphs. - oai:arXiv.org:2601.10624v1 - math.PR - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ritesh Goenka, Peter Keevash, Tomasz Przyby{\l}owski - - - Discrete-time maximally superintegrable systems and deformed symmetry algebras: the Calogero-Moser case - https://arxiv.org/abs/2601.10625 - arXiv:2601.10625v1 Announce Type: new -Abstract: We determine the complete structure of the symmetry algebras associated with the N-body Calogero-Moser system and its maximally superintegrable discretization. We prove that the discretization naturally leads to a nontrivial deformation of the continuous symmetry algebra, with the discretization parameter playing the r\^ole of a deformation parameter. This phenomenon illustrates how discrete superintegrable systems can be viewed as natural sources of deformed polynomial algebraic structures. As a byproduct of these results, we also reveal a connection between the above-mentioned symmetry algebras and the Bell polynomials, as a consequence of the trace properties. - oai:arXiv.org:2601.10625v1 - math-ph - math.MP - nlin.SI - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pavel Drozdov, Giorgio Gubbiotti, Danilo Latini - - - Differentially Private Inference for Longitudinal Linear Regression - https://arxiv.org/abs/2601.10626 - arXiv:2601.10626v1 Announce Type: new -Abstract: Differential Privacy (DP) provides a rigorous framework for releasing statistics while protecting individual information present in a dataset. Although substantial progress has been made on differentially private linear regression, existing methods almost exclusively address the item-level DP setting, where each user contributes a single observation. Many scientific and economic applications instead involve longitudinal or panel data, in which each user contributes multiple dependent observations. In these settings, item-level DP offers inadequate protection, and user-level DP - shielding an individual's entire trajectory - is the appropriate privacy notion. We develop a comprehensive framework for estimation and inference in longitudinal linear regression under user-level DP. We propose a user-level private regression estimator based on aggregating local regressions, and we establish finite-sample guarantees and asymptotic normality under short-range dependence. For inference, we develop a privatized, bias-corrected covariance estimator that is automatically heteroskedasticity- and autocorrelation-consistent. These results provide the first unified framework for practical user-level DP estimation and inference in longitudinal linear regression under dependence, with strong theoretical guarantees and promising empirical performance. - oai:arXiv.org:2601.10626v1 - math.ST - cs.CR - stat.ME - stat.ML - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Getoar Sopa, Marco Avella Medina, Cynthia Rush - - - On subradically sifted sums related to Alladi's higher order duality between prime factors - https://arxiv.org/abs/2601.10636 - arXiv:2601.10636v1 Announce Type: new -Abstract: In this paper, I utilize a variant of the Selberg--Delange method to find quantitative estimates of the sums \[M_{k,\omega}(x,y)=\sum_{\substack{p_{1}(n)> y\\ n\leq x} } \mu(n) {\omega(n)-1\choose k-1},\] where $y$ can grow with $x$ but we must have $y\leq Y_0\exp(\mathscr{p}\frac{\log x}{(\log\log (x+1))^{1+\epsilon}})$ with $Y_0,\mathscr{p},\epsilon>0$. Moreover, I give preliminary upper bounds for the general range $1.9\leq y\leq x^{\frac{1}{k}}$. In addition, I formalize the notions of subradical and radical dominance and discuss their relevance to the analytic approach of the study of arithmetic functions. Lastly, I give a fascinating formula related to the derivatives of the gamma function and the Hankel contour, which should be relevant for those employing the Selberg--Delange method to obtain higher-order terms. - oai:arXiv.org:2601.10636v1 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Yazan Alamoudi - - - Converse Bounds for Sun-Jafar-type Weak Private Information Retrieval - https://arxiv.org/abs/2601.10643 - arXiv:2601.10643v1 Announce Type: new -Abstract: Building on the well-established capacity-achieving schemes of Sun-Jafar (for replicated storage) and the closely related scheme of Banawan-Ulukus (for MDS-coded setting), a recent work by Chandan et al. proposed new classes of weak private information retrieval (WPIR) schemes for the collusion-free (replication and MDS-coded) setting, as well as for the $T$-colluding scenario. In their work, Chandan et al. characterized the expressions for the rate-privacy trade-offs for these classes of WPIR schemes, under the mutual information leakage and maximal leakage metrics. Explicit achievable trade-offs for the same were also presented, which were shown to be competitive or better than prior WPIR schemes. However, the class-wise optimality of the reported trade-offs were unknown. In this work, we show that the explicit rate-privacy trade-offs reported for the Sun-Jafar-type schemes by Chandan et al. are optimal for the non-colluding and replicated setting. Furthermore, we prove the class-wise optimality for Banawan-Ulukus-type MDS-WPIR and Sun-Jafar-type $T$-colluding WPIR schemes, under threshold-constraints on the system parameters. When these threshold-constraints do not hold, we present counter-examples which show that even higher rates than those reported before can be achieved. - oai:arXiv.org:2601.10643v1 - cs.IT - cs.CR - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chandan Anand, Jayesh Seshadri, Prasad Krishnan, Gowtham R. Kurri - - - Michael-Simon inequality for anisotropic energies close to the area via multilinear Kakeya-type bounds - https://arxiv.org/abs/2601.10647 - arXiv:2601.10647v1 Announce Type: new -Abstract: Given an anisotropic integrand $F:\text{Gr}_k(\mathbb R^n)\to(0,\infty)$, we can generalize the classical isotropic area by looking at the functional $$\mathcal{F}(\Sigma^k):=\int_\Sigma F(T_x\Sigma)\,d\mathcal{H}^k.$$ While a monotonicity formula is not available for critical points, when $k=2$ and $n=3$ we show that the Michael-Simon inequality holds if $F$ is convex and close to $1$ (in $C^1$), meaning that $\mathcal{F}$ is close to the usual area. - Our argument is partly based on some key ideas of Almgren, who proved this result in an unpublished manuscript, but we largely simplify his original proof by showing a new functional inequality for vector fields on the plane, which can be seen as a quantitative version of Alberti's rank-one theorem. - As another byproduct, we also show Michael-Simon for another class of integrands which includes the $\ell^p$ norms for $p\in(1,\infty)$. For a general $F$ satisfying the atomic condition, we also show that the validity of Michael-Simon is equivalent to compactness of rectifiable varifolds. - oai:arXiv.org:2601.10647v1 - math.AP - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Guido De Philippis, Alessandro Pigati - - - One-Shot Broadcast Joint Source-Channel Coding with Codebook Diversity - https://arxiv.org/abs/2601.10648 - arXiv:2601.10648v1 Announce Type: new -Abstract: We study a one-shot joint source-channel coding setting where the source is encoded once and broadcast to $K$ decoders through independent channels. Success is predicated on at least one decoder recovering the source within a maximum distortion constraint. We find that in the one-shot regime, utilizing disjoint codebooks at each decoder yields a codebook diversity gain, distinct from the channel diversity gain that may be expected when several decoders observe independent realizations of the channel's output but share the same codebook. Coding schemes are introduced that leverage this phenomenon, where first- and second-order achievability bounds are derived via an adaptation of the Poisson matching lemma (Li and Anantharam, 2021) which allows for multiple decoders using disjoint codebooks. We further propose a hybrid coding scheme that partitions decoders into groups to optimally balance codebook and channel diversity. Numerical results on the binary symmetric channel demonstrate that the hybrid approach outperforms strategies where the decoders' codebooks are either fully shared or disjoint. - oai:arXiv.org:2601.10648v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Joseph Rowan, Buu Phan, Ashish Khisti - - - Uniform stability of the inverse Sturm-Liouville problem on a star-shaped graph - https://arxiv.org/abs/2601.10652 - arXiv:2601.10652v1 Announce Type: new -Abstract: In this paper, we study the inverse spectral problem for the Sturm-Liouville operators on a star-shaped graph, which consists in the recovery of the potentials from specral data or several spectra. The uniform stability of these inverse problems on the whole graph is proved. - oai:arXiv.org:2601.10652v1 - math.SP - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - E. E. Chitorkin, N. P. Bondarenko - - - Symmetries of Borcherds algebras - https://arxiv.org/abs/2601.10653 - arXiv:2601.10653v1 Announce Type: new -Abstract: We give an overview of the construction of Borcherds algebras, particularly the Monstrous Lie algebras $\mathfrak m_g$ constructed by Carnahan, where $g$ is an element of the Monster finite simple group. When $g$ is the identity element, $\mathfrak m_g$ is the Monster Lie algebra of Borcherds. We discuss the appearance of the $\mathfrak m_g$ in compactified models of the Heterotic String. We also summarize recent work on associating Lie group analogs to the Lie algebras $\mathfrak m_g$. We include a discussion of some open problems. - oai:arXiv.org:2601.10653v1 - math.QA - hep-th - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lisa Carbone - - - A note on strong similarity and the Connes embedding problem - https://arxiv.org/abs/2601.10654 - arXiv:2601.10654v1 Announce Type: new -Abstract: We show that there exists a completely bounded (c.b. in short) homomorphism $u$ from a $C^*$-algebra $C$ with the lifting property (in short LP) into a QWEP von Neumann algebra $N$ that is not strongly similar to a $*$-homomorphism, i.e. the similarities that ``orthogonalize" $u$ (which exist since $u$ is c.b.) cannot belong to the von Neumann algebra $N$. Moreover, the map $u$ does not admit any c.b. lifting up into the WEP $C^*$-algebra of which $N$ is a quotient. We can take $C=C^*(G)$ (full $C^*$-algebra) where $G$ is any nonabelian free group and $N= B(H)\bar \otimes M$ where $M$ is the von Neumann algebra generated by the reduced $C^*$-algebra of $G$. - oai:arXiv.org:2601.10654v1 - math.OA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Gilles Pisier - - - Hyperk\"ahler Degenerations from Parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs Bundles Moduli Spaces on the Punctured Sphere to Hyperpolygon Spaces - https://arxiv.org/abs/2601.10656 - arXiv:2601.10656v1 Announce Type: new -Abstract: Complete hyperk\"ahler 4-manifolds of finite energy are grouped into ALE, ALF, ALG$^{(*)}$, ALH$^{(*)}$, each of these being further classified according to the Dynkin type of their noncompact end. A family of ALG-$D_4$ spaces are modeled by certain moduli spaces of strongly parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles on the Riemann sphere with $n=4$ punctures. Meanwhile, a family of ALE-$D_4$ spaces are modeled by certain Nakajima quiver varieties known as $n=4$ hyperpolygon spaces. There is a map from hyperpolygon space to the moduli space of strong parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles that is a diffeomorphism onto its open and dense image. We show that under a fine-tuned degenerate limit, the pullback of a family of ALG-$D_4$ metrics parameterized by $R$ converges pointwise to the ALE-$D_4$ metric as $R \to 0$. While the connection to gravitational instantons occurs in the $n=4$ case, we prove our result for any finite $n$. - oai:arXiv.org:2601.10656v1 - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Laura Fredrickson, Arya Yae - - - Stable evaluation of derivatives for barycentric and continued fraction representations of rational functions - https://arxiv.org/abs/2601.10667 - arXiv:2601.10667v1 Announce Type: new -Abstract: Fast algorithms for approximation by rational functions exist for both barycentric and Thiele continued fraction (TCF) representations. We present the first numerically stable methods for derivative evaluation in the barycentric representation, including an $O(n)$ algorithm for all derivatives. We also extend an earlier $O(n)$ algorithm for evaluation of the TCF first derivative to higher orders. Numerical experiments confirm the robustness and efficiency of the proposed methods. - oai:arXiv.org:2601.10667v1 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Tobin A. Driscoll, Yuxing Zhou - - - On Necessary and Sufficient Conditions for Fixed Point Convergence: A Contractive Iteration Principle - https://arxiv.org/abs/2601.10669 - arXiv:2601.10669v1 Announce Type: new -Abstract: While numerous extensions of Banach's fixed point theorem typically offer only sufficient conditions for the existence and uniqueness of a fixed point and the convergence of iterative sequences, this study introduces a generalization grounded in the iterative contraction principle in complete metric spaces. This generalization establishes both the necessary and sufficient conditions for the existence of a unique fixed point to which all iterative sequences converge, along with an accurate error estimate. Furthermore, we present and prove an additional theorem that characterizes the convergence of all iterative sequences to fixed points that may not be unique. Several examples are provided to illustrate the practical application of these results, including a case where the traditional and well-known generalizations of Banach's theorem, such as those by Banach, Kannan, Chatterjea, Hardy-Rogers, Meir-Keeler, and Guseman, are inapplicable. - oai:arXiv.org:2601.10669v1 - math.FA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Vasil Zhelinski - - - Real characters and real classes of $\mathrm{GL}_2$ and $\mathrm{GU}_2$ over discrete valuation rings - https://arxiv.org/abs/2601.10670 - arXiv:2601.10670v1 Announce Type: new -Abstract: Let $\mathfrak{o}$ be the ring of integers of a non-archimedean local field with residue field of odd characteristic, $\mathfrak{p}$ be its maximal ideal and let $\mathfrak{o}_\ell = \mathfrak{o}/\mathfrak{p}^\ell$ for $\ell\ge 2$. In this article, we study real-valued characters and real representations of the finite groups $\mathrm{GL}_2(\mathfrak{o}_\ell)$ and $\mathrm{GU}_2(\mathfrak{o}_\ell)$. We give a complete classification of real and strongly real classes of these groups and characterize the real-valued irreducible complex characters. We prove that every real-valued irreducible complex character of $\mathrm{GL}_2(\mathfrak{o}_\ell)$ is afforded by a representation over $\mathbb{R}$. In contrast, we show that $\mathrm{GU}_2(\mathfrak{o}_\ell)$ admits real-valued irreducible characters that are not realizable over $\mathbb{R}$. These results extend the parallel known phenomena for the finite groups $\mathrm{GL}_n(\mathbb{F}_q)$ and $\mathrm{GU}_n(\mathbb{F}_q)$. - oai:arXiv.org:2601.10670v1 - math.RT - math.GR - math.RA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Archita Gupta, Tejbir Lohan, Pooja Singla - - - Breaking the Storage-Bandwidth Tradeoff in Distributed Storage with Quantum Entanglement - https://arxiv.org/abs/2601.10676 - arXiv:2601.10676v1 Announce Type: new -Abstract: This work investigates the use of quantum resources in distributed storage systems. Consider an $(n,k,d)$ distributed storage system in which a file is stored across $n$ nodes such that any $k$ nodes suffice to reconstruct the file. When a node fails, any $d$ helper nodes transmit information to a newcomer to rebuild the system. In contrast to the classical repair, where helper nodes transmit classical bits, we allow them to send classical information over quantum channels to the newcomer. The newcomer then generates its storage by performing appropriate measurements on the received quantum states. In this setting, we fully characterize the fundamental tradeoff between storage and repair bandwidth (total communication cost). Compared to classical systems, the optimal storage--bandwidth tradeoff can be significantly improved with the enhancement of quantum entanglement shared only among the surviving nodes, particularly at the minimum-storage regenerating point. Remarkably, we show that when $d \geq 2k-2$, there exists an operating point at which \textit{both storage and repair bandwidth are simultaneously minimized}. This phenomenon breaks the tradeoff in the classical setting and reveals a fundamentally new regime enabled by quantum communication. - oai:arXiv.org:2601.10676v1 - cs.IT - cs.DC - cs.NI - eess.SP - math.IT - quant-ph - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lei Hu, Mohamed Nomeir, Alptug Aytekin, Sennur Ulukus - - - Synchronizing Probabilities in Model-Driven Lossless Compression - https://arxiv.org/abs/2601.10678 - arXiv:2601.10678v1 Announce Type: new -Abstract: It is well-known in the field of lossless data compression that probabilistic next-symbol prediction can be used to compress sequences of symbols. Deep neural networks are able to capture rich dependencies in data, offering a powerful means of estimating these probabilities and hence an avenue towards more effective compression algorithms. However, both compressor and decompressor must have exactly matching predictions; even small non-deterministic differences (which often happen with learned models due to hardware, software, or computation order) can lead to cascading decoding failures. In this paper, we formalize the problem of prediction mismatch in model-driven compression, and introduce Probability Matching Interval Coding (PMATIC), a model-agnostic algorithm that tolerates bounded prediction mismatch with low overhead. PMATIC works with the predicted probabilities, making it compatible as a drop-in replacement for the arithmetic encoder in model-driven compression tools. We show theoretical correctness and performance bounds for PMATIC, and validate these results on text data. These results confirm that, when paired an advanced prediction model, PMATIC is robust to prediction mismatch while achieving compression rates that out-perform standard modern compression tools. - oai:arXiv.org:2601.10678v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Aviv Adler, Jennifer Tang - - - Implementation of Oblivious Transfer over Binary-Input AWGN Channels by Polar Codes - https://arxiv.org/abs/2601.10682 - arXiv:2601.10682v1 Announce Type: new -Abstract: We develop a one-out-of-two-oblivious transfer protocol over the binary-input additive white Gaussian noise channel using polar codes. The scheme uses two decoder views linked by automorphisms of the polar transform and publicly draws the encoder at random from the corresponding automorphism group. This yields perfect receiver privacy at any finite blocklength, since the public encoder distribution is independent of the receiver's choice bit. Sender privacy is obtained asymptotically via channel polarization combined with privacy amplification. Because the construction deliberately injects randomness on selected bad bit-channels, we derive a relaxed reliability criterion and evaluate finite-blocklength performance. Finally, we characterize the polar-transform automorphisms as bit-level permutations of bit-channel indices, and exploit this structure to derive and optimize an achievable finite-blocklength OT rate. - oai:arXiv.org:2601.10682v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Pin-Hsun Lin, Hadi Aghaee, Christian Deppe, Eduard A. Jorswieck, Holger Boche - - - Improved Constructions of Reed-Solomon Codes with Optimal Repair Bandwidth - https://arxiv.org/abs/2601.10685 - arXiv:2601.10685v1 Announce Type: new -Abstract: Maximum-distance-separable (MDS) codes are widely used in distributed storage, yet naive repair of a single erasure in an $[n,k]$ MDS code downloads the entire contents of $k$ nodes. Minimum Storage Regenerating (MSR) codes (Dimakis et al., 2010) minimize repair bandwidth by contacting $d>k$ helpers and downloading only a fraction of data from each. Guruswami and Wootters first proposed a linear repair scheme for Reed-Solomon (RS) codes, showing that they can be repaired with lower bandwidth than the naive approach. The existence of RS codes achieving the MSR point (RS-MSR codes) nevertheless remained open until the breakthrough construction of Tamo, Barg, and Ye, which yields RS-MSR codes with subpacketization $\ell = s \prod_{i=1}^n p_i$, where $p_i$ are distinct primes satisfying $p_i \equiv 1 \pmod{s}$ and $s=d+1-k$. - In this paper, we present an improved construction of RS-MSR codes by eliminating the congruence condition $p_i \equiv 1 \pmod{s}$. Consequently, our construction reduces the subpacketization by a multiplicative factor of $\phi(s)^n$ ( $\phi(\cdot)$ is Euler's totient function) and broadens the range of feasible parameters for RS-MSR codes. - oai:arXiv.org:2601.10685v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jing Qiu, Weijun Fang, Shu-Tao Xia, Fang-Wei Fu - - - Vertex operator algebra bundles on modular curves and their associated modular forms - https://arxiv.org/abs/2601.10686 - arXiv:2601.10686v1 Announce Type: new -Abstract: This paper describes the vector bundle on the elliptic modular curve that is associated to a vertex operator algebra $V$ (VOA) or more generally a quasi-vertex operator algebra (QVOA), with a view towards future applications aimed at studying the characters of VOAs. We explain how the modes of sections of $V$ give rise naturally to $V$-valued quasi-modular forms. The space $Q(V)$ of $V$-valued quasi-modular forms is endowed with the structure of a doubled QVOA, and in particular the algebra $Q$ of quasi-modular forms is itself a doubled QVOA. $Q(V)$ also admits a natural derivative operator arising from the connection on the bundle defined by $V$ and the modular derivative, which we call the raising operator. We introduce an associated lowering operator $\Lambda$ on $Q(V)$ having the property that the $V$-valued modular forms $M(V)\subseteq Q(V)$ are the kernel of $\Lambda$. This extends the classical theory of scalar-valued quasi-modular forms. We exhibit an explicit isomorphism of $M(V)$ with $M \otimes V$. Finally, the coordinate invariance of vertex operators implies that $M(V)$ has a natural Hecke theory, and we use this isomorphism to fully describe the Hecke eigensystems: they are the same as the systems of eigenvalues that arise from scalar-valued quasi-modular forms. - oai:arXiv.org:2601.10686v1 - math.NT - math-ph - math.MP - math.QA - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Daniel Barake, Owen Chuchman, Cameron Franc, Geoffrey Mason, Brett Nasserden - - - Perfect Secret Key Generation for a class of Hypergraphical Sources - https://arxiv.org/abs/2601.10697 - arXiv:2601.10697v1 Announce Type: new -Abstract: Nitinawarat and Narayan proposed a perfect secret key generation scheme for the so-called \emph{pairwise independent network (PIN) model} by exploiting the combinatorial properties of the underlying graph, namely the spanning tree packing rate. This work considers a generalization of the PIN model where the underlying graph is replaced with a hypergraph, and makes progress towards designing similar perfect secret key generation schemes by exploiting the combinatorial properties of the hypergraph. - Our contributions are two-fold. We first provide a capacity achieving scheme for a complete $t$-uniform hypergraph on $m$ vertices by leveraging a packing of the complete $t$-uniform hypergraphs by what we refer to as star hypergraphs, and designing a scheme that gives $\binom{m-2}{t-2}$ bits of perfect secret key per star graph. Our second contribution is a 2-bit perfect secret key generation scheme for 3-uniform star hypergraphs whose projections are cycles. This scheme is then extended to a perfect secret key generation scheme for generic 3-uniform hypergraphs by exploiting star graph packing of 3-uniform hypergraphs and Hamiltonian packings of graphs. The scheme is then shown to be capacity achieving for certain classes of hypergraphs. - oai:arXiv.org:2601.10697v1 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Manuj Mukherjee, Sagnik Chatterjee, Alhad Sethi - - - Unbounded symbols, heat flow, and Toeplitz operators - https://arxiv.org/abs/2601.10711 - arXiv:2601.10711v1 Announce Type: new -Abstract: We disprove the natural domain extension of the Berger--Coburn heat-flow conjecture for Toeplitz operators on the Bargmann space and identify the failure mechanism as a gap between pointwise and uniform control of a Gaussian averaging of the squared modulus of the symbol, a gap that is invisible to the linear form $T_g$. We establish that the form-defined operator $T_g$ and the natural-domain operator $U_g$ decouple in the unbounded symbols regime: while $T_g$ is governed by linear averaging, $U_g$ is controlled by the quadratic intensity of $|g|^2$. We construct a smooth, nonnegative radial symbol $g$ satisfying the coherent-state admissibility hypothesis with bounded heat transforms for all time $t>0$; for this symbol, $T_g$ is bounded, yet $U_g$ is unbounded. This is a strictly global phenomenon: under the coherent-state hypothesis, local singularities are insufficient to cause unboundedness, leaving the ``geometry at infinity'' as the sole obstruction. Boundedness of $U_g$ is equivalent to the condition that $|g|^2 d\mu$ is a Fock--Carleson measure, a condition strictly stronger than the linear average $g d\mu$ governing $T_g$. Finally, regarding the gap between the known sub-critical sufficiency condition and the critical heat time, we prove that heat-flow regularity is irreversible in this context and show that bootstrapping strategies cannot resolve the gap between sufficiency and critical time. - oai:arXiv.org:2601.10711v1 - math.FA - math.AP - math.CV - Fri, 16 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sam Looi - - - Self-Organizing Dual-Buffer Adaptive Clustering Experience Replay (SODASER) for Safe Reinforcement Learning in Optimal Control - https://arxiv.org/abs/2601.06540 - arXiv:2601.06540v1 Announce Type: cross -Abstract: This paper proposes a novel reinforcement learning framework, named Self-Organizing Dual-buffer Adaptive Clustering Experience Replay (SODACER), designed to achieve safe and scalable optimal control of nonlinear systems. The proposed SODACER mechanism consisting of a Fast-Buffer for rapid adaptation to recent experiences and a Slow-Buffer equipped with a self-organizing adaptive clustering mechanism to maintain diverse and non-redundant historical experiences. The adaptive clustering mechanism dynamically prunes redundant samples, optimizing memory efficiency while retaining critical environmental patterns. The approach integrates SODASER with Control Barrier Functions (CBFs) to guarantee safety by enforcing state and input constraints throughout the learning process. To enhance convergence and stability, the framework is combined with the Sophia optimizer, enabling adaptive second-order gradient updates. The proposed SODACER-Sophia's architecture ensures reliable, effective, and robust learning in dynamic, safety-critical environments, offering a generalizable solution for applications in robotics, healthcare, and large-scale system optimization. The proposed approach is validated on a nonlinear Human Papillomavirus (HPV) transmission model with multiple control inputs and safety constraints. Comparative evaluations against random and clustering-based experience replay methods demonstrate that SODACER achieves faster convergence, improved sample efficiency, and a superior bias-variance trade-off, while maintaining safe system trajectories, validated via the Friedman test. - oai:arXiv.org:2601.06540v1 - eess.SY - cs.AI - cs.LG - cs.RO - cs.SY - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Roya Khalili Amirabadi, Mohsen Jalaeian Farimani, Omid Solaymani Fard - - - Absorption and fixation times for evolutionary processes on graphs - https://arxiv.org/abs/2601.09737 - arXiv:2601.09737v1 Announce Type: cross -Abstract: In this paper, we study the absorption and fixation times for evolutionary processes on graphs, under different updating rules. While in Moran process a single neighbour is randomly chosen to be replaced, in proliferation processes other neighbours can be replaced using Bernoulli or binomial draws depending on $0 < p \leq 1$. There is a critical value $p_c$ such that the proliferation is advantageous or disadvantageous in terms of fixation probability depending on whether $p > p_c$ or $p < p_c$. - We clarify the role of symmetries for computing the fixation time in Moran process. We show that the Maruyama-Kimura symmetry depend on the graph structure induced in each state, implying asymmetry for all graphs except cliques and cycles. There is a fitness value, not necessarily $1$, beyond which the fixation time decreases monotonically. - We apply Harris' graphical method to prove that the fixation time decreases monotonically depending on $p$. Thus there exists another value $p_t$ for which the proliferation is advantageous or disadvantageous in terms of time. However, at the critical level $p=p_c$, the proliferation is highly advantageous when $r \to +\infty$. - oai:arXiv.org:2601.09737v1 - q-bio.PE - math-ph - math.MP - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Fernando Alcalde Cuesta, Gustavo Guerberoff, \'Alvaro Lozano Rojo - - - Topological Percolation in Urban Dengue Transmission: A Multi-Scale Analysis of Spatial Connectivity - https://arxiv.org/abs/2601.09747 - arXiv:2601.09747v1 Announce Type: cross -Abstract: We investigate the spatial organization of dengue cases in the city of Recife, Brazil, from 2015 to 2024, using tools from statistical physics and topological data analysis. Reported cases are modeled as point clouds in a metric space, and their spatial connectivity is studied through Vietoris-Rips filtrations and zero-dimensional persistent homology, which captures the emergence and collapse of connected components across spatial scales. By parametrizing the filtration using percentiles of the empirical distance distribution, we identify critical percolation thresholds associated with abrupt growth of the largest connected component. These thresholds define distinct geometric regimes, ranging from fragmented spatial patterns to highly concentrated, percolated structures. Remarkably, years with similar incidence levels exhibit qualitatively different percolation behavior, demonstrating that case counts alone do not determine the spatial organization of transmission. Our analysis further reveals pronounced temporal heterogeneity in the percolation properties of dengue spread, including a structural rupture in 2020 characterized by delayed or absent spatial percolation. These findings highlight percolation-based topological observables as physically interpretable and sensitive descriptors of urban epidemic structure, offering a complementary perspective to traditional spatial and epidemiological analyses. - oai:arXiv.org:2601.09747v1 - q-bio.PE - math.GT - stat.AP - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Marc\'ilio Ferreira dos Santos, Cleiton de Lima Ricardo - - - Dedifferentiation stabilizes stem cell lineages: From CTMC to diffusion theory and thresholds - https://arxiv.org/abs/2601.09752 - arXiv:2601.09752v1 Announce Type: cross -Abstract: We study stem-terminally differentiated (TD) lineages in small niches where demographic noise from discrete division and death events is non-negligible. Starting from a mechanistic five-channel, density-dependent CTMC (symmetric self-renewal, symmetric differentiation, asymmetric division, dedifferentiation, TD death), we derive its mean-field limit and a functional CLT, obtaining a chemical Langevin diffusion whose explicit state-dependent covariance exactly matches the CTMC's aggregated channel-wise infinitesimal covariances. Within this diffusion approximation we remove the dedifferentiation flux and obtain a sharp dichotomy: in subcritical regimes the stem coordinate becomes extinct asymptotically almost surely, whereas in supercritical regimes polynomial moments diverge exponentially. This identifies, at the diffusion level, a structural failure mode of strictly hierarchical lineages under demographic fluctuations and clarifies how a cyclic return flux can rescue homeostasis. For interpretation we also derive an exact totals ODE backbone from a damage-structured transport model and obtain two steady-state constraints (ratio and equalization laws) linking compartment ratios to turnover and balancing dedifferentiation against fate bias. Numerical experiments corroborate the $\Omega^{-1/2}$ fluctuation scaling, illustrate the pathology, and contrast theorem-regime global convergence with threshold (Allee-type) behaviour outside the theorem hypotheses. - oai:arXiv.org:2601.09752v1 - physics.bio-ph - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Jiguang Yu, Louis Shuo Wang, Ye Liang - - - Limits of Rank Recovery in Bilinear Observation Problems - https://arxiv.org/abs/2601.09754 - arXiv:2601.09754v1 Announce Type: cross -Abstract: Bilinear observation problems arise in many physical and information-theoretic settings, where observables and states enter multiplicatively. Rank-based diagnostics are commonly used in such problems to assess the effective dimensionality accessible to observation, often under the implicit assumption that rank deficiency can be resolved through numerical refinement. Here we examine this assumption by analyzing the rank and nullity of a bilinear observation operator under systematic tolerance variation. Rather than focusing on a specific reconstruction algorithm, we study the operator directly and identify extended rank plateaus that persist across broad tolerance ranges. These plateaus indicate stable dimensional deficits that are not removed by refinement procedures applied within a fixed problem definition. To investigate the origin of this behavior, we resolve the nullspace into algebraic sectors defined by the block structure of the variables. The nullspace exhibits a pronounced but nonexclusive concentration in specific sectors, revealing an organized internal structure rather than uniform dimensional loss. Comparing refinement with explicit modification of the problem formulation further shows that rank recovery in the reported setting requires a change in the structure of the observation problem itself. Here, "problem modification" refers to changes that alter the bilinear observation structure (e.g., admissible operator/state families or coupling constraints), in contrast to refinements that preserve the original formulation such as tolerance adjustment and numerical reparameterizations. Together, these results delineate limits of rank recovery in bilinear observation problems and clarify the distinction between numerical refinement and problem modification in accessing effective dimensional structure. - oai:arXiv.org:2601.09754v1 - quant-ph - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Seungbeom Choi - - - Zero-Error List Decoding for Classical-Quantum Channels - https://arxiv.org/abs/2601.09786 - arXiv:2601.09786v1 Announce Type: cross -Abstract: The aim of this work is to study the zero-error capacity of pure-state classical-quantum channels in the setting of list decoding. We provide an achievability bound for list-size two and a converse bound holding for every fixed list size. The two bounds coincide for channels whose pairwise absolute state overlaps form a positive semi-definite matrix. Finally, we discuss a remarkable peculiarity of the classical-quantum case: differently from the fully classical setting, the rate at which the sphere-packing bound diverges might not be achievable by zero-error list codes, even when we take the limit of fixed but arbitrarily large list size. - oai:arXiv.org:2601.09786v1 - quant-ph - cs.IT - math-ph - math.IT - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Marco Dalai, Filippo Girardi, Ludovico Lami - - - Localization of quantum states within subspaces - https://arxiv.org/abs/2601.09817 - arXiv:2601.09817v1 Announce Type: cross -Abstract: A precise definition is proposed for the localization probability of a quantum state within a given subspace of the full Hilbert space of a quantum system. The corresponding localized component of the state is explicitly identified, and several mathematical properties are established. Applications and interpretations in the context of quantum information are also discussed. - oai:arXiv.org:2601.09817v1 - quant-ph - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - L. L. Salcedo - - - Forecasting Seasonal Peaks of Pediatric Respiratory Infections Using an Alert-Based Model Combining SIR Dynamics and Historical Trends in Santiago, Chile - https://arxiv.org/abs/2601.09821 - arXiv:2601.09821v1 Announce Type: cross -Abstract: Acute respiratory infections (ARI) are a major cause of pediatric hospitalization in Chile, producing marked winter increases in demand that challenge hospital planning. This study presents an alert-based forecasting model to predict the timing and magnitude of ARI hospitalization peaks in Santiago. The approach integrates a seasonal SIR model with a historical mobile predictor, activated by a derivative-based alert system that detects early epidemic growth. Daily hospitalization data from DEIS were smoothed using a 15-day moving average and Savitzky-Golay filtering, and parameters were estimated using a penalized loss function to reduce sensitivity to noise. Retrospective evaluation and real-world implementation in major Santiago pediatric hospitals during 2023 and 2024 show that peak date can be anticipated about one month before the event and predicted with high accuracy two weeks in advance. Peak magnitude becomes informative roughly ten days before the peak and stabilizes one week prior. The model provides a practical and interpretable tool for hospital preparedness. - oai:arXiv.org:2601.09821v1 - stat.AP - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Gloria Henr\'iquez, Jhoan B\'aez, V\'ictor Riquelme, Pedro Gajardo, Michel Royer, H\'ector Ram\'irez - - - A New Convergence Analysis of Plug-and-Play Proximal Gradient Descent Under Prior Mismatch - https://arxiv.org/abs/2601.09831 - arXiv:2601.09831v1 Announce Type: cross -Abstract: In this work, we provide a new convergence theory for plug-and-play proximal gradient descent (PnP-PGD) under prior mismatch where the denoiser is trained on a different data distribution to the inference task at hand. To the best of our knowledge, this is the first convergence proof of PnP-PGD under prior mismatch. Compared with the existing theoretical results for PnP algorithms, our new results removed the need for several restrictive and unverifiable assumptions. - oai:arXiv.org:2601.09831v1 - cs.LG - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Guixian Xu, Jinglai Li, Junqi Tang - - - The formation of periodic three-body orbits for Newtonian systems - https://arxiv.org/abs/2601.09843 - arXiv:2601.09843v1 Announce Type: cross -Abstract: Braids are periodic solutions to the general N-body problem in gravitational dynamics. These solutions seem special and unique, but they may result from rather usual encounters between four bodies. We aim at understanding the existence of braids in the Galaxy by reverse engineering the interactions in which they formed. We simulate self-gravitating systems of N particles, starting with the constructing of a specific braid, and bombard it with a single object. We study how frequently the bombarded braid dissolves in four singles, a triple and a single, a binary and 2 singles, or 2 binaries. The relative proportion of those events gives us insight into how easy it is to generate a braid through the reverse process. It turns out that braids are easily generated from encounters between 2 binaries, or a triple with a single object, independent on the braid's stability. We find that 3 of the explored braids are linearly stable against small perturbations, whereas one is unstable and short-lived. The shortest-lived braid appears the least stable and the most chaotic. nonplanar encounters also lead to braid formation, which, in our experiments, themselves are planar. The parameter space in azimuth and polar angle that lead to braid formation via binary-binary or triple-single encounters is anisotropic, and the distribution has a low fractal dimension. Since a substantial fraction of ~9% of our calculations lead to periodic 3-body systems, braids may be more common than expected. They could form in multi-body interactions. We do not expect many to exist for time, but they may be common as transients, as they survive for tens to hundreds of periodic orbits. We argue that braids are common in relatively shallow-potential background fields, such as the Oort cloud or the Galactic halo. If composed of compact objects, they potentially form interesting targets for gravitational wave detectors. - oai:arXiv.org:2601.09843v1 - astro-ph.GA - astro-ph.IM - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Simon Portegies Zwart, Arjen Doelman, Jelmer Sein - - - Accelerated Regularized Wasserstein Proximal Sampling Algorithms - https://arxiv.org/abs/2601.09848 - arXiv:2601.09848v1 Announce Type: cross -Abstract: We consider sampling from a Gibbs distribution by evolving a finite number of particles using a particular score estimator rather than Brownian motion. To accelerate the particles, we consider a second-order score-based ODE, similar to Nesterov acceleration. In contrast to traditional kernel density score estimation, we use the recently proposed regularized Wasserstein proximal method, yielding the Accelerated Regularized Wasserstein Proximal method (ARWP). We provide a detailed analysis of continuous- and discrete-time non-asymptotic and asymptotic mixing rates for Gaussian initial and target distributions, using techniques from Euclidean acceleration and accelerated information gradients. Compared with the kinetic Langevin sampling algorithm, the proposed algorithm exhibits a higher contraction rate in the asymptotic time regime. Numerical experiments are conducted across various low-dimensional experiments, including multi-modal Gaussian mixtures and ill-conditioned Rosenbrock distributions. ARWP exhibits structured and convergent particles, accelerated discrete-time mixing, and faster tail exploration than the non-accelerated regularized Wasserstein proximal method and kinetic Langevin methods. Additionally, ARWP particles exhibit better generalization properties for some non-log-concave Bayesian neural network tasks. - oai:arXiv.org:2601.09848v1 - stat.ML - cs.LG - math.OC - stat.CO - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hong Ye Tan, Stanley Osher, Wuchen Li - - - Model selection by cross-validation in an expectile linear regression - https://arxiv.org/abs/2601.09874 - arXiv:2601.09874v1 Announce Type: cross -Abstract: For linear models that may have asymmetric errors, we study variable selection by cross-validation. The data are split into training and validation sets, with the number of observations in the validation set much larger than in the training set. For the model coefficients, the expectile or adaptive LASSO expectile estimators are calculated on the training set. These estimators will be used to calculate the cross-validation mean score (CVS) on the validation set. We show that the model that minimizes CVS is consistent in two cases: when the number of explanatory variables is fixed or when it depends on the number of observations. Monte Carlo simulations confirm the theoretical results and demonstrate the superiority of our estimation method compared to two others in the literature. The usefulness of the CV expectile model selection technique is illustrated by applying it to real data sets. - oai:arXiv.org:2601.09874v1 - stat.ME - math.ST - stat.CO - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bilel Bousselmi, Gabriela Ciuperca - - - Learning about Treatment Effects with Prior Studies: A Bayesian Model Averaging Approach - https://arxiv.org/abs/2601.09888 - arXiv:2601.09888v1 Announce Type: cross -Abstract: We establish concentration rates for estimation of treatment effects in experiments that incorporate prior sources of information -- such as past pilots, related studies, or expert assessments -- whose external validity is uncertain. Each source is modeled as a Gaussian prior with its own mean and precision, and sources are combined using Bayesian model averaging (BMA), allowing data from the new experiment to update posterior weights. To capture empirically relevant settings in which prior studies may be as informative as the current experiment, we introduce a nonstandard asymptotic framework in which prior precisions grow with the experiment's sample size. In this regime, posterior weights are governed by an external-validity index that depends jointly on a source's bias and information content: biased sources are exponentially downweighted, while unbiased sources dominate. When at least one source is unbiased, our procedure concentrates on the unbiased set and achieves faster convergence than relying on new data alone. When all sources are biased, including a deliberately conservative (diffuse) prior guarantees robustness and recovers the standard convergence rate. - oai:arXiv.org:2601.09888v1 - econ.EM - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Frederico Finan, Demian Pouzo - - - Learning and Equilibrium under Model Misspecification - https://arxiv.org/abs/2601.09891 - arXiv:2601.09891v1 Announce Type: cross -Abstract: This chapter develops a unified framework for studying misspecified learning situations in which agents optimize and update beliefs within an incorrect model of their environment. We review the statistical foundations of learning from misspecified models and extend these insights to environments with endogenous, action-dependent data, including both single agent and strategic settings. - oai:arXiv.org:2601.09891v1 - econ.TH - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Conference Volume for 2025 World Congress of the Econometric Society, Chapter 6 - Ignacio Esponda, Demian Pouzo - - - Analytic approach to boundary integrability with application to mixed-flux $AdS_3 \times S^3$ - https://arxiv.org/abs/2601.09935 - arXiv:2601.09935v1 Announce Type: cross -Abstract: Boundary integrability provides rare analytic control over field theories in the presence of an interface, from quantum impurity problems to open string dynamics. We develop an analytic framework for integrable boundaries in two-dimensional sigma-models that determines admissible reflection maps directly from the meromorphic Lax connection. Applying it to open strings on $AdS_3\times S^3$ with mixed NSNS and RR flux, we find two branches of integrable boundary conditions. One branch admits D-branes wrapping twisted conjugacy classes on $SU(1,1)\times SU(2)$, with the mixed-flux deformation encoded entirely into dynamical boundary data. At the exactly solvable WZW point these coincide with the conformal D-branes, providing a natural link to conformal perturbation theory. - oai:arXiv.org:2601.09935v1 - hep-th - math-ph - math.MP - nlin.SI - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Julio Cabello Gil, Sibylle Driezen - - - A Level Set Method on Particle Flow Maps - https://arxiv.org/abs/2601.09939 - arXiv:2601.09939v1 Announce Type: cross -Abstract: This paper introduces a Particle Flow Map Level Set (PFM-LS) method for high-fidelity interface tracking. We store level-set values, gradients, and Hessians on particles concentrated in a narrow band around the interface, advecting them via bidirectional flow maps while using a conventional grid-based representation elsewhere. By interpreting the level set value as a 3-form and its gradient as a 1-form, PFM-LS achieves exceptional geometric fidelity during complex deformations and preserves sub-grid features that traditional methods cannot capture. Our dual-timescale approach utilizes long-range maps for values and gradients, with frequent reinitialization of short-range maps for the distortion-sensitive Hessian, alongside adaptive particle control that maintains sufficient density within the narrow band. We also develop a hybrid particle-grid quasi-Newton redistancing scheme that preserves fine-scale features while enforcing the signed-distance property. Benchmark comparisons in 2D and 3D demonstrate that PFM-LS achieves state-of-the-art volume preservation and shape fidelity against a broad range of existing level-set methods. - oai:arXiv.org:2601.09939v1 - physics.comp-ph - cs.NA - math.NA - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jinjin He, Taiyuan Zhang, Zhiqi Li, Junwei Zhou, Duowen Chen, Bo Zhu - - - Kinematic Tokenization: Optimization-Based Continuous-Time Tokens for Learnable Decision Policies in Noisy Time Series - https://arxiv.org/abs/2601.09949 - arXiv:2601.09949v1 Announce Type: cross -Abstract: Transformers are designed for discrete tokens, yet many real-world signals are continuous processes observed through noisy sampling. Discrete tokenizations (raw values, patches, finite differences) can be brittle in low signal-to-noise regimes, especially when downstream objectives impose asymmetric penalties that rationally encourage abstention. We introduce Kinematic Tokenization, an optimization-based continuous-time representation that reconstructs an explicit spline from noisy measurements and tokenizes local spline coefficients (position, velocity, acceleration, jerk). This is applied to financial time series data in the form of asset prices in conjunction with trading volume profiles. Across a multi-asset daily-equity testbed, we use a risk-averse asymmetric classification objective as a stress test for learnability. Under this objective, several discrete baselines collapse to an absorbing cash policy (the Liquidation Equilibrium), whereas the continuous spline tokens sustain calibrated, non-trivial action distributions and stable policies. These results suggest that explicit continuous-time tokens can improve the learnability and calibration of selective decision policies in noisy time series under abstention-inducing losses. - oai:arXiv.org:2601.09949v1 - cs.LG - cs.AI - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Griffin Kearney - - - Correspondences in computational and dynamical complexity II: forcing complex reductions - https://arxiv.org/abs/2601.09973 - arXiv:2601.09973v1 Announce Type: cross -Abstract: An algebraic telic problem is a decision problem in $\textsf{NP}_\mathbb{R}$ formalizing finite-time reachability questions for one-dimensional dynamical systems. We prove that the existence of "natural" mapping reductions between algebraic telic problems coming from distinct dynamical systems implies the two dynamical systems exhibit similar behavior (in a precise sense). As a consequence, we obtain explicit barriers for algorithms solving algebraic telic problems coming from complex dynamical systems, such as those with positive topological entropy. For example, some telic problems cannot be decided by uniform arithmetic circuit families with only $+$ and $\times$ gates. - oai:arXiv.org:2601.09973v1 - cs.CC - math.DS - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Samuel Everett - - - In-Context Operator Learning on the Space of Probability Measures - https://arxiv.org/abs/2601.09979 - arXiv:2601.09979v1 Announce Type: cross -Abstract: We introduce \emph{in-context operator learning on probability measure spaces} for optimal transport (OT). The goal is to learn a single solution operator that maps a pair of distributions to the OT map, using only few-shot samples from each distribution as a prompt and \emph{without} gradient updates at inference. We parameterize the solution operator and develop scaling-law theory in two regimes. In the \emph{nonparametric} setting, when tasks concentrate on a low-intrinsic-dimension manifold of source--target pairs, we establish generalization bounds that quantify how in-context accuracy scales with prompt size, intrinsic task dimension, and model capacity. In the \emph{parametric} setting (e.g., Gaussian families), we give an explicit architecture that recovers the exact OT map in context and provide finite-sample excess-risk bounds. Our numerical experiments on synthetic transports and generative-modeling benchmarks validate the framework. - oai:arXiv.org:2601.09979v1 - cs.LG - cs.NA - math.NA - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Frank Cole, Dixi Wang, Yineng Chen, Yulong Lu, Rongjie Lai - - - Nearest Kronecker Product Decomposition Based Subband Adaptive Filter: Algorithms and Applications - https://arxiv.org/abs/2601.10078 - arXiv:2601.10078v1 Announce Type: cross -Abstract: Recently, the nearest Kronecker product (NKP) decomposition-based normalized least mean square (NLMS-NKP) algorithm has demonstrated superior convergence performance compared to the conventional NLMS algorithm. However, its convergence rate exhibits significant degradation when processing highly correlated input signals. To address this problem, we propose a type-I NKP-based normalized subband adaptive filter (NSAF) algorithm, namely NSAF-NKP-I. Nevertheless, this algorithm incurs substantially higher computational overhead than the NLMS-NKP algorithm. Remarkably, our enhanced type-II NKP-based NSAF (NSAF-NKP-II) algorithm achieves equivalent convergence performance while substantially reducing computational complexity. Furthermore, to enhance robustness against impulsive noise interference, we develop two robust variants: the maximum correntropy criterion-based robust NSAF-NKP (RNSAF-NKP-MCC) and logarithmic criterion-based robust NSAF-NKP (RNSAF-NKP-LC) algorithms. Additionally, detailed analyses of computational complexity, step-size range, and theoretical steady-state performance are provided for theproposed algorithms. To enhance the practicability of the NSAF-NKP-II algorithm in complex nonlinear environments, we further devise two nonlinear implementations: the trigonometric functional link network-based NKP-NSAF (TFLN-NSAF-NKP) and Volterra series expansion-based NKP-NSAF (Volterra-NKP-NSAF) algorithms. In active noise control (ANC) systems, we further propose the filtered-x NSAF-NKP-II (NKP-FxNSAF) algorithm. Simulation experiments in echo cancellation, sparse system identification, nonlinear processing, and ANC scenarios are conducted to validate the superiority of the proposed algorithms over existing state-of-the-art counterparts. - oai:arXiv.org:2601.10078v1 - eess.AS - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1109/TASLPRO.2025.3649394 - IEEE Transactions on Audio, Speech and Language Processing, pp. 1-16, December 2025 - Jianhong Ye, Haiquan Zhao - - - Starfield: Demand-Aware Satellite Topology Design for Low-Earth Orbit Mega Constellations - https://arxiv.org/abs/2601.10083 - arXiv:2601.10083v1 Announce Type: cross -Abstract: Low-Earth orbit (LEO) mega-constellations are emerging as high-capacity backbones for next-generation Internet. Deployment of laser terminals enables high-bandwidth, low-latency inter-satellite links (ISLs); however, their limited number, slow acquisition, and instability make forming a stable satellite topology difficult. Existing patterns like +Grid and Motif ignore regional traffic, ground station placement, and constellation geometry. Given sparse population distribution on Earth and the isolation of rural areas, traffic patterns are inherently non-uniform, providing an opportunity to orient inter-satellite links (ISLs) according to these traffic patterns. In this paper, we propose Starfield, a novel demand-aware satellite topology design heuristic algorithm supported by mathematical analysis. We first formulate a vector field on the constellation's shell according to traffic flows and define a corresponding Riemannian metric on the spherical manifold of the shell. The metric, combined with the spatial geometry, is used to assign a distance to each potential ISL, which we then aggregate over all demand flows to generate a heuristic for each satellite's link selection. Inspired by +Grid, each satellite selects the link with the minimum Riemannian heuristic along with its corresponding angular links. To evaluate Starfield, we developed a custom, link-aware, and link-configurable packet-level simulator, comparing it against +Grid and Random topologies. For the Phase 1 Starlink, simulation results show up to a 30% reduction in hop count and a 15% improvement in stretch factor across multiple traffic distributions. Moreover, static Starfield, an inter-orbital link matching modification of Starfield, achieves a 20% improvement in stretch factor under realistic traffic patterns compared to +Grid. Experiments further demonstrate Starfield's robustness under traffic demand perturbations. - oai:arXiv.org:2601.10083v1 - cs.NI - cs.ET - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Shayan Hamidi Dehshali, Tzu-Hsuan Liao, Shaileshh Bojja Venkatakrishnan - - - A volume penalization method for solving conjugate scalar transport with interfacial jump conditions - https://arxiv.org/abs/2601.10134 - arXiv:2601.10134v1 Announce Type: cross -Abstract: Conjugate scalar transport with interfacial jump conditions on complex interfacial geometries is common in thermal and chemical processes, while its accurate and efficient simulations are still quite challenging. In the present study, a novel treatment of a two-phase interface in the volume penalization method, a kind of immersed boundary method, for solving conjugate scalar transport with general interfacial boundary conditions is developed. We first propose an interfacial treatment for solving an advection-diffusion equation with a Neumann boundary condition, and then extend it to general conjugate scalar transport with both interfacial flux and scalar jumps. A one-dimensional diffusion problem is solved to verify the present scheme and demonstrate the advantage of the present scheme in improving accuracy and unifying the governing equations in the two phases with an additional source term representing the local jump condition of the interfacial scalar flux. Then, the present scheme is further applied to fluid-solid coupled scalar diffusion and advection-diffusion problems with the scalar and its flux jumps across the interface. The simulation results of the present scheme generally show good agreement with reference results obtained by body-fitted mesh simulations with average relative deviations less than 3.0%. - oai:arXiv.org:2601.10134v1 - physics.comp-ph - cs.NA - math.NA - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Ming Liu, Yosuke Hasegawa - - - Distributed Linearly Separable Computation with Arbitrary Heterogeneous Data Assignment - https://arxiv.org/abs/2601.10177 - arXiv:2601.10177v1 Announce Type: cross -Abstract: Distributed linearly separable computation is a fundamental problem in large-scale distributed systems, requiring the computation of linearly separable functions over different datasets across distributed workers. This paper studies a heterogeneous distributed linearly separable computation problem, including one master and N distributed workers. The linearly separable task function involves Kc linear combinations of K messages, where each message is a function of one dataset. Distinguished from the existing homogeneous settings that assume each worker holds the same number of datasets, where the data assignment is carefully designed and controlled by the data center (e.g., the cyclic assignment), we consider a more general setting with arbitrary heterogeneous data assignment across workers, where `arbitrary' means that the data assignment is given in advance and `heterogeneous' means that the workers may hold different numbers of datasets. Our objective is to characterize the fundamental tradeoff between the computable dimension of the task function and the communication cost under arbitrary heterogeneous data assignment. Under the constraint of integer communication costs, for arbitrary heterogeneous data assignment, we propose a universal computing scheme and a universal converse bound by characterizing the structure of data assignment, where they coincide under some parameter regimes. We then extend the proposed computing scheme and converse bound to the case of fractional communication costs. - oai:arXiv.org:2601.10177v1 - cs.DC - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ziting Zhang, Kai Wan, Minquan Cheng, Shuo Shao, Giuseppe Caire - - - Discrete versus continuous -- lattice models and their exact continuous counterparts - https://arxiv.org/abs/2601.10184 - arXiv:2601.10184v1 Announce Type: cross -Abstract: We review and study the correspondence between discrete lattice/chain models of interacting particles and their continuous counterparts represented by partial differential equations. We study the correspondence problem for nearest neighbour interaction lattice models as well as for multiple-neighbour interaction lattice models, and we gradually proceed from infinite lattices to periodic lattices and finally to finite lattices with fixed ends/zero Dirichlet boundary conditions. The whole study is framed as systematic specialisation of Fourier analysis tools from the continuous to the discrete setting and vice versa, and the correspondence between the discrete and continuous models is examined primarily with regard to the dispersion relation. - oai:arXiv.org:2601.10184v1 - physics.class-ph - cs.NA - math.NA - physics.comp-ph - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lorenzo Fusi, Oliver K\v{r}enek, V\'it Pr\r{u}\v{s}a, Casey Rodriguez, Rebecca Tozzi, Martin Vejvoda - - - Adversarial Hypothesis Testing for Quantum Channels - https://arxiv.org/abs/2601.10243 - arXiv:2601.10243v1 Announce Type: cross -Abstract: This paper presents a systematic study of adversarial hypothesis testing for both quantum-quantum (QQ) and classical-quantum (CQ) channels. Unlike conventional channel discrimination, we consider a framework where the sender, Alice, selects the channel input adversarially to minimize Bob's distinguishability. We analyze this problem across four settings based on whether Alice employs i.i.d. or general inputs and whether the receiver, Bob, is informed of the specific input choice (allowing his measurement to depend on the input). We characterize the Stein exponents for each setting and reveal a striking distinction in behavior: for QQ channels with i.i.d. inputs, Bob's knowledge of the input significantly enhances distinguishability, yet this advantage vanishes when general inputs are permitted. In contrast, for CQ channels, Bob being informed provides a consistent advantage over the corresponding entanglement-breaking channels for both i.i.d. and general inputs. These results demonstrate a unique phenomenon in adversarial hypothesis testing where the CQ channel does not merely behave as a special case of the QQ channel. - oai:arXiv.org:2601.10243v1 - quant-ph - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Masahito Hayashi, Hao-Chung Cheng, Li Gao - - - Queueing-Aware Optimization of Reasoning Tokens for Accuracy-Latency Trade-offs in LLM Servers - https://arxiv.org/abs/2601.10274 - arXiv:2601.10274v1 Announce Type: cross -Abstract: We consider a single large language model (LLM) server that serves a heterogeneous stream of queries belonging to $N$ distinct task types. Queries arrive according to a Poisson process, and each type occurs with a known prior probability. For each task type, the server allocates a fixed number of internal thinking tokens, which determines the computational effort devoted to that query. The token allocation induces an accuracy-latency trade-off: the service time follows an approximately affine function of the allocated tokens, while the probability of a correct response exhibits diminishing returns. Under a first-in, first-out (FIFO) service discipline, the system operates as an $M/G/1$ queue, and the mean system time depends on the first and second moments of the resulting service-time distribution. We formulate a constrained optimization problem that maximizes a weighted average accuracy objective penalized by the mean system time, subject to architectural token-budget constraints and queue-stability conditions. The objective function is shown to be strictly concave over the stability region, which ensures existence and uniqueness of the optimal token allocation. The first-order optimality conditions yield a coupled projected fixed-point characterization of the optimum, together with an iterative solution and an explicit sufficient condition for contraction. Moreover, a projected gradient method with a computable global step-size bound is developed to guarantee convergence beyond the contractive regime. Finally, integer-valued token allocations are attained via rounding of the continuous solution, and the resulting performance loss is evaluated in simulation results. - oai:arXiv.org:2601.10274v1 - cs.LG - cs.AI - cs.IT - cs.NI - math.IT - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Emre Ozbas, Melih Bastopcu - - - Model-Agnostic and Uncertainty-Aware Dimensionality Reduction in Supervised Learning - https://arxiv.org/abs/2601.10357 - arXiv:2601.10357v1 Announce Type: cross -Abstract: Dimension reduction is a fundamental tool for analyzing high-dimensional data in supervised learning. Traditional methods for estimating intrinsic order often prioritize model-specific structural assumptions over predictive utility. This paper introduces predictive order determination (POD), a model-agnostic framework that determines the minimal predictively sufficient dimension by directly evaluating out-of-sample predictiveness. POD quantifies uncertainty via error bounds for over- and underestimation and achieves consistency under mild conditions. By unifying dimension reduction with predictive performance, POD applies flexibly across diverse reduction tasks and supervised learners. Simulations and real-data analyses show that POD delivers accurate, uncertainty-aware order estimates, making it a versatile component for prediction-centric pipelines. - oai:arXiv.org:2601.10357v1 - stat.ME - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yue Yu, Guanghui Wang, Liu Liu, Changliang Zou - - - Gene genealogies in diploid populations evolving according to sweepstakes reproduction - https://arxiv.org/abs/2601.10364 - arXiv:2601.10364v1 Announce Type: cross -Abstract: Recruitment dynamics, or the distribution of the number of offspring among individuals, is central for understanding ecology and evolution. Sweepstakes reproduction (heavy right-tailed offspring number distribution) is central for understanding the ecology and evolution of highly fecund natural populations. Sweepstakes reproduction can induce jumps in type frequencies and multiple mergers in gene genealogies of sampled gene copies. We take sweepstakes reproduction to be skewed offspring number distribution due to mechanisms not involving natural selection, such as in chance matching of broadcast spawning with favourable environmental conditions. Here, we consider population genetic models of sweepstakes reproduction in a diploid panmictic populations absent selfing and evolving in a random environment. Our main results are {\it (i)} continuous-time Beta and Poisson-Dirichlet coalescents, when combining the results the skewness parameter $\alpha$ of the Beta-coalescent ranges from $0$ to $2$, and the Beta-coalescents may be incomplete due to an upper bound on the number of potential offspring produced by any pair of parents; {\it (ii)} in large populations time is measured in units proportional to either $N/\log N$ or $N$ generations (where $2N$ is the population size when constant); {\it (iii)} it follows that incorporating population size changes leads to time-changed coalescents with the time-change independent of $\alpha$; {\it (iv)} using simulations we show that the ancestral process is not well approximated by the corresponding coalescent (as measured through certain functionals of the processes); {\it (v)} whenever the skewness of the offspring number distribution is increased the conditional (conditioned on the population ancestry) and the unconditional ancestral processes are not in good agreement. - oai:arXiv.org:2601.10364v1 - q-bio.PE - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Bjarki Eldon - - - Dynamic reinsurance via martingale transport - https://arxiv.org/abs/2601.10375 - arXiv:2601.10375v1 Announce Type: cross -Abstract: We formulate a dynamic reinsurance problem in which the insurer seeks to control the terminal distribution of its surplus while minimizing the L2-norm of the ceded risk. Using techniques from martingale optimal transport, we show that, under suitable assumptions, the problem admits a tractable solution analogous to the Bass martingale. We first consider the case where the insurer wants to match a given terminal distribution of the surplus process, and then relax this condition by only requiring certain moment or risk-based constraints. - oai:arXiv.org:2601.10375v1 - q-fin.RM - math.OC - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Beatrice Acciaio, Brandon Garcia Flores, Antonio Marini, Gudmund Pammer - - - A Predictive Model for Synergistic Oncolytic Virotherapy: Unveiling the Ping-Pong Mechanism and Optimal Timing of Combined Vesicular Stomatitis and Vaccinia Viruses - https://arxiv.org/abs/2601.10405 - arXiv:2601.10405v1 Announce Type: cross -Abstract: We present a mathematical model that describes the synergistic mechanism of combined Vesicular Stomatitis Virus (VSV) and Vaccinia Virus (VV). The model captures the dynamic interplay between tumor cells, viral replication, and the interferon-mediated immune response, revealing a `ping-pong' synergy where VV-infected cells produce B18R protein that neutralizes interferon-$\alpha$, thereby enhancing VSV replication within the tumor. Numerical simulations demonstrate that this combination achieves complete tumor clearance in approximately 50 days, representing an 11\% acceleration compared to VV monotherapy (56 days), while VSV alone fails to eradicate tumors. Through bifurcation analysis, we identify critical thresholds for viral burst size and B18R inhibition, while sensitivity analysis highlights infection rates and burst sizes as the most influential parameters for treatment efficacy. Temporal optimization reveals that therapeutic outcomes are maximized through immediate VSV administration followed by delayed VV injection within a 1-19 day window, offering a strategic approach to overcome the timing and dosing challenges inherent in OVT. - oai:arXiv.org:2601.10405v1 - q-bio.QM - math.DS - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Joseph Malinzi, Amina Eladdadi, Rachid Ouifki, Raluca Eftimie, Anotida Madzvamuse, Helen M. Byrne - - - The eigenvalues and eigenvectors of finite-rank normal perturbations of large rotationally invariant non-Hermitian matrices - https://arxiv.org/abs/2601.10427 - arXiv:2601.10427v1 Announce Type: cross -Abstract: We study finite-rank normal deformations of rotationally invariant non-Hermitian random matrices. Extending the classical Baik-Ben Arous-P\'ech\'e (BBP) framework, we characterize the emergence and fluctuations of outlier eigenvalues in models of the form $\mathbf{A} + \mathbf{T}$, where $\mathbf{A}$ is a large rotationally invariant non-Hermitian random matrix and $\mathbf{T}$ is a finite-rank normal perturbation. We also describe the corresponding eigenvector behavior. Our results provide a unified framework encompassing both Hermitian and non-Hermitian settings, thereby generalizing several known cases. - oai:arXiv.org:2601.10427v1 - cond-mat.dis-nn - math-ph - math.MP - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pierre Bousseyroux, Marc Potters - - - H-EFT-VA: An Effective-Field-Theory Variational Ansatz with Provable Barren Plateau Avoidance - https://arxiv.org/abs/2601.10479 - arXiv:2601.10479v1 Announce Type: cross -Abstract: Variational Quantum Algorithms (VQAs) are critically threatened by the Barren Plateau (BP) phenomenon. In this work, we introduce the H-EFT Variational Ansatz (H-EFT-VA), an architecture inspired by Effective Field Theory (EFT). By enforcing a hierarchical "UV-cutoff" on initialization, we theoretically restrict the circuit's state exploration, preventing the formation of approximate unitary 2-designs. We provide a rigorous proof that this localization guarantees an inverse-polynomial lower bound on the gradient variance: $Var[\partial \theta] \in \Omega(1/poly(N))$. Crucially, unlike approaches that avoid BPs by limiting entanglement, we demonstrate that H-EFT-VA maintains volume-law entanglement and near-Haar purity, ensuring sufficient expressibility for complex quantum states. Extensive benchmarking across 16 experiments -- including Transverse Field Ising and Heisenberg XXZ models -- confirms a 109x improvement in energy convergence and a 10.7x increase in ground-state fidelity over standard Hardware-Efficient Ansatze (HEA), with a statistical significance of $p < 10^{-88}$. - oai:arXiv.org:2601.10479v1 - quant-ph - cs.LG - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Eyad I. B Hamid - - - Twisted Cherednik spectrum as a $q,t$-deformation - https://arxiv.org/abs/2601.10500 - arXiv:2601.10500v1 Announce Type: cross -Abstract: The common eigenfunctions of the twisted Cherednik operators can be first analyzed in the limit of $q\longrightarrow 1$. Then, the polynomial eigenfunctions form a simple set originating from the symmetric ground state of non-vanishing degree and excitations over it, described by non-symmetric polynomials of higher degrees and enumerated by weak compositions. This pattern is inherited by the full spectrum at $q\neq 1$, which can be considered as a deformation. The whole story looks like a typical NP problem: the Cherednik equations are difficult to solve, but easy to check the solution once it is somehow found. - oai:arXiv.org:2601.10500v1 - hep-th - math-ph - math.CO - math.MP - math.QA - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - A. Mironov, A. Morozov, A. Popolitov - - - Coarsening Causal DAG Models - https://arxiv.org/abs/2601.10531 - arXiv:2601.10531v1 Announce Type: cross -Abstract: Directed acyclic graphical (DAG) models are a powerful tool for representing causal relationships among jointly distributed random variables, especially concerning data from across different experimental settings. However, it is not always practical or desirable to estimate a causal model at the granularity of given features in a particular dataset. There is a growing body of research on causal abstraction to address such problems. We contribute to this line of research by (i) providing novel graphical identifiability results for practically-relevant interventional settings, (ii) proposing an efficient, provably consistent algorithm for directly learning abstract causal graphs from interventional data with unknown intervention targets, and (iii) uncovering theoretical insights about the lattice structure of the underlying search space, with connections to the field of causal discovery more generally. As proof of concept, we apply our algorithm on synthetic and real datasets with known ground truths, including measurements from a controlled physical system with interacting light intensity and polarization. - oai:arXiv.org:2601.10531v1 - stat.ML - cs.LG - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Francisco Madaleno, Pratik Misra, Alex Markham - - - Hybrid Encryption with Certified Deletion in Preprocessing Model - https://arxiv.org/abs/2601.10542 - arXiv:2601.10542v1 Announce Type: cross -Abstract: Certified deletion allows Alice to outsource data to Bob and, at a later time, obtain a verifiable guarantee that the file has been irreversibly deleted at her request. The functionality, while impossible using classical information alone, can be achieved using quantum information. Existing approaches, rely on one-time pad (OTP) encryption, or use computational hardness assumptions that may be vulnerable to future advances in classical or quantum computing. In this work, we introduce and formalize hybrid encryption with certified deletion in the preprocessing model (pHE-CD) and propose two constructions. The constructions combine an information-theoretic key encapsulation mechanism (iKEM) with a data encapsulation mechanism that provides certified deletion (DEM-CD) and, respectively, provide {\em information-theoretic certified deletion}, where both confidentiality and deletion properties are provided against a computationally unbounded adversary; and {\em everlasting certified deletion}, where confidentiality is computational before deletion, and upon successful verification of the deletion certificate, the message becomes information-theoretically hidden from an adversary that is computationally unbounded. Our pHE-CD schemes provide IND-$q_e$-CPA notion of security and support encryption of arbitrarily long messages. In the second construction, using a computationally secure DEM-CD that is quantum-safe (i.e. constructed using quantum coding and AES), we obtain quantum-safe security with keys that are significantly shorter than the message. Instantiating the proposed framework using quantum enabled kem (qKEM) as the iKEM, is a future work. - oai:arXiv.org:2601.10542v1 - cs.CR - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Kunal Dey, Reihaneh Safavi-Naini - - - A Mirror-Descent Algorithm for Computing the Petz-R\'enyi Capacity of Classical-Quantum Channels - https://arxiv.org/abs/2601.10558 - arXiv:2601.10558v1 Announce Type: cross -Abstract: We study the computation of the $\alpha$-R\'enyi capacity of a classical-quantum (c-q) channel for $\alpha\in(0,1)$. We propose an exponentiated-gradient (mirror descent) iteration that generalizes the Blahut-Arimoto algorithm. Our analysis establishes relative smoothness with respect to the entropy geometry, guaranteeing a global sublinear convergence of the objective values. Furthermore, under a natural tangent-space nondegeneracy condition (and a mild spectral lower bound in one regime), we prove local linear (geometric) convergence in Kullback-Leibler divergence on a truncated probability simplex, with an explicit contraction factor once the local curvature constants are bounded. - oai:arXiv.org:2601.10558v1 - quant-ph - cs.IT - math.IT - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yu-Hong Lai, Hao-Chung Cheng - - - Rewriting Systems on Arbitrary Monoids - https://arxiv.org/abs/2601.10564 - arXiv:2601.10564v1 Announce Type: cross -Abstract: In this paper, we introduce monoidal rewriting systems (MRS), an abstraction of string rewriting in which reductions are defined over an arbitrary ambient monoid rather than a free monoid of words. This shift is partly motivated by logic: the class of free monoids is not first-order axiomatizable, so "working in the free setting" cannot be treated internally when applying first-order methods to rewriting presentations. - To analyze these systems categorically, we define $\mathbf{NCRS_2}$ as the 2-category of Noetherian Confluent MRS. We then prove the existence of a canonical biadjunction between $\mathbf{NCRS_2}$ and $\mathbf{Mon}$. - Finally, we classify all Noetherian Confluent MRS that present a given fixed monoid. For this, we introduce Generalized Elementary Tietze Transformations (GETTs) and prove that any two presentations of a monoid are connected by a (possibly infinite) sequence of these transformations, yielding a complete characterization of generating systems up to GETT-equivalence. - oai:arXiv.org:2601.10564v1 - cs.FL - cs.LO - math.CT - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Eduardo Magalh\~aes - - - Achievable Degrees of Freedom Analysis and Optimization in Massive MIMO via Characteristic Mode Analysis - https://arxiv.org/abs/2601.10576 - arXiv:2601.10576v1 Announce Type: cross -Abstract: Massive multiple-input multiple-output (MIMO) is esteemed as a critical technology in 6G communications, providing large degrees of freedom (DoF) to improve multiplexing gain. This paper introduces characteristic mode analysis (CMA) to derive the achievable DoF. Unlike existing works primarily focusing on the DoF of the wireless channel,the excitation and radiation properties of antennas are also involved in our DoF analysis, which influences the number of independent data streams for communication of a MIMO system. Specifically, we model the excitation and radiation properties of transceiver antennas using CMA to analyze the excitation and radiation properties of antennas. The CMA-based DoF analysis framework is established and the achievable DoF is derived. A characteristic mode optimization problem of antennas is then formulated to maximize the achievable DoF. A case study where the reconfigurable holographic surface (RHS) antennas are deployed at the transceiver is investigated, and a CMA-based genetic algorithm is later proposed to solve the above problem. By changing the characteristic modes electric field and surface current distribution of RHS, the achievable DoF is enhanced. Full-wave simulation verifies the theoretical analysis on the the achievable DoF and shows that, via the reconfiguration of RHS based on the proposed algorithm, the achievable DoF is improved. - oai:arXiv.org:2601.10576v1 - eess.SP - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Shaohua Yue, Siyu Miao, Shuhao Zeng, Fenghan Lin, Boya Di - - - Jordan-Segmentable Masks: A Topology-Aware definition for characterizing Binary Image Segmentation - https://arxiv.org/abs/2601.10577 - arXiv:2601.10577v1 Announce Type: cross -Abstract: Image segmentation plays a central role in computer vision. However, widely used evaluation metrics, whether pixel-wise, region-based, or boundary-focused, often struggle to capture the structural and topological coherence of a segmentation. In many practical scenarios, such as medical imaging or object delineation, small inaccuracies in boundary, holes, or fragmented predictions can result in high metric scores, despite the fact that the resulting masks fail to preserve the object global shape or connectivity. This highlights a limitation of conventional metrics: they are unable to assess whether a predicted segmentation partitions the image into meaningful interior and exterior regions. - In this work, we introduce a topology-aware notion of segmentation based on the Jordan Curve Theorem, and adapted for use in digital planes. We define the concept of a \emph{Jordan-segmentatable mask}, which is a binary segmentation whose structure ensures a topological separation of the image domain into two connected components. We analyze segmentation masks through the lens of digital topology and homology theory, extracting a $4$-curve candidate from the mask, verifying its topological validity using Betti numbers. A mask is considered Jordan-segmentatable when this candidate forms a digital 4-curve with $\beta_0 = \beta_1 = 1$, or equivalently when its complement splits into exactly two $8$-connected components. - This framework provides a mathematically rigorous, unsupervised criterion with which to assess the structural coherence of segmentation masks. By combining digital Jordan theory and homological invariants, our approach provides a valuable alternative to standard evaluation metrics, especially in applications where topological correctness must be preserved. - oai:arXiv.org:2601.10577v1 - cs.CV - cs.NA - math.AT - math.NA - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Serena Grazia De Benedictis, Amedeo Altavilla, Nicoletta Del Buono - - - Combinatorial Optimization Augmented Machine Learning - https://arxiv.org/abs/2601.10583 - arXiv:2601.10583v1 Announce Type: cross -Abstract: Combinatorial optimization augmented machine learning (COAML) has recently emerged as a powerful paradigm for integrating predictive models with combinatorial decision-making. By embedding combinatorial optimization oracles into learning pipelines, COAML enables the construction of policies that are both data-driven and feasibility-preserving, bridging the traditions of machine learning, operations research, and stochastic optimization. This paper provides a comprehensive overview of the state of the art in COAML. We introduce a unifying framework for COAML pipelines, describe their methodological building blocks, and formalize their connection to empirical cost minimization. We then develop a taxonomy of problem settings based on the form of uncertainty and decision structure. Using this taxonomy, we review algorithmic approaches for static and dynamic problems, survey applications across domains such as scheduling, vehicle routing, stochastic programming, and reinforcement learning, and synthesize methodological contributions in terms of empirical cost minimization, imitation learning, and reinforcement learning. Finally, we identify key research frontiers. This survey aims to serve both as a tutorial introduction to the field and as a roadmap for future research at the interface of combinatorial optimization and machine learning. - oai:arXiv.org:2601.10583v1 - cs.LG - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Maximilian Schiffer, Heiko Hoppe, Yue Su, Louis Bouvier, Axel Parmentier - - - Parametric RDT approach to computational gap of symmetric binary perceptron - https://arxiv.org/abs/2601.10628 - arXiv:2601.10628v1 Announce Type: cross -Abstract: We study potential presence of statistical-computational gaps (SCG) in symmetric binary perceptrons (SBP) via a parametric utilization of \emph{fully lifted random duality theory} (fl-RDT) [96]. A structural change from decreasingly to arbitrarily ordered $c$-sequence (a key fl-RDT parametric component) is observed on the second lifting level and associated with \emph{satisfiability} ($\alpha_c$) -- \emph{algorithmic} ($\alpha_a$) constraints density threshold change thereby suggesting a potential existence of a nonzero computational gap $SCG=\alpha_c-\alpha_a$. The second level estimate is shown to match the theoretical $\alpha_c$ whereas the $r\rightarrow \infty$ level one is proposed to correspond to $\alpha_a$. For example, for the canonical SBP ($\kappa=1$ margin) we obtain $\alpha_c\approx 1.8159$ on the second and $\alpha_a\approx 1.6021$ (with converging tendency towards $\sim 1.59$ range) on the seventh level. Our propositions remarkably well concur with recent literature: (i) in [20] local entropy replica approach predicts $\alpha_{LE}\approx 1.58$ as the onset of clustering defragmentation (presumed driving force behind locally improving algorithms failures); (ii) in $\alpha\rightarrow 0$ regime we obtain on the third lifting level $\kappa\approx 1.2385\sqrt{\frac{\alpha_a}{-\log\left ( \alpha_a \right ) }}$ which qualitatively matches overlap gap property (OGP) based predictions of [43] and identically matches local entropy based predictions of [24]; (iii) $c$-sequence ordering change phenomenology mirrors the one observed in asymmetric binary perceptron (ABP) in [98] and the negative Hopfield model in [100]; and (iv) as in [98,100], we here design a CLuP based algorithm whose practical performance closely matches proposed theoretical predictions. - oai:arXiv.org:2601.10628v1 - stat.ML - cond-mat.dis-nn - cs.IT - cs.LG - math.IT - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Mihailo Stojnic - - - Adjusted Similarity Measures and a Violation of Expectations - https://arxiv.org/abs/2601.10641 - arXiv:2601.10641v1 Announce Type: cross -Abstract: Adjusted similarity measures, such as Cohen's kappa for inter-rater reliability and the adjusted Rand index used to compare clustering algorithms, are a vital tool for comparing discrete labellings. These measures are intended to have the property of 0 expectation under a null distribution and maximum value 1 under maximal similarity to aid in interpretation. Measures are frequently adjusted with respect to the permutation distribution for historic and analytic reasons. There is currently renewed interest in considering other null models more appropriate for context, such as clustering ensembles permitting a random number of identified clusters. The purpose of this work is two -- fold: (1) to generalize the study of the adjustment operator to general null models and to a more general procedure which includes statistical standardization as a special case and (2) to identify sufficient conditions for the adjustment operator to produce the intended properties, where sufficient conditions are related to whether and how observed data are incorporated into null distributions. We demonstrate how violations of the sufficient conditions may lead to substantial breakdown, such as by producing a non-positive measure under traditional adjustment rather than one with mean 0, or by producing a measure which is deterministically 0 under statistical standardization. - oai:arXiv.org:2601.10641v1 - stat.ME - cs.LG - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - William L. Lippitt, Edward J. Bedrick, Nichole E. Carlson - - - Transforming Crises into Opportunities: From Chaos to Urban Antifragility - https://arxiv.org/abs/2601.10658 - arXiv:2601.10658v1 Announce Type: cross -Abstract: Urban crises - floods, pandemics, economic shocks, and conflicts - function as accelerators of urban change, exposing structural vulnerabilities while creating windows for reinvention. Building on a prior theoretical contribution that identified fifteen principles of urban antifragility, this paper tests and operationalizes the framework through an empirical assessment of 26 cities selected for their post-crisis adaptation trajectories. Using a tailored diagnostic methodology, we benchmark cities' Stress Response Strategies (SRS) and then evaluate Urban Development Trajectories (UDT) across four weighted dimensions, positioning each case along a fragility-robustness-resilience-antifragility continuum and applying a balanced-threshold rule to confirm antifragile status. Results show that "resilience enhanced by innovation and technology" is the most effective response typology (86.9/100), and that six cities meet the antifragile trajectory criteria. By mapping best practices to activated principles and analysing co-activations, the study identifies a robust "hard core" of principles - Sustainable Resilience (O), Strategic Diversity (F), Proactive Innovation (I), and Active Prevention (N) - supplemented by operational enablers (e.g., anticipation, mobilization, shock absorption). The paper concludes by proposing an evidence-based, SDG-aligned operational model that links high-impact principle pairings to measurable indicators, offering a practical roadmap for cities seeking to convert crises into sustained transformation. Keywords: Post-crisis strategies, Urban antifragility, Sustainable cities and communities, Disaster resilience and urban regeneration, Risk governance and Black Swan adaptation. - oai:arXiv.org:2601.10658v1 - physics.soc-ph - cs.CY - econ.GN - math.ST - q-fin.EC - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Joseph Uguet, Nicola Tollin, Jordi Morato - - - Optimal lower bound for quantum channel tomography in away-from-boundary regime - https://arxiv.org/abs/2601.10683 - arXiv:2601.10683v1 Announce Type: cross -Abstract: Consider quantum channels with input dimension $d_1$, output dimension $d_2$ and Kraus rank at most $r$. Any such channel must satisfy the constraint $rd_2\geq d_1$, and the parameter regime $rd_2=d_1$ is called the boundary regime. In this paper, we show an optimal query lower bound $\Omega(rd_1d_2/\varepsilon^2)$ for quantum channel tomography to within diamond norm error $\varepsilon$ in the away-from-boundary regime $rd_2\geq 2d_1$, matching the existing upper bound $O(rd_1d_2/\varepsilon^2)$. In particular, this lower bound fully settles the query complexity for the commonly studied case of equal input and output dimensions $d_1=d_2=d$ with $r\geq 2$, in sharp contrast to the unitary case $r=1$ where Heisenberg scaling $\Theta(d^2/\varepsilon)$ is achievable. - oai:arXiv.org:2601.10683v1 - quant-ph - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kean Chen, Zhicheng Zhang, Nengkun Yu - - - Madelung hydrodynamics of spin-orbit coupling: action principles, currents, and correlations - https://arxiv.org/abs/2601.10698 - arXiv:2601.10698v1 Announce Type: cross -Abstract: We exploit the variational and Hamiltonian structures of quantum hydrodynamics with spin to unfold the correlation and torque mechanisms accompanying spin-orbit coupling (SOC) in electronic motion. Using Hamilton's action principle for the Pauli equation, we isolate SOC-induced quantum forces that act on the orbital Madelung--Bohm trajectories and complement the usual force terms known to appear in quantum hydrodynamics with spin. While the latter spin-hydrodynamic forces relate to the quantum geometric tensor (QGT), SOC-induced orbital forces originate from a particular current operator that contributes prominently to the spin current and whose contribution was overlooked in the past. The distinction between different force terms reveals two fundamentally different mechanisms generating quantum spin-orbit correlations. Leveraging the Hamiltonian structure of the hydrodynamic system, we also elucidate spin transport features such as the current shift in the spin Hall effect and the correlation-induced quantum torques. Finally, we illustrate the framework via the Madelung--Rashba equations for planar SOC configurations and propose a particle-based scheme for numerical implementation. - oai:arXiv.org:2601.10698v1 - quant-ph - cond-mat.other - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Cesare Tronci - - - High-accuracy and dimension-free sampling with diffusions - https://arxiv.org/abs/2601.10708 - arXiv:2601.10708v1 Announce Type: cross -Abstract: Diffusion models have shown remarkable empirical success in sampling from rich multi-modal distributions. Their inference relies on numerically solving a certain differential equation. This differential equation cannot be solved in closed form, and its resolution via discretization typically requires many small iterations to produce \emph{high-quality} samples. - More precisely, prior works have shown that the iteration complexity of discretization methods for diffusion models scales polynomially in the ambient dimension and the inverse accuracy $1/\varepsilon$. In this work, we propose a new solver for diffusion models relying on a subtle interplay between low-degree approximation and the collocation method (Lee, Song, Vempala 2018), and we prove that its iteration complexity scales \emph{polylogarithmically} in $1/\varepsilon$, yielding the first ``high-accuracy'' guarantee for a diffusion-based sampler that only uses (approximate) access to the scores of the data distribution. In addition, our bound does not depend explicitly on the ambient dimension; more precisely, the dimension affects the complexity of our solver through the \emph{effective radius} of the support of the target distribution only. - oai:arXiv.org:2601.10708v1 - cs.LG - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Khashayar Gatmiry, Sitan Chen, Adil Salim - - - Quantum Maxwell Erasure Decoder for qLDPC codes - https://arxiv.org/abs/2601.10713 - arXiv:2601.10713v1 Announce Type: cross -Abstract: We introduce a quantum Maxwell erasure decoder for CSS quantum low-density parity-check (qLDPC) codes that extends peeling with bounded guessing. Guesses are tracked symbolically and can be eliminated by restrictive checks, giving a tunable tradeoff between complexity and performance via a guessing budget: an unconstrained budget recovers Maximum-Likelihood (ML) performance, while a constant budget yields linear-time decoding and approximates ML. We provide theoretical guarantees on asymptotic performance and demonstrate strong performance on bivariate bicycle and quantum Tanner codes. - oai:arXiv.org:2601.10713v1 - quant-ph - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bruno Costa Alves Freire, Fran\c{c}ois-Marie Le R\'egent, Anthony Leverrier - - - $\mathfrak{B}$-free integers in number fields and dynamics - https://arxiv.org/abs/1507.00855 - arXiv:1507.00855v2 Announce Type: replace -Abstract: In 2010, Sarnak initiated the study of the dynamics of the system determined by the square of the M\"obius function (the characteristic function of the square-free integers). We deal with his program in the more general context of $\mathfrak{B}$-free integers in number fields, suggested 5 years later by Baake and Huck. This setting encompasses the classical square-free case and its generalizations. Given a number field $K$, let $\mathfrak{B}$ be a family of pairwise coprime ideals in its ring of integers $\mathcal{O}_K$, such that $\sum_{\mathfrak{b}\in\mathfrak{B}}1/|\mathcal{O}_K / \mathfrak{b}|<\infty$. We study the dynamical system determined by the set $\mathcal{F}_\mathfrak{B}=\mathcal{O}_K\setminus \bigcup_{\mathfrak{b}\in\mathfrak{B}}\mathfrak{b}$ of $\mathfrak{B}$-free integers in $\mathcal{O}_K$. We show that the characteristic function $\mathbb{1}_{\mathcal{F}_\mathfrak{B}}$ of $\mathcal{F}_\mathfrak{B}$ is generic along the natural F\o{}lner sequence for a probability measure on $\{0,1\}^{\mathcal{O}_K}$, invariant under the multidimensional shift. The corresponding measure-theoretical dynamical system is proved to be isomorphic to an ergodic rotation on a compact Abelian group. In particular, it is of zero Kolmogorov entropy. Moreover, we provide a description of ``patterns'' appearing in $\mathcal{F}_\mathfrak{B}$ and compute the topological entropy of the orbit closure of $\mathbb{1}_{\mathcal{F}_\mathfrak{B}}$. Finally, we show that this topological dynamical system has a non-trivial topological joining with an ergodic rotation on a compact Abelian group. - oai:arXiv.org:1507.00855v2 - math.DS - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Francisco Ara\'ujo, Aurelia Dymek, Joanna Ku{\l}aga-Przymus - - - An invitation to Alexandrov geometry: CAT(0) spaces - https://arxiv.org/abs/1701.03483 - arXiv:1701.03483v4 Announce Type: replace -Abstract: Our goal is to show the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation. - The topics include - 1. Reshetnyak's gluing theorem, - 2. Estimates on the number of collisions in billiards, - 3. Reshetnyak's majorization theorem, - 4. Hadamard--Cartan globalization theorem, - 5. Polyhedral spaces, - 6. Construction of exotic aspherical manifolds, - 7. The geometry of two-convex sets in Euclidean space, - 8. Barycenters and dimension theory. - oai:arXiv.org:1701.03483v4 - math.DG - math.AT - math.MG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-sa/4.0/ - Stephanie Alexander, Vitali Kapovitch, Anton Petrunin - - - Elastodynamical properties of Sturmian structured media - https://arxiv.org/abs/2105.02548 - arXiv:2105.02548v2 Announce Type: replace -Abstract: In this paper, wave propagation in structured media with quasiperiodic patterns is investigated. We propose a methodology based on Sturmian sequences for the generation of structured mechanical systems from a given parameter. The approach is presented in a general form so that it can be applied to waveguides of different nature, as long as they can be modeled with the transfer matrix method. The bulk spectrum is obtained and its fractal nature analyzed. For validation of the theoretical results, three numerical examples are presented. The obtained bulk spectra show different shapes for the studied examples, but they share features which can be explained from the proposed theoretical setting. - oai:arXiv.org:2105.02548v2 - math-ph - math.MP - physics.comp-ph - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - 10.1016/j.jsv.2021.116539 - Mario L\'azaro, Agnieszka Niemczynowicz, Artur Siemaszko, Luis M. Garc\'ia-Raffi - - - Meta-nilpotent knot invariants and symplectic automorphism groups of free nilpotent groups - https://arxiv.org/abs/2105.14414 - arXiv:2105.14414v4 Announce Type: replace -Abstract: We develop nilpotently $p$-localization of knot groups in terms of the (symplectic) automorphism groups of free nilpotent groups. We show that any map from the set of conjugacy classes of the outer automorphism groups yields a knot invariant. We also investigate the automorphism groups and compute the resulting knot invariants. - oai:arXiv.org:2105.14414v4 - math.GT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Takefumi Nosaka - - - Decreasing subsequences and Viennot for oscillating tableaux - https://arxiv.org/abs/2108.11528 - arXiv:2108.11528v2 Announce Type: replace -Abstract: We establish an extension of Viennot's geometric (shadow line) construction to the setting of oscillating tableaux. We then use this to give a new proof of the Type $C$ analogue of Schensted's theorem on longest decreasing subsequences. This pairs with our results from arXiv:2103.14997v1 [math.RT] on Type $C$ webs to give a direct proof of a result of Sundaram and Stanley: that the dimension of the space of invariant vectors in a $2k$-fold tensor product of the vector representation of $\mathfrak{sp}_{2n}$ equals the number of $(n+1)$-avoiding matchings of $2k$ points. - oai:arXiv.org:2108.11528v2 - math.CO - math.RT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Elijah Bodish, Ben Elias, David E. V. Rose, Logan Tatham - - - Steenrod Lengths and a Problem of Vakil - https://arxiv.org/abs/2110.01672 - arXiv:2110.01672v3 Announce Type: replace -Abstract: We give an explicit combinatorial description of the function $f(n)$ governing the Steenrod length of real projective spaces $\mathbb{RP}^n$. This function arises in stable homotopy theory through the action of Steenrod squares on mod-$2$ cohomology and is closely related to the ghost length, which measures the minimal number of spheres required to construct a space up to homotopy. Building on the directed graphs $T_n$ introduced by Vakil to encode degree constraints for Steenrod operations, we interpret $f(n)$ as the length of the longest directed path starting at $n$. Using this framework, we resolve a question posed by Vakil by deriving concrete combinatorial formulas for $f(n)$ in terms of binary classes and a distinguished family of integers, which we call Vakil numbers. - oai:arXiv.org:2110.01672v3 - math.RT - math.AT - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Khanh Nguyen Duc - - - Surface slices and homology spheres - https://arxiv.org/abs/2202.02696 - arXiv:2202.02696v3 Announce Type: replace -Abstract: We develop the theory of the diagrammatics of surface cross sections to prove that there are an infinite number of homology 3-spheres smoothly embeddable in a homology 4-sphere but not in a homotopy 4-sphere. Our primary obstruction comes from work of Taubes. - oai:arXiv.org:2202.02696v3 - math.GT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Clayton McDonald - - - On The large Time Asymptotics of Schr\"odinger type equations with General Data - https://arxiv.org/abs/2203.00724 - arXiv:2203.00724v5 Announce Type: replace -Abstract: For the Schr\"odinger equation with a general interaction term, which may be linear or nonlinear, time dependent and including charge transfer potentials, we prove the global solutions are asymptotically given by the sum of a free wave and a weakly localized part. The proof is based on constructing in a new way the Free Channel Wave Operator, and further tools from the recent works \cite{Liu-Sof1,Liu-Sof2,SW2020}. This work generalizes the results of the first part of \cite{Liu-Sof1,Liu-Sof2} to arbitrary dimension, and non-radial data. - oai:arXiv.org:2203.00724v5 - math.AP - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Avy Soffer, Xiaoxu Wu - - - A Modern Theory for High-dimensional Cox Regression Models - https://arxiv.org/abs/2204.01161 - arXiv:2204.01161v2 Announce Type: replace -Abstract: The proportional hazards model has been extensively used in many fields such as biomedicine to estimate and perform statistical significance testing on the effects of covariates influencing the survival time of patients. The classical theory of maximum partial-likelihood estimation (MPLE) is used by most software packages to produce inference, e.g., the coxph function in R and the PHREG procedure in SAS. In this paper, we investigate the asymptotic behavior of the MPLE in the regime in which the number of parameters p is of the same order as the number of samples n. The main results are (i) existence of the MPLE undergoes a sharp 'phase transition'; (ii) the classical MPLE theory leads to invalid inference in the high-dimensional regime. We show that the asymptotic behavior of the MPLE is governed by a new asymptotic theory. These findings are further corroborated through numerical studies. The main technical tool in our proofs is the Convex Gaussian Min-max Theorem (CGMT), which has not been previously used in the analysis of partial likelihood. Our results thus extend the scope of CGMT and shed new light on the use of CGMT for examining the existence of MPLE and non-separable objective functions. - oai:arXiv.org:2204.01161v2 - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Hanxuan Ye, Xianyang Zhang, Huijuan Zhou - - - Hamiltonicity in generalized quasi-dihedral groups - https://arxiv.org/abs/2204.05484 - arXiv:2204.05484v2 Announce Type: replace -Abstract: Witte Morris showed in [21] that every connected Cayley graph of a finite (generalized) dihedral group has a Hamiltonian path. The infinite dihedral group is defined as the free product with amalgamation $\mathbb Z_2 \ast \mathbb Z_2$. We show that every connected Cayley graph of the infinite dihedral group has both a Hamiltonian double ray, and extend this result to all two-ended generalized quasi-dihedral groups. - oai:arXiv.org:2204.05484v2 - math.CO - math.GR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Babak Miraftab, Konstantinos Stavropoulos - - - Extension of Lipschitz maps definable in Hensel minimal structures - https://arxiv.org/abs/2204.05900 - arXiv:2204.05900v4 Announce Type: replace -Abstract: In this paper, we establish a theorem on extension of Lipschitz maps $f$ definable in Hensel minimal, non-trivially valued fields $K$ of equicharacteristic zero. This may be regarded as a definable, non-Archimedean, non-locally compact version of Kirszbraun's extension theorem. We proceed with double induction with respect to the dimensions of the ambient space and of the domain of $f$. To this end we introduce the concept of a definable open cell package with a skeleton, which along with the concept of a risometry plays a key role in our induction procedure. - oai:arXiv.org:2204.05900v4 - math.LO - math.AG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Krzysztof Jan Nowak - - - Planar site percolation via tree embeddings - https://arxiv.org/abs/2304.00923 - arXiv:2304.00923v4 Announce Type: replace -Abstract: We prove that if $G$ is an infinite, connected, planar graph properly embedded in $\mathbb{R}^2$ with minimum degree at least $7$, then i.i.d.\ Bernoulli$(p)$ site percolation on $G$ almost surely has infinitely many infinite open (1-)clusters for every \[ p \in \bigl(p_c^{\mathrm{site}},\, 1-p_c^{\mathrm{site}}\bigr). \] Moreover, we show that $p_c^{\mathrm{site}}<\tfrac12$, so this non-uniqueness interval is nonempty. This verifies Conjecture~7 of Benjamini and Schramm~\cite{bs96} for this class of properly embedded planar graphs. - Our proof introduces a new construction of embedded trees in $G$. These trees yield infinitely many infinite clusters for percolation parameters near $\tfrac12$, and they also enable exponential decay of two-point connection probabilities by partitioning $G$ using infinitely many disjoint trees. Variants of this approach were later used in~\cite{ZL26} to construct a counterexample to Conjecture~7 of~\cite{bs96} for planar graphs with uncountably many ends. - Finally, the methods developed here have further applications: in~\cite{perc24} they are used to prove a vertex-cut characterization of $p_c^{\mathrm{site}}$ (conjectured by Kahn in~\cite{JK03}) and to refute an edge-cut characterization proposed by Lyons and Peres~\cite{LP16} and Tang (\cite{Tang2023}). - oai:arXiv.org:2304.00923v4 - math.PR - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Zhongyang Li - - - Differential characterization of quadratic surfaces - https://arxiv.org/abs/2304.08073 - arXiv:2304.08073v3 Announce Type: replace -Abstract: Let $f\in W^{3,1}_{\mathrm{loc}}(\Omega)$ be a function defined on a connected open subset $\Omega\subseteq\mathbb R^2$. We will show that its graph is contained in a quadratic surface if and only if $f$ is a weak solution to a certain system of third-order partial differential equations unless the Hessian determinant of $f$ is non-positive everywhere on $\Omega$. Moreover, we will prove that the system is, in a sense, the simplest possible in a wide class of differential equations, which will lead to the classification of all polynomial partial differential equations satisfied by parametrizations of generic quadratic surfaces. Although we will mainly use the tools of linear and commutative algebra, the theorem itself is also somewhat related to holomorphic functions. - oai:arXiv.org:2304.08073v3 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1007/s13366-024-00735-0 - Bart{\l}omiej Zawalski - - - Higher Lie theory in positive characteristic - https://arxiv.org/abs/2306.07829 - arXiv:2306.07829v4 Announce Type: replace -Abstract: The main goal of this article is to develop integration theory for absolute partition $L_\infty$-algebras, which are point-set models for the (spectral) partition Lie algebras of Brantner-Mathew where infinite sums of operations are well-defined by definition. We construct a Quillen adjunction between absolute partition $L_\infty$-algebras and simplicial sets, and show that the right adjoint is a well-behaved integration functor. Points in this simplicial set are given by solutions to a Maurer-Cartan equation, and we give explicit formulas for gauge equivalences between them. We construct the analogue of the Baker-Campbell-Hausdorff formula in this setting and show it produces an isomorphic group to the classical one over a characteristic zero field. - We apply these constructions to show that absolute partition $L_\infty$-algebras encode the $p$-adic homotopy types of pointed connected finite nilpotent spaces, up to certain equivalences which we describe by explicit formulas. In particular, these formulas also allow us to give a combinatorial description of the homotopy groups of the $p$-completed spheres as solutions to a certain equation in a given degree, up to an equivalence relation imposed by elements one degree above. Finally, we construct absolute partition $L_\infty$ models for $p$-adic mapping spaces, which combined with the description of the homotopy groups gives an algebraic description of the homotopy type of these $p$-adic mapping spaces parallel to the unstable Adams spectral sequence. - oai:arXiv.org:2306.07829v4 - math.AT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Victor Roca i Lucio - - - The cohomology of $BPU(p^m)$ and invariant polynomials - https://arxiv.org/abs/2306.17599 - arXiv:2306.17599v5 Announce Type: replace -Abstract: Let $p$ be an odd prime. For a compact Lie group $G$ and an elementary abelian $p$-group $A$ of $G$, one may define the Weyl group $W_A$ of $A$ in a similar fashion as defining the Weyl group of a maximal torus, such that $W_A$ acts on $H^*(BA;R)$ for any coefficient ring $R$, and the image of the restriction $H^*(BG;R)\to H^*(BA;R)$ lies in $H^*(BA;R)^{W_A}$, the sub-algebra of $H^*(BA:R)$ of $W_A$-invariant elements. - In this paper, we consider the projective unitary group $PU(p^m)$ and one of its maximal elementary abelian $p$-subgroup $A_m$, of which the Weyl group is isomorphic to $Sp_{2m}(\mathbb{F}_p)$. Then the theory of $Sp_{2m}(\mathbb{F}_p)$-invariant polynomials over $\mathbb{F}_p$ may be applied to study the cohomology of $BPU(p^m)$, the classifying space of $PU(p^m)$. Following a theorem by Quillen, we deduce several theorems on $H^*(BPU(p^m);\mathbb{F}_p)$ modulo the nilradical from results on invariant polynomials. - oai:arXiv.org:2306.17599v5 - math.AT - math.RT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1007/s00209-025-03912-6 - Xing Gu - - - A non-Archimedean Arens--Eells isometric embedding theorem on valued fields - https://arxiv.org/abs/2309.06704 - arXiv:2309.06704v2 Announce Type: replace -Abstract: In 1959, Arens and Eells proved that every metric space can be isometrically embedded into a normed linear space as a closed subset. In later years, in the paper on a short proof of the Arens--Eells theorem, Michael implicitly pointed out that the Arens--Eells theorem follows from the statement that every metric space can be isometrically embedded into a normed linear space as a linearly independent subset. In this paper, we prove a non-Archimedean analogue of the Arens--Eells isometric embedding theorem, which states that for every non-Archimedean valued field $K$, every ultrametric space can be isometrically embedded into a non-Archimedean valued field that is a valued field extension of $K$ such that the image of the embedding is algebraically independent over $K$. - oai:arXiv.org:2309.06704v2 - math.MG - math.AC - math.GN - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Yoshito Ishiki - - - Reuniting $\chi$-boundedness with polynomial $\chi$-boundedness - https://arxiv.org/abs/2310.11167 - arXiv:2310.11167v4 Announce Type: replace -Abstract: A class $\mathcal{F}$ of graphs is $\chi$-bounded if there is a function $f$ such that $\chi(H)\le f(\omega(H))$ for all induced subgraphs $H$ of a graph in $\mathcal{F}$. If $f$ can be chosen to be a polynomial, we say that $\mathcal{F}$ is polynomially $\chi$-bounded. Esperet proposed a conjecture that every $\chi$-bounded class of graphs is polynomially $\chi$-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are $\chi$-bounded but not polynomially $\chi$-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class $\mathcal{C}$ of graphs is Pollyanna if $\mathcal{C}\cap \mathcal{F}$ is polynomially $\chi$-bounded for every $\chi$-bounded class $\mathcal{F}$ of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna. - oai:arXiv.org:2310.11167v4 - math.CO - cs.DM - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.jctb.2025.08.002 - J. Combin. Theory Ser. B, 176:30-73, 2026 - Maria Chudnovsky, Linda Cook, James Davies, Sang-il Oum - - - Reformulation of the stable Adams conjecture - https://arxiv.org/abs/2310.14425 - arXiv:2310.14425v3 Announce Type: replace -Abstract: We revisit methods of proof of the Adams Conjecture in order to correct and supplement earlier efforts to prove analogous conjectures in the stable homotopy category. We utilize simplicial schemes over an algebraically closed field of positive characteristic and a rigid version of Artin-Mazur \'etale homotopy theory. Consideration of special $\mathcal F$-spaces and together with Bousfield-Kan $\mathbb Z/\ell$-completion enables us to employ an "\'etale functor" which commutes up to homotopy with products of simplicial schemes. In order to prove the Stable Adams Conjecture, we construct the universal $\mathbb Z/\ell$-completed $X$-fibrations for various pointed simplicial sets $X$. Thus, two maps from a given $\mathcal F$-space $\underline{\mathcal B}$ to the base $\mathcal F$-space of the universal $\mathbb Z/\ell$-completed $X$-fibration $\pi_{X,\ell}: \underline {\mathcal B} (G_\ell(X),X_\ell) \to \underline {\mathcal B} G_\ell(X)$ determine homotopy equivalent maps of spectra if and only they correspond via pull-back of $\pi_{X,\ell}$ to fiber homotopy equivalent $\mathbb Z/\ell$-completed $X$-fibrations over $\underline {\mathcal B}$. For the proof of the Stable Adams Conjecture, we consider maps of $\mathcal F$-spaces $\underline {\mathcal B }\to \underline {\mathcal B} G_\ell(S^2)$ where $\underline {\mathcal B}$ is an $\mathcal F$-space model of connective $\ell$-completed connective $K$-theory. - oai:arXiv.org:2310.14425v3 - math.AT - math.AG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Eric M. Friedlander - - - Distribution-uniform anytime-valid sequential inference and the Robbins-Siegmund distributions - https://arxiv.org/abs/2311.03343 - arXiv:2311.03343v3 Announce Type: replace -Abstract: This paper develops a theory of distribution- and time-uniform asymptotics, culminating in the first large-sample anytime-valid inference procedures that are shown to be uniformly valid in a rich class of distributions. Historically, anytime-valid methods -- including confidence sequences, anytime $p$-values, and sequential hypothesis tests -- have been justified nonasymptotically. By contrast, large-sample inference procedures such as those based on the central limit theorem occupy an important part of statistical toolbox due to their simplicity, universality, and the weak assumptions they make. While recent work has derived asymptotic analogues of anytime-valid methods, they were not distribution-uniform (also called \emph{honest}), meaning that their type-I errors may not be uniformly upper-bounded by the desired level in the limit. The theory and methods we outline resolve this tension, and they do so without imposing assumptions that are any stronger than the distribution-uniform fixed-$n$ (non-anytime-valid) counterparts or distribution-pointwise anytime-valid special cases. It is shown that certain ``Robbins-Siegmund'' probability distributions play roles in anytime-valid asymptotics analogous to those played by Gaussian distributions in standard asymptotics. As an application, we derive the first anytime-valid test of conditional independence without the Model-X assumption. - oai:arXiv.org:2311.03343v3 - math.ST - stat.ME - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ian Waudby-Smith, Edward H. Kennedy, Aaditya Ramdas - - - Euler Product Asymptotics for $L$-functions of Elliptic Curves - https://arxiv.org/abs/2312.05236 - arXiv:2312.05236v4 Announce Type: replace -Abstract: Let $E/\mathbb Q$ be an elliptic curve and for each prime $p$, let $N_p$ denote the number of points of $E$ modulo $p$. The original version of the Birch and Swinnerton-Dyer conjecture asserts that $\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x) ^{\text{rank}(E(\mathbb Q))}$ as $x \to \infty$. Goldfeld (1982) showed that this conjecture implies both the Riemann Hypothesis for $L(E, s)$ and the modern formulation of the conjecture i.e. that $\text{ord}_{s=1} L(E, s)= \text{rank}(E(\mathbb Q))$. In this paper, we prove that if we let $r=\text{ord} _{s=1}L(E, s)$, then under the assumption of the Riemann Hypothesis for $L(E, s)$, we have that $\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x)^r$ for all $x$ outside a set of finite logarithmic measure. As corollaries, we recover not only Goldfeld's result, but we also prove a result in the direction of the converse. Our method of proof is based on establishing the asymptotic behaviour of partial Euler products of $L(E, s)$ in the right-half of the critical strip. - oai:arXiv.org:2312.05236v4 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - 10.1093/imrn/rnaf214 - International Mathematics Research Notices, Volume 2025, Issue 14, July 2025, rnaf214 - Arshay Sheth - - - Mixed-Integer Linear Optimization for Semi-Supervised Optimal Classification Trees - https://arxiv.org/abs/2401.09848 - arXiv:2401.09848v2 Announce Type: replace -Abstract: Decision trees are one of the most popular methods for solving classification problems, mainly because of their good interpretability properties. Moreover, due to advances in recent years in mixed-integer optimization, several models have been proposed to formulate the problem of computing optimal classification trees. The goal is, given a set of labeled points, to split the feature spacewith hyperplanes and assign a class to each part of the resulting partition. In certain scenarios, however, labels are only available for a subset of the given points. Additionally, this subset may be non-representative, such as in the case of self-selection in a survey. Semi-supervised decision trees tackle the setting of labeled and unlabeled data and often contribute to enhancing the reliability of the results. Furthermore, undisclosed sources may provide extra information about the size of the classes. We propose a mixed-integer linear optimization model for computing semi-supervised optimal classification trees that cover the setting of labeled and unlabeled data points as well as the overall number of points in each class for a binary classification. Our numerical results show that our approach leads to a better accuracy and a better Matthews correlation coefficient for biased samples compared to other optimal classification trees, even if onlyfew labeled points are available. - oai:arXiv.org:2401.09848v2 - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Jan Pablo Burgard, Maria Eduarda Pinheiro, Martin Schmidt - - - From higher-order rewriting systems to higher-order categorial algebras and higher-order Curry-Howard isomorphisms - https://arxiv.org/abs/2402.12051 - arXiv:2402.12051v2 Announce Type: replace -Abstract: This ongoing project aims to define and investigate, from the standpoint of category theory, order theory and universal algebra, the notions of higher-order many-sorted rewriting system and of higher-order many-sorted categorial algebra and their relationships, via the higher-order Curry-Howard isomorphisms. The ultimate goal, to be developed in future versions of this work, is to define and investigate the category of towers, whose objects will consist of families, indexed by $\mathbb{N}$, of higher-order many-sorted rewriting systems and of higher-order many-sorted categorial algebras, including higher-order Curry-Howard type results for the latter, together with an additional structure that intertwines such $\mathbb{N}$-families; and whose morphism from a tower to another will be families, indexed by $\mathbb{N}$, of morphisms between its higher-order many-sorted rewriting systems and of higher-order many-sorted categorial algebras compatible with their structures. All feedback is appreciated. - oai:arXiv.org:2402.12051v2 - math.CT - cs.FL - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Juan Climent Vidal, Enric Cosme Ll\'opez, Ra\'ul Ruiz Mora - - - Higher-dimensional multifractal analysis for the cusp winding process on hyperbolic surfaces - https://arxiv.org/abs/2402.16418 - arXiv:2402.16418v2 Announce Type: replace -Abstract: We perform a multifractal analysis of the growth rate of the number of cusp windings for the geodesic flow on hyperbolic surfaces with $m \geq 1$ cusps. Our main theorem establishes a conditional variational principle for the Hausdorff dimension spectrum of the multi-cusp winding process. Moreover, we show that the dimension spectrum defined on $\mathbb{R}_{>0}^m$ is real analytic. To prove the main theorem we use a countable Markov shift with a finitely primitive transition matrix and thermodynamic formalism. - oai:arXiv.org:2402.16418v2 - math.DS - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tokyo J. Math. 48(1): 247-273 (June 2025) - Yuya Arima - - - A refinement of the Ewens sampling formula - https://arxiv.org/abs/2403.05077 - arXiv:2403.05077v3 Announce Type: replace -Abstract: We consider an infinitely-many neutral allelic model of population genetics where all alleles are divided into a finite number of classes, and each class is characterized by its own mutation rate. For this model the allelic composition of a sample taken from a very large population of genes is characterized by a random matrix, and the problem is to describe the joint distribution of the matrix entries. The answer is given by a new generalization of the classical Ewens sampling formula called the refined Ewens sampling formula in the present paper. We discuss a Poisson approximation for the refined Ewens sampling formula, and present its derivation by several methods. As an application we obtain limit theorems for the numbers of alleles in different asymptotic regimes. - oai:arXiv.org:2403.05077v3 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Eugene Strahov - - - Collapsing regular Riemannian foliations with flat leaves - https://arxiv.org/abs/2403.11602 - arXiv:2403.11602v2 Announce Type: replace -Abstract: In this manuscript we present how to collapse a manifold equipped with a closed flat regular Riemannian foliation with leaves of positive dimension, while keeping the sectional curvature uniformly bounded from above and below. From this deformation, we show that in the case when the manifold is compact and simply connected the foliation is given by torus actions. This gives a geometric characterization of aspherical regular Riemannian foliations given by torus actions - oai:arXiv.org:2403.11602v2 - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Diego Corro - - - Large sieve inequalities for exceptional Maass forms and the greatest prime factor of $n^2+1$ - https://arxiv.org/abs/2404.04239 - arXiv:2404.04239v3 Announce Type: replace -Abstract: We prove new large sieve inequalities for the Fourier coefficients $\rho_{j\mathfrak{a}}(n)$ of exceptional Maass forms of a given level, weighted by sequences $(a_n)$ with sparse Fourier transforms - including two key types of sequences that arise in the dispersion method. These give the first savings in the exceptional spectrum for the critical case of sequences as long as the level, and lead to improved bounds for various multilinear forms of Kloosterman sums. - As an application, we show that the greatest prime factor of $n^2+1$ is infinitely often greater than $n^{1.3}$, improving Merikoski's previous threshold of $n^{1.279}$. We also announce applications to the exponents of distribution of primes and smooth numbers in arithmetic progressions. - oai:arXiv.org:2404.04239v3 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexandru Pascadi - - - On the solutions of the generalized Fermat equation over totally real number fields - https://arxiv.org/abs/2404.09171 - arXiv:2404.09171v3 Announce Type: replace -Abstract: Let $K$ be a totally real number field and $\mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the asymptotic solutions of the generalized Fermat equation $Ax^p+By^p+Cz^p=0$ over $K$ with prime exponent $p$, where $A,B,C \in \mathcal{O}_K \setminus \{0\}$ with $ABC$ is even. For certain class of fields $K$, we prove that the equation $Ax^p+By^p+Cz^p=0$ has no asymptotic solution $(a,b,c) \in \mathcal{O}_K^3$ with $2|abc$. Then, under some assumptions on $A,B,C$, we also prove that $Ax^p+By^p+Cz^p=0$ has no asymptotic solution in $K^3$. Finally, we give several purely local criteria of $K$ such that $Ax^p+By^p+Cz^p=0$ has no asymptotic solutions in $K^3$, and calculate the density of such fields $K$ when $K$ is a real quadratic field. - oai:arXiv.org:2404.09171v3 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Satyabrat Sahoo - - - On the asymptotics of Kempner-Irwin sums - https://arxiv.org/abs/2404.13763 - arXiv:2404.13763v4 Announce Type: replace -Abstract: Let $I(b,d,k)$ be the subseries of the harmonic series keeping the integers having exactly $k$ occurrences of the digit $d$ in base $b$. We prove the existence of an asymptotic expansion to all orders in descending powers of $b$, for fixed $d$ and $k$, of $I(b,d,k)-b\log(b)$. We explicitly give, depending on cases, either four or five terms. The coefficients involve the values of the zeta function at the integers. - oai:arXiv.org:2404.13763v4 - math.NT - math.CA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-sa/4.0/ - Jean-Fran\c{c}ois Burnol - - - Preservation under Reduced Products in Continuous Logic - https://arxiv.org/abs/2405.12720 - arXiv:2405.12720v2 Announce Type: replace -Abstract: We introduce a fragment of continuous first-order logic, analogue of Palyutin formulas (or h-formulas) in classical model theory, which is preserved under reduced products in both directions. We use it to extend classical results on complete theories which are preserved under reduced product and their stability. We also characterize the set of Palyutin sentences, Palyutin theories and other related fragments in terms of their preservation properties, both in the classical setting and the metric one. - oai:arXiv.org:2405.12720v2 - math.LO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ivory Fronteau - - - Analytic Extended Dynamic Mode Decomposition - https://arxiv.org/abs/2405.15945 - arXiv:2405.15945v4 Announce Type: replace -Abstract: We develop a novel EDMD-type algorithm that captures the spectrum of the Koopman operator defined on a reproducing kernel Hilbert space of analytic functions. This method, which we call analytic EDMD, relies on an orthogonal projection on polynomial subspaces, which is equivalent to a data-driven Taylor approximation. In the case of dynamics with a hyperbolic equilibrium, analytic EDMD demonstrates excellent performance to capture the lattice-structured Koopman spectrum based on the eigenvalues of the linearized system at the equilibrium. Moreover, it yields the Taylor approximation of associated principal eigenfunctions. Since the method preserves the triangular structure of the operator, it does not suffer from spectral pollution and, moreover, arbitrary accuracy on the spectrum can be reached with a fixed finite dimension of the approximation and with a (possibly non-uniform) sampling over an arbitrary set of nonzero measure. The performance of analytic EDMD is illustrated with numerical examples and is assessed through a comparative study with related methods. Finally, the method is complemented with theoretical results, proving strong convergence of the eigenfunctions and providing error bounds on the spectrum estimation. - oai:arXiv.org:2405.15945v4 - math.DS - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexandre Mauroy, Igor Mezic - - - Moduli of rank two semistable sheaves on rational Fano threefolds of the main series - https://arxiv.org/abs/2405.20460 - arXiv:2405.20460v2 Announce Type: replace -Abstract: In this paper we investigate the moduli spaces of semistable coherent sheaves of rank two on the projective space $\mathbb{P}^3$ and the following rational Fano manifolds of the main series - the three-dimensional quadric $X_2$, the intersection of two 4-dimensional quadrics $X_4$ and the Fano manifold $X_5$ of degree 5. For the quadric $X_2$, the boundedness of the third Chern class $c_3$ of rank two semistable objects in $\mathrm{D}^b(X_2)$, including sheaves, is proved. An explicit description is given of all the moduli spaces of semistable sheaves of rank two on $X_2$, including reflexive ones, with a maximal third class $c_3\ge0$. These spaces turn out to be irreducible smooth rational manifolds in all cases, except for the following two: $(c_1,c_2,c_3)=(0,2,2)$ or (0,4,8). Several new infinite series of rational components of the moduli spaces of semistable sheaves of rank two on $\mathbb{P}^3$, $X_2$, $X_4$ and $X_5$ are constructed, as well as a new infinite series of irrational components on $X_4$. The boundedness of the class $c_3$ is proved for $c_1=0$ and any $c_2>0$ for stable reflexive sheaves of general type on manifolds $X_4$ and $X_5$. - oai:arXiv.org:2405.20460v2 - math.AG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Alexander S. Tikhomirov, Danil A. Vassiliev - - - Cross-Dimensional Mathematics: A Foundation For STP/STA - https://arxiv.org/abs/2406.12920 - arXiv:2406.12920v5 Announce Type: replace -Abstract: A new mathematical structure, called the cross-dimensional mathematics (CDM), is proposed. The CDM considered in this paper consists of three parts: hyper algebra, hyper geometry, and hyper Lie group/Lie algebra. Hyper algebra proposes some new algebraic structures such as hyper group, hyper ring, and hyper module over matrices and vectors with mixed dimensions (MVMDs). They have sets of classical groups, rings, and modules as their components and cross-dimensional connections among their components. Their basic properties are investigated. Hyper geometry starts from mixed dimensional Euclidian space, and hyper vector space. Then the hyper topological vector space, hyper inner product space, and hyper manifold are constructed. They have a joined cross-dimensional geometric structure. Finally, hyper metric space, topological hyper group and hyper Lie algebra are built gradually, and finally, the corresponding hyper Lie group is introduced. All these concepts are built over MVMDs, and to reach our purpose in addition to existing semi-tensor products (STPs) and semi-tensor additions (STAs), a couple of most general STP and STA are introduced. Some existing structures/results about STPs/STAs have also been resumed and integrated into this CDM. - oai:arXiv.org:2406.12920v5 - math.RA - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - 10.1007/s11425-024-2528-4 - Daizhan Cheng - - - Polynomial convergence rate at infinity for the cusp winding spectrum of generalized Schottky groups - https://arxiv.org/abs/2407.12398 - arXiv:2407.12398v2 Announce Type: replace -Abstract: We show that the convergence rate of the cusp winding spectrum to the Hausdorff dimension of the limit set of a generalized Schottky group with one parabolic generator is polynomial. Our main theorem provides the new phenomenon in which differences in the Hausdorff dimension of the limit set generated by a Markov system cause essentially different results on multifractal analysis. This paper also provides a new characterization of the geodesic flow on the Poinca\'re disc model of two-dimensional hyperbolic space and the limit set of a generalized Schottky group. To prove our main theorem we use thermodynamic formalism on a countable Markov shift, gamma function, and zeta function. - oai:arXiv.org:2407.12398v2 - math.DS - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yuya Arima - - - Nearly-linear solution to the word problem for 3-manifold groups - https://arxiv.org/abs/2407.18029 - arXiv:2407.18029v2 Announce Type: replace -Abstract: We show that the word problem for any 3-manifold group is solvable in time $O(n\log^3 n)$. Our main contribution is the proof that the word problem for admissible graphs of groups, in the sense of Croke and Kleiner, is solvable in $O(n\log n)$; this covers fundamental groups of non-geometric graph manifolds. Similar methods also give that the word problem for free products can be solved ``almost as quickly'' as the word problem in the factors. - oai:arXiv.org:2407.18029v2 - math.GR - math.GT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alessandro Sisto, Stefanie Zbinden - - - Littlewood-Offord problems for Ising models - https://arxiv.org/abs/2408.05720 - arXiv:2408.05720v2 Announce Type: replace -Abstract: We consider the one-dimensional Littlewood-Offord problem for general Ising models. More precisely, we consider the concentration function \[Q_n(x,v)=P\left(\sum_{i=1}^{n}\varepsilon_iv_i\in(x-1,x+1)\right),\] where $x\in\mathbb{R}$, $v_1,v_2,\ldots,v_n$ are real numbers such that $|v_1|\geq 1, |v_2|\geq 1,\ldots, |v_n|\geq 1$, and $(\varepsilon_i)_{i=1,2,\ldots,n}\in\{-1,1\}^{n}$ are random spins of some Ising model. Let $Q_n=\sup_{x,v}Q_n(x,v)$. Under natural assumptions, we show that there exists a universal constant $C$ such that for all $n\geq 1$, \[\binom{n}{[n/2]}2^{-n}\leq Q_n\leq Cn^{-\frac{1}{2}}.\] As an application of the method, under the same assumption, we give a lower bound on the smallest eigenvalue of the truncated correlation matrix of the Ising model. - oai:arXiv.org:2408.05720v2 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yinshan Chang - - - Persistent Homology via Ellipsoids - https://arxiv.org/abs/2408.11450 - arXiv:2408.11450v3 Announce Type: replace -Abstract: Persistent homology is one of the most popular methods in topological data analysis. An initial step in its use involves constructing a nested sequence of simplicial complexes. There is an abundance of different complexes to choose from, with \v{C}ech, Rips, alpha, and witness complexes being popular choices. In this manuscript, we build a novel type of geometrically informed simplicial complex, called a Rips-type ellipsoid complex. This complex is based on the idea that ellipsoids aligned with tangent directions better approximate the data compared to conventional (Euclidean) balls centered at sample points, as used in the construction of Rips and Alpha complexes. We use Principal Component Analysis to estimate tangent spaces directly from samples and present an algorithm for computing Rips-type ellipsoid barcodes, i.e., topological descriptors based on Rips-type ellipsoid complexes. Additionally, we show that the ellipsoid barcodes depend continuously on the input data so that small perturbations of a k-generic point cloud lead to proportionally small changes in the resulting ellipsoid barcodes. This provides a theoretical guarantee analogous, if somewhat weaker, to the classical stability results for Rips and \v{C}ech filtrations. We also conduct extensive experiments and compare Rips-type ellipsoid barcodes with standard Rips barcodes. Our findings indicate that Rips-type ellipsoid complexes are particularly effective for estimating the homology of manifolds and spaces with bottlenecks from samples. In particular, the persistence intervals corresponding to ground-truth topological features are longer compared to those obtained using the Rips complex of the data. Furthermore, Rips-type ellipsoid barcodes lead to better classification results in sparsely sampled point clouds. Finally, we demonstrate that Rips-type ellipsoid barcodes outperform Rips barcodes in classification tasks. - oai:arXiv.org:2408.11450v3 - math.AT - cs.LG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Niklas Canova, Sara Kali\v{s}nik, Aaron Moser, Bastian Rieck, Ana \v{Z}egarac - - - Generalized Fruit Diophantine equation over number fields - https://arxiv.org/abs/2408.12278 - arXiv:2408.12278v3 Announce Type: replace -Abstract: Let $K$ be a number field and $\mathcal{O}_K$ be the ring of integers of $K$. In this article, we study the solutions of the generalized fruit Diophantine equation $ax^d-y^2-z^2 +xyz-c=0$ over $K$, where $d \geq 3$ is an integer and $a,c\in \mathcal{O}_K\setminus \{0\}$. Subsequently, we provide explicit values of square-free integers $t$ such that the equation $ax^d-y^2-z^2 +xyz-c=0$ has no solution $(x_0, y_0, z_0) \in \mathcal{O}_{\mathbb{Q}(\sqrt{t})}^3$ with $2 | x_0$, and demonstrate that the set of all such square-free integers $t$ with $t \geq 2$ has density exactly $\frac{1}{6}$. As an application, we construct infinitely many elliptic curves $E$ defined over number fields $K$ having no integral point $(x_0,y_0) \in \mathcal{O}_K^2$ with $2|x_0$. - oai:arXiv.org:2408.12278v3 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Satyabrat Sahoo, Shanta Laishram - - - Uniform Approximation of Eigenproblems of a Large-Scale Parameter-Dependent Hermitian Matrix - https://arxiv.org/abs/2409.05791 - arXiv:2409.05791v4 Announce Type: replace -Abstract: We consider the uniform approximation of the smallest eigenvalue of a large parameter-dependent Hermitian matrix by that of a smaller counterpart obtained through projections. The projection subspaces are constructed iteratively by means of a greedy strategy; at each iteration the parameter where a surrogate error is maximal is computed and the eigenvectors associated with the smallest eigenvalues at the maximizing parameter value are added to the subspace. Unlike the classical approaches, such as the successive constraint method, that maximize such surrogate errors over a discrete and finite set, we maximize the surrogate error over the continuum of all permissible parameter values globally. We formally prove that the projected eigenvalue function converges to the actual eigenvalue function uniformly. In the second part, we focus on the uniform approximation of the smallest singular value of a large parameter-dependent matrix, in case it is non-Hermitian. The proposed frameworks on numerical examples, including those arising from discretizations of parametric PDEs, reduce the size of the large matrix-valued function drastically, while retaining a high accuracy over all permissible parameter values. - oai:arXiv.org:2409.05791v4 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Mattia Manucci, Emre Mengi, Nicola Guglielmi - - - Graphs missing a connected partition - https://arxiv.org/abs/2409.12934 - arXiv:2409.12934v2 Announce Type: replace -Abstract: We prove that a graph with a cut vertex whose deletion produces at least five connected components must be missing a connected partition of some type. We prove that this also holds if there are four connected components that each have at least two vertices. In particular, the chromatic symmetric function of such a graph cannot be $e$-positive. This brings us very close to the conjecture by Dahlberg, She, and van Willigenburg of non-$e$-positivity for all trees with a vertex of degree at least four. We also prove that spiders with four legs cannot have an $e$-positive chromatic symmetric function. - oai:arXiv.org:2409.12934v2 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - 10.1016/j.aam.2026.103044 - Adv. in Appl. Math. Vol. 175 103044 (2026) - Foster Tom - - - K\"ahler metrics of negative holomorphic (bi)sectional curvature on a compact relative K\"ahler fibration - https://arxiv.org/abs/2409.14650 - arXiv:2409.14650v2 Announce Type: replace -Abstract: For a compact relative K\"ahler fibration over a compact K\"ahler manifold with negative holomorphic sectional curvature, if the relative K\"ahler form on each fiber also exhibits negative holomorphic sectional curvature, we can construct K\"ahler metrics with negative holomorphic sectional curvature on the total space. Additionally, if this form induces a Griffiths negative Hermitian metric on the relative tangent bundle, and the base admits a K\"ahler metric with negative holomorphic bisectional curvature, we can also construct K\"ahler metrics with negative holomorphic bisectional curvature on the total space. As an application, for a non-trivial fibration where both the fibers and base have K\"ahler metrics with negative holomorphic bisectional curvature, and the fibers are one-dimensional, we can explicitly construct K\"ahler metrics of negative holomorphic bisectional curvature on the total space, thus resolving a question posed by To and Yeung for the case where the fibers have dimension one. - oai:arXiv.org:2409.14650v2 - math.DG - math.CV - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xueyuan Wan - - - Finite Element Approximations of Stochastic Linear Schr\"{o}dinger equation driven by additive Wiener noise - https://arxiv.org/abs/2410.06006 - arXiv:2410.06006v2 Announce Type: replace -Abstract: In this article, we have analyzed semi-discrete finite element approximations of the Stochastic linear Schr\"{o}dinger equation in a bounded convex polygonal domain driven by additive Wiener noise. We use the finite element method for spatial discretization and derive an error estimate with respect to the discretization parameter of the finite element approximation. Numerical experiments have also been performed to support theoretical bounds. - oai:arXiv.org:2410.06006v2 - math.NA - cs.NA - math-ph - math.AP - math.MP - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Suprio Bhar, Mrinmay Biswas, Mangala Prasad - - - The five-color hypercube Adinkra and the Jacobian of a generalized Fermat curve - https://arxiv.org/abs/2410.11137 - arXiv:2410.11137v2 Announce Type: replace -Abstract: Adinkras are highly structured graphs developed to study 1-dimensional supersymmetry algebras. A cyclic ordering of the edge colors of an Adinkra, or rainbow, determines a Riemann surface and a height function on the vertices of the Adinkra determines a divisor on this surface. We study the induced map from height functions to divisors on the Jacobian of the Riemann surface. In the first nontrivial case, a 5-dimensional hypercube corresponding to a Jacobian given by a product of 5 elliptic curves each with $j$-invariant 2048, we develop and characterize a purely combinatorial algorithm to compute height function images. We show that when restricted to a single elliptic curve, every height function is a multiple of a specified generating divisor, and raising and lowering vertices corresponds to adding or subtracting this generator. We also give strict bounds on the coefficients of this generator that appear in the collection of all divisors of height functions. - oai:arXiv.org:2410.11137v2 - math.AG - math.CO - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Amanda E. Francis, Ursula A. Whitcher - - - A $\Lambda$-adic Kudla lift - https://arxiv.org/abs/2410.19992 - arXiv:2410.19992v2 Announce Type: replace -Abstract: The Kudla lift studied in this article is a classical version for Picard modular forms of the automorphic theta lift between $\text{GU}(2)$ and $\text{GU}(3)$. We construct an explicit $p$-adic analytic family of Picard modular forms varying with respect to the weight and level, which interpolates a so-called $p$-modification of the lift at arithmetic weights, by exploiting a formula of Finis for the Fourier-Jacobi coefficients of a lifted form. - oai:arXiv.org:2410.19992v2 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Francesco Maria Iudica - - - Convex optimization with $p$-norm oracles - https://arxiv.org/abs/2410.24158 - arXiv:2410.24158v2 Announce Type: replace -Abstract: In recent years, there have been significant advances in efficiently solving $\ell_s$-regression using linear system solvers and $\ell_2$-regression [Adil-Kyng-Peng-Sachdeva, J. ACM'24]. Would efficient smoothed $\ell_p$-norm solvers lead to even faster rates for solving $\ell_s$-regression when $2 \leq p < s$? In this paper, we give an affirmative answer to this question and show how to solve $\ell_s$-regression using $\tilde{O}(n^{\frac{\nu}{1+\nu}})$ iterations of solving smoothed $\ell_p$ regression problems, where $\nu := \frac{1}{p} - \frac{1}{s}$. To obtain this result, we provide improved accelerated rates for convex optimization problems when given access to an $\ell_p^s(\lambda)$-proximal oracle, which, for a point $c$, returns the solution of the regularized problem $\min_{x} f(x) + \lambda ||x-c||_p^s$. Additionally, we show that these rates for the $\ell_p^s(\lambda)$-proximal oracle are optimal for algorithms that query in the span of the outputs of the oracle, and we further apply our techniques to settings of high-order and quasi-self-concordant optimization. - oai:arXiv.org:2410.24158v2 - math.OC - cs.DS - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Deeksha Adil, Brian Bullins, Arun Jambulapati, Aaron Sidford - - - Complexity and curvature of (pairs of) Cohen-Macaulay modules, and their applications - https://arxiv.org/abs/2411.17622 - arXiv:2411.17622v2 Announce Type: replace -Abstract: The complexity and curvature of a module, introduced by Avramov, measure the growth of Betti and Bass numbers of a module, and distinguish the modules of infinite homological dimension. The notion of complexity was extended by Avramov-Buchweitz to pairs of modules that measure the growth of Ext modules. The related notion of Tor complexity was first studied by Dao. Inspired by these notions, we define Ext and Tor curvature of pairs of modules. The aim of this article is to study (Ext and Tor) complexity and curvature of pairs of certain CM (Cohen-Macaulay) modules, and establish lower bounds of complexity and curvature of pairs of modules in terms of that of a single module. It is known that among all modules, the residue field has maximal complexity and curvature, moreover they characterize complete intersection local rings. As applications of our results, we provide some upper bounds of the curvature of the residue field in terms of curvature and multiplicity of any nonzero CM module. As a final upshot, these allow us to characterize complete intersection local rings (including hypersurfaces and regular rings) in terms of complexity and curvature of pairs of certain CM modules. In particular, under some additional hypotheses, we characterize complete intersection and regular local rings via injective curvature of the ring and that of the module of K\"{a}hler differentials respectively. Thus, we make partial progress towards a question of Christensen-Striuli-Veliche, as well as another by Vasconcelos. - oai:arXiv.org:2411.17622v2 - math.AC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Souvik Dey, Dipankar Ghosh, Aniruddha Saha - - - A note on the $L_p$-Brunn-Minkowski inequality for intrinsic volumes and the $L_p$-Christoffel-Minkowski problem - https://arxiv.org/abs/2411.17896 - arXiv:2411.17896v4 Announce Type: replace -Abstract: The first goal of this paper is to improve some of the results in \cite{BCPR}. Namely, we establish the $L_p$-Brunn-Minkwoski inequality for intrinsic volumes for origin-symmetric convex bodies that are close to the ball in the $C^2$ sense for a certain range of $p<1$ (including negative values) and we prove that this inequality does not hold true in the entire class of origin-symmetric convex bodies for any $p<1$. The second goal is to establish a uniqueness result for the (closely related) $L_p$-Christoffel-Minkowski problem. More specifically, we show uniqueness in the symmetric case when $p\in[0,1)$ and the data function $g$ in the right hand side is sufficiently close to the constant 1. One of the main ingredients of the proof is the existence of upper and lower bounds for the (convex) solution, that depend only $\|\log g\|_{L^\infty}$, a fact that might be of independent interest. - oai:arXiv.org:2411.17896v4 - math.MG - math.AP - math.FA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Konstantinos Patsalos, Christos Saroglou - - - Test properties of some Cohen-Macaulay modules and criteria for local rings via finite vanishing of Ext or Tor - https://arxiv.org/abs/2412.01636 - arXiv:2412.01636v2 Announce Type: replace -Abstract: In this article, we show test properties, in the sense of finitely many vanishing of Ext or Tor, of CM (Cohen-Macaulay) modules whose multiplicity and number of generators (resp., type) are related by certain inequalities. We apply these test behaviour, along with other results, to characterize various kinds of local rings, including hypersurface rings of multiplicity at most two, surprisingly requiring only finitely many vanishing of Ext or Tor involving such CM modules. As further applications, we verify the long-standing (Generalized) Auslander-Reiten Conjecture for every CM module of minimal multiplicity over a Noetherian local ring, thus vastly extending a result of Huneke-\c{S}ega-Vraciu. - oai:arXiv.org:2412.01636v2 - math.AC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Souvik Dey, Dipankar Ghosh, Aniruddha Saha - - - The John inclusion for log-concave functions - https://arxiv.org/abs/2412.18444 - arXiv:2412.18444v2 Announce Type: replace -Abstract: John's inclusion states that a convex body in $\mathbb{R}^d$ can be covered by the $d$-dilation of its maximal volume ellipsoid. We obtain a certain John-type inclusion for log-concave functions. As a byproduct of our approach, we establish the following asymptotically tight inequality: \\ \noindent For any log-concave function $f$ with finite, positive integral, there exist a positive definite matrix $A$, a point $a \in \mathbb{R}^d$, and a positive constant $\alpha$ such that \[ \chi_{\mathbf{B}^{d}}(x) \leq \alpha f\!\!\left(A(x-a)\right) \leq \sqrt{d+1} \cdot e^{-\frac{\left|x\right|}{d+2} + (d+1)}, \] where $\chi_{\mathbf{B}^{d}}$ is the indicator function of the unit ball $\mathbf{B}^{d}$. - oai:arXiv.org:2412.18444v2 - math.MG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - G. Ivanov - - - Distributionally Robust Fault Detection Trade-off Design with Prior Fault Information - https://arxiv.org/abs/2412.20237 - arXiv:2412.20237v5 Announce Type: replace -Abstract: The robustness of fault detection algorithms against uncertainty is crucial in the real-world industrial environment. Recently, a new probabilistic design scheme called distributionally robust fault detection (DRFD) has emerged and received immense interest. Despite its robustness against unknown distributions in practice, current DRFD focuses on the overall detectability of all possible faults rather than the detectability of critical faults that are a priori known. Henceforth, a new DRFD trade-off design scheme is put forward in this work by utilizing prior fault information. The key contribution includes a novel distributional robustness metric of detecting a known fault and a new relaxed distributionally robust chance constraint that ensures robust detectability. Then, a new DRFD design problem of fault detection under unknown probability distributions is proposed, and this offers a flexible balance between the robustness of detecting known critical faults and the overall detectability against all possible faults. To address the resulting semi-infinite chance-constrained problem, we first reformulate it to a finite-dimensional problem characterized by bilinear matrix inequalities. Subsequently, a tailored heuristic solution algorithm is developed, which includes a sequential minimization procedure and an initialization strategy. Finally, case studies on a simulated three-tank system and a real-world battery cell are carried out to showcase the effectiveness of the proposed heuristic algorithm and the advantages of our DRFD method. - oai:arXiv.org:2412.20237v5 - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yulin Feng, Hailang Jin, Steven X. Ding, Hao Ye, Chao Shang - - - Counting the number of integral fixed points of a discrete dynamical system with applications from arithmetic statistics, I - https://arxiv.org/abs/2501.04026 - arXiv:2501.04026v3 Announce Type: replace -Abstract: In this first article of a multi-part series, we inspect a surprising relationship between the set of fixed points of a polynomial map $\varphi_{d, c}$ defined by $\varphi_{d, c}(z) = z^d + c$ for all $c, z \in \mathbb{Z}$ and the coefficient $c$, where $d > 2$ is an integer. Inspired greatly by the elegant counting problems along with the very striking results of Bhargava-Shankar-Tsimerman and their collaborators in arithmetic statistics, and also by interesting point-counting result of Narkiewicz on rational periodic points of any odd degree map $\varphi_{d, c}$ in arithmetic dynamics, we then first prove that for any prime $p\geq 3$, the average number of distinct integral fixed points of any $\varphi_{p, c}$ modulo $p$ is $3$ or $0$ as $c$ tends to infinity. Inspired further by a conjecture of Hutz on rational periodic points of $\varphi_{p-1, c}$ for any prime $p\geq 5$ in arithmetic dynamics, we then also prove that the average number of distinct integral fixed points of any $\varphi_{p-1, c}$ modulo $p$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply density and number field-counting results from arithmetic statistics, and as a result obtain counting and statistical results on the irreducible integer polynomials and number fields arising naturally in our polynomial discrete dynamical settings. - oai:arXiv.org:2501.04026v3 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Brian Kintu - - - Non-Expansive Mappings in Two-Time-Scale Stochastic Approximation: Finite-Time Analysis - https://arxiv.org/abs/2501.10806 - arXiv:2501.10806v3 Announce Type: replace -Abstract: Two-time-scale stochastic approximation algorithms are iterative methods used in applications such as optimization, reinforcement learning, and control. Finite-time analysis of these algorithms has primarily focused on fixed point iterations where both time-scales have contractive mappings. In this work, we broaden the scope of such analyses by considering settings where the slower time-scale has a non-expansive mapping. For such algorithms, the slower time-scale can be viewed as a stochastic inexact Krasnoselskii-Mann iteration. We also study a variant where the faster time-scale has a projection step which leads to non-expansiveness in the slower time-scale. We show that the last-iterate mean square residual error for such algorithms decays at a rate $O(1/k^{1/4-\epsilon})$, where $\epsilon>0$ is arbitrarily small. We further establish almost sure convergence of iterates to the set of fixed points. We demonstrate the applicability of our framework by applying our results to minimax optimization, linear stochastic approximation, and Lagrangian optimization. - oai:arXiv.org:2501.10806v3 - math.OC - cs.LG - cs.SY - eess.SY - stat.ML - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Siddharth Chandak - - - Hypercyclicity of Toeplitz operators with smooth symbols - https://arxiv.org/abs/2502.03303 - arXiv:2502.03303v3 Announce Type: replace -Abstract: This paper is devoted to the study of the dynamics of Toeplitz operators $T_F$ with smooth symbols $F$ on the Hardy spaces of the unit disk $H^p$, $p>1$. Building on a model theory for Toeplitz operators on $H^2$ developed by Yakubovich in the 90's, we carry out an in-depth study of hypercyclicity properties of such operators. Under some rather general smoothness assumptions on the symbol, we provide some necessary/sufficient/necessary and sufficient conditions for $T_F$ to be hypercyclic on $H^p$. In particular, we extend previous results on the subject by Baranov-Lishanskii and Abakumov-Baranov-Charpentier-Lishanskii. We also study some other dynamical properties for this class of operators. - oai:arXiv.org:2502.03303v3 - math.FA - math.CV - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Emmanuel Fricain, Sophie Grivaux, Ma\"eva Ostermann - - - Improved regularity estimates for degenerate or singular fully nonlinear dead-core systems and H\'{e}non-type equations - https://arxiv.org/abs/2502.10099 - arXiv:2502.10099v3 Announce Type: replace -Abstract: In this paper, we study the degenerate or singular fully nonlinear dead-core systems coupled with strong absorption terms. We establish several properties, including improved regularity of viscosity solutions along the free boundary, non-degeneracy, a measure estimate of the free boundary, Liouville-type results, and the behavior of blow-up solution. We also derive sharp regularity estimates for viscosity solutions to H\'{e}non-type equations with a degenerate weight and strong absorption, governed by a degenerate fully nonlinear operator. Our results are new even for the model equations involving degenerate Laplacian operators. - oai:arXiv.org:2502.10099v3 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jiangwen Wang, Feida Jiang - - - Isometries of spacetimes without observer horizons - https://arxiv.org/abs/2502.13904 - arXiv:2502.13904v3 Announce Type: replace -Abstract: We study the isometry groups of (non-compact) Lorentzian manifolds with well-behaved causal structure, aka causal spacetimes satisfying the ``no observer horizons'' condition. Our main result is that the group of time orientation-preserving isometries acts properly on the spacetime. As corollaries, we obtain the existence of an invariant Cauchy temporal function, and a splitting of the isometry group into a compact subgroup and a subgroup roughly corresponding to time translations. The latter can only be the trivial group, $\mathbb{Z}$, or $\mathbb{R}$. - oai:arXiv.org:2502.13904v3 - math.DG - gr-qc - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Leonardo Garc\'ia-Heveling, Abdelghani Zeghib - - - Abelian congruences and similarity in varieties with a weak difference term - https://arxiv.org/abs/2502.20517 - arXiv:2502.20517v2 Announce Type: replace -Abstract: This is the first of three papers motivated by the author's desire to understand and explain "algebraically" one aspect of Dmitriy Zhuk's proof of the CSP Dichotomy Theorem. In this paper we study abelian congruences in varieties having a weak difference term. Each class of the congruence supports an abelian group structure; if the congruence is minimal, each class supports the structure of a vector space over a division ring determined by the congruence. A construction due to J. Hagemann, C. Herrmann and R. Freese in the congruence modular setting extends to varieties with a weak difference term, and provides a "universal domain" for the abelian groups or vector spaces that arise from the classes of the congruence within a single class of the annihilator of the congruence. The construction also supports an extension of Freese's similarity relation (between subdirectly irreducible algebras) from the congruence modular setting to varieties with a weak difference term. - oai:arXiv.org:2502.20517v2 - math.LO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ross Willard - - - Global solutions for supersonic flow of a Chaplygin gas past a conical wing with a shock wave detached from the leading edges - https://arxiv.org/abs/2503.03406 - arXiv:2503.03406v4 Announce Type: replace -Abstract: In this paper, we first investigate the mathematical aspects of supersonic flow of a Chaplygin gas past a conical wing with diamond-shaped cross sections in the case of a shock wave detached from the leading edges. The flow under consideration is governed by the three-dimensional steady compressible Euler equations. For the Chaplygin gas, all characteristics are linearly degenerate, and shocks are reversible and characteristic. Using these properties, we can determine the location of the shock in advance and reformulate our problem as an oblique derivative problem for a nonlinear degenerate elliptic equation in conical coordinates. By establishing a Lipschitz estimate, we show that the equation is uniformly elliptic in any subdomain strictly away from the degenerate boundary, and then further prove the existence of a solution to the problem via the continuity method and vanishing viscosity method. - oai:arXiv.org:2503.03406v4 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Bingsong Long - - - Probabilistic Insights for Efficient Exploration Strategies in Reinforcement Learning - https://arxiv.org/abs/2503.03565 - arXiv:2503.03565v2 Announce Type: replace -Abstract: We investigate efficient exploration strategies of environments with unknown stochastic dynamics and sparse rewards. Specifically, we analyze first the impact of parallel simulations on the probability of reaching rare states within a finite time budget. Using simplified models based on random walks and L\'evy processes, we provide analytical results that demonstrate a phase transition in reaching probabilities as a function of the number of parallel simulations. We identify an optimal number of parallel simulations that balances exploration diversity and time allocation. Additionally, we analyze a restarting mechanism that exponentially enhances the probability of success by redirecting efforts toward more promising regions of the state space. Our findings contribute to a more qualitative and quantitative theory of some exploration schemes in reinforcement learning, offering insights into developing more efficient strategies for environments characterized by rare events. - oai:arXiv.org:2503.03565v2 - math.PR - cs.LG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ernesto Garcia, Paola Bermolen, Matthieu Jonckheere, Seva Shneer - - - Counting the number of $\mathcal{O}_{K}$-fixed points of a discrete dynamical system with applications from arithmetic statistics, II - https://arxiv.org/abs/2503.11393 - arXiv:2503.11393v3 Announce Type: replace -Abstract: In this follow-up paper, we again inspect a surprising connection between the set of fixed points of a polynomial map $\varphi_{d,c}$ defined by $\varphi_{d,c}(z) = z^d + c$ for all $c, z \in \mathcal{O}_{K}$ and the coefficient $c$, where $K$ is any number field of degree $n > 1$ and $d > 2$ is an integer. As before, we wish to study counting problems which are inspired by exciting advances in arithmetic statistics, and again partly by point-counting result of Narkiewicz on real $K$-rational periodic points of any odd degree map $\varphi_{d,c}$ in arithmetic dynamics. In doing so, we then first prove that for any real algebraic number field $K$ of degree $n \geq 2$, and for any prime $p \geq 3$ and integer $\ell \geq 1$, the average number of distinct integral fixed points of any $\varphi_{p^{\ell},c}$ modulo prime ideal $p\mathcal{O}_{K}$ is $3$ or $0$ as $c\to \infty$. Motivated further by $K$-rational periodic point-counting result of Benedetto on any $\varphi_{(p-1)^{\ell},c}$ for any prime $p \geq 5$ and integer $\ell \in \mathbb{Z}_{\geq 1}$ in arithmetic dynamics, we then also prove unconditionally that for any number field (not necessarily real) $K$ of degree $n \geq 2$, the average number of distinct integral fixed points of any $\varphi_{(p-1)^{\ell},c}$ modulo prime $p\mathcal{O}_{K}$ is $1$ or $2$ or $0$ as $c\to \infty$. Finally, we then apply density and number field-counting results from arithmetic statistics, and as a result obtain counting and statistical results on irreducible polynomials and number fields arising naturally in our polynomial discrete dynamical settings. - oai:arXiv.org:2503.11393v3 - math.NT - math.DS - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Brian Kintu - - - On a polygon version of Wiegmann-Zabrodin formula - https://arxiv.org/abs/2503.13718 - arXiv:2503.13718v2 Announce Type: replace -Abstract: Let $P$ be a convex polygon in ${\mathbb C}$ and let $\Delta_{D, P}$ be the operator of the Dirichlet boundary value problem for the Lapalcian $\Delta=-4\partial_z\partial_{\bar z}$ in $P$. We derive a variational formula for the logarithm of the $\zeta$-regularized determinant of $\Delta_{D, P}$ for arbitrary infinitesimal deformations of the polygon $P$ in the class of polygons (with the same number of vertices). For a simply connected domain with smooth boundary such a formula was recently discovered by Wiegmann and Zabrodin as a non obvious corollary of the Alvarez variational formula, for domains with corners this approach is unavailable (at least for those deformations that do not preserve the corner angles) and we have to develop another one. - oai:arXiv.org:2503.13718v2 - math.SP - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexey Kokotov, Dmitrii Korikov - - - Quasi-convex Splittings of Acylindrical Graphs of Locally Finite-Height Groups - https://arxiv.org/abs/2503.15459 - arXiv:2503.15459v2 Announce Type: replace -Abstract: We find a condition on the acylindrical action of a finitely presented group on a simplicial tree which guarantees that this action will be dominated by an acylindrical action with finitely generated edge stabilisers, and find the first example of an action of a finitely presented group where there is no such dominating action. As a consequence, we show that any finitely presented group that admits a decomposition as an acylindrical graph of (possibly infinitely generated) free groups is virtually compact special, and that every finitely generated subgroup of a one-relator group with an acylindrical Magnus hierarchy is virtually compact special. - oai:arXiv.org:2503.15459v2 - math.GR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - William D. Cohen - - - Information-theoretic coordinate subset and partition selection of multivariate Markov chains via submodular optimization - https://arxiv.org/abs/2503.23340 - arXiv:2503.23340v2 Announce Type: replace -Abstract: We study the problem of optimally projecting the transition matrix of a finite ergodic multivariate Markov chain onto a lower-dimensional state space, as well as the problem of finding an optimal partition of coordinates such that the factorized Markov chain gives minimal information loss compared to the original multivariate chain. Specifically, we seek to construct a Markov chain that optimizes various information-theoretic criteria under cardinality constraints. These criteria include entropy rate, information-theoretic distance to factorizability, independence, and stationarity. We formulate these tasks as best subset or partition selection problems over multivariate Markov chains and leverage the (k-)submodular (or (k-)supermodular) structures of the objective functions to develop efficient greedy-based algorithms with theoretical guarantees. Along the way, we introduce a generalized version of the distorted greedy algorithm, which may be of independent interest. Finally, we illustrate the theory and algorithms through extensive numerical experiments with publicly available code on multivariate Markov chains associated with the Bernoulli--Laplace and Curie--Weiss models. - oai:arXiv.org:2503.23340v2 - math.PR - cs.IT - math.CO - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Zheyuan Lai, Michael C. H. Choi - - - Talenti comparison results for solutions to $p$-Laplace equation on multiply connected domains - https://arxiv.org/abs/2504.06103 - arXiv:2504.06103v2 Announce Type: replace -Abstract: In the last years comparison results of Talenti type for Elliptic Problems have been widely investigated. In this paper we obtain a comparison result for the $p$-Laplace operator in multiply connected domains with Robin boundary condition on the exterior boundary and non-homogeneous Dirichlet boundary conditions on the interior one, generalizing the results obtained in \cite{ANT, AGM} to this type of domains. This will be a generalization to Robin boundary condition of the results obtained in \cite{B, B2}, with an improvement of the $L^2$ comparison in the case $p=2$. As a consequence, we obtain a Bossel-Daners and Saint-Venant type inequalities for multiply connected domains. - oai:arXiv.org:2504.06103v2 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Luca Barbato, Francesco Salerno - - - On the uniqueness of a generalized quadrangle of order (4,16) - https://arxiv.org/abs/2504.09372 - arXiv:2504.09372v4 Announce Type: replace -Abstract: In the manuscript [v4], we prove the uniqueness of a generalized quadrangle of order (4,16). - oai:arXiv.org:2504.09372v4 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Koichi Inoue - - - Magnetic Thomas-Fermi theory for 2D abelian anyons - https://arxiv.org/abs/2504.13481 - arXiv:2504.13481v2 Announce Type: replace -Abstract: Two-dimensional abelian anyons are, in the magnetic gauge picture, represented as fermions coupled to magnetic flux tubes. For the ground state of such a system in a trapping potential, we theoretically and numerically investigate a Hartree approximate model, obtained by restricting trial states to Slater determinants and introducing a self-consistent magnetic field, locally proportional to matter density. This leads to a fermionic variant of the Chern-Simons-Schr{\"o}dinger system. We find that for dense systems, a semi-classical approximation yields qualitatively good results. Namely, we derive a density functional theory of magnetic Thomas-Fermi type, which correctly captures the trends of our numerical results. In particular, we explore the subtle dependence of the ground state with respect to the fraction of magnetic flux units attached to particles. - oai:arXiv.org:2504.13481v2 - math.AP - cond-mat.mes-hall - cond-mat.quant-gas - cond-mat.str-el - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Antoine Levitt (LMO), Douglas Lundholm (UMPA-ENSL), Nicolas Rougerie (UMPA-ENSL) - - - Non-solutions to mixed equations in acylindrically hyperbolic groups coming from random walks - https://arxiv.org/abs/2504.15456 - arXiv:2504.15456v2 Announce Type: replace -Abstract: A mixed equation in a group $G$ is given by a non-trivial element $w (x)$ of the free product $G \ast \mathbb{Z}$, and a solution is some $g\in G$ such that $w(g)$ is the identity. For $G$ acylindrically hyperbolic with trivial finite radical (e.g. torsion-free) we show that any mixed equation of length $n$ has a non-solution of length comparable to $\log(n)$, which is the best possible bound. Similarly, we show that there is a common non-solution of length $O(n)$ to all mixed equations of length $n$, again the best possible bound. In fact, in both cases we show that a random walk of appropriate length yields a non-solution with positive probability. - oai:arXiv.org:2504.15456v2 - math.GR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Henry Bradford, Alessandro Sisto - - - Topological triviality and link-constancy in deformations of inner Khovanskii non-degenerate maps - https://arxiv.org/abs/2504.18816 - arXiv:2504.18816v5 Announce Type: replace -Abstract: For real and mixed polynomial maps $f=(f^1,\dots,f^p)$ satisfying $f(0)=0$, we introduce the notion of Inner Khovanskii Non-Degeneracy (IKND), that generalizes a previous non-degeneracy condition introduced by Wall for complex polynomial functions (J. Reine Angew. Math. 509 (1999), 1-19). We prove that IKND is a sufficient condition ensuring that the link of the singularity of $f$ at the origin is smooth and well-defined. We study one-parameter deformations of an IKND map $f$, given by $F(\boldsymbol{x},\varepsilon)=f(\boldsymbol{x})+\theta(\boldsymbol{x},\varepsilon)$, with $ F(0,\varepsilon)=0$. We prove that the deformation is \textit{link-constant} under suitable conditions on $f$ and $\theta$, meaning that the ambient isotopy type of the link remains unchanged along the deformation. Furthermore, by employing a strong version of this non-degeneracy, Strong Inner Khovanskii Non-Degeneracy (SIKND), we obtain results on topological triviality. In the final section, we present link-constant deformations for IKND mixed polynomial functions of two variables. We also explore several applications motivated by the recent findings of Ara\'ujo dos Santos, Bode, and Sanchez Quiceno (Bull. Braz. Math. Soc. (N.S.) 55 (2024), no. 3, Paper No. 34). - oai:arXiv.org:2504.18816v5 - math.AG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/publicdomain/zero/1.0/ - Julian D. Espinel Leal, Eder L. Sanchez Quiceno - - - Monoidal Relative Categories Model Monoidal $\infty$-Categories - https://arxiv.org/abs/2504.20606 - arXiv:2504.20606v3 Announce Type: replace -Abstract: We prove that the homotopy theory of monoidal relative categories is equivalent to that of monoidal $\infty$-categories, and likewise in the symmetric monoidal setting. As an application, we give a concise and complete proof of the fact that every presentably monoidal or presentably symmetric monoidal $\infty$-category is presented by a monoidal or symmetric monoidal model category, which, in the monoidal case, was sketched by Lurie, and in the symmetric monoidal case, was proved by Nikolaus--Sagave. - oai:arXiv.org:2504.20606v3 - math.CT - math.AT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kensuke Arakawa - - - A Note On Generalized $L_p$ Inequalities for the polar derivative of a polynomial - https://arxiv.org/abs/2505.06539 - arXiv:2505.06539v3 Announce Type: replace -Abstract: Let \( P(z) \) be a polynomial of degree \( n \) and $\alpha \in \mathbb{C}$. The polar derivative of \( P(z) \), denoted by \( D_\alpha P(z) \) and is defined by $D_\alpha P(z): = nP(z) + (\alpha -z)P'(z)$. The polar derivative \( D_\alpha P(z) \) is a polynomial of degree at most \( n - 1 \) and it generalizes the ordinary derivative \( P'(z) \). In this paper, we establish some \( L_p \) inequalities for the polar derivative of a polynomial with all its zeros located within a prescribed disk. Our results refine and generalize previously known findings. - oai:arXiv.org:2505.06539v3 - math.CV - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - N. A. Rather, Danish Rashid Bhat, Tanveer Bhat - - - The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics - https://arxiv.org/abs/2505.07506 - arXiv:2505.07506v3 Announce Type: replace -Abstract: We study a two-dimensional variational model for ferronematics -- composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes~$\mathbf{Q}$-tensor for the liquid crystal component and a magnetisation vector field~$\mathbf{M}$, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between~$\mathbf{Q}$ and~$\mathbf{M}$. We analyse the asymptotic behaviour of (not necessarily minimizing) critical points as a small parameter~$\varepsilon$ tends to zero. Our main results show that the energy concentrates along distinct singular sets: the (rescaled) energy density for the~$\mathbf{Q}$-component concentrates, to leading order, on a finite number of singular points, while the energy density for the~$\mathbf{M}$-component concentrate along a one-dimensional rectifiable set. Moreover, we prove that the curvature of the singular set for the $\M$-component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e.~the singular set for the~$\Q$-component. - oai:arXiv.org:2505.07506v3 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Giacomo Canevari, Federico Luigi Dipasquale, Bianca Stroffolini - - - On the extremal length of the hyperbolic metric - https://arxiv.org/abs/2505.12400 - arXiv:2505.12400v2 Announce Type: replace -Abstract: For any closed hyperbolic Riemann surface $X$, we show that the extremal length of the Liouville current is determined solely by the topology of \(X\). This confirms a conjecture of Mart\'inez-Granado and Thurston. We also obtain an upper bound, depending only on $X$, for the diameter of extremal metrics on $X$ with area one. - oai:arXiv.org:2505.12400v2 - math.GT - math.CV - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Hidetoshi Masai - - - Invariant random subgroups on certain orbits - https://arxiv.org/abs/2506.09723 - arXiv:2506.09723v2 Announce Type: replace -Abstract: Let $G$ be a connected Lie group and $\text{Sub}_G$ be the space of closed subgroups of $G$ equipped with the Chabauty topology. In this article, we investigate the existence of invariant random subgroups of $G$ supported on various orbits of the conjugation action of $G$ on $\text{Sub}_G$. - oai:arXiv.org:2506.09723v2 - math.DS - math.GN - math.GR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Manoj Choudhuri, C. R. E. Raja - - - The rational homotopy groups of virtual spheres for rank 1 compact Lie groups - https://arxiv.org/abs/2506.18085 - arXiv:2506.18085v4 Announce Type: replace -Abstract: We calculate the rational representation-ring-graded stable stems for rank 1 groups, SU(2), SO(3), Pin (2), O(2), Spin(2) and - SO(2), in the same spirit as the calculations for finite groups in arXiv:2205.02382 with J.D.Quigley. This illustrates the effectiveness - of the algebraic models for these categories of G-spectra, and the way tom Dieck splitting fails for desuspensions. - [v4: typos and tweaks in wording] - oai:arXiv.org:2506.18085v4 - math.AT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - J. P. C. Greenlees - - - Local H\"older Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights - https://arxiv.org/abs/2507.01899 - arXiv:2507.01899v2 Announce Type: replace -Abstract: We establish local boundedness and local H\"older continuity of weak solutions to the following prototype problem: - $$ -\operatorname{div}\left(|x|^{-2 \beta}|\nabla u|^{\mathbf{q}-2} \nabla u\right)+(-\Delta)_{p(\cdot, \cdot), \beta}^{s(\cdot, \cdot)} u=0 \quad \text { in } \quad \Omega, $$ - where $\Omega \subset \mathbb{R}^n, n \geq 2$, is a bounded domain. The nonlocal operator is defined by - $$ (-\Delta)_{p(\cdot, \cdot), \beta}^{s(\cdot, \cdot)} u(x):=\mathrm{P} . \mathrm{V} . \int_{\Omega} \frac{|u(x)-u(y)|^{p(x, y)-2}(u(x)-u(y))}{|x-y|^{n+s(x, y) p(x, y)}} \frac{1}{|x|^\beta|y|^\beta} \mathrm{d} y $$ - Here, $p: \Omega \times \Omega \rightarrow(1, \infty)$ and $s: \Omega \times \Omega \rightarrow(0,1)$ are measurable functions, $\mathbf{q}:=\operatorname{ess}_{\Omega \times \Omega} p$, and $0 \leq \beta<n$. Our approach is analytic and relies on an adaptation of the De Giorgi-Nash-Moser theory to a mixed local-nonlocal framework with variable exponents and weights. - oai:arXiv.org:2507.01899v2 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.jmaa.2026.130417 - Juan Pablo Alcon Apaza - - - P\'olya's conjecture up to $\epsilon$-loss and quantitative estimates for the remainder of Weyl's law - https://arxiv.org/abs/2507.04307 - arXiv:2507.04307v3 Announce Type: replace -Abstract: Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. For any $\epsilon\in (0,1)$ we show that for any Dirichlet eigenvalue $\lambda_k(\Omega)>\Lambda(\epsilon,\Omega)$, it holds \begin{align*} k&\le (1+\epsilon)\frac{|\Omega|\omega(n)}{(2\pi)^n}\lambda_k(\Omega)^{n/2}, \end{align*} where $\Lambda(\epsilon,\Omega)$ is given explicitly. This reduces the $\epsilon$-loss version of P\'olya's conjecture to a computational problem. This estimate is based on quantitative estimates on the remainder of the Weyl law with explicit constants, which we give a new proof without using Neumann eigenvalues. Our arguments in deriving such uniform estimates yield also, in all dimensions $n\ge 2$, classes of domains that may even have rather irregular shapes or boundaries but satisfy P\'olya's conjecture. Another key observation is that on strip-tiling domains (and therefore any triangles for instance) one actually has better eigenvalue estimates than P\'olya conjectured. - oai:arXiv.org:2507.04307v3 - math.SP - math-ph - math.AP - math.CA - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Renjin Jiang, Fanghua Lin - - - Keep the beat going: Automatic drum transcription with momentum - https://arxiv.org/abs/2507.12596 - arXiv:2507.12596v2 Announce Type: replace -Abstract: How can we process a piece of recorded music to detect and visualize the onset of each instrument? A simple, interpretable approach is based on partially fixed nonnegative matrix factorization (NMF). Yet despite the method's simplicity, partially fixed NMF is challenging to apply because the associated optimization problem is high-dimensional and non-convex. This paper explores two optimization approaches that preserve the nonnegative structure, including a multiplicative update rule and projected gradient descent with momentum. These techniques are derived from the previous literature, but they have not been fully developed for partially fixed NMF before now. Results indicate that projected gradient descent with momentum leads to the higher accuracy among the two methods, and it satisfies stronger local convergence guarantees. - oai:arXiv.org:2507.12596v2 - math.NA - cs.NA - cs.SD - eess.AS - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alisha L. Foster, Robert J. Webber - - - A Framework of Distributed Source Encryption using Mutual Information Security Criterion and the Strong Converse Theorem - https://arxiv.org/abs/2507.13294 - arXiv:2507.13294v4 Announce Type: replace -Abstract: We reinvestigate the general distributed secure source coding based on the common key cryptosystem proposed by Oohama and Santoso (ITW 2021). They proposed a framework of distributed source encryption and derived the necessary and sufficient conditions to have reliable and secure transmission. However, the bounds of the rate region, which specifies both necessary and sufficient conditions to have reliable and secure transmission under the proposed cryptosystem, were derived based on a self-tailored non-standard} security criterion. In this paper we adopt the standard security criterion, i.e., standard mutual information. We successfully establish the bounds of the rate region based on this security criterion. Information spectrum method and a variant of Birkhoff-von Neumann theorem play an important role in deriving the result. - oai:arXiv.org:2507.13294v4 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Yasutada Oohama, Bagus Santoso - - - Cohen-Macaulay approximations and the $\text{SC}_r$-condition - https://arxiv.org/abs/2507.14424 - arXiv:2507.14424v3 Announce Type: replace -Abstract: We study the relation between MCM approximations and FID hulls of modules over a Cohen-Macaulay local ring $R$ with canonical module, specifically when $R$ is generically Gorenstein. We then generalize a result of Kato, who proved that a Gorenstein complete local ring $R$ satisfies the $\text{SC}_{2}$-condition if and only if $R$ is a UFD. For $r \geq 3$, we prove a criterion for when an MCM $R$-module $M$ satisfies the $\text{SC}_{r}$-condition, assuming that its first syzygy $\Omega_{R}^{1}(M)$ satisfies the $\text{SC}_{r-1}$-condition. - oai:arXiv.org:2507.14424v3 - math.AC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Richard F. Bartels - - - 1/2 order convergence rate of Euler-type methods for time-changed stochastic differential equations with super-linearly growing drift and diffusion coefficients - https://arxiv.org/abs/2507.14562 - arXiv:2507.14562v4 Announce Type: replace -Abstract: This paper investigates the strong convergence properties of two Euler-type methods for a class of time-changed stochastic differential equations (TCSDEs) with super-linearly growing drift and diffusion coefficients. Building upon existing research, we propose a backward Euler method (BEM) and introduce its explicit counterpart -- the projected Euler method (PEM). We prove that both methods converge strongly in the $L_2$-sense at the optimal rate of 1/2. This result extends the applicability of both the BEM and the PEM to a broader class of TCSDEs. Moreover, the two methods offer complementary strengths: while BEM possesses wide applicability, PEM is computationally more efficient. Numerical simulations confirm our theoretical findings and illustrate practical performance of both schemes. - oai:arXiv.org:2507.14562v4 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Shuai Wang, Yuanling Niu, Ying Zhang - - - Some reverse inequalities for scalar Birkhoff weak integrable functions - https://arxiv.org/abs/2507.16332 - arXiv:2507.16332v2 Announce Type: replace -Abstract: Some inequalities and reverses of classic H\"{o}lder and Minkowski types are obtained for scalar Birkhoff weak integrable functions with respect to a non-additive measure. - oai:arXiv.org:2507.16332v2 - math.FA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Anca Croitoru, Alina Iosif, Anna Rita Sambucini, Luca Zampogni - - - Matrix convex sets over the Euclidean ball and polar duals of real free spectrahedra - https://arxiv.org/abs/2507.20325 - arXiv:2507.20325v2 Announce Type: replace -Abstract: We show that the free spectrahedron determined by universal anticommuting self-adjoint unitaries is not equal to the minimal matrix convex set over the ball in dimension three or higher. This example, as well as other matrix convex sets over the ball, then provides context for structure results on the extreme points of coordinate projections. In particular, we show that the free polar dual of a real free spectrahedron is rarely the projection of a real free spectrahedron, contrasting a prior result of Helton, Klep, and McCullough over the complexes. We use this to show that spanning results for free spectrahedra that are closed under complex conjugation do not extend to free spectrahedrops that meet the same assumption. These results further clarify the role of the coefficient field. - oai:arXiv.org:2507.20325v2 - math.FA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Eric Evert, Benjamin Passer - - - Towards the classification of maximum scattered linear sets of $\mathrm{PG}(1,q^5)$ - https://arxiv.org/abs/2507.23409 - arXiv:2507.23409v3 Announce Type: replace -Abstract: Every maximum scattered linear set in $\mathrm{PG}(1,q^5)$ is the projection of an $\mathbb{F}_q$-subgeometry $\Sigma$ of $\mathrm{PG}(4,q^5)$ from a plane $\Gamma$ external to the secant variety to $\Sigma$. The pair $(\Gamma,\Sigma)$ will be called a projecting configuration for the linear set. The projecting configurations for the only known maximum scattered linear sets in $\mathrm{PG}(1,q^5)$, namely those of pseudoregulus and LP type, have been characterized in the literature by B. Csajb\'{o}k, C. Zanella in 2016 and by C. Zanella, F. Zullo in 2020. Let $(\Gamma,\Sigma)$ be a projecting configuration for a maximum scattered linear set in $\mathrm{PG}(1,q^5)$, let $\sigma$ be a generator of $\mathbb{G}=\mathrm{P}\Gamma \mathrm{L}(5,q^5)_\Sigma$, and $A=\Gamma\cap\Gamma^{\sigma^4}$, $B=\Gamma\cap\Gamma^{\sigma^3}$. If $A$ and $B$ are not both points, then the projected linear set is of pseudoregulus type. Then, suppose that they are points. The rank of a point $X$ is the vectorial dimension of the span of the orbit of $X$ under the action of $\mathbb{G}$. In this paper, by investigating the geometric properties of projecting configurations, it is proved that if at least one of the points $A$ and $B$ has rank 5, the associated maximum scattered linear set must be of LP type. Then, if a maximum scattered linear set of a new type exists, it must be such that $\mathrm{rk} A=\mathrm{rk} B=4$. In this paper we derive two possible polynomial forms that such a linear set must have. An exhaustive analysis by computer shows that for $q\leq 25$, no new maximum scattered linear set exists. - oai:arXiv.org:2507.23409v3 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Stefano Lia, Giovanni Longobardi, Corrado Zanella - - - On the characteristic form of $\mathfrak{g}$-valued zero-curvature representations - https://arxiv.org/abs/2508.01224 - arXiv:2508.01224v2 Announce Type: replace -Abstract: We study $\mathfrak{g}$-valued zero-curvature representations (ZCRs) for partial differential equations in two independent variables from the perspective of their extension to the entire infinite jet space, focusing on their characteristic elements. Since conservation laws -- more precisely, conserved currents -- and their generating functions for a given equation are precisely the $\mathbb{R}$-valued ZCRs and their characteristic elements, a natural question arises: to what extent can results known for conservation laws be extended to general $\mathfrak{g}$-valued ZCRs. - For a fixed matrix Lie algebra $\mathfrak{g} \subset \mathfrak{gl}(n)$, we formulate ZCRs as equivalence classes of $\mathfrak{g}$-valued function pairs on the infinite jet space that satisfy the Maurer--Cartan condition. Our main result establishes that every such ZCR admits a characteristic representative -- i.e., a representative in which the Maurer--Cartan condition takes a characteristic form -- generalizing the characteristic form known for conservation laws. This form is preserved under gauge transformations and can thus be regarded as a kind of normal form for ZCRs. We derive a new necessary condition, independent of the Maurer--Cartan equation, that must be satisfied by any characteristic representative. This condition is trivial in the abelian case but nontrivial whenever $\mathfrak{g}$ is nonabelian. These findings not only confirm structural assumptions used in previous works but also suggest potential applications in the classification and computation of ZCRs. - oai:arXiv.org:2508.01224v2 - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-sa/4.0/ - Jirina Jahnova - - - Canonical Frames for Bracket Generating Rank 2 Distributions which are not Goursat - https://arxiv.org/abs/2508.09307 - arXiv:2508.09307v5 Announce Type: replace -Abstract: We complete a uniform construction of canonical absolute parallelism for bracket generating rank $2$ distributions with $5$-dimensional cube on $n$-dimensional manifold with $n\geq 5$ by showing that the condition of maximality of class that was assumed previously by Doubrov-Zelenko for such a construction holds automatically at generic points. This also gives analogous constructions in the case when the cube is not $5$-dimensional but the distribution is not Goursat through the procedure of iterative Cartan deprolongation. This together with the classical theory of Goursat distributions covers in principle the local geometry of all bracket generating rank 2 distributions in a neighborhood of generic points. As a byproduct, for any $n\geq 5$ we describe the maximally symmetric germs among bracket generating rank $2$ distributions with $5$-dimensional cube, as well as among those which reduce to such a distribution under a fixed number of Cartan deprolongations. Another consequence of our results on maximality of class is for optimal control problems with constraint given by a rank $2$ distribution with $5$-dimensional cube: it implies that for a generic point $q_0$ of $M$, there are plenty abnormal extremal trajectories of corank $1$ (which is the minimal possible corank) starting at $q_0$. The set of such points contains all points where the distribution is equiregular. - oai:arXiv.org:2508.09307v5 - math.DG - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Nicklas Day, Igor Zelenko - - - Geometric inequalities for electrostatic systems with boundary - https://arxiv.org/abs/2508.10258 - arXiv:2508.10258v3 Announce Type: replace -Abstract: In this article, we investigate electrostatic systems with a nonzero cosmological constant on compact manifolds with boundary. We establish new geometric properties for electrostatic manifolds in higher dimensions, extending previous results in the literature. Moreover, we prove sharp boundary estimates and isoperimetric-type inequalities for electrostatic manifolds, as well as volume and boundary inequalities involving the Brown-York and Hawking masses. - oai:arXiv.org:2508.10258v3 - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Allan Freitas, Benedito Leandro, Ernani Ribeiro Jr, Guilherme Sabo - - - False Data-Injection Attack Detection in Cyber-Physical Systems: A Wasserstein Distributionally Robust Reachability Optimization Approach - https://arxiv.org/abs/2508.12402 - arXiv:2508.12402v2 Announce Type: replace -Abstract: Cyber-physical system (CPS) is the foundational backbone of modern critical infrastructures, so ensuring its security and resilience against cyber-attacks is of pivotal importance. This paper addresses the challenge of designing anomaly detectors for CPS under false-data injection (FDI) attacks and stochastic disturbances governed by unknown probability distribution. By using the Wasserstein ambiguity set, a prevalent data-driven tool in distributionally robust optimization (DRO), we first propose a new security metric to deal with the absence of disturbance distribution. This metric is designed by asymptotic reachability analysis of state deviations caused by stealthy FDI attacks and disturbance in a distributionally robust confidence set. We then formulate the detector design as a DRO problem that optimizes this security metric while controlling the false alarm rate robustly under a set of distributions. This yields a trade-off between robustness to disturbance and performance degradation under stealthy attacks. The resulting design problem turns out to be a challenging semi-infinite program due to the existence of distributionally robust chance constraints. We derive its exact albeit non-convex reformulation and develop an effective solution algorithm based on sequential minimization. Finally, a case study on a simulated three-tank is illustrated to demonstrate the efficiency of our design in robustifying against unknown disturbance distribution. - oai:arXiv.org:2508.12402v2 - math.DS - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yulin Feng, Dapeng Lan, Chao Shang - - - New weighted Riesz-type pointwise inequalities and applications to generalized Sobolev estimates - https://arxiv.org/abs/2508.12771 - arXiv:2508.12771v2 Announce Type: replace -Abstract: In this article we study some new pointwise inequalities between rough singular integral operators, weighted maximal functions of the gradient and weighted Morrey spaces. These pointwise estimates will naturally lead us to a new class of weighted Sobolev-type inequalities. - oai:arXiv.org:2508.12771v2 - math.FA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Diego Chamorro (LaMME), Anca-Nicoleta Marcoci, Liviu-Gabriel Marcoci - - - Some new properties of the PamPa scheme - https://arxiv.org/abs/2508.17147 - arXiv:2508.17147v2 Announce Type: replace -Abstract: In this paper, we provide a few new properties of Active Flux (AF)/Point-Average-Moment PolynomiAl-interpreted (\pampa) schemes. First, we show, in full generality, that the AF/pampa schemes can be interpreted in such a way that the discontinuous Galerkin (dG) scheme is one of their building blocks. Secondly we provide intrinsic bound preserving properties of the current variant of pampa. This is also illustrated numerically. Last, we show, at least in one dimension, that the pampa scheme has the summation by part (SBP) property. - oai:arXiv.org:2508.17147v2 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - R\'emi Abgrall, Philipp \"Offner, Yongle Liu - - - Voter Model stability with respect to conservative noises - https://arxiv.org/abs/2509.02717 - arXiv:2509.02717v2 Announce Type: replace -Abstract: The notions of noise sensitivity and stability were recently extended for the voter model. In this model, the vertices of a graph have opinions that are updated by uniformly selecting edges. We further extend stability results to different classes of perturbations. We consider two different types of noise: in the first one, an exclusion process is performed on the edge selections, while in the second, independent Brownian motions are applied to such a sequence. In both cases, we prove stability of the consensus opinion provided the noise is run for a short amount of time, depending on the underlying graph structure. This is done by analyzing the expected size of the pivotal set, whose definition differs from the usual one in order to reflect the change associated with these noises. - oai:arXiv.org:2509.02717v2 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gideon Amir, Omer Angel, Rangel Baldasso, Daniel de la Riva - - - Sequential Change Detection with Differential Privacy - https://arxiv.org/abs/2509.02768 - arXiv:2509.02768v2 Announce Type: replace -Abstract: Sequential change detection is a fundamental problem in statistics and signal processing, with the CUSUM procedure widely used to achieve minimax detection delay under a prescribed false-alarm rate when pre- and post-change distributions are fully known. However, releasing CUSUM statistics and the corresponding stopping time directly can compromise individual data privacy. We therefore introduce a differentially private (DP) variant, called DP-CUSUM, that injects calibrated Laplace noise into both the vanilla CUSUM statistics and the detection threshold, preserving the recursive simplicity of the classical CUSUM statistics while ensuring per-sample differential privacy. We derive closed-form bounds on the average run length to false alarm and on the worst-case average detection delay, explicitly characterizing the trade-off among privacy level, false-alarm rate, and detection efficiency. Our theoretical results imply that under a weak privacy constraint, our proposed DP-CUSUM procedure achieves the same first-order asymptotic optimality as the classical, non-private CUSUM procedure. Numerical simulations are conducted to demonstrate the detection efficiency of our proposed DP-CUSUM under different privacy constraints, and the results are consistent with our theoretical findings. - oai:arXiv.org:2509.02768v2 - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - 10.1109/TIT.2025.3644744 - Liyan Xie, Ruizhi Zhang - - - Chebyshev's bias for modular forms - https://arxiv.org/abs/2509.04187 - arXiv:2509.04187v2 Announce Type: replace -Abstract: We study Chebyshev's bias for the signs of Fourier coefficients of cuspidal newforms on $\Gamma_0(N)$. Our main result shows that the bias towards either sign is completely determined by the order of vanishing of the $L$-function $L(s, f)$ at the central point of the critical strip. We then give several examples of modular forms where we explicitly compute the order of vanishing of $L(s, f)$ at the central point and as a by-product, verify the super-positivity property, in the sense of Yun--Zhang (2017), for these examples. - oai:arXiv.org:2509.04187v2 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Shin-ya Koyama, Arshay Sheth - - - Polynomial Stability of Non-Linearly Damped Contraction Semigroups - https://arxiv.org/abs/2509.04275 - arXiv:2509.04275v2 Announce Type: replace -Abstract: We investigate the stability properties of an abstract class of semi-linear systems. Our main result establishes rational rates of decay for classical solutions assuming a certain non-uniform observability estimate for the linear part and suitable conditions on the non-linearity. We illustrate the strength of our abstract results by applying them to a one-dimensional wave equation with weak non-linear damping and to an Euler-Bernoulli beam with a tip mass subject to non-linear damping. - oai:arXiv.org:2509.04275v2 - math.FA - math.AP - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Lassi Paunonen, David Seifert - - - On the geometry of punctual Hilbert schemes on singular curves and their motivic zeta functions - https://arxiv.org/abs/2509.06761 - arXiv:2509.06761v4 Announce Type: replace -Abstract: Inspired by the work of Soma and Watari, we define a tree structure on certain subsemimodules of the semigroup $\Gamma$ associated with an irreducible plane curve singularity $(C,O)$. Building on results of Oblomkov, Rasmussen, and Shende, we show that for specific classes of singularities, this tree encodes key aspects of the geometry of the punctual Hilbert schemes of $(C,O)$. As an application, we compute the motivic Hilbert zeta function for a family of singular curves. \vskip 0.1cm A point in the Hilbert scheme corresponds to an ideal in the local ring $\mathcal{O}_{C,O}$ of the singularity. We study the stratification of these Hilbert schemes induced by constraints on the minimal number of generators of the defining ideals, and we describe geometric properties of these strata, including their dimension and closure relations.\vskip 0.1cm More importantly, we study their motivic zeta functions, particularly the motivic Hilbert zeta function, which encodes the classes of all punctual Hilbert schemes in the Grothendieck ring of varieties. - oai:arXiv.org:2509.06761v4 - math.AG - math.AC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mounir Hajli, Hussein Mourtada, Wenhao Zhu - - - Timelike conjugate points in Lorentzian length spaces - https://arxiv.org/abs/2509.12855 - arXiv:2509.12855v3 Announce Type: replace -Abstract: We study notions of conjugate points along timelike geodesics in the synthetic setting of Lorentzian (pre-)length spaces, inspired by earlier work for metric spaces by Shankar--Sormani. After preliminary considerations on convergence of timelike and causal geodesics, we introduce and compare one-sided, symmetric, unreachable and ultimate conjugate points along timelike geodesics. We show that all such notions are compatible with the usual one in the smooth (strongly causal) spacetime setting. As applications, we prove a timelike Rauch comparison theorem, as well as a result closely related to the recently established Lorentzian Cartan--Hadamard theorem by Er\"{o}s--Gieger. In the appendix, we give a detailed treatment of the Fr\'{e}chet distance on the space of non-stopping curves up to reparametrization, a technical tool used throughout the paper. - oai:arXiv.org:2509.12855v3 - math.DG - math.MG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - James D. E. Grant, Michael Kunzinger, Argam Ohanyan, Yasmin Schinnerl, Roland Steinbauer - - - The strange story of an almost unknown prime number counter: The Rafael Barrett formula - https://arxiv.org/abs/2509.19324 - arXiv:2509.19324v5 Announce Type: replace -Abstract: In this brief article, we present the formula created by Rafael Barrett in 1903 in a note to Henri Poincar\'e, which remained unknown for decades. Discovered in the 1930s by a Uruguayan mathematician, this formula was published and analyzed in a journal published in Montevideo in 1935. In this study, we present Barrett's formula and analyze a challenge it could pose. - oai:arXiv.org:2509.19324v5 - math.HO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Eduardo Mizraji - - - On Rapid mixing for random walks on nilmanifolds - https://arxiv.org/abs/2510.00398 - arXiv:2510.00398v3 Announce Type: replace -Abstract: We prove rapid mixing for almost all random walks generated by $m$ translations on an arbitrary nilmanifold under mild assumptions on the size of $m$. For several classical classes of nilmanifolds, we show $m=2$ suffices. This provides a partial answer to the question raised in \cite{D02} about the prevalence of rapid mixing for random walks on homogeneous spaces. - oai:arXiv.org:2510.00398v3 - math.DS - math.DG - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Dmitry Dolgopyat, Spencer Durham, Minsung Kim - - - Data selection: at the interface of PDE-based inverse problem and randomized linear algebra - https://arxiv.org/abs/2510.01567 - arXiv:2510.01567v2 Announce Type: replace -Abstract: All inverse problems rely on data to recover unknown parameters, yet not all data are equally informative. This raises the central question of data selection. A distinctive challenge in PDE-based inverse problems is their inherently infinite-dimensional nature: both the parameter space and the design space are infinite, which greatly complicates the selection process. Somewhat unexpectedly, randomized numerical linear algebra (RNLA), originally developed in very different contexts, has provided powerful tools for addressing this challenge. These methods are inherently probabilistic, with guarantees typically stating that information is preserved with probability at least 1-p when using N randomly selected, weighted samples. Here, the notion of "information" can take different mathematical forms depending on the setting. In this review, we survey the problem of data selection in PDE-based inverse problems, emphasize its unique infinite-dimensional aspects, and highlight how RNLA strategies have been adapted and applied in this context. - oai:arXiv.org:2510.01567v2 - math.NA - cs.NA - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-sa/4.0/ - Kathrin Hellmuth, Ruhui Jin, Qin Li, Stephen J. Wright - - - A flux-based approach for analyzing the disguised toric locus of reaction networks - https://arxiv.org/abs/2510.03621 - arXiv:2510.03621v2 Announce Type: replace -Abstract: Dynamical systems with polynomial right-hand sides are very important in various applications, e.g., in biochemistry and population dynamics. The mathematical study of these dynamical systems is challenging due to the possibility of multistability, oscillations, and chaotic dynamics. One important tool for this study is the concept of reaction systems, which are dynamical systems generated by reaction networks for some choices of parameter values. Among these, disguised toric systems are remarkably stable: they have a unique attracting fixed point, and cannot give rise to oscillations or chaotic dynamics. The computation of the set of parameter values for which a network gives rise to disguised toric systems (i.e., the disguised toric locus of the network) is an important but difficult task. We introduce new ideas based on network fluxes for studying the disguised toric locus. We prove that the disguised toric locus of any network $G$ is a contractible manifold with boundary, and introduce an associated graph $G^{\max}$ that characterizes its interior. These theoretical tools allow us, for the first time, to compute the full disguised toric locus for many networks of interest. - oai:arXiv.org:2510.03621v2 - math.DS - q-bio.MN - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bal\'azs Boros, Gheorghe Craciun, Oskar Henriksson, Jiaxin Jin, Diego Rojas La Luz - - - A recursive approach to the construction and enumeration of self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic - https://arxiv.org/abs/2510.06069 - arXiv:2510.06069v2 Announce Type: replace -Abstract: Let $\mathcal{R}_{e,m}$ be a finite commutative chain ring of even characteristic with maximal ideal $\langle u \rangle$ of nilpotency index $e \geq 2,$ Teichm$\ddot{u}$ller set $\mathcal{T}_{m},$ and residue field $\mathcal{R}_{e,m}/\langle u \rangle$ of order $2^m.$ Suppose that $2 \in \langle u^{\kappa}\rangle \setminus \langle u^{\kappa+1}\rangle$ for some even positive integer $ \kappa \leq e.$ In this paper, we provide a recursive method to construct a self-orthogonal code $\mathcal{C}_e$ of type $\{\lambda_1, \lambda_2, \ldots, \lambda_e\}$ and length $n$ over $\mathcal{R}_{e,m}$ from a chain $\mathcal{D}^{(1)}\subseteq \mathcal{D}^{(2)} \subseteq \cdots \subseteq \mathcal{D}^{(\lceil \frac{e}{2} \rceil)}$ of self-orthogonal codes of length $n$ over $\mathcal{T}_{m},$ and vice versa, where $\dim \mathcal{D}^{(i)}=\lambda_1+\lambda_2+\cdots+\lambda_i$ for $1 \leq i \leq \lceil \frac{e}{2} \rceil,$ the codes $\mathcal{D}^{(\lfloor \frac{e+1}{2} \rfloor-\kappa)},\mathcal{D}^{(\lfloor \frac{e+1}{2} \rfloor -\kappa+1)},\ldots,\mathcal{D}^{(\lfloor \frac{e}{2}\rfloor-\lfloor \frac{\kappa}{2} \rfloor)}$ satisfy certain additional conditions, and $\lambda_1,\lambda_2,\ldots,\lambda_e$ are non-negative integers satisfying $2\lambda_1+2\lambda_2+\cdots+2\lambda_{e-i+1}+\lambda_{e-i+2}+\lambda_{e-i+3}+\cdots+\lambda_i \leq n$ for $\lceil \frac{e+1}{2} \rceil \leq i\leq e.$ This construction guarantees that $Tor_i(\mathcal{C}_e)=\mathcal{D}^{(i)}$ for $1 \leq i \leq \lceil \frac{e}{2} \rceil.$ By employing this recursive construction method, together with the results from group theory and finite geometry, we derive explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over $\mathcal{R}_{e,m}.$ We also demonstrate these results through examples. - oai:arXiv.org:2510.06069v2 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Monika Yadav, Anuradha Sharma - - - Recursive construction and enumeration of self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic - https://arxiv.org/abs/2510.06082 - arXiv:2510.06082v2 Announce Type: replace -Abstract: Let $\mathscr{R}_{e,m}$ denote a finite commutative chain ring of even characteristic with maximal ideal $\langle u \rangle$ of nilpotency index $e \geq 3,$ Teichm$\ddot{u}$ller set $\mathcal{T}_{m},$ and residue field $\mathscr{R}_{e,m}/\langle u \rangle$ of order $2^m.$ Suppose that $2 \in \langle u^{\kappa}\rangle \setminus \langle u^{\kappa+1}\rangle$ for some odd integer $\kappa$ with $3 \leq \kappa \leq e.$ In this paper, we first develop a recursive method to construct a self-orthogonal code $\mathscr{D}_e$ of type $\{\lambda_1, \lambda_2, \ldots, \lambda_e\}$ and length $n$ over $\mathscr{R}_{e,m}$ from a chain $\mathcal{C}^{(1)}\subseteq \mathcal{C}^{(2)} \subseteq \cdots \subseteq \mathcal{C}^{(\lceil \frac{e}{2} \rceil)} $ of self-orthogonal codes of length $n$ over $\mathcal{T}_{m},$ and vice versa, subject to certain conditions, where $\lambda_1,\lambda_2,\ldots,\lambda_e$ are non-negative integers satisfying $2\lambda_1+2\lambda_2+\cdots+2\lambda_{e-i+1}+\lambda_{e-i+2}+\lambda_{e-i+3}+\cdots+\lambda_i \leq n$ for $\lceil \frac{e+1}{2} \rceil \leq i\leq e,$ and - $\lfloor \cdot \rfloor$ and $\lceil \cdot \rceil$ denote the floor and ceiling functions, respectively. This construction ensures that $Tor_i(\mathscr{D}_e)=\mathcal{C}^{(i)}$ for $1 \leq i \leq \lceil \frac{e}{2} \rceil.$ - With the help of this recursive construction method and by applying results from group theory and finite geometry, we obtain explicit enumeration formulae for all self-orthogonal and self-dual codes of an arbitrary length over $\mathscr{R}_{e,m}.$ We also illustrate these results with some examples. - oai:arXiv.org:2510.06082v2 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Monika Yadav, Anuradha Sharma - - - Diameter and mixing time of the giant component in the percolated hypercube - https://arxiv.org/abs/2510.13348 - arXiv:2510.13348v3 Announce Type: replace -Abstract: We consider bond percolation on the $d$-dimensional binary hypercube with $p=c/d$ for fixed $c>1$. We prove that the typical diameter of the giant component $L_1$ is of order $\Theta(d)$, and the typical mixing time of the lazy random walk on $L_1$ is of order $\Theta(d^2)$. This resolves long-standing open problems of Bollob\'as, Kohayakawa and {\L}uczak from 1994, and of Benjamini and Mossel from 2003. - A key component in our approach is a new tight large deviation estimate on the number of vertices in $L_1$ whose proof includes several novel ingredients: a structural description of the residue outside the giant component after sprinkling, a tight quantitative estimate on the spread of the giant in the hypercube, and a stability principle which rules out the disintegration of large connected sets under thinning. This toolkit further allows us to obtain optimal bounds on the expansion in $L_1$. - oai:arXiv.org:2510.13348v3 - math.PR - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Michael Anastos, Sahar Diskin, Lyuben Lichev, Maksim Zhukovskii - - - Universal Maximum Likelihood (List) Decoding via Fast Vector-Matrix Multiplication - https://arxiv.org/abs/2510.21414 - arXiv:2510.21414v2 Announce Type: replace -Abstract: Maximum-likelihood (ML) decoding for arbitrary block codes remains fundamentally hard, with worst-case time complexity-measured by the total number of multiplications-being no better than straightforward exhaustive search, which requires $q^{k} n$ operations for an $[n,k]_q$ code. This paper introduces a simple, code-agnostic framework that reduces the worst-case complexity by a factor of $n$, down to $q^{k}$ operations, a highly desirable reduction in practice. The result holds for both linear and nonlinear block codes over general memoryless channels and under both hard-decision and soft-decision decoding. It naturally extends to intersymbol-interference (ISI) channels and ML list decoding with only a negligible increase in complexity. Our core insight is that, upon receipt of each sequence at the receiver, the conditional probability of that sequence for each codeword in the codebook (i.e., the \emph{likelihood}) can be expressed as the inner product of two carefully constructed vectors -- the first depending on the received sequence, and the second on that codeword itself. As a result, evaluating the likelihoods for all codewords in the codebook reduces to a single vector-matrix multiplication, and ML decoding (MLD) becomes the simple task of picking the maximum entry in the resulting vector. The only non-trivial cost lies in the vector-matrix product. However, our matrix construction allows the use of the Mailman algorithm to reduce this cost. This time reduction is achieved at the cost of high space complexity, requiring $\mathcal{O}(q^{k+1} n)$ space to store the pre-computed codebook matrix. - oai:arXiv.org:2510.21414v2 - cs.IT - cs.DS - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Hoang Ly, Emina Soljanin, Michael Schleppy - - - INTHOP: A Second-Order Globally Convergent Method for Nonconvex Optimization - https://arxiv.org/abs/2510.22342 - arXiv:2510.22342v3 Announce Type: replace -Abstract: Second-order Newton-type algorithms that leverage the exact Hessian or its approximation are central to solve nonlinear optimization problems. However, their applications in solving large-scale nonconvex problems are hindered by three primary challenges: (1) the high computational cost associated with Hessian evaluations, (2) its inversion, and (3) ensuring descent direction at points where the Hessian becomes indefinite. We propose INTHOP, an interval Hessian-based optimization algorithm for nonconvex problems to deal with these primary challenges. The proposed search direction is based on approximating the original Hessian matrix by a positive definite matrix. The novelty of the proposed method is that the proposed search direction is guaranteed to be descent and requires approximation of Hessian and its inversion only at specific iterations. We prove that the difference between the calculated approximate and exact Hessian is bounded within an interval. Accordingly, the approximate Hessian matrix is reused if the iterates are in that chosen interval while computing the gradients at each iteration. We develop various algorithm variants based on the interval size updating methods and minimum eigenvalue computation methods. We also prove the global convergence of the proposed algorithm. Further, we apply the algorithm to an extensive set of test problems and compare its performance with the existing methods such as steepest descent, quasi-Newton, and Newton method. We show empirically that the proposed method solves more problems in fewer function and gradient evaluations than steepest descent and the quasi-Newton method. While in the comparison to the Newton method, we illustrate that for nonconvex optimization problems, we require substantially less $O(n^3)$ operations. - oai:arXiv.org:2510.22342v3 - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Krishan Kumar, Ashutosh Sharma, Gauransh Dingwani, Nikhil Gupta, Vaishnavi Gupta, Ishan Bajaj - - - A class of forward-backward regularizations of the Perona-Malik equation with variable exponent - https://arxiv.org/abs/2510.23982 - arXiv:2510.23982v3 Announce Type: replace -Abstract: This paper investigates a novel class of regularizations of the Perona-Malik equation with variable exponents, of forward-backward parabolic type, which possess a variational structure and have potential applications in image processing. The existence of Young measure solutions to the Neumann initial-boundary value problem for the proposed equation is established via Sobolev approximation and the vanishing viscosity limit. The proofs rely on Rothe's method, variational principles, and Young measure theory. The theoretical results confirm numerical observations concerning the generic behavior of solutions with suitably chosen variable exponents. - oai:arXiv.org:2510.23982v3 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yihui Tong, Wenjie Liu, Zhichang Guo, Wenjuan Yao - - - Monotone Sobolev extensions in metric surfaces and applications to uniformization - https://arxiv.org/abs/2510.26458 - arXiv:2510.26458v2 Announce Type: replace -Abstract: We prove a monotone Sobolev extension theorem for maps to Jordan domains with rectifiable boundary in metric surfaces of locally finite Hausdorff 2-measure. This is then used to prove a uniformization result for compact metric surfaces by minimizing energy in the class of monotone Sobolev maps. - oai:arXiv.org:2510.26458v2 - math.MG - math.CV - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Damaris Meier, Noa Vikman, Stefan Wenger - - - The free bifibration on a functor - https://arxiv.org/abs/2511.07314 - arXiv:2511.07314v3 Announce Type: replace -Abstract: We consider the problem of constructing the free bifibration generated by a functor of categories $p : D \to C$. This problem was previously considered by Lamarche, and is closely related to the problem, considered by Dawson, Par\'e, and Pronk, of ``freely adjoining adjoints'' to a category. We develop a proof-theoretic approach to the problem, beginning with a construction of the free bifibration $\Lambda_p : Bifib(p)\to C$ in which objects of $Bifib(p)$ are formulas of a primitive ``bifibrational logic'', and arrows are derivations in a cut-free sequent calculus modulo a notion of permutation equivalence. We show that instantiating the construction to the identity functor generates a _zigzag double category_ $\mathbb{Z}(C)$, which is also the free double category with companions and conjoints (or fibrant double category) on $C$. The approach adapts smoothly to the more general task of building $(P,N)$-fibrations, where one only asks for pushforwards along arrows in $P$ and pullbacks along arrows in $N$ for some subsets of arrows; this encompasses Kock and Joyal's notion of _ambifibration_ when $(P,N)$ form a factorization system. We establish a series of progressively stronger normal forms, guided by ideas of _focusing_ from proof theory, and obtain a canonicity result under assumption that the base category is factorization preordered relative to $P$ and $N$. This canonicity result allows us to decide the word problem and to enumerate relative homsets without duplicates. Finally, we describe several examples of a combinatorial nature, including a category of plane trees generated as a free bifibration over $\omega$, and a category of increasing forests generated as a free ambifibration over $\Delta$, which contains the lattices of noncrossing partitions as quotients of its fibers by the Beck-Chevalley condition for bicartesian squares. - oai:arXiv.org:2511.07314v3 - math.CT - cs.LO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Bryce Clarke, Gabriel Scherer, Noam Zeilberger - - - A round of Pintz to celebrate oscillations in sums - https://arxiv.org/abs/2511.09978 - arXiv:2511.09978v2 Announce Type: replace -Abstract: We explore a method, going back to Landau and developed by Pintz, for connecting sums of arithmetic functions with zero-free regions for $L$-functions. In particular, we make explicit a general result of Pintz of this form; showing how one can use arithmetical information to deduce information about zeroes of $L$-functions, rather than the other way around. As a prototype, we work through an example with the Riemann zeta-function and sums of the M\"obius function, but we also outline the utility of this method in general. - oai:arXiv.org:2511.09978v2 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Daniel R. Johnston, Tim Trudgian - - - L^1 data fitting for Inverse Problems yields optimal rates of convergence in case of discretized white Gaussian noise - https://arxiv.org/abs/2511.11321 - arXiv:2511.11321v2 Announce Type: replace -Abstract: It is well-known in practice, that L^1 data fitting leads to improved robustness compared to standard L^2 data fitting. However, it is unclear whether resulting algorithms will perform as well in case of regular data without outliers. In this paper, we therefore analyze generalized Tikhonov regularization with L^1 data fidelity for Inverse Problems F(u) = g in a general setting, including general measurement errors and errors in the forward operator. The derived results are then applied to the situation of discretized Gaussian white noise, and we show that the resulting error bounds allow for order-optimal rates of convergence. These findings are also investigated in numerical simulations. - oai:arXiv.org:2511.11321v2 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Kristina B\"atz, Frank Werner - - - Complexity and curvature of pairs of Burch modules and ideals - https://arxiv.org/abs/2511.13258 - arXiv:2511.13258v2 Announce Type: replace -Abstract: The complexity and curvature of a module were first introduced by Avramov to distinguish modules of infinite homological dimension. Later, Avramov-Buchweitz extended the notion of complexity from a single module to that of pairs of modules, which measures the polynomial growth rate of the minimal number of generators of their Ext modules. Dao studied a similar notion of Tor-complexity. Recently, Dey-Ghosh-Saha initiated the study of Ext and Tor curvature of a pair of modules, which measure the exponential growth rates of the corresponding Ext and Tor, respectively. On the other hand, the concept of Burch ideals was introduced by Dao-Kobayashi-Takahashi, motivated by the classical work of Burch, and subsequently extended to modules by Dey-Kobayashi. This class includes several large and well-studied families of modules and ideals over a Noetherian local ring $(R,\mathfrak{m},k)$. For example, these include the residue field $k$ as an $R$-module, every nonzero module of the form $\mathfrak{m} M$ (e.g., $\mathfrak{m}^n$ for $n\ge 1$), and under mild conditions every integrally closed ideal $I$ with $\rm{depth}(R/I)=0$. Suppose $I$ and $J$ are Burch ideals such that $I$ is $\mathfrak{m}$-primary. Motivated by Avramov's result that Burch modules exhibit extremal complexity and curvature, we establish in this article that $\rm{cx}_R(I,J)=\rm{tcx}_R(I,J)=\rm{cx}_R(k)$. Moreover, we show that $R$ is complete intersection if and only if $\rm{cx}_R(I,J)$ or $\rm{tcx}_R(I,J)$ is finite if and only if $\rm{curv}_R(I,J)$ or $\mathrm{tcurv}_R(I,J)$ is at most $1$. We deduce these results from the corresponding more general results on Burch modules. - oai:arXiv.org:2511.13258v2 - math.AC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Souvik Dey, Dipankar Ghosh, Mouma Samanta - - - Segregated Solutions to Critical Elliptic Systems in High Dimensions ($N \geq 5$) - https://arxiv.org/abs/2511.14115 - arXiv:2511.14115v2 Announce Type: replace -Abstract: We study the existence of multiple segregated solutions to the critical coupled Schr\"odinger system \[ \begin{cases} -\Delta u_{1} = K_1(| y|) | u_{1}|^{2^*-2}u_{1}+\beta | u_{2}|^{\frac{2^{*}}{2}}| u_{1}|^{\frac{2^{*}}{2}-2}u_{1}, & y\in \mathbb R^N,\\ -\Delta u_{2} = K_2(| y|) | u_{2}|^{2^*-2}u_{2}+\beta | u_{1}|^{\frac{2^{*}}{2}}| u_{2}|^{\frac{2^{*}}{2}-2}u_{2}, & y\in\mathbb R^N,\\ u_{1},u_{2}\geq0, u_{1},u_{2}\in C_0(\mathbb R^{N})\cap D^{1,2}(\mathbb R^N), \end{cases} \] with $N \geq 5$, $2^* = \frac{2N}{N-2}$, radial potentials $K_1, K_2 > 0$,and repulsive coupling $\beta < 0$.Under the assumption that $K_1$ and $K_2$ attain local maxima at distinct radii $r_0 \ne \rho_0$ with precise asymptotic expansions near these points, we prove the existence of infinitely many non-radial segregated solutions $(u_{1,k}, u_{2,k})$ for all sufficiently large integers $k$. These solutions exhibit multiple bumps concentrating on two separate circles of radius $r_0$ and $\rho_0$ respectively. Moreover, each component develops a "dead core'' near the concentration points of the other. The proof overcomes the sublinear and non-smooth nature of the coupling term ($2^*/2 -1 < 1$) by constructing a tailored complete metric space and combining a finite-dimensional reduction with a novel tail minimization argument. - oai:arXiv.org:2511.14115v2 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zijuan Gao, Qing Guo, Chengxiang Zhang - - - Genus two embedded minimal surfaces in $\mathbb{S}^3$ with bidihedral symmetry - https://arxiv.org/abs/2511.16295 - arXiv:2511.16295v2 Announce Type: replace -Abstract: The isometry group of the classical Lawson embedded minimal surface $\xi_{2,1}\subset \mathbb{S}^3$ of genus 2 is isomorphic to the product $S_3\times D_4$ of the permutation group of three elements and the dihedral group of order 8 (symmetries of a square). $S_3\times D_4$ has a subgroup of index 3 isomorphic to the bidihedral group $D_{4h}=\mathbb{Z}_2\times D_4$, where $D_4$ is the dihedral group of order 8. We prove that $\xi_{2,1}$ is the unique closed embedded minimal surface of genus 2 in $\mathbb{S}^3$ whose isometry group contains $D_{4h}$. - oai:arXiv.org:2511.16295v2 - math.DG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Jos\'e M. Espinar, Joaqu\'in P\'erez - - - Exceptions to the Erd\H os--Straus--Schinzel conjecture - https://arxiv.org/abs/2511.16817 - arXiv:2511.16817v2 Announce Type: replace -Abstract: A famous conjecture of Erd\H os and Straus is that for every integer $n\ge2$, $4/n$ can be represented as $1/x+1/y+1/z$, where $x,y,z$ are positive integers. This conjecture was generalized to $5/n$ by Sierpi\'nski, and then Schinzel conjectured that for every integer $m\ge4$ there is a bound $n_m$ such that the fraction $m/n$ is the sum of 3 unit fractions for all integers $n\ge n_m$. Leveraging and generalizing work of Elsholtz and Tao, we show that if $n_m$ exists it must be at least $\exp(m^{1/3+o(1)})$; that is, there are numbers $n$ this large for which $m/n$ is not the sum of 3 unit fractions. We prove a weaker, but numerically explicit version of this theorem, showing that for $m\ge 6.52\times10^9$ there is a prime $p\in(m^2,2m^2)$ with $m/p$ not the sum of 3 unit fractions, and report on some extensive numerical calculations that support this assertion with the much smaller bound $m\ge20$. A result of Vaughan is that for each $m$, most $n$'s have $m/n$ representable; we make the dependence on $m$ in this result explicit. In addition, we prove a result generalizing the problem to the sum of $j$ unit fractions. - oai:arXiv.org:2511.16817v2 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Carl Pomerance, Andreas Weingartner - - - Higher-dimensional Teter rings via the canonical trace - https://arxiv.org/abs/2512.06761 - arXiv:2512.06761v2 Announce Type: replace -Abstract: We study Puthenpurakal's higher-dimensional Teter rings via the canonical trace ideal. We give a sufficient criterion for Teterness and show that, in the standard graded case, it is also necessary, yielding a characterization. Consequently, several nearly Gorenstein families are Teter; moreover, under certain hypotheses, the Cohen--Macaulay type of nearly Gorenstein rings is bounded by the codimension. We also analyze Teterness for fiber products, Veronese subrings, and numerical semigroup rings. - oai:arXiv.org:2512.06761v2 - math.AC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sora Miyashita, Taiga Ozaki - - - Higher walks and squares - https://arxiv.org/abs/2512.08633 - arXiv:2512.08633v2 Announce Type: replace -Abstract: We continue the development of the theory of higher dimensional walks on ordinals began recently by Bergfalk. In particular we identify natural coherence conditions on higher dimensional $C$-sequences that entail coherence of the resultant higher rho-functions. We also introduce various higher square principles by adding non-triviality conditions to these coherent higher $C$-sequences and investigate basic properties of said square principles. For example, in analogy with the classical case, we prove that these higher square principles abound in the constructible universe but can be forced to fail, modulo large cardinals. Finally, we prove that certain higher rho-functions obtained by walking along higher square sequences exhibit non-triviality in addition to coherence. In particular, it follows that higher square principles on a cardinal $\lambda$ entail certain non-vanishing \v{C}ech cohomology groups for $\lambda$ considered with the order topology. - oai:arXiv.org:2512.08633v2 - math.LO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Chris Lambie-Hanson, Pedro Marun - - - Skew-symmetrizable cluster algebras from surfaces and symmetric quivers - https://arxiv.org/abs/2512.12247 - arXiv:2512.12247v2 Announce Type: replace -Abstract: We study skew-symmetrizable cluster algebras $\mathcal{A}$ associated with unpunctured surfaces $\tilde{\mathbf{S}}$ endowed with an orientation-preserving involution $\sigma$. We give a geometric realization of such cluster algebras by showing that cluster variables of $\mathcal{A}$ correspond to $\sigma$-orbits of arcs of $\tilde{\mathbf{S}}$, while clusters are given by admissible $\sigma$-invariant triangulations. We establish a ring homomorphism from $\mathcal{A}$ to a skew-symmetric cluster algebra of the same rank, which is combinatorially derived from $\mathcal{A}$. We use this result to provide a cluster expansion formula for any $\sigma$-orbit $[\gamma]$ in terms of perfect matchings of some labeled modified snake graphs constructed from the arcs of $[\gamma]$. Then, we associate a symmetric finite-dimensional algebra $A$ to any seed of $\mathcal{A}$, such that non-initial cluster variables bijectively correspond to orthogonal indecomposable $A$-modules. Finally, we exhibit a purely representation-theoretic map from the category of orthogonal $A$-modules to $\mathcal{A}$, providing a Caldero-Chapoton map in this setting. - oai:arXiv.org:2512.12247v2 - math.RT - math.CO - math.RA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Azzurra Ciliberti - - - Spectral properties of Toeplitz operators with harmonic function symbols on the Bergman space - https://arxiv.org/abs/2512.16952 - arXiv:2512.16952v2 Announce Type: replace -Abstract: This paper investigates the spectral properties of Toeplitz operators on the Bergman space of unit disk. We present an integral representation of $ T^*_{z^m}$, which establishes a connection between the Bergman functions and the solutions of PDE theory. In fact, by leveraging the Poincar\'e theorem in difference equations and the solution forms of differential equations, this paper describes the kernels of certain Toeplitz operators with harmonic polynomial symbols, and further gives the sufficient conditions for the connectedness of the spectra of these Toeplitz operators. The spectral properties of $ T_\varphi$ with $\varphi (z) =\overline{z}^{m} + \alpha z^m + \beta$ are characterized, such as $\sigma(T_\varphi)= \overline{\varphi (\mathbb {D})}$, Fredholm index of $T_\varphi$ can only be one of $m,-m$ and $0$, $T_\varphi$ satisfies Coburn's theorem. These findings offer an illuminating example for the essential projective spectra of non-commuting operators. - oai:arXiv.org:2512.16952v2 - math.FA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Puyu Cui, Yufeng Lu, Rongwei Yang, Chao Zu - - - A bound on the equivariant unknotting number - https://arxiv.org/abs/2512.17700 - arXiv:2512.17700v2 Announce Type: replace -Abstract: We study how the equivariant signature of strongly invertible knots changes when one of the Boyle-Chen equivariant unknotting moves is applied. It follows form our results that the absolute value of the equivariant signature introduced by Alfieri-Boyle gives a lower bound to three times the equivariant unknotting number. - oai:arXiv.org:2512.17700v2 - math.GT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sarah Zampa - - - Milstein-type Schemes for Hyperbolic SPDEs - https://arxiv.org/abs/2512.19647 - arXiv:2512.19647v2 Announce Type: replace -Abstract: This article studies the temporal approximation of hyperbolic semilinear stochastic evolution equations with multiplicative Gaussian noise by Milstein-type schemes. We take the term hyperbolic to mean that the leading operator generates a contractive, not necessarily analytic $C_0$-semigroup. Optimal convergence rates are derived for the pathwise uniform strong error \[ - E_h^\infty := \Big(\mathbb{E}\Big[\max_{1\le j \le M}\|U_{t_j}-u_j\|_X^p\Big]\Big)^{1/p} \] on a Hilbert space $X$ for $p\in [2,\infty)$. Here, $U$ is the mild solution and $u_j$ its Milstein approximation at time $t_j=jh$ with step size $h>0$ and final time $T=Mh>0$. For sufficiently regular nonlinearity and noise, we establish strong convergence of order one, with the error satisfying $E_h^\infty\lesssim h\sqrt{\log(T/h)}$ for rational Milstein schemes and $E_h^\infty \lesssim h$ for exponential Milstein schemes. This extends previous results from parabolic to hyperbolic SPDEs and from exponential to rational Milstein schemes. Moreover, root-mean-square error estimates are strengthened to pathwise uniform estimates. Numerical experiments validate the convergence rates for the stochastic Schr\"odinger equation. Further applications to Maxwell's and transport equations are included. - oai:arXiv.org:2512.19647v2 - math.NA - cs.NA - math.AP - math.FA - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Felix Kastner, Katharina Klioba - - - Relative center construction for $G$-graded C$^*$-tensor categories and Longo-Rehren inclusions - https://arxiv.org/abs/2512.21485 - arXiv:2512.21485v2 Announce Type: replace -Abstract: Gelaki-Naidu-Nikshych and Turaev-Virelizier showed the existence of $G$-braiding on the - relative Drinfeld center of a $G$-graded tensor category. We will explain this concept - from the viewpoint of Longo-Rehren inclusions. - oai:arXiv.org:2512.21485v2 - math.OA - math.CT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Toshihiko Masuda - - - Asymmetry in Spectral Graph Theory: Harmonic Analysis on Directed Networks via Biorthogonal Bases (Random-Walk Laplacian Formulation) - https://arxiv.org/abs/2512.21770 - arXiv:2512.21770v2 Announce Type: replace -Abstract: The operator-theoretic dichotomy underlying diffusion on directed networks is \emph{symmetry versus non-self-adjointness} of the Markov transition operator. In the reversible (detailed-balance) regime, a directed random walk $P$ is self-adjoint in a stationary $\pi$-weighted inner product and admits orthogonal spectral coordinates; outside reversibility, $P$ is genuinely non-self-adjoint (often non-normal), and stability is governed by biorthogonal geometry and eigenvector conditioning. In this paper we develop a harmonic-analysis framework for directed graphs anchored on the random-walk transition matrix $P=D_{\mathrm{out}}^{-1}A$ and the random-walk Laplacian $L_{\mathrm{rw}}=I-P$. Using biorthogonal left/right eigenvectors we define a \emph{Biorthogonal Graph Fourier Transform} (BGFT) adapted to directed diffusion, propose a diffusion-consistent frequency ordering based on decay rates $\Re(1-\lambda)$, and derive operator-norm stability bounds for iterated diffusion and for BGFT spectral filters. We prove sampling and reconstruction theorems for $P$-bandlimited (equivalently $L_{\mathrm{rw}}$-bandlimited) signals and quantify noise amplification through the conditioning of the biorthogonal eigenbasis. A simulation protocol on directed cycles and perturbed non-normal digraphs demonstrates that asymmetry alone does not dictate instability; rather, non-normality and eigenvector ill-conditioning drive reconstruction sensitivity, making BGFT a natural analytical language for directed diffusion processes. - oai:arXiv.org:2512.21770v2 - math.RA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chandrasekhar Gokavarapu (Lecturer in Mathematics, Government College) - - - On the number of words of $N=3 \,n$ letters with a three-letter alphabet - https://arxiv.org/abs/2512.22362 - arXiv:2512.22362v2 Announce Type: replace -Abstract: In this paper we address the well-known problem of counting the number of $3n$-letter words that can be formed from a three-letter alphabet by decomposing it into four possible cases based on its remainder when divided by three. The solution to the problem also gives us some sums of trinomial coefficients. - oai:arXiv.org:2512.22362v2 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Pablo Serra - - - A Proximal-Gradient Method for Solving Regularized Optimization Problems with General Constraints - https://arxiv.org/abs/2512.23166 - arXiv:2512.23166v2 Announce Type: replace -Abstract: We propose, analyze, and test a proximal-gradient method for solving regularized optimization problems with general constraints. The method employs a decomposition strategy to compute trial steps and uses a merit function to determine step acceptance or rejection. Under various assumptions, we establish a worst-case iteration complexity result, prove that limit points are first-order KKT points, and show that manifold identification and active-set identification properties hold. Preliminary numerical experiments on a subset of the CUTEst test problems and sparse canonical correlation analysis problems demonstrate the promising performance of our approach. - oai:arXiv.org:2512.23166v2 - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Frank E. Curtis, Xiaoyi Qu, Daniel P. Robinson - - - Chamber zeta function and closed galleries in the standard non-uniform complex from $\operatorname{PGL}_3$ - https://arxiv.org/abs/2512.23276 - arXiv:2512.23276v2 Announce Type: replace -Abstract: We introduce the \emph{chamber zeta function} for a complex of groups, defined via an Euler product over primitive tailless chamber galleries, extending the Ihara--Bass framework from weighted graphs to higher-rank settings. Let $\mathcal{B}$ be the Bruhat--Tits building of $\mathrm{PGL}_{3}(F)$ for a non-archimedean local field $F$ with residue field $\mathbb{F}_{q}$. For the standard arithmetic quotient $\Gamma\backslash\mathcal{B}$ with $\Gamma=\mathrm{PGL}_{3}(\mathbb{F}_{q}[t])$, we prove an Ihara--Bass type \emph{determinant formula} expressing the chamber zeta function as the reciprocal of a characteristic polynomial of a naturally defined chamber transfer operator. In particular, the chamber zeta function is \emph{rational} in its complex parameter. As an application of the determinant formula, we obtain explicit counting results for closed gallery classes arising from tailless galleries in $\mathcal{B}$, including exact identities and spectral asymptotics governed by the chamber operator. - oai:arXiv.org:2512.23276v2 - math.NT - math.CO - math.DS - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Soonki Hong, Sanghoon Kwon - - - Explicit bounds for the graphicality of the prime gap sequence - https://arxiv.org/abs/2512.24230 - arXiv:2512.24230v2 Announce Type: replace -Abstract: We establish explicit unconditional results on the graphic properties of the prime gap sequence. Let $p_n$ denote the $n$-th prime number (with $p_0=1$) and $\mathrm{PD}_n = (p_\ell - p_{\ell-1})_{\ell=1}^n$ be the sequence of the first $n$ prime gaps. Building upon the recent work by Erd\H{o}s \emph{et al}, which proved the graphic nature of $\mathrm{PD}_n$ for large $n$ unconditionally, and for all $n$ under RH, we provide the first explicit unconditional threshold such that: (1) For all $n \geq \exp\exp(30.5)$, $\mathrm{PD}_n$ is graphic. (2) For all $n \geq \exp\exp(34.5)$, every realization $G_n$ of $\mathrm{PD}_n$ satisfies that $(G_n, p_{n+1}-p_n)$ is DPG-graphic. - Our proofs utilize a more refined criterion for when a sequence is graphic, and better estimates for the first moment of large prime gaps proven through an explicit zero-free region and explicit zero-density estimate for the Riemann zeta function. - oai:arXiv.org:2512.24230v2 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Keshav Aggarwal, Robin Frot, Haozhe Gou, Hui Wang - - - Poincar\'e duality for singular tropical hypersurfaces - https://arxiv.org/abs/2512.24548 - arXiv:2512.24548v2 Announce Type: replace -Abstract: We establish a partial extension of the Poincar\'e duality theorem of Jell-Rau-Shaw to tropical hypersurfaces arising from non-primitive triangulations. We introduce a notion of level of primitivity for triangulations of lattice polytopes and show that tropical hypersurfaces satisfy a partial form of Poincar\'e duality determined by this level. This notion of primitivity is defined modulo a fixed integral domain and is weaker than the classical notion of primitivity. Moreover, we obtain a generalization of complete Poincar\'e duality over this integral domain for tropical hypersurfaces whose underlying triangulations are primitive modulo the integral domain. As a corollary, we show that any tropical hypersurface obtained by patchworking from a triangualtion of a simple lattice polytope satisfies complete Poincar\'e duality over the field of rational numbers, providing a converse to a theorem of Aksnes. Throughout, we allow triangulations that are not necessarily convex. - oai:arXiv.org:2512.24548v2 - math.AG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Samuel Dentan - - - Rational Angle Bisection Problem in Higher Dimensional Spaces and Incenters of Simplices over Fields - https://arxiv.org/abs/2512.24660 - arXiv:2512.24660v3 Announce Type: replace -Abstract: In this article, we generalize the following problem, which is called the rational angle bisection problem, to the $n$-dimensional space $k^n$ over a subfield $k$ of $\mathbb R$: in the coordinate plane, for which rational numbers $a$ and $b$ are the slopes of the angle bisectors between two lines with slopes $a$ and $b$ rational? First, we give a few characterizations of when the angle bisectors between two lines with direction vectors in $k^n$ have direction vectors in $k^n.$ To find solutions to the problem in the case when $k = \mathbb Q,$ we also give a formula for the integral solutions of $x_1{}^2+\dots +x_n{}^2 = dx_{n+1}{}^2,$ which is a generalization of the negative Pell's equation $x^2-dy^2 = -1,$ where $d$ is a square-free positive integer. Second, by applying the above characterizations, we give a necessary and sufficient condition for the incenter of a given $n$-simplex with $k$-rational vertices to be $k$-rational. In the coordinate plane, we prove that every triangle with $k$-rational vertices and incenter can be obtained by scaling a triangle with $k$-rational side lengths and area, which is a generalization of a Heronian triangle. We also state certain fundamental properties of a few centers of a given triangle with $k$-rational vertices. - oai:arXiv.org:2512.24660v3 - math.NT - math.MG - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Takashi Hirotsu - - - Polygons in Polygons with a Twist - https://arxiv.org/abs/2601.00899 - arXiv:2601.00899v3 Announce Type: replace -Abstract: This is a study of the construction of particular regular sub-n-gons T in regular n-gons P using a special system of chords of P. In particular, some of these sub-n-gons have areas which are integer divisors of the area of the given n-gon P. Initially, the study will concentrate on chords which are from a vertex to special points of one of the opposite sides of P. Several examples are explored. However, it will become apparent that a much more general situation exists. A Dynamic Geometry software, such as Sketchpad, or GeoGebra, is the key to investigating this new relationship. - oai:arXiv.org:2601.00899v3 - math.HO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - James M Parks - - - Physics-Constrained Learning of Energy-Preserving Stencils for Maxwell's Equations - https://arxiv.org/abs/2601.01902 - arXiv:2601.01902v3 Announce Type: replace -Abstract: We study data-driven construction of spatial discretizations for the one-dimensional Maxwell system. Using high-fidelity training data from a spectral discretization, we learn a \emph{linear convolution stencil} that approximates the spatial derivative operator in Maxwell's equations. We formulate a convex quadratic program for the stencil coefficients with linear constraints that enforce skew-adjointness of the discrete derivative; these constraints guarantee a semi-discrete electromagnetic energy identity and yield a CFL condition expressed directly in terms of the stencil's Fourier symbol. We compare several convex solvers for the resulting quadratic program -- projected gradient, Nesterov-accelerated gradient, ADMM, and an interior-point reference implemented in CVXPY -- and evaluate the learned operators in time-dependent Maxwell simulations using a Crank--Nicolson (CN) discretization. Numerical experiments, including cases with nonstandard target operators and noisy training data, show that (i) energy-constrained learned stencils achieve accuracy comparable to standard central differences while exactly preserving the discrete electromagnetic energy under CN time-stepping, and (ii) ADMM and interior-point solvers produce nearly identical operators, with ADMM offering a favorable tradeoff between accuracy, constraint satisfaction, and runtime. - oai:arXiv.org:2601.01902v3 - math.NA - cs.NA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Victory Obieke - - - On the forward self-similar solutions to the two-dimensional Navier-Stokes equations - https://arxiv.org/abs/2601.03833 - arXiv:2601.03833v2 Announce Type: replace -Abstract: We establish the global existence of forward self-similar solutions to the two-dimensional incompressible Navier-Stokes equations for any divergence-free initial velocity that is homogeneous of degree $-1$ and locally H\"older continuous. This result requires no smallness assumption on the initial data. In sharp contrast to the three-dimensional case, where $(-1)$-homogeneous vector fields are locally square-integrable, the major difficulty for the 2D problem is the criticality in the sense that the initial kinetic energy is locally infinite at the origin, and the initial vorticity fails to be locally integrable, so that the classical local energy estimates are not available. Our key ideas are to decompose the solution into a linear part solving the heat equation and a finite-energy perturbation part, and to exploit a kind of inherent cancellation relation between the linear part and the perturbation part. These, together with suitable choices of multipliers, enable us to control the interaction terms and to establish the $H^1$-estimates for the perturbation part. Furthermore, we can get an optimal pointwise estimate via investigating the corresponding Leray equations in weighted Sobolev spaces.This gives the faster decay of the perturbation part at infinity and compactness, which play important roles in proving the existence of global-in-time self-similar solutions. - oai:arXiv.org:2601.03833v2 - math.AP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Changfeng Gui, Hao Liu, Chunjing Xie - - - Constrained dynamics for searching saddle points on general Riemannian manifolds - https://arxiv.org/abs/2601.03931 - arXiv:2601.03931v2 Announce Type: replace -Abstract: Finding constrained saddle points on Riemannian manifolds is significant for analyzing energy landscapes arising in physics and chemistry. Existing works have been limited to special manifolds that admit global regular level-set representations, excluding applications such as electronic excited-state calculations. In this paper, we develop a constrained saddle dynamics applicable to smooth functions on general Riemannian manifolds. Our dynamics is formulated compactly over the Grassmann bundle of the tangent bundle. By analyzing the Grassmann bundle geometry, we achieve universality via incorporating the second fundamental form, which captures variations of tangent spaces along the trajectory. We rigorously establish the local linear stability of the dynamics and the local linear convergence of the resulting algorithms. Remarkably, our analysis provides the first convergence guarantees for discretized saddle-search algorithms in manifold settings. Moreover, by respecting the intrinsic quotient structure, we remove unnecessary nondegeneracy assumptions on the eigenvalues of the Riemannian Hessian that are present in existing works. We also point out that locating saddle points can be more ill-conditioning than finding local minimizers, and requires using nonredundant parametrizations. Finally, numerical experiments on linear eigenvalue problems and electronic excited-state calculations showcase the effectiveness of the proposed algorithms and corroborate the established local theory. - oai:arXiv.org:2601.03931v2 - math.NA - cs.NA - math.OC - physics.chem-ph - physics.comp-ph - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Yukuan Hu, Laura Grazioli - - - Ergodic Theorems and Equivalence of Green's Kernel for Random Walks in Random Environments - https://arxiv.org/abs/2601.04161 - arXiv:2601.04161v2 Announce Type: replace -Abstract: We study the Ergodic Properties of Random Walks in stationary ergodic environments without uniform ellipticity under a minimal assumption. There are two main components in our work. The first step is to adopt the arguments of Lawler to first prove a uniqueness principle. We use a more general definition of environments using~\textit{Environment Functions}. As a corollary, we can deduce an invariance principle under these assumptions for balanced environments under some assumptions. We also use the uniqueness principle to show that any balanced, elliptic random walk must have the same transience behaviour as the simple symmetric random walk. The second is to transfer the results we deduce in balanced environments to general ergodic environments(under some assumptions) using a control technique to derive a measure under which the \textit{local process} is stationary and ergodic. As a consequence of our results, we deduce the Law of Large Numbers for the Random Walk and an Invariance Principle under our assumptions. - oai:arXiv.org:2601.04161v2 - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ayan Ghosh - - - Linear identities for partition pairs with $5$-cores - https://arxiv.org/abs/2601.04743 - arXiv:2601.04743v2 Announce Type: replace -Abstract: We prove an infinite family of linear identities for the number $A_5(n)$ of partition pairs of $n$ with $5$-cores by using certain theta function identities involving the Ramanujan's parameter $k(q)$ due to Cooper, and Lee and Park. Consequently, we deduce an infinite family of congruences for $A_5(n)$ using these linear identities. - oai:arXiv.org:2601.04743v2 - math.NT - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Russelle Guadalupe - - - A Unified Spectral Framework for Aging, Heterogeneous, and Distributed Order Systems via Weighted Weyl-Sonine Operators - https://arxiv.org/abs/2601.05423 - arXiv:2601.05423v2 Announce Type: replace -Abstract: While General Fractional Calculus has successfully expanded the scope of memory operators beyond power-laws, standard formulations remain predominantly restricted to the half-line via Riemann-Liouville or Caputo definitions. This constraint artificially truncates the system's history, limiting the thermodynamic consistency required for modeling processes on unbounded domains. To overcome these barriers, we construct the \textbf{Weighted Weyl-Sonine Framework}, a generalized formalism that extends non-local theory to the entire real line without history truncation. - Unlike recent algebraic approaches based on conjugation for finite intervals, we develop a rigorous harmonic analysis framework. Our central contribution is the \textbf{Generalized Spectral Mapping Theorem}, which establishes the Weighted Fourier Transform as a unitary diagonalization map for these operators. This result allows us to rigorously classify and solve distinct physical regimes under a single algebraic structure. We explicitly derive exact solutions for \textit{diffusive relaxation} (governed by Complete Bernstein Functions), \textit{inertial wave propagation} (exhibiting oscillatory dynamics), and \textit{retarded aging} (via distributed order), proving that our framework unifies the description of anomalous transport and wave mechanics in complex, time-deformed media. - oai:arXiv.org:2601.05423v2 - math.FA - math-ph - math.AP - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gustavo Dorrego - - - Composition Ax-Kochen/Ershov principles and tame fields of mixed characteristic - https://arxiv.org/abs/2601.05790 - arXiv:2601.05790v2 Announce Type: replace -Abstract: We study in which settings we have a composition AKE principle, i.e. when the theory of the coarsening $(K,w)$ and the theory of the induced valuation $(Kw,\overline{v})$ determine the theory of the composition $(K,v)$. We show that this is the case when $(K,w)$ is tame of equal characteristic, and provide counterexamples in mixed characteristic. We further show that, for a tame field of mixed characteristic, the theory of the valued field cannot, in general, be determined solely by the theories of its underlying field, its residue field, and its value group. - oai:arXiv.org:2601.05790v2 - math.LO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Margarete Ketelsen, Philip Dittmann - - - Global Optimization for Combinatorial Geometry Problems Revisited in the Era of LLMs - https://arxiv.org/abs/2601.05943 - arXiv:2601.05943v2 Announce Type: replace -Abstract: Recent progress in LLM-driven algorithm discovery, exemplified by DeepMind's AlphaEvolve, has produced new best-known solutions for a range of hard geometric and combinatorial problems. This raises a natural question: to what extent can modern off-the-shelf global optimization solvers match such results when the problems are formulated directly as nonlinear optimization problems (NLPs)? - We revisit a subset of problems from the AlphaEvolve benchmark suite and evaluate straightforward NLP formulations with two state-of-the-art solvers, the commercial FICO Xpress and the open-source SCIP. Without any solver modifications, both solvers reproduce, and in several cases improve upon, the best solutions previously reported in the literature, including the recent LLM-driven discoveries. Our results not only highlight the maturity of generic NLP technology and its ability to tackle nonlinear mathematical problems that were out of reach for general-purpose solvers only a decade ago, but also position global NLP solvers as powerful tools that may be exploited within LLM-driven algorithm discovery. - oai:arXiv.org:2601.05943v2 - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Timo Berthold, Dominik Kamp, Gioni Mexi, Sebastian Pokutta, Imre P\'olik - - - Applications of an identity of Bat{\i}r - https://arxiv.org/abs/2601.06210 - arXiv:2601.06210v2 Announce Type: replace -Abstract: Based on an interesting identity of Bat{\i}r we derive new identities for double sums involving famous number sequences. We also prove some double sum identities for binomial transform pairs. - oai:arXiv.org:2601.06210v2 - math.CO - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kunle Adegoke, Robert Frontczak - - - Profinite genus of HNN-extensions with finite associated subgroups - https://arxiv.org/abs/2601.06934 - arXiv:2601.06934v2 Announce Type: replace -Abstract: We study the profinite genus of HNN-extensions whose associated subgroups are finite. We give precise formulas for the number of isomorphism classes of HNN(G,H,K,t,f) and of its profinite completion and compute the profinite genus of such an HNN-extension HNN(G,H,K,t,f). We also list various situations when HNN(G,H,K,t,f) is determined by its profinite completion. - oai:arXiv.org:2601.06934v2 - math.GR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - V. R. de Bessa, A. L. P. Porto, P. A. Zalesskii - - - The Greedy Algorithm for Dissociated Sets - https://arxiv.org/abs/2601.07068 - arXiv:2601.07068v2 Announce Type: replace -Abstract: A set $\mathcal S\subset \mathbb N$ is said to be a subset-sum-distinct or dissociated if all of its finite subsets have different sums. Alternately, an equivalent classification is if any equality of the form $$\sum_{s\in \mathcal S} \varepsilon_s \cdot s =0$$ where $\varepsilon_s \in \{-1,0,+1\}$ implies that all the $\varepsilon_s$'s are $0$. For a dissociated set $\mathcal S$, we prove that for $c_\ast = \frac 12 \log_2 \left(\frac \pi 2\right)$ and any $c_\ast-1<C<c_\ast$, we have $$\mathcal S(n) \,:=\, \mathcal S\cap [1,n] \,\le\, \log_2 n +\frac 12 \log_2\log_2 n + C$$ for all $n\in \mathcal N_C$ with asymptotic density $\mathbf d\left(\mathcal N_C\right)=2-2^{c_\ast-C}$. Further, we consider the greedy algorithm for generating these sets and prove that this algorithm always eventually doubles. Finally, we also consider some generalizations of dissociated sets and prove similar results about them. - oai:arXiv.org:2601.07068v2 - math.CO - math.NT - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Sayan Dutta - - - Rate-distortion Theory with Lower Semi-continuous Distortion: A Concentration-compactness Approach - https://arxiv.org/abs/2601.07246 - arXiv:2601.07246v2 Announce Type: replace -Abstract: In this paper, we study rate-distortion theory for general sources with an emphasis on the existence of optimal reconstruction distributions. Classical existence results rely on compactness assumptions with continuous distortion that are often violated in general settings. By introducing the concentration-compactness principle into the analysis of the rate-distortion functional, we establish the existence of optimal reconstructions under mild coercivity and lower semi-continuity conditions on the distortion function. Our results provide a unified and transparent existence theorem for rate-distortion problems with lower semi-continuous distortion. - oai:arXiv.org:2601.07246v2 - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jiayang Zou, Luyao Fan, Jiayang Gao, Jia Wang - - - Resolution of Erd\H{o}s Problem #728: a writeup of Aristotle's Lean proof - https://arxiv.org/abs/2601.07421 - arXiv:2601.07421v3 Announce Type: replace -Abstract: We provide a writeup of a resolution of Erd\H{o}s Problem #728; this is the first Erd\H{o}s problem (a problem proposed by Paul Erd\H{o}s which has been collected in the Erd\H{o}s Problems website) regarded as fully resolved autonomously by an AI system. The system in question is a combination of GPT-5.2 Pro by OpenAI and Aristotle by Harmonic, operated by Kevin Barreto. The final result of the system is a formal proof written in Lean, which we translate to informal mathematics in the present writeup for wider accessibility. - The proved result is as follows. We show a logarithmic-gap phenomenon regarding factorial divisibility: For any constants $0<C_1<C_2$ and $0 < \varepsilon < 1/2$ there exist infinitely many triples $(a,b,n)\in\mathbb N^3$ with $\varepsilon n \le a,b \le (1-\varepsilon)n$ such that \[ a!\,b!\mid n!\,(a+b-n)!\qquad\text{and}\qquad C_1\log n < a+b-n < C_2\log n. \] The argument reduces this to a binomial divisibility $\binom{m+k}{k}\mid\binom{2m}{m}$ and studies it prime-by-prime. By Kummer's theorem, $\nu_p\binom{2m}{m}$ translates into a carry count for doubling $m$ in base $p$. We then employ a counting argument to find, in each scale $[M,2M]$, an integer $m$ whose base-$p$ expansions simultaneously force many carries when doubling $m$, for every prime $p\le 2k$, while avoiding the rare event that one of $m+1,\dots,m+k$ is divisible by an unusually high power of $p$. These "carry-rich but spike-free" choices of $m$ force the needed $p$-adic inequalities and the divisibility. The overall strategy is similar to results regarding divisors of $\binom{2n}{n}$ studied earlier by Erd\H{o}s and by Pomerance. - oai:arXiv.org:2601.07421v3 - math.NT - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Nat Sothanaphan - - - A Non-Renormalization Theorem for Local Functionals in Ghost-Free Vector Field Theories Coupled to Dynamical Geometry - https://arxiv.org/abs/2601.08081 - arXiv:2601.08081v2 Announce Type: replace -Abstract: We establish a non-renormalization theorem for a class of ghost-free local functionals describing massive vector field theories coupled to dynamical geometry. Under the assumptions of locality, Lorentz invariance, and validity of the effective field theory expansion below a fixed cutoff, we show that quantum corrections do not generate local operators that renormalize the classical derivative self-interactions responsible for the constraint structure of the theory. The proof combines an operator-level analysis of the space of allowed local counterterms with a systematic decoupling-limit argument, which isolates the leading contributions to the effective action at each order in the derivative expansion. As a consequence, all radiatively induced local functionals necessarily involve additional derivatives per field and are suppressed by the intrinsic strong-coupling scales of the theory. In particular, the classical interactions defining ghost-free vector field theories are stable under renormalization, and any additional degrees of freedom arising from quantum corrections appear only above the effective field theory cutoff. This result extends known non-renormalization properties of flat-space vector theories to the case of dynamical geometry and provides a structural explanation for their perturbative stability to all loop orders. - oai:arXiv.org:2601.08081v2 - math-ph - gr-qc - hep-th - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Lavinia Heisenberg, Shayan Hemmatyar, Nadine Nussbaumer - - - Wasserstein-p Central Limit Theorem Rates: From Local Dependence to Markov Chains - https://arxiv.org/abs/2601.08184 - arXiv:2601.08184v2 Announce Type: replace -Abstract: Finite-time central limit theorem (CLT) rates play a central role in modern machine learning. In this paper, we study CLT rates for multivariate dependent data in Wasserstein-$p$ ($W_p$) distance, for general $p \geq 1$. We focus on two fundamental dependence structures that commonly arise in machine learning: locally dependent sequences and geometrically ergodic Markov chains. In both settings, we establish the first optimal $O(n^{-1/2})$ rate in $W_1$, as well as the first $W_p$ ($p\ge 2$) CLT rates under mild moment assumptions, substantially improving the best previously known bounds in these dependent-data regimes. As an application of our optimal $W_1$ rate for locally dependent sequences, we further obtain the first optimal $W_1$-CLT rate for multivariate $U$-statistics. - On the technical side, we derive a tractable auxiliary bound for $W_1$ Gaussian approximation errors that is well suited for studying dependent data. For Markov chains, we further prove that the regeneration time of the split chain associated with a geometrically ergodic chain has a geometric tail without assuming strong aperiodicity or other restrictive conditions. These tools may be of independent interests and enable our optimal $W_1$ rates and underpin our $W_p$ ($p\ge 2$) results. - oai:arXiv.org:2601.08184v2 - math.PR - cs.LG - stat.ML - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Yixuan Zhang, Qiaomin Xie - - - The signless Laplacian spectral Tur\'an problems for hypergraphs - https://arxiv.org/abs/2601.08595 - arXiv:2601.08595v2 Announce Type: replace -Abstract: Let $\mathcal{H}=(V, E)$ be an $r$-uniform hypergraph on $n$ vertices. The signless Laplacian spectral radius of $\mathcal{H}$ is defined as the maximum modulus of the eigenvalues of the tensor $\mathcal{Q}(\mathcal{H})=\mathcal{D}(\mathcal{H})+\mathcal{A}(\mathcal{H})$, where $\mathcal{D}(\mathcal{H})$ and $\mathcal{A}(\mathcal{H})$ are the degree diagonal tensor and the adjacency tensor of $\mathcal{H}$, respectively. In this paper, we establish a general theorem that extends the spectral Tur\'an result of Keevash, Lenz and Mubayi [SIAM J. Discrete Math., 28 (4) (2014)] - to the setting of signless Laplacian spectral Tur\'an problems. We prove that if a family $\mathcal{F}$ of $r$-uniform hypergraphs is degree-stable with respect to a family $\mathcal{H}_n$ of $r$-uniform hypergraphs and its extremal constructions satisfy certain natural assumptions, then the signless Laplacian spectral Tur\'an problem for $\mathcal{F}$ can be effectively reduced to the corresponding problem restricted to the family $\mathcal{H}_n$. As a concrete application, we completely determine the extremal hypergraph that maximizes the signless Laplacian spectral radius among all Fano plane-free $3$-uniform hypergraphs, showing that the unique extremal hypergraph is the balanced complete bipartite $3$-uniform hypergraph. - oai:arXiv.org:2601.08595v2 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yongchun Lu, Jiadong Wu, Liying Kang - - - The Spectral Geometry of Ternary Gamma Schemes:Sheaf-Theoretic Foundations and Laplacian Clustering - https://arxiv.org/abs/2601.09268 - arXiv:2601.09268v2 Announce Type: replace -Abstract: This article develops a self-contained affine $\Gamma$-scheme theory for a class of commutative ternary $\Gamma$-semirings. By establishing all geometric and spectral results internally, the work provides a unified framework for triadic symmetry and spectral analysis. The central thesis is that a triadic $\Gamma$-algebra canonically induces two primary structures: (i) an intrinsic triadic symmetry in the sense of a Nambu--Filippov-type fundamental identity on the structure sheaf, and (ii) a canonical Laplacian on the finite $\Gamma$-spectrum whose spectral decomposition detects the clopen (connected-component) decomposition of the underlying space. We define $\Gamma$-ideals and prime $\Gamma$-ideals, endow $\SpecG(T)$ with a $\Gamma$-Zariski topology, construct localizations and the structure sheaf on the basis of principal opens, and prove the affine anti-equivalence between commutative ternary $\Gamma$-semirings and affine $\Gamma$-schemes. Furthermore, we demonstrate that the triadic bracket on sections is invariant under $\Gamma$-automorphisms and compatible with localization. The main spectral theorem establishes the block-diagonalization of the Laplacian under topological decompositions and provides an algebraic-connectivity criterion. The theory is verified through explicit computations of finite $\Gamma$-spectra and their corresponding Laplacian spectra - oai:arXiv.org:2601.09268v2 - math.RA - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chandrasekhar Gokavarapu - - - Proof of a Conjecture on Young Tableaux with Walls - https://arxiv.org/abs/2601.09551 - arXiv:2601.09551v2 Announce Type: replace -Abstract: Banderier, Marchal, and Wallner considered Young tableaux with walls, which are similar to standard Young tableaux, except that local decreases are allowed at some walls. In this work, we prove a conjecture of Fuchs and Yu concerning the enumeration of two classes of three-row Young tableaux with walls. Combining with the work by Chang, Fuchs, Liu, Wallner, and Yu leads to the verification of a conjecture on tree-child networks proposed by Pons and Batle. This conjecture was regarded as a specific and challenging problem in the Phylogenetics community until it was finally resolved by the present work. - oai:arXiv.org:2601.09551v2 - math.CO - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Zhicong Lin, Feihu Liu, Jiahang Liu, Jing Liu, Guoce Xin - - - The Addition Theorem for the Algebraic Entropy of Torsion Nilpotent Groups - https://arxiv.org/abs/2601.09643 - arXiv:2601.09643v2 Announce Type: replace -Abstract: The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved by Dikranjan, Goldsmith, Salce and Zanardo. It was later extended by Shlossberg to torsion nilpotent groups of class 2. As our main result, we prove the Addition Theorem for endomorphisms of torsion nilpotent groups of arbitrary nilpotency class. As an application, we show that if $G$ is a torsion nilpotent group, then for every $\phi\in \mathrm{End}(G)$ either the entropy $h(\phi)$ is infinite or $h(\phi)=\log(\alpha)$ for some $\alpha\in\mathbb N$. We further obtain, for automorphisms of locally finite groups, the Addition Theorem with respect to all terms of the upper central series; in particular, the Addition Theorem holds for automorphisms of $\omega$-hypercentral groups. Finally, we establish a reduction principle: if $\mathfrak X$ is a variety of locally finite groups, then the Addition Theorem for endomorphisms holds in $\mathfrak X$ if and only if it holds for locally finite groups generated by bounded sets. - oai:arXiv.org:2601.09643v2 - math.GR - Fri, 16 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Menachem Shlossberg - - - A GMM approach to estimate the roughness of stochastic volatility - https://arxiv.org/abs/2010.04610 - arXiv:2010.04610v5 Announce Type: replace-cross -Abstract: We develop a GMM approach for estimation of log-normal stochastic volatility models driven by a fractional Brownian motion with unrestricted Hurst exponent. We show that a parameter estimator based on the integrated variance is consistent and, under stronger conditions, asymptotically normally distributed. We inspect the behavior of our procedure when integrated variance is replaced with a noisy measure of volatility calculated from discrete high-frequency data. The realized estimator contains sampling error, which skews the fractal coefficient toward "illusive roughness." We construct an analytical approach to control the impact of measurement error without introducing nuisance parameters. In a simulation study, we demonstrate convincing small sample properties of our approach based both on integrated and realized variance over the entire memory spectrum. We show the bias correction attenuates any systematic deviance in the parameter estimates. Our procedure is applied to empirical high-frequency data from numerous leading equity indexes. With our robust approach the Hurst index is estimated around 0.05, confirming roughness in stochastic volatility. - oai:arXiv.org:2010.04610v5 - q-fin.ST - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.jeconom.2022.06.009 - Anine E. Bolko, Kim Christensen, Mikko S. Pakkanen, Bezirgen Veliyev - - - Genetic Algorithm Based Combinatorial Optimization for the Optimal Design of Water Distribution Network of Gurudeniya Service Zone, Sri Lanka - https://arxiv.org/abs/2304.09720 - arXiv:2304.09720v4 Announce Type: replace-cross -Abstract: This paper brings an in detail Genetic Algorithm (GA) based combinatorial optimization method used for the optimal design of the water distribution network (WDN) of Gurudeniya Service Zone, Sri Lanka. Genetic Algorithm (GA) mimics the survival of the fittest principle of nature to develop a search process. Methodology employs fuzzy combinations of pipe diameters to check their suitability to be considered as the cost effective optimal design solutions. Furthermore, the hydraulic constraints were implicitly evaluated within the GA itself in its aim to reaching the global optimum solution. Upon analysis, the results of this approach delivered agreeable design outputs. In addition, the comparison made between the results obtained by a previous study inspired by the Honey Bee Mating Optimization (HBMO) Algorithm and results obtained by the GA based approach, proves competency of GA for the optimal design of water distribution network in Gurudeniya Service Zone, Sri Lanka. - oai:arXiv.org:2304.09720v4 - cs.NE - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - K. H. M. R. N. Senavirathna, C. K. Walgampaya - - - Directed univalence in simplicial homotopy type theory - https://arxiv.org/abs/2407.09146 - arXiv:2407.09146v2 Announce Type: replace-cross -Abstract: Simplicial type theory extends homotopy type theory with a directed path type which internalizes the notion of a homomorphism within a type. This concept has significant applications both within mathematics -- where it allows for synthetic (higher) category theory -- and programming languages -- where it leads to a directed version of the structure identity principle. In this work, we construct the first types in simplicial type theory with non-trivial homomorphisms. We extend simplicial type theory with modalities and new reasoning principles to obtain triangulated type theory in order to construct the universe of discrete types $\mathcal{S}$. We prove that homomorphisms in this type correspond to ordinary functions of types i.e., that $\mathcal{S}$ is directed univalent. The construction of $\mathcal{S}$ is foundational for both of the aforementioned applications of simplicial type theory. We are able to define several crucial examples of categories and to recover important results from category theory. Using $\mathcal{S}$, we are also able to define various types whose usage is guaranteed to be functorial. These provide the first complete examples of the proposed directed structure identity principle. - oai:arXiv.org:2407.09146v2 - cs.LO - math.AT - math.CT - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz - - - Mathematical theory of deep learning - https://arxiv.org/abs/2407.18384 - arXiv:2407.18384v4 Announce Type: replace-cross -Abstract: This book provides an introduction to the mathematical analysis of deep learning. It covers fundamental results in approximation theory, optimization theory, and statistical learning theory, which are the three main pillars of deep neural network theory. Serving as a guide for students and researchers in mathematics and related fields, the book aims to equip readers with foundational knowledge on the topic. It prioritizes simplicity over generality, and presents rigorous yet accessible results to help build an understanding of the essential mathematical concepts underpinning deep learning. - oai:arXiv.org:2407.18384v4 - cs.LG - math.HO - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Philipp Petersen, Jakob Zech - - - A nonparametric test for diurnal variation in spot correlation processes - https://arxiv.org/abs/2408.02757 - arXiv:2408.02757v2 Announce Type: replace-cross -Abstract: The association between log-price increments of exchange-traded equities, as measured by their spot correlation estimated from high-frequency data, exhibits a pronounced upward-sloping and almost piecewise linear relationship at the intraday horizon. There is notably lower-on average less positive-correlation in the morning than in the afternoon. We develop a nonparametric testing procedure to detect such variation in a correlation process. The test statistic has a known distribution under the null hypothesis, whereas it diverges under the alternative. We run a Monte Carlo simulation to discover the finite sample properties of the test statistic, which are close to the large sample predictions, even for small sample sizes and realistic levels of diurnal variation. In an application, we implement the test on a high-frequency dataset covering the stock market over an extended period. The test leads to rejection of the null most of the time. This suggests diurnal variation in the correlation process is a nontrivial effect in practice. We show how conditioning information about macroeconomic news and corporate earnings announcements affect the intraday correlation curve. - oai:arXiv.org:2408.02757v2 - econ.EM - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kim Christensen, Ulrich Hounyo, Zhi Liu - - - An unbounded intensity model for point processes - https://arxiv.org/abs/2408.06519 - arXiv:2408.06519v2 Announce Type: replace-cross -Abstract: We develop a model for point processes on the real line, where the intensity can be locally unbounded without inducing an explosion. In contrast to an orderly point process, for which the probability of observing more than one event over a short time interval is negligible, the bursting intensity causes an extreme clustering of events around the singularity. We propose a nonparametric approach to detect such bursts in the intensity. It relies on a heavy traffic condition, which admits inference for point processes over a finite time interval. With Monte Carlo evidence, we show that our testing procedure exhibits size control under the null, whereas it has high rejection rates under the alternative. We implement our approach on high-frequency data for the EUR/USD spot exchange rate, where the test statistic captures abnormal surges in trading activity. We detect a nontrivial amount of intensity bursts in these data and describe their basic properties. Trading activity during an intensity burst is positively related to volatility, illiquidity, and the probability of observing a drift burst. The latter effect is reinforced if the order flow is imbalanced or the price elasticity of the limit order book is large. - oai:arXiv.org:2408.06519v2 - econ.EM - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.jeconom.2024.105840 - Kim Christensen, Alexei Kolokolov - - - A response-adaptive multi-arm design for continuous endpoints based on a weighted information measure - https://arxiv.org/abs/2409.04970 - arXiv:2409.04970v2 Announce Type: replace-cross -Abstract: Multi-arm trials are gaining interest in practice given the statistical and logistical advantages they can offer. The standard approach uses a fixed allocation ratio, but there is a call for making it adaptive and skewing the allocation of patients towards better-performing arms. However, it is well-known that these approaches might suffer from lower statistical power. We present a response-adaptive design for continuous endpoints which explicitly allows to control the trade-off between the number of patients allocated to the "optimal" arm and the statistical power. Such a balance is achieved through the calibration of a tuning parameter, and we explore robust procedures to select it. The proposed criterion is based on a context-dependent information measure which gives greater weight to treatment arms with characteristics close to a pre-specified clinical target. We establish conditions under which the procedure consistently selects the target arm and derive the corresponding limiting allocation ratios. We also introduce a simulation-based hypothesis testing procedure which focuses on selecting the target arm and discuss strategies to effectively control the type-I error rate. The practical implementation of the proposed criterion and its potential advantage over currently used alternatives are illustrated in the context of early Phase IIa proof-of-concept oncology trials. - oai:arXiv.org:2409.04970v2 - stat.ME - cs.IT - math.IT - stat.AP - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Gianmarco Caruso, Pavel Mozgunov - - - Error-Minimizing Measurements in Postselected One-Shot Symmetric Quantum State Discrimination and Acceptance as a Performance Metric - https://arxiv.org/abs/2409.13379 - arXiv:2409.13379v2 Announce Type: replace-cross -Abstract: In hypothesis testing with quantum states, given a black box containing one of the two possible states, measurement is performed to detect in favor of one of the hypotheses. In postselected hypothesis testing, a third outcome is added, corresponding to not selecting any of the hypotheses. In postselected scenario, minimum error one-shot symmetric hypothesis testing is characterized in literature conditioned on the fact that one of the selected outcomes occur. We proceed further in this direction to give the set of all possible measurements that lead to the minimum error. We have given an arbitrary error-minimizing measurement in a parametric form. Note that not selecting any of the hypotheses decimates the quality of testing. We further give an example to show that these measurements vary in quality. There is a need to discuss the quality of postselected hypothesis testing. We then characterize the quality of postselected hypothesis testing by defining a new metric acceptance and give expression of acceptance for an arbitrary error-minimizing measurement in terms of some parameters of the measurement. On the set of measurements that achieve minimum error, we have maximized the acceptance, and given an example which achieves that, thus giving an example of the best possible measurement in terms of acceptance. - oai:arXiv.org:2409.13379v2 - quant-ph - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Saurabh Kumar Gupta, Abhishek K. Gupta - - - Global dynamical structures from infinitesimal data - https://arxiv.org/abs/2410.02111 - arXiv:2410.02111v2 Announce Type: replace-cross -Abstract: Scientists and engineers alike target modeling of complex, high dimensional, and nonlinear dynamical systems as a central goal. Machine learning breakthroughs alongside mounting computation and data advance the efficacy of learning from trajectory measurements. However scientifically interpreting data-driven models, e.g., localizing attracting sets and their basins, remains elusive. Such limitations particularly afflict identification of system-level regulatory mechanisms characteristic of living systems, e.g., stabilizing control for whole-body locomotion, where discontinuous, transient, and multiscale phenomena are common and prior models are rare. As a next step towards theory-grounded discovery of behavioral mechanisms in biology and beyond, we introduce VERT, a framework for discovering attracting sets from trajectories without recourse to any global model. Our infinitesimal-local-global (ILG) pipeline estimates the proximity of any sampled state to an attracting set, if one exists, with formal accuracy guarantees. We demonstrate our approach on phenomenological and physical oscillators with hierarchical and impulsive dynamics, finding sensitivity to both global and intermediate attractors composed in sequence and parallel. Application of VERT to human running kinematics data reveals insight into control modules that stabilize task-level dynamics, supporting a longstanding neuromechanical control hypothesis. The VERT framework promotes rigorous inference of underlying dynamical structure even for systems where learning a global dynamics model is impractical or impossible. - oai:arXiv.org:2410.02111v2 - q-bio.QM - math.DS - nlin.CD - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Benjamin McInroe, Robert J. Full, Daniel E. Koditschek, Yuliy Baryshnikov - - - Nonlinear stability of extremal Reissner-Nordstr\"om black holes in spherical symmetry - https://arxiv.org/abs/2410.16234 - arXiv:2410.16234v2 Announce Type: replace-cross -Abstract: In this paper, we prove the codimension-one nonlinear asymptotic stability of the extremal Reissner-Nordstr\"om family of black holes in the spherically symmetric Einstein-Maxwell-neutral scalar field model, up to and including the event horizon. More precisely, we show that there exists a teleologically defined, codimension-one "submanifold" $\mathfrak M_\mathrm{stab}$ of the moduli space of spherically symmetric characteristic data for the Einstein-Maxwell-scalar field system lying close to the extremal Reissner-Nordstr\"om family, such that any data in $\mathfrak M_\mathrm{stab}$ evolve into a solution with the following properties as time goes to infinity: (i) the metric decays to a member of the extremal Reissner-Nordstr\"om family uniformly up to the event horizon, (ii) the scalar field decays to zero pointwise and in an appropriate energy norm, (iii) the first translation-invariant ingoing null derivative of the scalar field is approximately constant on the event horizon $\mathcal H^+$, (iv) for "generic" data, the second translation-invariant ingoing null derivative of the scalar field grows linearly along the event horizon. Due to the coupling of the scalar field to the geometry via the Einstein equations, suitable components of the Ricci tensor exhibit non-decay and growth phenomena along the event horizon. Points (i) and (ii) above reflect the "stability" of the extremal Reissner-Nordstr\"om family and points (iii) and (iv) verify the presence of the celebrated "Aretakis instability" for the linear wave equation on extremal Reissner-Nordstr\"om black holes in the full nonlinear Einstein-Maxwell-scalar field model. - oai:arXiv.org:2410.16234v2 - gr-qc - math-ph - math.AP - math.DG - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yannis Angelopoulos, Christoph Kehle, Ryan Unger - - - Characterising memory in quantum channel discrimination via constrained separability problems - https://arxiv.org/abs/2411.08110 - arXiv:2411.08110v2 Announce Type: replace-cross -Abstract: Quantum memories are a crucial precondition in many protocols for processing quantum information. A fundamental problem that illustrates this statement is given by the task of channel discrimination, in which an unknown channel drawn from a known random ensemble should be determined by applying it for a single time. In this paper, we characterise the quality of channel discrimination protocols when the quantum memory, quantified by the auxiliary dimension, is limited. This is achieved by formulating the problem in terms of separable quantum states with additional affine constraints that all of their factors in each separable decomposition obey. We discuss the computation of upper and lower bounds to the solutions of such problems which allow for new insights into the role of memory in channel discrimination. In addition to the single-copy scenario, this methodological insight allows to systematically characterise quantum and classical memories in adaptive channel discrimination protocols. Especially, our methods enabled us to identify channel discrimination scenarios where classical or quantum memory is required, and to identify the hierarchical and non-hierarchical relationships within adaptive channel discrimination protocols. - oai:arXiv.org:2411.08110v2 - quant-ph - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Ties-A. Ohst, Shijun Zhang, Hai Chau Nguyen, Martin Pl\'avala, Marco T\'ulio Quintino - - - VICON: Vision In-Context Operator Networks for Multi-Physics Fluid Dynamics Prediction - https://arxiv.org/abs/2411.16063 - arXiv:2411.16063v5 Announce Type: replace-cross -Abstract: In-Context Operator Networks (ICONs) have demonstrated the ability to learn operators across diverse partial differential equations using few-shot, in-context learning. However, existing ICONs process each spatial point as an individual token, severely limiting computational efficiency when handling dense data in higher spatial dimensions. We propose Vision In-Context Operator Networks (VICON), which integrates vision transformer architectures to efficiently process 2D data through patch-wise operations while preserving ICON's adaptability to multiphysics systems and varying timesteps. Evaluated across three fluid dynamics benchmarks, VICON significantly outperforms state-of-the-art baselines: DPOT and MPP, reducing the averaged last-step rollout error by 37.9% compared to DPOT and 44.7% compared to MPP, while requiring only 72.5% and 34.8% of their respective inference times. VICON naturally supports flexible rollout strategies with varying timestep strides, enabling immediate deployment in imperfect measurement systems where sampling frequencies may differ or frames might be dropped - common challenges in real-world settings - without requiring retraining or interpolation. In these realistic scenarios, VICON exhibits remarkable robustness, experiencing only 24.41% relative performance degradation compared to 71.37%-74.49% degradation in baseline methods, demonstrating its versatility for deploying in realistic applications. Our scripts for processing datasets and code are publicly available at https://github.com/Eydcao/VICON. - oai:arXiv.org:2411.16063v5 - cs.LG - cs.NA - math.NA - physics.flu-dyn - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Yadi Cao, Yuxuan Liu, Liu Yang, Rose Yu, Hayden Schaeffer, Stanley Osher - - - Rydberg Atomic Quantum Receivers for Classical Wireless Communications and Sensing: Their Models and Performance - https://arxiv.org/abs/2412.05554 - arXiv:2412.05554v3 Announce Type: replace-cross -Abstract: The significant progress of quantum sensing technologies offer numerous radical solutions for measuring a multitude of physical quantities at an unprecedented precision. Among them, Rydberg atomic quantum receivers (RAQRs) emerge as an eminent solution for detecting the electric field of radio frequency (RF) signals, exhibiting great potential in assisting classical wireless communications and sensing. So far, most experimental studies have aimed for the proof of physical concepts to reveal its promise, while the practical signal model of RAQR-aided wireless communications and sensing remained under-explored. Furthermore, the performance of RAQR-based wireless receivers and their advantages over classical RF receivers have not been fully characterized. To fill these gaps, we introduce the RAQR to the wireless community by presenting an end-to-end reception scheme. We then develop a corresponding equivalent baseband signal model relying on a realistic reception flow. Our scheme and model provide explicit design guidance to RAQR-aided wireless systems. We next study the performance of RAQR-aided wireless systems based on our model, and compare them to classical RF receivers. The results show that Doppler broadening-free RAQRs are capable of achieving a substantial received signal-to-noise ratio (SNR) gain of over $27$ decibel (dB) and $40$ dB in the photon shot limit and standard quantum limit regimes, respectively. - oai:arXiv.org:2412.05554v3 - eess.SP - cs.IT - math.IT - quant-ph - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tierui Gong, Jiaming Sun, Chau Yuen, Guangwei Hu, Yufei Zhao, Yong Liang Guan, Chong Meng Samson See, M\'erouane Debbah, Lajos Hanzo - - - Rational RG flow, extension, and Witt class - https://arxiv.org/abs/2412.08935 - arXiv:2412.08935v3 Announce Type: replace-cross -Abstract: Consider a renormalization group flow preserving a pre-modular fusion category $\mathcal S_1$. If it flows to a rational conformal field theory, the surviving symmetry $\mathcal S_1$ flows to a pre-modular fusion category $\mathcal S_2$ with monoidal functor $F:\mathcal S_1\to\mathcal S_2$. By clarifying mathematical (especially category theoretical) meaning of renormalization group domain wall/interface or boundary condition, we find the hidden extended vertex operator (super)algebra gives a unique (up to braided equivalence) completely $(\mathcal S_1\boxtimes\mathcal S_2)'$-anisotropic representative of the Witt equivalence class $[\mathcal S_1\boxtimes\mathcal S_2]$. The mathematical conjecture is supported physically, and passes various tests in concrete examples including non/unitary minimal models, and Wess-Zumino-Witten models. In particular, the conjecture holds beyond diagonal cosets. - The picture also establishes the conjectured half-integer condition, which fixes infrared conformal dimensions mod $\frac12$. It further leads to the double braiding relation, namely braiding structures jump at conformal fixed points. As an application, we solve the flow from the $E$-type minimal model $(A_{10},E_6)\to M(4,3)$. - oai:arXiv.org:2412.08935v3 - hep-th - cond-mat.str-el - math-ph - math.CT - math.MP - math.QA - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ken Kikuchi - - - Adaptive Economic Model Predictive Control: Performance Guarantees for Nonlinear Systems - https://arxiv.org/abs/2412.13046 - arXiv:2412.13046v3 Announce Type: replace-cross -Abstract: We consider the problem of optimizing the economic performance of nonlinear constrained systems subject to uncertain time-varying parameters and bounded disturbances. In particular, we propose an adaptive economic model predictive control (MPC) framework that: (i) directly minimizes transient economic costs, (ii) addresses parametric uncertainty through online model adaptation, (iii) determines optimal setpoints online, and (iv) ensures robustness by using a tube-based approach. The proposed design ensures recursive feasibility, robust constraint satisfaction, and a transient performance bound. In case the disturbances have a finite energy and the parameter variations have a finite path length, the asymptotic average performance is (approximately) not worse than the performance obtained when operating at the best reachable steady-state. We highlight performance benefits in a numerical example involving a chemical reactor with unknown time-invariant and time-varying parameters. - oai:arXiv.org:2412.13046v3 - eess.SY - cs.SY - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Maximilian Degner, Raffaele Soloperto, Melanie N. Zeilinger, John Lygeros, Johannes K\"ohler - - - Encrypted Qubits can be Cloned - https://arxiv.org/abs/2501.02757 - arXiv:2501.02757v3 Announce Type: replace-cross -Abstract: We show that encrypted cloning of unknown quantum states is possible. Any number of encrypted clones of a qubit can be created through a unitary transformation, and each of the encrypted clones can be decrypted through a unitary transformation. The decryption of an encrypted clone consumes the decryption key, i.e., only one decryption is possible, in agreement with the no-cloning theorem. Encrypted cloning represents a new paradigm that provides a form of redundancy, parallelism or scalability where direct duplication is forbidden by the no-cloning theorem. For example, a possible application of encrypted cloning is to enable encrypted quantum multi-cloud storage. - oai:arXiv.org:2501.02757v3 - quant-ph - gr-qc - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1103/y4y1-1ll6 - Phys. Rev. Lett. 136, 010801 (2026) - Koji Yamaguchi, Achim Kempf - - - Online Scheduling for LLM Inference with KV Cache Constraints - https://arxiv.org/abs/2502.07115 - arXiv:2502.07115v5 Announce Type: replace-cross -Abstract: Large Language Model (LLM) inference, where a trained model generates text one word at a time in response to user prompts, is a computationally intensive process requiring efficient scheduling to optimize latency and resource utilization. A key challenge in LLM inference is the management of the Key-Value (KV) cache, which reduces redundant computations but introduces memory constraints. In this work, we model LLM inference with KV cache constraints theoretically and propose a novel batching and scheduling algorithm that minimizes inference latency while effectively managing the KV cache's memory. - More specifically, we make the following contributions. First, to evaluate the performance of online algorithms for scheduling in LLM inference, we introduce a hindsight optimal benchmark, formulated as an integer program that computes the minimum total inference latency under full future information. Second, we prove that no deterministic online algorithm can achieve a constant competitive ratio when the arrival process is arbitrary. Third, motivated by the computational intractability of solving the integer program at scale, we propose a polynomial-time online scheduling algorithm and show that under certain conditions it can achieve a constant competitive ratio. We also demonstrate our algorithm's strong empirical performance by comparing it to the hindsight optimal in a synthetic dataset. Finally, we conduct empirical evaluations on a real-world public LLM inference dataset, simulating the Llama2-70B model on A100 GPUs, and show that our algorithm significantly outperforms the benchmark algorithms. Overall, our results offer a path toward more sustainable and cost-effective LLM deployment. - oai:arXiv.org:2502.07115v5 - cs.LG - cs.AI - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Patrick Jaillet, Jiashuo Jiang, Konstantina Mellou, Marco Molinaro, Chara Podimata, Zijie Zhou - - - Exploring specialization and sensitivity of convolutional neural networks in the context of simultaneous image augmentations - https://arxiv.org/abs/2503.03283 - arXiv:2503.03283v2 Announce Type: replace-cross -Abstract: Drawing parallels with the way biological networks are studied, we adapt the treatment--control paradigm to explainable artificial intelligence research and enrich it through multi-parametric input alterations. In this study, we propose a framework for investigating the internal inference impacted by input data augmentations. The internal changes in network operation are reflected in activation changes measured by variance, which can be decomposed into components related to each augmentation, employing Sobol indices and Shapley values. These quantities enable one to visualize sensitivity to different variables and use them for guided masking of activations. In addition, we introduce a way of single-class sensitivity analysis where the candidates are filtered according to their matching to prediction bias generated by targeted damaging of the activations. Relying on the observed parallels, we assume that the developed framework can potentially be transferred to studying biological neural networks in complex environments. - oai:arXiv.org:2503.03283v2 - stat.ML - cs.AI - cs.LG - cs.NA - math.NA - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pavel Kharyuk, Sergey Matveev, Ivan Oseledets - - - A vector bundle approach to Nash equilibria - https://arxiv.org/abs/2504.03456 - arXiv:2504.03456v2 Announce Type: replace-cross -Abstract: We use vector bundles to study the locus of totally mixed Nash equilibria of an $n$-player game in normal form, which we call the Nash equilibrium scheme. When the payoff tensor format is balanced, we study the Nash discriminant variety, i.e., the algebraic variety of games whose Nash equilibrium scheme is nonreduced or has a positive dimensional component. We prove that this variety has codimension one. We classify all possible components of the Nash equilibrium scheme for a binary three-player game. We prove that if the payoff tensor is of boundary format, then the Nash discriminant variety has two components: an irreducible hypersurface and a larger-codimensional component. A generic game with an unbalanced payoff tensor format does not admit totally mixed Nash equilibria. We define the Nash resultant variety of games admitting a positive number of totally mixed Nash equilibria. We prove that it is irreducible and determine its codimension and degree. - oai:arXiv.org:2504.03456v2 - cs.GT - math.AG - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.aam.2025.103028 - Hirotachi Abo, Irem Portakal, Luca Sodomaco - - - Bosonization of Noise Effects in Nonlocal Quantum Dynamics - https://arxiv.org/abs/2504.20891 - arXiv:2504.20891v2 Announce Type: replace-cross -Abstract: Quantum systems that interact non-locally with an environment are paradigms for exploring collective phenomena. They naturally emerge in various physical contexts involving long-range, many-body interactions. We consider a general class of such open systems characterized by a coupling to the environment which is inversely proportional to the square root of the environment size. We show that the induced system dynamics has a universal bosonic nature: the same evolution arises from coupling the system to a collection of noninteracting bosonic modes, independently of the microscopic structure of the original environment. This emergent "bosonization" of the environment's influence results from the scaling of the coupling in the thermodynamic limit and is a manifestation of the quantum central limit theorem. While the effect has been observed in specific models before, we show that it is, in fact, a universal feature. - oai:arXiv.org:2504.20891v2 - quant-ph - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - 10.1103/ls2g-mkhm - Michele Fantechi, Marco Merkli - - - Deep Learning for Continuous-Time Stochastic Control with Jumps - https://arxiv.org/abs/2505.15602 - arXiv:2505.15602v3 Announce Type: replace-cross -Abstract: In this paper, we introduce a model-based deep-learning approach to solve finite-horizon continuous-time stochastic control problems with jumps. We iteratively train two neural networks: one to represent the optimal policy and the other to approximate the value function. Leveraging a continuous-time version of the dynamic programming principle, we derive two different training objectives based on the Hamilton-Jacobi-Bellman equation, ensuring that the networks capture the underlying stochastic dynamics. Empirical evaluations on different problems illustrate the accuracy and scalability of our approach, demonstrating its effectiveness in solving complex high-dimensional stochastic control tasks. - oai:arXiv.org:2505.15602v3 - cs.LG - cs.SY - eess.SY - math.OC - q-fin.PM - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Patrick Cheridito, Jean-Loup Dupret, Donatien Hainaut - - - Data-Driven Dynamic Factor Modeling via Manifold Learning - https://arxiv.org/abs/2506.19945 - arXiv:2506.19945v2 Announce Type: replace-cross -Abstract: We introduce a data-driven dynamic factor framework for modeling the joint evolution of high-dimensional covariates and responses without parametric assumptions. Standard factor models applied to covariates alone often lose explanatory power for responses. Our approach uses anisotropic diffusion maps, a manifold learning technique, to learn low-dimensional embeddings that preserve both the intrinsic geometry of the covariates and the predictive relationship with responses. For time series arising from Langevin diffusions in Euclidean space, we show that the associated graph Laplacian converges to the generator of the underlying diffusion. We further establish a bound on the approximation error between the diffusion map coordinates and linear diffusion processes, and we show that ergodic averages in the embedding space converge under standard spectral assumptions. These results justify using Kalman filtering in diffusion-map coordinates for predicting joint covariate-response evolution. We apply this methodology to equity-portfolio stress testing using macroeconomic and financial variables from Federal Reserve supervisory scenarios, achieving mean absolute error improvements of up to 55% over classical scenario analysis and 39% over principal component analysis benchmarks. - oai:arXiv.org:2506.19945v2 - stat.ML - cs.LG - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Graeme Baker, Agostino Capponi, J. Antonio Sidaoui - - - COALA: Numerically Stable and Efficient Framework for Context-Aware Low-Rank Approximation - https://arxiv.org/abs/2507.07580 - arXiv:2507.07580v2 Announce Type: replace-cross -Abstract: Recent studies suggest that context-aware low-rank approximation is a useful tool for compression and fine-tuning of modern large-scale neural networks. In this type of approximation, a norm is weighted by a matrix of input activations, significantly improving metrics over the unweighted case. Nevertheless, existing methods for neural networks suffer from numerical instabilities due to their reliance on classical formulas involving explicit Gram matrix computation and their subsequent inversion. We demonstrate that this can degrade the approximation quality or cause numerically singular matrices. - To address these limitations, we propose a novel inversion-free regularized framework that is based entirely on stable decompositions and overcomes the numerical pitfalls of prior art. Our method can handle possible challenging scenarios: (1) when calibration matrices exceed GPU memory capacity, (2) when input activation matrices are nearly singular, and even (3) when insufficient data prevents unique approximation. For the latter, we prove that our solution converges to a desired approximation and derive explicit error bounds. - oai:arXiv.org:2507.07580v2 - cs.LG - cs.CL - cs.NA - math.NA - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Uliana Parkina, Maxim Rakhuba - - - Quantum circuit complexity and unsupervised machine learning of topological order - https://arxiv.org/abs/2508.04486 - arXiv:2508.04486v2 Announce Type: replace-cross -Abstract: Inspired by the close relationship between Kolmogorov complexity and unsupervised machine learning, we explore quantum circuit complexity, an important concept in quantum computation and quantum information science, as a pivot to understand and to build interpretable and efficient unsupervised machine learning for topological order in quantum many-body systems. We argue that Nielsen's quantum circuit complexity represents an intrinsic topological distance between topological quantum many-body phases of matter, and as such plays a central role in interpretable manifold learning of topological order. To span a bridge from conceptual power to practical applicability, we present two theorems that connect Nielsen's quantum circuit complexity for the quantum path planning between two arbitrary quantum many-body states with quantum Fisher complexity (Bures distance) and entanglement generation, respectively. Leveraging these connections, fidelity-based and entanglement-based similarity measures or kernels, which are more practical for implementation, are formulated. Using the two proposed distance measures, unsupervised manifold learning of quantum phases of the bond-alternating XXZ spin chain, the ground state of Kitaev's toric code and random product states, is conducted, demonstrating their superior performance. Moreover, we find that the entanglement-based approach, which captures the long-range structure of quantum entanglement of topological orders, is more robust to local Haar random noises. Relations with classical shadow tomography and shadow kernel learning are also discussed, where the latter can be naturally understood from our approach. Our results establish connections between key concepts and tools of quantum circuit computation, quantum complexity, quantum metrology, and machine learning of topological quantum order. - oai:arXiv.org:2508.04486v2 - quant-ph - cond-mat.dis-nn - cs.CC - cs.IT - cs.LG - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by-sa/4.0/ - Yanming Che, Clemens Gneiting, Xiaoguang Wang, Franco Nori - - - Morse sequences on stacks and flooding sequences - https://arxiv.org/abs/2509.01384 - arXiv:2509.01384v2 Announce Type: replace-cross -Abstract: This paper builds upon the framework of \emph{Morse sequences}, a simple and effective approach to discrete Morse theory. A Morse sequence on a simplicial complex consists of a sequence of nested subcomplexes generated by expansions and fillings-two operations originally introduced by Whitehead. Expansions preserve homotopy, while fillings introduce critical simplexes that capture essential topological features. We extend the notion of Morse sequences to \emph{stacks}, which are monotonic functions defined on simplicial complexes, and define \emph{Morse sequences on stacks} as those whose expansions preserve the homotopy of all sublevel sets. This extension leads to a generalization of the fundamental collapse theorem to weighted simplicial complexes. Within this framework, we focus on a refined class of sequences called \emph{flooding sequences}, which exhibit an ordering behavior similar to that of classical watershed algorithms. Although not every Morse sequence on a stack is a flooding sequence, we show that the gradient vector field associated with any Morse sequence can be recovered through a flooding sequence. Finally, we present algorithmic schemes for computing flooding sequences using cosimplicial complexes. - oai:arXiv.org:2509.01384v2 - cs.DM - math.AT - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gilles Bertrand (LIGM) - - - Learning Regularization Functionals for Inverse Problems: A Comparative Study - https://arxiv.org/abs/2510.01755 - arXiv:2510.01755v2 Announce Type: replace-cross -Abstract: In recent years, a variety of learned regularization frameworks for solving inverse problems in imaging have emerged. These offer flexible modeling together with mathematical insights. The proposed methods differ in their architectural design and training strategies, making direct comparison challenging due to non-modular implementations. We address this gap by collecting and unifying the available code into a common framework. This unified view allows us to systematically compare the approaches and highlight their strengths and limitations, providing valuable insights into their future potential. We also provide concise descriptions of each method, complemented by practical guidelines. - oai:arXiv.org:2510.01755v2 - cs.LG - cs.NA - math.NA - math.OC - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Johannes Hertrich, Hok Shing Wong, Alexander Denker, Stanislas Ducotterd, Zhenghan Fang, Markus Haltmeier, \v{Z}eljko Kereta, Erich Kobler, Oscar Leong, Mohammad Sadegh Salehi, Carola-Bibiane Sch\"onlieb, Johannes Schwab, Zakhar Shumaylov, Jeremias Sulam, German Sh\^ama Wache, Martin Zach, Yasi Zhang, Matthias J. Ehrhardt, Sebastian Neumayer - - - Non-uniqueness of the steady state for run-and-tumble particles with a double-well interaction potential - https://arxiv.org/abs/2510.07212 - arXiv:2510.07212v3 Announce Type: replace-cross -Abstract: We study $N$ run-and-tumble particles (RTPs) in one dimension interacting via a double-well potential $W(r)=-k_0 \, r^2/2+g \, r^4/4$, which is repulsive at short interparticle distance $r$ and attractive at large distance. At large time, the system forms a bound state where the density of particles has a finite support. We focus on the determination of the total density of particles in the stationary state $\rho_s(x)$, in the limit $N\to+\infty$. We obtain an explicit expression for $\rho_s(x)$ as a function of the ''renormalized" interaction parameter $k=k_0-3m_2$ where $m_2$ is the second moment of $\rho_s(x)$. Interestingly, this stationary solution exhibits a transition between a connected and a disconnected support for a certain value of $k$, which has no equivalent in the case of Brownian particles. Analyzing in detail the expression of the stationary density in the two cases, we find a variety of regimes characterized by different behaviors near the edges of the support and around $x=0$. Furthermore, we find that the mapping $k_0\to k$ becomes multi-valued below a certain value of the tumbling rate $\gamma$ of the RTPs for some range of values of $k_0$ near the transition, implying the existence of two stable solutions. Finally, we show that in the case of a disconnected support, it is possible to observe steady states where the density $\rho_s(x)$ is not symmetric. All our analytical predictions are in good agreement with numerical simulations already for systems of $N = 100$ particles. The non-uniqueness of the stationary state is a particular feature of this model in the presence of active (RTP) noise, which contrasts with the uniqueness of the Gibbs equilibrium for Brownian particles. We argue that these results are also relevant for a class of more realistic interactions with both an attractive and a repulsive part, but which decay at infinity. - oai:arXiv.org:2510.07212v3 - cond-mat.stat-mech - cond-mat.soft - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1103/9b6g-gmdk - L\'eo Touzo, Pierre Le Doussal - - - Pulse Shaping Filter Design for Integrated Sensing & Communication with Zak-OTFS - https://arxiv.org/abs/2510.15195 - arXiv:2510.15195v2 Announce Type: replace-cross -Abstract: Zak-OTFS provides a framework for integrated sensing & communication (ISAC) in high delay and Doppler spread environments. Pulse shaping filter design enables joint optimization of sensing and communication performance. For sensing, a localized pulse shaping filter enables input-output (I/O) relation estimates close to the physical scattering channel. For communication, orthogonality of the pulse shape on the information lattice prevents inter-symbol interference, and no time and bandwidth expansion enables full spectral efficiency. A filter simultaneously meeting all three objectives is ideal for ISAC. Existing filter designs achieve two, but not all three objectives. In this work, we design pulse shaping filters meeting all three objectives via the Isotropic Orthogonal Transform Algorithm. The proposed filters have improved spectral efficiency, data detection and sensing performance over existing filter choices. - oai:arXiv.org:2510.15195v2 - eess.SP - cs.IT - math.IT - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Nishant Mehrotra, Sandesh Rao Mattu, Robert Calderbank - - - Orbit Elements from Kepler Solutions in Projective Coordinates - https://arxiv.org/abs/2511.12957 - arXiv:2511.12957v3 Announce Type: replace-cross -Abstract: Closed-Form Kepler solutions in projective coordinates are used to define a corresponding set of eight orbit elements and obtain their governing equations for arbitrarily-perturbed two-body dynamics. The elements and their dynamics are singularity-free in all cases besides rectilinear motion (when angular momentum vanishes). The classic J2-perturbed two-body problem is developed and used for numerical verification. - oai:arXiv.org:2511.12957v3 - astro-ph.EP - math.DS - physics.class-ph - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Joseph T. A. Peterson, Manoranjan Majji, John L. Junkins - - - Eventually LIL Regret: Almost Sure $\ln\ln T$ Regret for a sub-Gaussian Mixture on Unbounded Data - https://arxiv.org/abs/2512.12325 - arXiv:2512.12325v2 Announce Type: replace-cross -Abstract: We prove that a classic sub-Gaussian mixture proposed by Robbins in a stochastic setting actually satisfies a path-wise (deterministic) regret bound. For every path in a natural ``Ville event'' $E_\alpha$, this regret till time $T$ is bounded by $\ln^2(1/\alpha)/V_T + \ln (1/\alpha) + \ln \ln V_T$ up to universal constants, where $V_T$ is a nonnegative, nondecreasing, cumulative variance process. (The bound reduces to $\ln(1/\alpha) + \ln \ln V_T$ if $V_T \geq \ln(1/\alpha)$.) If the data were stochastic, then one can show that $E_\alpha$ has probability at least $1-\alpha$ under a wide class of distributions (eg: sub-Gaussian, symmetric, variance-bounded, etc.). In fact, we show that on the Ville event $E_0$ of probability one, the regret on every path in $E_0$ is eventually bounded by $\ln \ln V_T$ (up to constants). We explain how this work helps bridge the world of adversarial online learning (which usually deals with regret bounds for bounded data), with game-theoretic statistics (which can handle unbounded data, albeit using stochastic assumptions). In short, conditional regret bounds serve as a bridge between stochastic and adversarial betting. - oai:arXiv.org:2512.12325v2 - cs.LG - math.ST - stat.ML - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Shubhada Agrawal, Aaditya Ramdas - - - Uniqueness of invariant measures as a structural property of markov kernels - https://arxiv.org/abs/2601.04900 - arXiv:2601.04900v2 Announce Type: replace-cross -Abstract: We identify indecomposability as a key measure-theoretic underlying uniqueness of invariant probability measures for discrete-time Markov kernels on general state spaces. The argument relies on the mutual singularity of distinct invariant ergodic measures and on the observation that uniqueness follows whenever all invariant probability measures are forced to charge a common reference measure. - Once existence of invariant probability measures is known, indecomposability alone is sufficient to rule out multiplicity. On standard Borel spaces, this viewpoint is consistent with the classical theory: irreducibility appears as a convenient sufficient condition ensuring indecomposability, rather than as a structural requirement for uniqueness. - The resulting proofs are purely measure-theoretic and do not rely on recurrence, regeneration, return-time estimates, or regularity assumptions on the transition kernel. - oai:arXiv.org:2601.04900v2 - q-fin.MF - math.PR - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jean-Gabriel Attali - - - Inference for Multiple Change-points in Piecewise Locally Stationary Time Series - https://arxiv.org/abs/2601.07400 - arXiv:2601.07400v2 Announce Type: replace-cross -Abstract: Change-point detection and locally stationary time series modeling are two major approaches for the analysis of non-stationary data. The former aims to identify stationary phases by detecting abrupt changes in the dynamics of a time series model, while the latter employs (locally) time-varying models to describe smooth changes in dependence structure of a time series. However, in some applications, abrupt and smooth changes can co-exist, and neither of the two approaches alone can model the data adequately. In this paper, we propose a novel likelihood-based procedure for the inference of multiple change-points in locally stationary time series. In contrast to traditional change-point analysis where an abrupt change occurs in a real-valued parameter, a change in locally stationary time series occurs in a parameter curve, and can be classified as a jump or a kink depending on whether the curve is discontinuous or not. We show that the proposed method can consistently estimate the number, locations, and the types of change-points. Two different asymptotic distributions corresponding respectively to jump and kink estimators are also established. Extensive simulation studies and a real data application to financial time series are provided. - oai:arXiv.org:2601.07400v2 - stat.ME - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Wai Leong Ng, Xinyi Tang, Mun Lau Cheung, Jiacheng Gao, Chun Yip Yau, Holger Dette - - - Riesz Representer Fitting under Bregman Divergence: A Unified Framework for Debiased Machine Learning - https://arxiv.org/abs/2601.07752 - arXiv:2601.07752v2 Announce Type: replace-cross -Abstract: Estimating the Riesz representer is central to debiased machine learning for causal and structural parameter estimation. We propose generalized Riesz regression, a unified framework that estimates the Riesz representer by fitting a representer model via Bregman divergence minimization. This framework includes the squared loss and the Kullback--Leibler (KL) divergence as special cases: the former recovers Riesz regression, while the latter recovers tailored loss minimization. Under suitable model specifications, the dual problems correspond to covariate balancing, which we call automatic covariate balancing. Moreover, under the same specifications, outcome averages weighted by the estimated Riesz representer satisfy Neyman orthogonality even without estimating the regression function, a property we call automatic Neyman orthogonalization. This property not only reduces the estimation error of Neyman orthogonal scores but also clarifies a key distinction between debiased machine learning and targeted maximum likelihood estimation. Our framework can also be viewed as a generalization of density ratio fitting under Bregman divergences to Riesz representer estimation, and it applies beyond density ratio estimation. We provide convergence analyses for both reproducing kernel Hilbert space (RKHS) and neural network model classes. A Python package for generalized Riesz regression is available at https://github.com/MasaKat0/grr. - oai:arXiv.org:2601.07752v2 - econ.EM - cs.LG - math.ST - stat.ME - stat.ML - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Masahiro Kato - - - A Nonlinear Mechanism for Transient Anomalous Diffusion - https://arxiv.org/abs/2601.08083 - arXiv:2601.08083v2 Announce Type: replace-cross -Abstract: Diffusion is a fundamental physical phenomenon with critical applications in fields such as metallurgy, cell biology, and population dynamics. While standard diffusion is well-understood, anomalous diffusion often requires complex non-local models. This paper investigates a nonlinear diffusion equation where the diffusion coefficient is linearly dependent on concentration. We demonstrate through a perturbative analysis that this physically-grounded model exhibits transient anomalous diffusion. The system displays a clear crossover from an initial subdiffusive regime to standard Fickian behavior at long times. This result establishes an important mechanism for trasient anomalous diffusion that arises purely from local interactions, providing an intuitive alternative to models based on fractional calculus or non-local memory effects. - oai:arXiv.org:2601.08083v2 - cond-mat.stat-mech - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gabriel Barreiro, Vladimir P\'erez-Veloz - - - Sample Complexity of Composite Quantum Hypothesis Testing - https://arxiv.org/abs/2601.08588 - arXiv:2601.08588v2 Announce Type: replace-cross -Abstract: This paper investigates symmetric composite binary quantum hypothesis testing (QHT), where the goal is to determine which of two uncertainty sets contains an unknown quantum state. While asymptotic error exponents for this problem are well-studied, the finite-sample regime remains poorly understood. We bridge this gap by characterizing the sample complexity -- the minimum number of state copies required to achieve a target error level. Specifically, we derive lower bounds that generalize the sample complexity of simple QHT and introduce new upper bounds for various uncertainty sets, including of both finite and infinite cardinalities. Notably, our upper and lower bounds match up to universal constants, providing a tight characterization of the sample complexity. Finally, we extend our analysis to the differentially private setting, establishing the sample complexity for privacy-preserving composite QHT. - oai:arXiv.org:2601.08588v2 - quant-ph - cs.IT - cs.LG - math.IT - math.ST - stat.TH - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Jacob Paul Simpson, Efstratios Palias, Sharu Theresa Jose - - - Bogomol'nyi Equations in Two-Species Born--Infeld Theories Governing Vortices and Antivortices - https://arxiv.org/abs/2601.09091 - arXiv:2601.09091v2 Announce Type: replace-cross -Abstract: We derive several new Bogomol'nyi (self-dual) equations in two-species $U(1)\times U(1)$ gauge theories governed by the Born--Infeld nonlinear electrodynamics. By identifying appropriate Born--Infeld type Higgs potentials, we show that the highly nonlinear energy functionals admit exact topological lower bounds saturated by coupled first-order equations. The resulting models accommodate both vortex-vortex and vortex-antivortex configurations and generalize previously known single-species Born--Infeld systems to interacting multi-component settings. - Beyond the derivation of the Bogomol'nyi equations, we develop an exact thermodynamic theory for pinned multivortex configurations in both the full plane and compact doubly periodic domains. Owing to the linear dependence of the Bogomol'nyi energy spectrum on topological charges, we obtain closed-form expressions for the canonical partition function, internal energy, heat capacity, and magnetization. In compact domains, the Bradlow type geometric bounds constrain admissible vortex numbers and lead to qualitatively new high-temperature behavior. In particular, vortex-only systems exhibit spontaneous magnetization, while vortex-antivortex systems do not, reflecting the underlying symmetry between opposite topological charges. These results provide a rare analytically solvable framework for studying thermodynamics in nonlinear multi-component gauge theories regulated by the Born--Infeld electrodynamics. - oai:arXiv.org:2601.09091v2 - hep-th - hep-ph - math-ph - math.MP - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Aonan Xu, Yisong Yang - - - Discrete Solution Operator Learning for Geometry-Dependent PDEs - https://arxiv.org/abs/2601.09143 - arXiv:2601.09143v2 Announce Type: replace-cross -Abstract: Neural operator learning accelerates PDE solution by approximating operators as mappings between continuous function spaces. Yet in many engineering settings, varying geometry induces discrete structural changes, including topological changes, abrupt changes in boundary conditions or boundary types, and changes in the computational domain, which break the smooth-variation premise. Here we introduce Discrete Solution Operator Learning (DiSOL), a complementary paradigm that learns discrete solution procedures rather than continuous function-space operators. DiSOL factorizes the solver into learnable stages that mirror classical discretizations: local contribution encoding, multiscale assembly, and implicit solution reconstruction on an embedded grid, thereby preserving procedure-level consistency while adapting to geometry-dependent discrete structures. Across geometry-dependent Poisson, advection-diffusion, linear elasticity, as well as spatiotemporal heat conduction problems, DiSOL produces stable and accurate predictions under both in-distribution and strongly out-of-distribution geometries, including discontinuous boundaries and topological changes. These results highlight the need for procedural operator representations in geometry-dominated problems and position discrete solution operator learning as a distinct, complementary direction in scientific machine learning. - oai:arXiv.org:2601.09143v2 - cs.LG - cs.NA - math.NA - physics.comp-ph - Fri, 16 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jinshuai Bai, Haolin Li, Zahra Sharif Khodaei, M. H. Aliabadi, YuanTong Gu, Xi-Qiao Feng -