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" @@ -7,12 +7,4106 @@ http://www.rssboard.org/rss-specification en-us - Sun, 04 Jan 2026 05:00:01 +0000 + Wed, 07 Jan 2026 05:00:15 +0000 rss-help@arxiv.org - Sun, 04 Jan 2026 00:00:00 -0500 + Wed, 07 Jan 2026 00:00:00 -0500 - Sunday Saturday + Sunday + + Breaking Rank - A Novel Unscented Kalman Filter for Parameter Estimations of a Lumped-Parameter Cardiovascular Model + https://arxiv.org/abs/2601.02390 + arXiv:2601.02390v1 Announce Type: new +Abstract: We make modifications to the unscented Kalman filter (UKF) which bestow almost complete practical identifiability upon a lumped-parameter cardiovascular model with 10 parameters and 4 output observables - a highly non-linear, stiff problem of clinical significance. The modifications overcome the challenging problems of rank deficiency when applying the UKF to parameter estimation. Rank deficiency usually means only a small subset of parameters can be estimated. Traditionally, pragmatic compromises are made, such as selecting an optimal subset of parameters for estimation and fixing non-influential parameters. Kalman filters are typically used for dynamical state tracking, to facilitate the control u at every time step. However, for the purpose of parameter estimation, this constraint no longer applies. Our modification has transformed the utility of UKF for the parameter estimation purpose, including minimally influential parameters, with excellent robustness (i.e., under severe noise corruption, challenging patho-physiology, and no prior knowledge of parameter distributions). The modified UKF algorithm is robust in recovering almost all parameters to over 98% accuracy, over 90% of the time, with a challenging target data set of 50, 10-parameter samples. We compare this to the original implementation of the UKF algorithm for parameter estimation and demonstrate a significant improvement. + oai:arXiv.org:2601.02390v1 + cs.IT + math.IT + stat.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Alex Thornton, Ian Halliday, Harry Saxton, Xu Xu + + + Mean Field Variational Bayesian Inference and Statistical Mechanics of Gaussian Mixture Model + https://arxiv.org/abs/2601.02418 + arXiv:2601.02418v1 Announce Type: new +Abstract: One of the main modeling in many data science applications is the Gaussian Mixture Model (GMM), and Mean Field Variational Bayesian Inference (MFVBI) is classically used for approximate fast computation. In this paper, we provide a definitive answer to the fundamental inquiry about the uncertainty quantification of the MFVBI applied to the GMM. It turns out that GMM can be considered as a generalization of Curie--Weiss model in statistical mechanics. The standard quantities like partition function and free energy appear naturally in the process of our analysis. + oai:arXiv.org:2601.02418v1 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Alireza Bahraini, Saeed Sadeghi + + + Thrust Regulation in a Solid Fuel Ramjet using Dynamic Mode Adaptive Control + https://arxiv.org/abs/2601.02429 + arXiv:2601.02429v1 Announce Type: new +Abstract: This paper presents the application of a novel data-driven adaptive control technique, called dynamic mode adaptive control (DMAC), for regulating thrust in a solid fuel ramjet (SFRJ). A high-fidelity computational model incorporating compressible flow theory and equilibrium chemistry is used to simulate the combustion dynamics. An adaptive tracking controller is designed using the DMAC framework, which leverages dynamic mode decomposition to approximate the local system behavior, followed by a tracking controller synthesized around the identified model. Simulation results demonstrate that DMAC provides an effective and reliable approach for thrust regulation in SFRJs. In addition, a systematic hyperparameter sensitivity study is conducted by varying the tuning parameters over several orders of magnitude. The resulting responses show that the closed-loop performance and tracking error remain stable across wide parameter variations, indicating that DMAC exhibits strong robustness to hyper parameter tuning. + oai:arXiv.org:2601.02429v1 + math.OC + physics.comp-ph + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Parham Oveissi, Gohar T. Khokhar, Kyle Hanquist, Ankit Goel + + + An Analogue of Heyde's Theorem for Discrete Torsion Abelian Groups with Cyclic $p$-Components + https://arxiv.org/abs/2601.02489 + arXiv:2601.02489v1 Announce Type: new +Abstract: According to the well-known Heyde theorem, the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of $n$ independent random variables given another. In the article, we prove an analogue of this theorem for two independent random variables taking values in a discrete torsion Abelian group $X$ with cyclic $p$-components. In doing so, we do not impose any restrictions on coefficients of the linear forms and the characteristic functions of random variables. The proof uses methods of abstract harmonic analysis and is based on the solution some functional equation on the character group of the group $X$. + oai:arXiv.org:2601.02489v1 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/publicdomain/zero/1.0/ + Gennadiy Feldman + + + Variational (Energy-Based) Spectral Learning: A Machine Learning Framework for Solving Partial Differential Equations + https://arxiv.org/abs/2601.02492 + arXiv:2601.02492v1 Announce Type: new +Abstract: We introduce variational spectral learning (VSL), a machine learning framework for solving partial differential equations (PDEs) that operates directly in the coefficient space of spectral expansions. VSL offers a principled bridge between variational PDE theory, spectral discretization, and contemporary machine learning practice. The core idea is to recast a given PDE \[ \mathcal{L}u = f \quad \text{in} \quad Q=\Omega\times(0,T), \] together with boundary and initial conditions, into differentiable space--time energies built from strong-form least-squares residuals and weak (Galerkin) formulations. The solution is represented as a finite spectral expansion \[ u_N(x,t)=\sum_{n=1}^{N} c_n\,\phi_n(x,t), \] where $\phi_n$ are tensor-product Chebyshev bases in space and time, with Dirichlet-satisfying spatial modes enforcing homogeneous boundary conditions analytically. This yields a compact linear parameterization in the coefficient vector $\mathbf{c}$, while all PDE complexity is absorbed into the variational energy. We show how to construct strong-form and weak-form space-time functionals, augment them with initial-condition and Tikhonov regularization terms, and minimize the resulting objective with gradient-based optimization. In practice, VSL is implemented in TensorFlow using automatic differentiation and Keras cosine-decay-with-restarts learning-rate schedules, enabling robust optimization of moderately sized coefficient vectors. Numerical experiments on benchmark elliptic and parabolic problems, including one- and two-dimensional Poisson, diffusion, and Burgers-type equations, demonstrate that VSL attains accuracy comparable to classical spectral collocation with Crank-Nicolson time stepping, while providing a differentiable objective suitable for modern optimization tooling. + oai:arXiv.org:2601.02492v1 + math.NA + cs.LG + cs.NA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + M. M. Hammad + + + Long Time Asymptotics for the Stochastic Follow-the-Leader System + https://arxiv.org/abs/2601.02501 + arXiv:2601.02501v1 Announce Type: new +Abstract: We introduce and analyze a class of interacting particle systems on the real line that combine features of the stochastic rat race and (deterministic) follow-the-leader models. The particle system evolves as a continuous-time pure jump process: the leading particle moves independently, at Exponential jump times, with constant jump rate and iid jump sizes distributed according to a law $\theta$, while each of the remaining particles jumps forward, at Exponential times, at rate equal to its distance from the particle immediately ahead, with jump sizes drawn uniformly from the corresponding gap. The dynamics thus encode competition for leadership together with distance-dependent stochastic interactions. Our main focus is the associated gap process, representing the vector of inter-particle distances. We establish the existence of a unique stationary distribution for the gap process and prove uniform geometric ergodicity. Further, when the leader's jump sizes follow an Exponential distribution, we identify the stationary law explicitly as a product of independent Exponential laws, and show that the associated mixing time scales between $\Theta(n)$ and $O(n(\log n)^2)$ for an $n$-particle system. As an application of the mixing time results we establish a functional limit theorem that characterizes fluctuations of particle states at large time, under a suitable spatial and temporal scaling and large particle limit. Finally, when the leader's jumps have heavy but integrable tails, we show that each gap has at least one additional finite moment under stationarity than that of the leader's jump size distribution. The model offers a tractable setting for exploring ergodicity, explicit invariant laws, and mixing behavior in non-diffusive particle systems. + oai:arXiv.org:2601.02501v1 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Sayan Banerjee, Amarjit Budhiraja, Dilshad Imon + + + On well-posed energy/entropy stable boundary conditions for the rotating shallow water equations + https://arxiv.org/abs/2601.02513 + arXiv:2601.02513v1 Announce Type: new +Abstract: We derive and analyze well-posed, energy- and entropy-stable boundary conditions (BCs) for the two-dimensional linear and nonlinear rotating shallow water equations (RSWE) in vector invariant form. The focus of the study is on subcritical flows, which are commonly observed in atmospheric, oceanic, and geostrophic flow applications. We consider spatial domains with smooth boundaries and formulate both linear and nonlinear BCs using mass flux, Riemann's invariants, and Bernoulli's potential, ensuring that the resulting initial boundary value problem (IBVP) is provably entropy- and energy-stable. The linear analysis is comprehensive, providing sufficient conditions to establish the existence, uniqueness, and energy stability of solutions to the linear IBVP. For the nonlinear IBVP, which admits more general solutions, our goal is to develop nonlinear BCs that guarantee entropy stability. We introduce the concepts of linear consistency and linear stability for nonlinear IBVPs, demonstrating that if a nonlinear IBVP is both linearly consistent and linearly stable, then, for sufficiently regular initial and boundary data over a finite time interval, a unique smooth solution exists. Both the linear and nonlinear IBVPs can be efficiently solved using high-order accurate numerical methods. By employing high-order summation-by-parts operators to discretize spatial derivatives and implementing weak enforcement of BCs via penalty techniques, we develop provably energy- and entropy-stable numerical schemes on curvilinear meshes. Extensive numerical experiments are presented to verify the accuracy of the methods and to demonstrate the robustness of the proposed BCs and numerical schemes. + oai:arXiv.org:2601.02513v1 + math.NA + cs.NA + physics.ao-ph + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Kenneth Duru, Chuqiao Xu + + + Diffusion Computation versus Quantum Computation: A Comparative Model for Order Finding and Factoring + https://arxiv.org/abs/2601.02518 + arXiv:2601.02518v1 Announce Type: new +Abstract: We study a hybrid computational model for integer factorization in which the only non-classical resource is access to an \emph{iterated diffusion process} on a finite graph. Concretely, a \emph{diffusion step} is defined to be one application of a symmetric stochastic matrix (the half-lazy walk operator) to an $\ell^{1}$--normalized state vector, followed by an optional readout of selected coordinates. + Let $N\ge 3$ be an odd integer which is neither prime nor a prime power, and let $b\in(\mathbb{Z}/N\mathbb{Z})^\ast$ have odd multiplicative order $r={\rm ord}_N(b)$. We construct, without knowing $r$ in advance, a weighted Cayley graph whose vertex set is the cyclic subgroup $\langle b\rangle$ and whose edges correspond to the powers $b^{\pm 2^t}$ for $t\le \lfloor \log_2 N\rfloor+1$. Using an explicit spectral decomposition together with an elementary doubling lemma, we show that $r$ can be recovered from a single heat-kernel value after at most $O((\log_2 N)^2)$ diffusion steps, with an effective bound. + We then combine this order-finding model with the standard reduction from factoring to order finding (in the spirit of Shor's framework) to obtain a randomized factorization procedure whose success probability depends only on the number $m$ of distinct prime factors of $N$. Our comparison with Shor's algorithm is \emph{conceptual and model-based}. We replace unitary $\ell^2$ evolution by Markovian $\ell^1$ evolution, and we report complexity in two cost measures: digital steps and diffusion steps. Finally, we include illustrative examples and discussion of practical implementations. + oai:arXiv.org:2601.02518v1 + math.SP + cs.CR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Carlos A. Cadavid, Paulina Hoyos, Jay Jorgenson, Lejla Smajlovi\'c, J. D. V\'elez + + + First Provably Optimal Asynchronous SGD for Homogeneous and Heterogeneous Data + https://arxiv.org/abs/2601.02523 + arXiv:2601.02523v1 Announce Type: new +Abstract: Artificial intelligence has advanced rapidly through large neural networks trained on massive datasets using thousands of GPUs or TPUs. Such training can occupy entire data centers for weeks and requires enormous computational and energy resources. Yet the optimization algorithms behind these runs have not kept pace. Most large scale training still relies on synchronous methods, where workers must wait for the slowest device, wasting compute and amplifying the effects of hardware and network variability. Removing synchronization seems like a simple fix, but asynchrony introduces staleness, meaning updates computed on outdated models. This makes analysis difficult, especially when delays arise from system level randomness rather than algorithmic choices. As a result, the time complexity of asynchronous methods remains poorly understood. This dissertation develops a rigorous framework for asynchronous first order stochastic optimization, focusing on the core challenge of heterogeneous worker speeds. Within this framework, we show that with proper design, asynchronous SGD can achieve optimal time complexity, matching guarantees previously known only for synchronous methods. Our first contribution, Ringmaster ASGD, attains optimal time complexity in the homogeneous data setting by selectively discarding stale updates. The second, Ringleader ASGD, extends optimality to heterogeneous data, common in federated learning, using a structured gradient table mechanism. Finally, ATA improves resource efficiency by learning worker compute time distributions and allocating tasks adaptively, achieving near optimal wall clock time with less computation. Together, these results establish asynchronous optimization as a theoretically sound and practically efficient foundation for distributed learning, showing that coordination without synchronization can be both feasible and optimal. + oai:arXiv.org:2601.02523v1 + math.OC + cs.DC + cs.LG + stat.ML + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + 10.25781/KAUST-WH234 + Artavazd Maranjyan + + + Lagrangian slice disks with symplectomorphic exteriors + https://arxiv.org/abs/2601.02524 + arXiv:2601.02524v1 Announce Type: new +Abstract: By modifying a construction of Abe and Tange, we exhibit arbitrarily large families of Lagrangian slice disks with Weinstein deformation equivalent exteriors. This answers a Lagrangian version of a question of Hitt and Sumners. We raise other open questions related to Lagrangian slice disks and their exteriors. + oai:arXiv.org:2601.02524v1 + math.SG + math.GT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Joseph Breen + + + Lee-Yang phenomena in edge-coloured graph counting + https://arxiv.org/abs/2601.02525 + arXiv:2601.02525v1 Announce Type: new +Abstract: We study the accumulation of zeros of a polynomial arising from the enumeration of edge-coloured graphs along certain limit curves. The polynomial is a variant of an edge-chromatic polynomial, which specialises to the partition function of the ferromagnetic Ising model on a random regular graph. We call this accumulation behaviour a Lee-Yang phenomenon in analogy with the Lee-Yang theorem. The limiting loci are semialgebraic and arise from anti-Stokes curves of an exponential integral. + oai:arXiv.org:2601.02525v1 + math.CO + cond-mat.stat-mech + math-ph + math.AG + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Maximilian Wiesmann + + + H\"older estimates of weak solutions to chemotaxis systems of fast diffusion type + https://arxiv.org/abs/2601.02528 + arXiv:2601.02528v1 Announce Type: new +Abstract: We study a quasilinear chemotaxis system of singular type, where the diffusion operator is given by $\Delta u^m$ with $0<m<1$, corresponding to the fast diffusion regime, and where the chemotactic drift is nonlinear. Since H\"older continuity constitutes the optimal regularity class for weak solutions to the porous medium equation, we establish analogous regularity results for bounded solutions of parabolic--parabolic chemotaxis systems in this setting. The proof is based on a refined De Giorgi--Di Benedetto iteration scheme adapted to the coupled structure of the system. These results advance the understanding of the fine regularity properties of chemotaxis models with nonlinear diffusion, and demonstrate that the interplay between singular diffusion and aggregation exhibits a regularizing mechanism consistent with the porous medium paradigm. + oai:arXiv.org:2601.02528v1 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-sa/4.0/ + M. Marras, F. Ragnedda, S. Vernier-Piro, V. Vespri + + + GPU-Accelerated Energy-Conserving Methods for the Hyperbolized Serre-Green-Naghdi Equations in 2D + https://arxiv.org/abs/2601.02540 + arXiv:2601.02540v1 Announce Type: new +Abstract: We develop energy-conserving numerical methods for a two-dimensional hyperbolic approximation of the Serre-Green-Naghdi equations with variable bathymetry for both periodic and reflecting boundary conditions. The hyperbolic formulation avoids the costly inversion of an elliptic operator present in the classical model. Our schemes combine split forms with summation-by-parts (SBP) operators to construct semidiscretizations that conserve the total water mass and the total energy. We provide analytical proofs of these conservation properties and also verify them numerically. While the framework is general, our implementation focuses on second-order finite-difference SBP operators. The methods are implemented in Julia for CPU and GPU architectures (AMD and NVIDIA) and achieve substantial speedups on modern accelerators. We validate the approach through convergence studies based on solitary-wave and manufactured-solution tests, and by comparisons to analytical, experimental, and existing numerical results. All source code to reproduce our results is available online. + oai:arXiv.org:2601.02540v1 + math.NA + cs.NA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Collin Wittenstein (Massachusetts Institute of Technology, Cambridge, USA, Johannes Gutenberg University Mainz, Germany), Vincent Marks (Johannes Gutenberg University Mainz, Germany), Mario Ricchiuto (INRIA Bordeaux, France), Hendrik Ranocha (Johannes Gutenberg University Mainz, Germany) + + + Affine mappings of translation surfaces: shrinking targets and Diophantine properties + https://arxiv.org/abs/2601.02541 + arXiv:2601.02541v1 Announce Type: new +Abstract: Let $(X,\omega)$ be a translation surface whose Veech group $\Gamma$ is a lattice. We prove that the generic orbit of the group of affine homeomorphisms of $(X,\omega)$ can be used to approximate each point of $X$ with Diophantine precision. The proof utilizes an induced $SL_2(\mathbb{R})$-action on a fiber bundle $Y$ whose base is $SL_2(\mathbb{R})/\Gamma$ and whose fiber is $X$. We observe that this bundle embeds as an $SL_2(\mathbb{R})$-orbit closure in the moduli space of once marked translation surfaces, and hence we may invoke the spectral gap results of Avila and Gou\"ezel and a quantitative mean ergodic theorem for the $SL_2(\mathbb{R})$-action on the mean-zero, square-integrable functions on $Y$. + oai:arXiv.org:2601.02541v1 + math.DS + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Chris Judge, Josh Southerland + + + The fine spectral expansion of the Rankin-Selberg period + https://arxiv.org/abs/2601.02542 + arXiv:2601.02542v1 Announce Type: new +Abstract: We state and prove the spectral expansion of the theta series attached to the Rankin-Selberg spherical variety $(\mathrm{GL}_{n+1} \times \mathrm{GL}_n)/\mathrm{GL}_n$. This is a key result towards the fine spectral expansion of the Jacquet-Rallis trace formula. Our expansion is written in terms of regularized Rankin--Selberg periods for non-tempered automorphic representations, which we show compute special values of $L$-functions. The proof relies on shifts of contours of integration \`a la Langlands. We also establish two technical but crucial results on bounds and singularities for discrete Eisenstein series of $\mathrm{GL}_n$ in the positive Weyl chamber. + oai:arXiv.org:2601.02542v1 + math.NT + math.RT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Paul Boisseau + + + Construction of groups with triality and their corresponding code loops + https://arxiv.org/abs/2601.02546 + arXiv:2601.02546v1 Announce Type: new +Abstract: We generalize the global construction of code loops introduced by Nagy, which is based on the connection between Moufang loops and groups with triality. This follows from the construction of a nilpotent group $G_n$ of class 3 with triality and $2n$ generators, based on embeddings of $G_n$ into direct products of copies of $G_3$. In the finite case, where $G_n$ is a group such that $|G_n| = 2^{4n+m}$ with $n \ge 3$ and $m = 3 {n \choose 2} + 2 {n \choose 3}$, we prove that the corresponding Moufang loop is the free loop $F_n$ with $n$ generators in the variety generated by code loops. The result depends on a construction similar to that of $G_n$, namely, embedding $F_n$ into direct products of copies of $F_3$, the free code loop associated with $G_3$. + oai:arXiv.org:2601.02546v1 + math.GR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Rosemary Miguel Pires, Alexandre Grishkov, Rodrigo Lucas Rodrigues, Marina Rasskazova + + + Tree metrics and log-concavity for matroids + https://arxiv.org/abs/2601.02547 + arXiv:2601.02547v1 Announce Type: new +Abstract: We show that a set function $\nu$ satisfies the gross substitutes property if and only if its homogeneous generating polynomial $Z_{q,\nu}$ is a Lorentzian polynomial for all positive $q \le 1$, answering a question of Eur-Huh. We achieve this by giving a rank 1 upper bound for the distance matrix of an ultrametric tree, refining a classical result of Graham-Pollak. This characterization enables us to resolve two open problems that strengthen Mason's log-concavity conjectures for the number of independent sets of a matroid: one posed by Giansiracusa-Rinc\'on-Schleis-Ulirsch for valuated matroids, and another posed by Pak for ordinary matroids. + oai:arXiv.org:2601.02547v1 + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Federico Ardila-Mantilla, Sergio Cristancho, Graham Denham, Christopher Eur, June Huh, Botong Wang + + + Lamperti scaling for fractional Gaussian processes with non-stationary increments + https://arxiv.org/abs/2601.02558 + arXiv:2601.02558v1 Announce Type: new +Abstract: The Lamperti transform offers a powerful bridge between self-similar processes and stationary dynamics, making it especially useful for analyzing anomalous diffusion models that lack stationary increments. In this paper we examine the Lamperti transforms of scaled sub-fractional and bi-fractional Brownian motions, deriving explicit covariance formulas, asymptotic behaviour, and precise exponential mixing rates. We also introduce Langevin type integral processes driven by these Gaussian fields, identify their self-similarity exponents, and show that their Lamperti images again form stationary Gaussian processes with rapid decorrelation. Through inverse Lamperti relations and Birkhoff's theorem, we establish rigorous single trajectory reconstruction of ensemble quantities for the original non-stationary processes. The results extend the scope of the scaled Lamperti framework to Gaussian processes with non-stationary increments and richer dependence structures. + oai:arXiv.org:2601.02558v1 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Foad Shokrollahi, Saeed Vahdati + + + A Schr\"odinger-Based Dispersive Regularization Approach for Numerical Simulation of One-Dimensional Shallow Water Equations + https://arxiv.org/abs/2601.02561 + arXiv:2601.02561v1 Announce Type: new +Abstract: We propose a novel dispersive regularization framework for the numerical simulation of the one-dimensional shallow water equations (SWE). The classical hyperbolic system is regularized by a third-order dispersive term in the momentum equation, which renders the system equivalent, via the Madelung transform, to a defocusing cubic nonlinear Schr\"odinger equation with a drift term induced by bottom topography. + Instead of solving the shallow water equations directly, we solve the associated Schr\"odinger equation and recover the hydrodynamic variables through a simple postprocessing procedure. This approach transforms the original nonlinear hyperbolic system into a semilinear complex-valued equation, which can be efficiently approximated using a Strang time-splitting method combined with a spectral element discretization in space. + Numerical experiments demonstrate that, in subcritical regimes without shock formation, the Schr\"odinger regularization provides an $O(\varepsilon)$ approximation to the classical shallow water solution, where $\varepsilon$ denotes the regularization parameter. Importantly, we observe that this convergence behavior persists even in the presence of moving wetting--drying interfaces, where vacuum states emerge and standard shallow water solvers often encounter difficulties. These results suggest that the Schr\"odinger-based formulation offers a robust and promising alternative framework for the numerical simulation of shallow water flows with dry states. + oai:arXiv.org:2601.02561v1 + math.NA + cs.NA + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Guosheng Fu, Chun Liu + + + Travelling Waves in a Mathematical Model for Oncolytic Virotherapy + https://arxiv.org/abs/2601.02568 + arXiv:2601.02568v1 Announce Type: new +Abstract: Oncolytic virotherapy (OVT) is a promising cancer treatment strategy in which engineered viruses selectively infect and destroy tumor cells. Motivated by the biological mechanisms underlying viral spread and tumor invasion into the tissue, we analyze a non-cooperative reaction-diffusion model capturing the invasion of tumor tissue by oncolytic viruses. Using carefully constructed upper and lower solutions together with Schauder's fixed point theorem, we establish the existence of positive travelling-wave solutions. In particular, we identify a minimal wave speed value $\bar c$ such that positive travelling waves exist for all $c \geq \bar c$ . Our analysis also highlights parameter regions where the existence of travelling waves remains ambiguous, suggesting new mathematical questions about the propagation of viral treatments through tumor environments. + oai:arXiv.org:2601.02568v1 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Negar Mohammadnejad, Thomas Hillen + + + A modern perspective on Tutte's homotopy theorem + https://arxiv.org/abs/2601.02582 + arXiv:2601.02582v1 Announce Type: new +Abstract: We begin with a review of Tutte's homotopy theory, which concerns the structure of certain graph associated to a matroid (together with some extra data). Concretely, Tutte's path theorem asserts that this graph is connected, and his homotopy theorem asserts that every cycle in the graph is a composition of ''elementary cycles'', which come in four different flavors. We present an extended version of the homotopy theorem, in which we give a more refined classification of the different types of elementary cycles. We explain in detail how the path theorem allows one to prove that the foundation of a matroid (in the sense of Baker--Lorscheid) is generated by universal cross-ratios, and how the extended homotopy theorem allows one to classify all algebraic relations between universal cross-ratios. The resulting ''fundamental presentation'' of the foundation was previously established in [Baker--Lorscheid], but the argument here is more self-contained. We then recall a few applications of the fundamental presentation to the representation theory of matroids. Finally, in the most novel but also the most speculative part of the paper, we discuss what a ''higher Tutte homotopy theorem'' might look like, and we present some preliminary computations along these lines. + oai:arXiv.org:2601.02582v1 + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Matthew Baker, Tong Jin, Oliver Lorscheid + + + The fiber product of the Torelli map with any product $\mathcal{A}_{g_1}\times \dots \times \mathcal{A}_{g_k}\to\mathcal{A}_g$ is reduced + https://arxiv.org/abs/2601.02592 + arXiv:2601.02592v1 Announce Type: new +Abstract: We prove that the fiber product of the Torelli map $t\colon \mathcal{M}^{ct}_g \to \mathcal{A}_g$ with any product $\mathcal{A}_{g_1}\times\dots\times \mathcal{A}_{g_k} \to \mathcal{A}_g$ for $g=g_1+\dots+g_k$ has a reduced scheme structure. As a consequence, letting $d=\text{codim}(t^*[\mathcal{A}_{g_1}\times\dots\times \mathcal{A}_{g_k}])$, we find that the class $t^*[\mathcal{A}_{g_1}\times\dots\times \mathcal{A}_{g_k}]\in \mathsf{CH}^{d}(\mathcal{M}^{ct}_g)$ is tautological. In particular, we obtain $t^*[\mathcal{A}_{g_1}\times\dots\times \mathcal{A}_{g_k}] = 0$ for $d > 2g-3.$ + oai:arXiv.org:2601.02592v1 + math.AG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Lycka Drakengren + + + Volumetric locking-free Mixed Virtual Element Methods for Contact Problems + https://arxiv.org/abs/2601.02595 + arXiv:2601.02595v1 Announce Type: new +Abstract: We consider the approximation of the 2D frictionless contact problem in elasticity using the Virtual Element Methods (VEMs). To overcome the volumetric locking phenomenon in the nearly incompressible case, we adopt a mixed displacement/pressure ($u/p$) variational formulation, where pressure is introduced as an independent unknown. We present the VEM discretization and develop a general error analysis, keeping explicit track of the constants involved in the error estimates, thus allowing to consider meshes with "small edges". As examples, we consider two possible VEM schemes: a first-order scheme and a second-order scheme. The numerical results confirm the theoretical predictions, specifically both schemes show: 1) robustness with respect to the volumetric parameter $\lambda$, thus preventing the occurrence of the volumetric locking phenomenon; 2) good behavior even in the presence of "small edges"; 3) achievement of the expected theoretical convergence rates. + oai:arXiv.org:2601.02595v1 + math.NA + cs.NA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + C. Lovadina, L. Molinari + + + Extremum Seeking Control for Wave-PDE Actuation with Distributed Effects + https://arxiv.org/abs/2601.02607 + arXiv:2601.02607v1 Announce Type: new +Abstract: This paper deals with the gradient-based extremum seeking control (ESC) with actuation dynamics governed by distributed wave partial differential equations (PDEs). To achieve the control objective of real-time optimization for this class of infinite-dimensional systems, we first solve the trajectory generation problem to re-design the additive perturbation signal of the ESC system. Then, we develop a boundary control law through the backstepping method to compensate for the wave PDE with distributed effects, which ensures the exponential stability of the average closed-loop system by means of a Lyapunov-based analysis. At last, by employing the averaging theory for infinite-dimensional systems, we prove that the closed-loop trajectories converge to a small neighborhood surrounding the optimal point. Numerical simulations are presented to illustrate the effectiveness of the proposed method. + oai:arXiv.org:2601.02607v1 + math.OC + cs.SY + eess.SY + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Elisio Juvenal Muchave, Pedro Henrique Silva Coutinho, Tiago Roux Oliveira, Miroslav Krsti\'c + + + Weights on finite fields and failures of the MacWilliams identities + https://arxiv.org/abs/2601.02608 + arXiv:2601.02608v1 Announce Type: new +Abstract: In the 1960s, MacWilliams proved that the Hamming weight enumerator of a linear code over a finite field completely determines, and is determined by, the Hamming weight enumerator of its dual code. In particular, if two linear codes have the same Hamming weight enumerator, then their dual codes have the same Hamming weight enumerator. + In contrast, there is a wide class of weights on finite fields whose weight enumerators have the opposite behavior: there exist two linear codes having the same weight enumerator, but their dual codes have different weight enumerators. + oai:arXiv.org:2601.02608v1 + cs.IT + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jay A. Wood + + + Antidiagonal Initial Complexes of Infinite Matrix Schubert Varieties are Cohen-Macaulay + https://arxiv.org/abs/2601.02612 + arXiv:2601.02612v1 Announce Type: new +Abstract: We show that, under certain constraints, the Stanley-Reisner ring of an infinite simplicial complex is Cohen-Macaulay in the sense of ideals and weak Bourbaki unmixed. We apply this result to prove the wanted claim -- that initial complexes of matrix Schubert varieties corresponding to infinite permutations in $S_{\infty}$ with respect to an antidiagonal term order are Cohen-Macaulay (in the same sense), giving rise to new examples of non-Noetherian Cohen-Macaulay rings. + oai:arXiv.org:2601.02612v1 + math.AC + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Anna Natalie Chlopecki, Nathaniel Gallup, Jason Meintjes + + + A spectral product formula for repunits via a tridiagonal Toeplitz similarity + https://arxiv.org/abs/2601.02615 + arXiv:2601.02615v1 Announce Type: new +Abstract: For $b>0$ and $n\geqslant 1$, we consider the $n\times n$ tridiagonal matrix $V_n(b)$ with diagonal entries $b+1$, superdiagonal entries $1$, and subdiagonal entries $b$. A diagonal similarity reduces $V_n(b)$ to a symmetric tridiagonal Toeplitz matrix and hence makes its spectrum explicit. Since $\det\left(V_n(b)\right)$ equals the geometric sum $1+b+\cdots+b^{n}$, taking determinants yields a finite cosine product evaluation for this quantity. As further consequences, we derive sharp bounds from the extremal eigenvalues, write down explicit eigenvectors with respect to a natural weighted inner product, and obtain a closed formula for $V_n(b)^{-1}$. + oai:arXiv.org:2601.02615v1 + math.SP + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Johann Verwee + + + Mass splitting in the time-discrete generalized Euler equations and non-Monge solutions in multi-marginal optimal transport + https://arxiv.org/abs/2601.02616 + arXiv:2601.02616v1 Announce Type: new +Abstract: The time-discretized, spatially continuous generalized Euler equations are a prototype example of multi-marginal optimal transport, yet the question whether they exhibit mass-splitting (or equivalently, whether they have solutions that are not of Monge form) has remained open. Here we resolve this question by giving a mass-splitting example in one spatial dimension. Moreover we present a related and very simple fully discrete example of mass-splitting which reveals a transparent underlying mechanism. + oai:arXiv.org:2601.02616v1 + math.AP + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Gero Friesecke + + + $K$-Bad Spheres + https://arxiv.org/abs/2601.02620 + arXiv:2601.02620v1 Announce Type: new +Abstract: In this paper we look at the $E$-completion of topological spaces where $E$ is a $p$-local ring spectrum. After a brief review of the concept of $E$-completion, we specialize to the case where $E=K$, $p$-local complex periodic $K$-theory, and consider the $K$-theory of the unstable sphere $S^{2n+1}$. We show that for certain values of $n$ and an odd prime $p$, the $K$-homology of the $K$-completion is not isomorphic to the $K$-homology of the sphere itself, thus in the terminology of Bousfield and Kan, these spheres are '$K$-bad'. + oai:arXiv.org:2601.02620v1 + math.AT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Martin Bendersky, Robert Thompson + + + Local Asymptotic Normality for Mixed Fractional Brownian Motion with $H>3/4$ Under High-Frequency Observation + https://arxiv.org/abs/2601.02622 + arXiv:2601.02622v1 Announce Type: new +Abstract: In this paper we will consider the LAN property for both the Hurst parameter $H>3/4$ and the variance of the fractional Brownian motion plus an independent standard Brownian motion (called mixed fractional Brownian motion) with high-frequency observation. We will first remove the $H$-score linear term and orthogonalize the remainder through two non-diagonal transformations, then we can construct the CLT for the quadratic form base on $\| \cdot \|_{\mathrm{op}}/\|\cdot\|_F\to0$. At last we obtain a diagonal Gaussian LAN expansion with an explicit information matrix. + oai:arXiv.org:2601.02622v1 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Chunhao Cai + + + Joint extreme values of the Riemann zeta function at harmonic points + https://arxiv.org/abs/2601.02623 + arXiv:2601.02623v1 Announce Type: new +Abstract: Using the resonance method, we obtain refined estimates for joint extreme values of the Riemann zeta function at harmonic points, improving upon Levinson's 1972 results and providing new insight into the behavior of the Riemann zeta function. Our proof is primarily based on Dirichlet series theory and the truncated Euler product for the Riemann zeta function. As a corollary, we can recover some previously known extreme value results for the zeta function. + oai:arXiv.org:2601.02623v1 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Qiyu Yang, Shengbo Zhao + + + New approach for elastic collisions with singular stress functions + https://arxiv.org/abs/2601.02639 + arXiv:2601.02639v1 Announce Type: new +Abstract: A collision of a rubber rod to a hard floor is regarded as a simple example of obstacle problems for elastic material. In this article we have proposed a new mathematical model for the collision phenomenon by applying beam equations with singular stress functions, which is investigated in our recent works. As in the works we have established a mathematical method to deal with the singular stress function. Here, we demonstrate the validity of our modeling through observation to the numerical results. Also, we present existence and uniqueness results of the model given as initial boundary value problems. + oai:arXiv.org:2601.02639v1 + math.AP + math-ph + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Toyohiko Aiki, Chiharu Kosugi + + + Quasiconvexity in the Riemannian setting + https://arxiv.org/abs/2601.02642 + arXiv:2601.02642v1 Announce Type: new +Abstract: We introduce a notion of quasiconvexity for continuous functions $f$ defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold $(M,g)$ and $\mathbb{R}^m$, naturally generalizing the classical Euclidean definition. We prove that this condition characterizes the sequential lower semicontinuity of the associated integral functional \[ F(u, \Omega) = \int_{\Omega} f(du) \, d\mu \] with respect to the weak$^*$ topology of $W^{1,\infty}(\Omega, \mathbb{R}^m)$, for every bounded open subset $\Omega\subseteq M$. + oai:arXiv.org:2601.02642v1 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Aurora Corbisiero, Chiara Leone, Carlo Mantegazza + + + A Derivative-Free Saddle-search Algorithm With Linear Convergence Rate + https://arxiv.org/abs/2601.02650 + arXiv:2601.02650v1 Announce Type: new +Abstract: We propose a derivative-free saddle-search algorithm designed to locate transition states using only function evaluations. The algorithm employs a nested architecture consisting of an inner eigenvector search and an outer saddle-point search. Through rigorous numerical analysis, we prove the almost sure convergence of the inner step under suitable assumptions. Furthermore, we establish the convergence of the outer search using a decaying step size, while demonstrating linear convergence under constant step size and boundedness conditions. Numerical experiments are provided to validate our theoretical results and demonstrate the algorithm's practical applicability. + oai:arXiv.org:2601.02650v1 + math.NA + cs.NA + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Qiang Du, Baoming Shi, Lei Zhang, Xiangcheng Zheng + + + Incubulable hyperbolic 3-pseudomanifold groups + https://arxiv.org/abs/2601.02655 + arXiv:2601.02655v1 Announce Type: new +Abstract: We construct compact hyperbolic 3-manifolds with totally geodesic boundary, such that the closed 3-pseudomanifolds obtained by coning off the boundary components are negatively curved and contain locally convex subspaces whose fundamental groups have property (T). In particular, the fundamental groups of these 3-pseudomanifolds are word hyperbolic but not cubulable. + oai:arXiv.org:2601.02655v1 + math.GT + math.GR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jason Manning, Lorenzo Ruffoni + + + Powerful Fibonacci polynomials over finite fields + https://arxiv.org/abs/2601.02664 + arXiv:2601.02664v1 Announce Type: new +Abstract: Bugeaud, Mignotte, and Siksek proved that the only perfect powers in Fibonacci sequence are 0, 1, 8, and 144. In this paper, we study the polynomial analogue of the problem. Especially, we give a complete characterization of the Fibonacci polynomials that are perfect powers or powerful over finite fields, where there are infinitely many of them. We also give similar characterizations for some of Horadam's generalized Lucas polynomial sequences, which include Fibonacci, Lucas, Chebyshev, and Jacobsthal polynomials. + oai:arXiv.org:2601.02664v1 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Graeme Bates, Ryan Jesubalan, Seewoo Lee, Jane Lu, Hyewon Shim + + + The weighted Forman and Lin-Lu-Yau Ricci flow on graphs + https://arxiv.org/abs/2601.02673 + arXiv:2601.02673v1 Announce Type: new +Abstract: In this paper, we propose a type of Ricci flow on graphs where the probability distribution for the Lin-Lu-Yau curvature remains constant over time, and also study the related Forman curvature flow. These two curvature flows coincide on trees. We first prove the existence and uniqueness of solutions for both curvature flows in general graphs. Then, we obtain that the normalized curvature flow on trees converges to a constant curvature metric, and under the uniform measure, a complete classification of trees can be obtained based on the convergence results. + oai:arXiv.org:2601.02673v1 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Shuliang Bai, Shuang Liu, Xin Lai + + + Branching $k$-path vertex cover of forests + https://arxiv.org/abs/2601.02685 + arXiv:2601.02685v1 Announce Type: new +Abstract: We define a set $P$ to be a branching $k$-path vertex cover of an undirected forest $F$ if all leaves and isolated vertices (vertices of degree at most $1$) of $F$ belong to $P$ and every path on $k$ vertices (of length $k-1$) contains either a branching vertex (a vertex of degree at least $3$) or a vertex belonging to $P$. We define the branching $k$-path vertex cover number of an undirected forest $F$, denoted by $\psi_b(F,k)$, to be the number of vertices in the smallest branching $k$-path vertex cover of $F$. These notions for a rooted directed forest are defined similarly, with natural adjustments. We prove the lower bound $\psi_b(F,k) \geq \frac{n+3k-1}{2k}$ for undirected forests, the lower bound $\psi_b(F,k) \geq \frac{n+k}{2k}$ for rooted directed forests, and that both of them are tight. + oai:arXiv.org:2601.02685v1 + math.CO + cs.DM + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mikhail Makarov + + + Revisiting a Fast Newton Solver for a 2-D Spectral Estimation Problem: Computations with the Full Hessian + https://arxiv.org/abs/2601.02690 + arXiv:2601.02690v1 Announce Type: new +Abstract: Spectral estimation plays a fundamental role in frequency-domain identification and related signal processing problems. This paper revisits a 2-D spectral estimation problem formulated in terms of convex optimization. More precisely, we work with the dual optimization problem and show that the full Hessian of the dual function admits a Toeplitz-block Toeplitz structure which is consistent with our finding in a previous work. This particular structure of the Hessian enables a fast inversion algorithm in the solution of the dual optimization problem via Newton's method whose superior speed of convergence is illustrated via simulations. + oai:arXiv.org:2601.02690v1 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ji Cheng, Bin Zhu + + + A Classification of Fractal Squares + https://arxiv.org/abs/2601.02696 + arXiv:2601.02696v1 Announce Type: new +Abstract: Let $\lambda_K:\bbR^2\rightarrow\{0,1,\ldots\}\cup\{\infty\}$ be the lambda function of a planar comapctum $K$, as defined in MR4488162. It is known that a planar continuum is locally connected if and only if its lambda function vanishes everywhere, or equivalently, $\lambda_K(K)=\{0\}$. In this article we show that every fractal square $K$ satisfies $\lambda_K(K)\subset\{0,1\}$ and find criterions to classify when $\lambda_K(K)$ equals $\{0\}$, $\{1\}$ or $\{0,1\}$. Here for any integer $N\ge2$ and any set $\Dc=\left\{(i,j): 0\le i,j\le N-1\right\}$ with cardinality $\ge2$, if we set $K^{(0)}=[0,1]^2$ and $\displaystyle K^{(n)}=\left\{\frac{x+d}{N}: x\in K^{(n-1)}, d\in\Dc\right\}(n\ge1)$ then $K=\bigcap_nK^{(n)}$ is called a fractal square. + oai:arXiv.org:2601.02696v1 + math.GN + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Gregory Conner, Curtis Kent, Jun Luo, Yi Yang + + + Diffusion limit for the stationary distribution of a history-dependent two-level M/M/1 queue + https://arxiv.org/abs/2601.02705 + arXiv:2601.02705v1 Announce Type: new +Abstract: Recently, Atar and Miyazawa [2] introduced a multi-level GI/G/1 queue with a finite number of levels, where both the arrival and service rates depend on the level corresponding to the current queue length. For this model, they proved that the diffusion limit of its queue length process in heavy traffic is the level-dependent reflected Brownian motion of [6]. In a subsequent study, Kobayashi et al. [4] derived the corresponding diffusion limit of the stationary distribution. These studies are motivated by the control of service capacity depending on the queue length. We are interested in the more general case where this control may also depend on the history of the queue length. As the first step toward such a generalization, we specialize the multi-level GI/G/1 queue to a two-level M/M/1 queue. We then extend the dynamics of this model so that its arrival and service rates depend not only on the current queue length but also on the recent history of queue lengths. Under the stability condition for this model, we first compute its stationary distribution in closed form, then derive its diffusion limit in heavy traffic. Finally, using this diffusion limit, we derive approximation formulas for the stationary distribution and then numerically assess their accuracy. + oai:arXiv.org:2601.02705v1 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Masahiro Kobayashi, Masakiyo Miyazawa, Yutaka Sakuma + + + The Effective Ehrenpreis Conjecture + https://arxiv.org/abs/2601.02710 + arXiv:2601.02710v1 Announce Type: new +Abstract: Let $M$ and $N$ be two closed hyperbolic Riemann surfaces. The Ehrenpreis Conjecture (proved by Kahn-Markovic) asserts that for any $\epsilon>0$ there are finite covers $M_\epsilon \to M$, and $N_\epsilon \to N$, such that the Teichmuller distance (in the suitable moduli space) between $M_\epsilon$ and $N_\epsilon$ is less than $\epsilon$. It is natural to ask how large the degrees of these coverings need to be to achieve that the distance between $M_\epsilon$ and $N_\epsilon$ is less than $\epsilon$. In this paper we show that there exists a constant $k>0$, depending only on $M$ and $N$, so that the covers $M_\epsilon \to M$, and $N_\epsilon \to N$, can be chosen to have the degrees less than $\epsilon^{-k}$. We show that this bound is optimal by considering the case when $M$ and $N$ are arithmetic Riemann surfaces with the same invariant trace field which are not commensurable to each other. + oai:arXiv.org:2601.02710v1 + math.GT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Qiliang Luo + + + Symmetric quiver varieties and critical stable envelopes + https://arxiv.org/abs/2601.02719 + arXiv:2601.02719v1 Announce Type: new +Abstract: Symmetric quiver varieties with potentials are natural generalizations of Nakajima quiver varieties, and their equivariant critical cohomologies provide more flexible settings for geometric representation theory and enumerative geometry. In this paper, we study their geometric properties and show that they behave like universally deformed Nakajima quiver varieties. Based on this, we provide a new proof of the existence of critical stable envelopes on them. Following an idea of Nakajima, we give a sheaf theoretic interpretation of critical stable envelopes by the hyperbolic restriction in the affinization of symmetric quiver varieties. The associativity of hyperbolic restrictions implies the triangle lemma of critical stable envelopes. + oai:arXiv.org:2601.02719v1 + math.AG + hep-th + math-ph + math.MP + math.RT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Yalong Cao, Andrei Okounkov, Yehao Zhou, Zijun Zhou + + + Manifolds with harmonic curvature and curvature operator of the second kind + https://arxiv.org/abs/2601.02722 + arXiv:2601.02722v1 Announce Type: new +Abstract: We prove that compact Riemannian manifolds of dimension $n\ge3$ with harmonic curvature and $\frac{n(n+2)}{2(n+1)}$-nonnegative curvature operator of the second kind must be Einstein. In particular, Building upon Dai-Fu's work \cite{DF}, it follows that if the curvature operator of the second kind is $\min\{\frac{n(n+2)}{2(n+1)},\max\{4,\frac{(n+2)}{4}\}\}$-nonnegative , then such a manifold must be of constant curvature. + oai:arXiv.org:2601.02722v1 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Haiping Fu, Yao Lu, Zhilin Dai + + + Sampling non-log-concave densities via Hessian-free high-resolution dynamics + https://arxiv.org/abs/2601.02725 + arXiv:2601.02725v1 Announce Type: new +Abstract: We study the problem of sampling from a target distribution $\pi(q)\propto e^{-U(q)}$ on $\mathbb{R}^d$, where $U$ can be non-convex, via the Hessian-free high-resolution (HFHR) dynamics, which is a second-order Langevin-type process that has $e^{-U(q)-\frac12|p|^2}$ as its unique invariant distribution, and it reduces to kinetic Langevin dynamics (KLD) as the resolution parameter $\alpha\to0$. The existing theory for HFHR dynamics in the literature is restricted to strongly-convex $U$, although numerical experiments are promising for non-convex settings as well. We focus on studying the convergence of HFHR dynamics when $U$ can be non-convex, which bridges a gap between theory and practice. Under a standard assumption of dissipativity and smoothness on $U$, we adopt the reflection/synchronous coupling method. This yields a Lyapunov-weighted Wasserstein distance in which the HFHR semigroup is exponentially contractive for all sufficiently small $\alpha>0$ whenever KLD is. We further show that, under an additional assumption that asymptotically $\nabla U$ has linear growth at infinity, the contraction rate for HFHR dynamics is strictly better than that of KLD, with an explicit gain. As a case study, we verify the assumptions and the resulting acceleration for three examples: a multi-well potential, Bayesian linear regression with $L^p$ regularizer and Bayesian binary classification. We conduct numerical experiments based on these examples, as well as an additional example of Bayesian logistic regression with real data processed by the neural networks, which illustrates the efficiency of the algorithms based on HFHR dynamics and verifies the acceleration and superior performance compared to KLD. + oai:arXiv.org:2601.02725v1 + math.PR + stat.ML + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Xiaoyu Wang, Yingli Wang, Lingjiong Zhu + + + Remark about scalar curvature on certain noncompact manifolds + https://arxiv.org/abs/2601.02726 + arXiv:2601.02726v1 Announce Type: new +Abstract: We give a sufficient condition to rule out complete Riemannian metrics with nonnegative scalar curvature on the interior of handlebodies. In higher dimensions, we give examples of ends of manifolds with positive scalar curvature metrics. + oai:arXiv.org:2601.02726v1 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + John Lott + + + Almost complex totally geodesic surfaces in the nearly K\"ahler $\frac{\text{SL}(3,\mathbb R)}{\mathbb R\times \text{SO}(2)}$ + https://arxiv.org/abs/2601.02733 + arXiv:2601.02733v1 Announce Type: new +Abstract: We give a detailed description of the nearly K\"ahler $\frac{\mathrm{SL}(3,\mathbb R)}{\mathbb R\times \mathrm{SO}(2)}$, which is one of the pseudo-Riemannian counterparts of the flag manifold $F(\mathbb{C}^3)$. The main result is the classification of totally geodesic almost complex surfaces in this space. + oai:arXiv.org:2601.02733v1 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mateo Anarella, Xiuxiu Cheng, Marie D'haene, Zejun Hu, Luc Vrancken + + + Stacks of p-adic shtukas and spatial kimberlites + https://arxiv.org/abs/2601.02741 + arXiv:2601.02741v1 Announce Type: new +Abstract: The main purpose of this article is to show that the special Newton polygon map from the stack of p-adic shtukas to the stack of G-bundles on the Fargues--Fontaine curve is representable in diamonds and sufficiently nice for cohomological considerations (i.e. fdcs). The second purpose is to show that the $\bar{\mathbb{F}}_p$-fibers of the special Newton polygon map behave like formal schemes, and in particular, satisfy henselianity properties with respect to their reduced locus. These two goals achieved in this article are two of the crucial ingredients used in our collaboration with Hamman, Ivanov, Louren\c{c}o and Zou to construct the equivalence that compares the schematic and analytic local Langlands categories of Zhu and of Fargues--Scholze. To achieve these goals, we introduce and study spatial kimberlites, which is a better behaved variant of the theory previously developed by the author. + oai:arXiv.org:2601.02741v1 + math.AG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ian Gleason + + + Generalized Double Duals of the Riemann Tensor in Geometry and Gravity + https://arxiv.org/abs/2601.02742 + arXiv:2601.02742v1 Announce Type: new +Abstract: The Riemann curvature tensor fully encodes local geometry, but its Ricci contraction retains only limited information: only the Ricci tensor and the scalar curvature survive, while the Weyl curvature vanishes identically. We show that contracting instead the double dual of the Riemann tensor unlocks the full curvature structure, producing a canonical hierarchy of symmetric, divergence--free $(p,p)$ double forms. These tensors satisfy the first Bianchi identity and obey a hereditary contraction relation interpolating between the double dual tensor and the Einstein tensor. + We prove that, in a generic geometric setting, each tensor in this hierarchy is the unique divergence--free $(p,p)$ double form depending linearly on the Riemann curvature tensor, thereby providing canonical higher--rank parents of the Einstein tensor. + Their sectional curvatures coincide with the $p$--curvatures; notably, the $2$--curvature determines the full Riemann curvature tensor and forces the $\hat A$--genus of a compact spin manifold to vanish when nonnegative, a property not shared by Ricci or scalar curvature. Finally, we extend the construction to Gauss--Kronecker curvature tensors and Lovelock theory, showing in particular that the second Lovelock tensor $T_4$ admits a genuine four--index parent tensor. + oai:arXiv.org:2601.02742v1 + math.DG + math-ph + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mohammed Larbi Labbi + + + Counting Polynomial-type Exceptional Units on Algebraic Varieties over Number Fields + https://arxiv.org/abs/2601.02743 + arXiv:2601.02743v1 Announce Type: new +Abstract: Previous research on exceptional units has primarily focused on the ring of rational integers or abstract finite rings, often restricted to linear or quadratic constraints. In this paper, we extend the concept of polynomial-type exceptional units to the ring of integers of an arbitrary algebraic number field. We investigate the number of these polynomial-type exceptional units on general algebraic varieties. By employing the Chinese Remainder Theorem and Hensel's lifting technique, we derive an exact counting formula for the number of these exceptional units on a smooth closed subscheme under the assumption of good reduction. Furthermore, using the Lang-Weil inequality, we establish an asymptotic estimate for the counting function. In particular, we prove that for varieties of degree at most two, the error term can be significantly improved, yielding a sharper asymptotic bound. + oai:arXiv.org:2601.02743v1 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Chen Lin, Kaihan Tang + + + Affirmative Results on a Conjecture on the Column Space of the Adjacency Matrix + https://arxiv.org/abs/2601.02746 + arXiv:2601.02746v1 Announce Type: new +Abstract: The Akbari-Cameron-Khosrovshahi (ACK) conjecture, which appears to be unresolved, states that for any simple graph $G$ with at least one edge, there exists a nonzero {$\{0,1\}$}-vector in the row space of its adjacency matrix that is not a row of the matrix itself. In this talk, we present a unified framework that includes several families and operations of graphs that satisfy the ACK conjecture. Using these fundamental results, we introduce new graph constructions and demonstrate, through graph structural and linear algebraic arguments, that these constructions adhere to the conjecture. Further, we show that certain graph operations preserve the ACK property. These results collectively expand the known classes of graphs satisfying the conjecture and provide insight into its structural invariance under composition and extension. + oai:arXiv.org:2601.02746v1 + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/publicdomain/zero/1.0/ + S. Akansha, K. C. Sivakumar + + + Data-Driven Output-Based Approach to the Output Regulation Problem of Unknown Linear Systems via Value Iteration + https://arxiv.org/abs/2601.02748 + arXiv:2601.02748v1 Announce Type: new +Abstract: The output regulation problem for unknown linear systems has been studied using state-based and output-based internal model approaches in the special case with no disturbances. This paper further investigates the output regulation problem for unknown linear systems using a data-driven output-based approach via value iteration. For this purpose, we first develop a novel output-feedback control law that does not explicitly rely on the observer gain to solve the output regulation problem. We then show that the data-driven approach for designing an output-feedback control law for the given plant can be reduced to the data-driven design of a state-feedback control law for a well-defined augmented auxiliary system. As a result, we develop a systematic data-driven approach to solve the output regulation problem for unknown linear systems via value iteration. Finally, we establish a relation between the data-driven state-feedback control law and the data-driven output-feedback control law in the LQR sense. + oai:arXiv.org:2601.02748v1 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Haoyan Lin, Jie Huang + + + Boundary operators in the Brownian loop soup + https://arxiv.org/abs/2601.02755 + arXiv:2601.02755v1 Announce Type: new +Abstract: We obtain infinitely many boundary operators in the Brownian loop soup in the subcritical phase by analyzing the conformal block expansion of the two-point function that computes the probability of having two marked points on the upper half-plane being separated by Brownian loops. The resulting boundary operators are primary operators in a 2D CFT with central charge $c\leq1$ and have conformal dimensions that are non-negative integers. By comparing the above-mentioned conformal block expansion with probabilities in the Brownian loop soup, we provide a physical interpretation of the boundary operators of even dimensions as operators that insert multiple outer boundaries of Brownian loops at points on the real axis. + oai:arXiv.org:2601.02755v1 + math-ph + cond-mat.stat-mech + hep-th + math.MP + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Federico Camia, Rongvoram Nivesvivat + + + The automorphism groups of generalized Kausz compactifications and spaces of complete collineations + https://arxiv.org/abs/2601.02768 + arXiv:2601.02768v1 Announce Type: new +Abstract: In this paper, we determine the automorphism groups of generalized Kausz compactifications $\mathcal T_{s,p,n}$. By establishing the (semi-)positivity of the anticanonical bundles of $\mathcal T_{s,p,n}$, we also determine the automorphism groups of generalized spaces of complete collineations $\mathcal M_{s,p,n}$. The results in this paper are partially taken from the author's earlier arxiv post (Canonical blow-ups of grassmann manifolds, arxiv:2007.06200). + oai:arXiv.org:2601.02768v1 + math.CV + math.AG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Hanlong Fang + + + On rates of convergence in central limit theorems of Selberg and Bourgade + https://arxiv.org/abs/2601.02781 + arXiv:2601.02781v1 Announce Type: new +Abstract: Based on the recent works of Radziwill-Soundararajan and Roberts, we establish a rate of convergence in Bourgade's central limit theorem for shifted Dirichlet $L$-functions. Our results also indicate that the dependence structure in the components of a random vector could have a dramatic impact on the rate of convergence in such a multivariate central limit theorem. + oai:arXiv.org:2601.02781v1 + math.NT + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Po-Han Hsu, Peng-Jie Wong + + + Minimal Sets of Involution Generators for Big Mapping Class Groups + https://arxiv.org/abs/2601.02784 + arXiv:2601.02784v1 Announce Type: new +Abstract: Let $S(n)$, for $n \in \mathbb{N}$, be the infinite-type surface of infinite genus with $n$ ends, each of which is accumulated by genus. The mapping class groups of these types of surfaces are not countably generated. However, they are Polish groups, so they can be topologically countably generated. This paper focuses on finding minimal topological generating sets of involutions for $\mathrm{Map}(S(n))$. We establish that for $n \geq 16$, $\mathrm{Map}(S(n))$ can be topologically generated by four involutions. Furthermore, we establish that the the mapping class groups of the Loch Ness Monster surface ($n=1$) and the Jacob's Ladder surface ($n=2$) can be topologically generated by three involutions. + oai:arXiv.org:2601.02784v1 + math.GT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + T\"ulin Altun\"oz, Celal Can Bellek, Emir G\"ul, Mehmetcik Pamuk, O\u{g}uz Y{\i}ld{\i}z + + + Approximate Birkhoff-James orthogonality preserver on Lebesgue-Bochner spaces + https://arxiv.org/abs/2601.02786 + arXiv:2601.02786v1 Announce Type: new +Abstract: In this article, we examine an approximate version of Koldobsky-Blanco-Turn\v{s}ek theorem (namely, Property P) in the space of vector-valued integrable functions. More precisely, we prove that the Lebesgue-Bochner spaces $L^p(\mu,X),\;(1\leq p<\infty)$, do not have Property P under certain conditions on $\mu$ and the Banach space $X$. + oai:arXiv.org:2601.02786v1 + math.FA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Mohit, Ranjana Jain + + + On Liouiville Type Theorem for the 3D Isentropic Navier-Stokes System without D-condition + https://arxiv.org/abs/2601.02791 + arXiv:2601.02791v1 Announce Type: new +Abstract: In this paper, we establish Liouville-type theorems for the steady compressible Navier-Stokes system. Assuming a smooth solution \(u \in L^p(\mathbb{R}^3)\), \(3 \le p \le \frac{9}{2}\), with bounded density, one obtains \(u \equiv0\). This generalizes the result of Li-Yu \cite{Li-Yu} by removing the Dirichlet condition \(\int_{\mathbb{R}^3} |\nabla u|^2 \, dx < \infty\). If \(\frac{9}{2} < p < 6\), Liouville-type theorem holds under the additional oscillation condition for momentum \(\rho u \in \dot{B}^{\frac{3}{p} - \frac{3}{2}}_{\infty,\infty}(\mathbb{R}^3)\). For the marginal case \(u \in L^6(\mathbb{R}^3)\), the oscillation condition can be replaced by \(\rho u \in BMO^{-1}(\mathbb{R}^3)\). We also present results in Morrey-type spaces: \(u \in \dot{M}^{s,6}(\mathbb{R}^3)\) and \(\rho u \in \dot{M}_w^{q,3}(\mathbb{R}^3)\) for \(2 \le s \le 6\) and \(\frac{3}{2} < q \le 3\). + oai:arXiv.org:2601.02791v1 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Quansen Jiu, Jie Tan, Zhihong Yan + + + Adapting Polyhedral Dominance Cones to Ordinal Preference Structures + https://arxiv.org/abs/2601.02796 + arXiv:2601.02796v1 Announce Type: new +Abstract: In combinatorial optimization, ordinal costs can be used to model the quality of elements whenever numerical values are not available. When considering, for example, routing problems for cyclists, the safety of a street can be ranked in ordered categories like safe (separate bike lane), medium safe (street with a bike lane) and unsafe (street without a bike lane). However, ordinal optimization may suggest unrealistic solutions with huge detours to avoid unsafe street segments. In this paper, we investigate how partial preference information regarding the relative quality of the ordinal categories can be used to improve the relevance of the computed solutions. By introducing preference weights which describe how much better a category is at least or at most, compared to the subsequent category, we enlarge the ordinal dominance cone. This leads to a smaller set of alternatives, i. e., of ordinally efficient solutions. We show that the corresponding weighted ordinal ordering cone is a polyhedral cone and provide descriptions via its extreme rays and via its facets. The latter implies a linear transformation to an associated multi-objective optimization problem. This paves the way for the application of standard multi-objective solution approaches. We demonstrate the usefulness of the weighted ordinal ordering cone by investigating a safest path problem with different preference weights. Moreover, we investigate the interrelation between the weighted ordering cone to standard dominance concepts of multi-objective optimization, like, e.g., Pareto dominance, lexicographic dominance and weighted sum dominance. + oai:arXiv.org:2601.02796v1 + math.OC + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Kathrin Klamroth, Michael Stiglmayr, Julia Sudhoff Santos + + + Log-Polynomial Optimization + https://arxiv.org/abs/2601.02797 + arXiv:2601.02797v1 Announce Type: new +Abstract: We study an optimization problem in which the objective is given as a sum of logarithmic-polynomial functions. This formulation is motivated by statistical estimation principles such as maximum likelihood estimation, and by loss functions including cross-entropy and Kullback-Leibler divergence. We propose a hierarchy of moment relaxations based on the truncated $K$-moment problems to solve log-polynomial optimization. We provide sufficient conditions for the hierarchy to be tight and introduce a numerical method to extract the global optimizers when the tightness is achieved. In addition, we modify relaxations with optimality conditions to better fit log-polynomial optimization with convenient Lagrange multipliers expressions. Various applications and numerical experiments are presented to show the efficiency of our method. + oai:arXiv.org:2601.02797v1 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jiyoung Choi, Jiawang Nie, Xindong Tang, Suhan Zhong + + + An extended symmetric union with multiple tangle regions and its Alexander polynomial + https://arxiv.org/abs/2601.02800 + arXiv:2601.02800v1 Announce Type: new +Abstract: The authors recently introduced a new construction of a knot as an extended symmetric union of a knot with a single tangle region. In this paper, we generalize the construction to include multiple tangle regions. The constructed knot $K$ with a partial knot $\hat{K}$ and multiple tangle regions satisfies the following two properties: its Alexander polynomial is the product of the Alexander polynomials of the numerators of these tangles and the square of the Alexander polynomial of the partial knot $\hat{K}$, and there exists a surjective homomorphism from the knot group of $K$ to that of $\hat{K}$ which maps the longitude of $K$ to the trivial element. + oai:arXiv.org:2601.02800v1 + math.GT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Teruaki Kitano, Yasuharu Nakae + + + State-Dependent Fading Gaussian Channel with Common Reconstruction Constraints + https://arxiv.org/abs/2601.02802 + arXiv:2601.02802v1 Announce Type: new +Abstract: The task of jointly communicating a message and reconstructing a common estimate of the channel state is examined for a fading Gaussian model with additive state interference. The state is an independent and identically distributed Gaussian sequence known noncausally at the transmitter, and the instantaneous fading coefficient is perfectly known at both the transmitter and the receiver. The receiver is required to decode the transmitted message and, in addition, reconstruct the state under a common reconstruction constraint ensuring that its estimate coincides with that at the transmitter. A complete characterization of the optimal rate distortion tradeoff region for this setting is the main result of our work. The analytical results are also validated through numerical examples illustrating the rate distortion and power distortion tradeoffs. + oai:arXiv.org:2601.02802v1 + cs.IT + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Viswanathan Ramachandran + + + Diffusion on homogeneous ultrametric spaces: the contributions of Alessandro Fig\`a-Talamanca + https://arxiv.org/abs/2601.02809 + arXiv:2601.02809v1 Announce Type: new +Abstract: Alessandro Fig\`a-Talamanca (1938-2023) was an influential Italian mathematician, scientific leader of the Italian group of harmonic analysis for many years. Since the late 1970ies, his interest focussed on harmonic analysis on free groups and trees. In the later years of his scientific work he became also interested in diffusion processes on homogeneous ultrametric spaces such as local fields and totally disconnected Abelian groups. This is related with the close connection of those spaces with trees and their boundaries and concerns, in particular, the construction of such processes via discrete-time walks on trees. The present notes provide rather detailed comments on this part of his work and the related, quite abundant literature. This is intended to become part of a volume of selected papers by Fig\`a-Talamanca, accompanied by comments such as the present text. + oai:arXiv.org:2601.02809v1 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Wolfgang Woess + + + Decision-Theoretic Robustness for Network Models + https://arxiv.org/abs/2601.02811 + arXiv:2601.02811v1 Announce Type: new +Abstract: Bayesian network models (Erdos Renyi, stochastic block models, random dot product graphs, graphons) are widely used in neuroscience, epidemiology, and the social sciences, yet real networks are sparse, heterogeneous, and exhibit higher-order dependence. How stable are network-based decisions, model selection, and policy recommendations to small model misspecification? We study local decision-theoretic robustness by allowing the posterior to vary within a small Kullback-Leibler neighborhood and choosing actions that minimize worst-case posterior expected loss. Exploiting low-dimensional functionals available under exchangeability, we (i) adapt decision-theoretic robustness to exchangeable graphs via graphon limits and derive sharp small-radius expansions of robust posterior risk; under squared loss the leading inflation is controlled by the posterior variance of the loss, and for robustness indices that diverge at percolation/fragmentation thresholds we obtain a universal critical exponent describing the explosion of decision uncertainty near criticality. (ii) Develop a nonparametric minimax theory for robust model selection between sparse Erdos-Renyi and block models, showing-via robustness error exponents-that no Bayesian or frequentist method can uniformly improve upon the decision-theoretic limits over configuration models and sparse graphon classes for percolation-type functionals. (iii) Propose a practical algorithm based on entropic tilting of posterior or variational samples, and demonstrate it on functional brain connectivity and Karnataka village social networks. + oai:arXiv.org:2601.02811v1 + math.ST + stat.ME + stat.TH + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Marios Papamichalis, Regina Ruane, Simon Lunagomez, Swati Chandna + + + An introduction of Berezin sectorial operators and its application to Berezin number inequalities + https://arxiv.org/abs/2601.02817 + arXiv:2601.02817v1 Announce Type: new +Abstract: We introduce a new class of operators, called Berezin sectorial operators, which generalizes classical sectorial operators. We provide examples on the Hardy-Hilbert space showing that there exist operators that are Berezin sectorial but not sectorial and that the Berezin sectorial index can be strictly smaller than the classical one. We derive Berezin number inequalities for this class, including a weak version of the power inequality, and study geometric properties of the Berezin range for finite-rank and weighted shift operators on the Dirichlet space. We also raise the question of whether similar constructions are possible for composition-differentiation operators on the Dirichlet space. + oai:arXiv.org:2601.02817v1 + math.FA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Saikat Mahapatra, Sweta Mukherjee, Anirban Sen, Riddhick Birbonshi, Kallol Paul + + + Effective Disjunction and Effective Interpolation in Suffciently Strong Proof Systems + https://arxiv.org/abs/2601.02821 + arXiv:2601.02821v1 Announce Type: new +Abstract: In this article, we deal with the uniform effective disjunction property and the uniform effective interpolation property, which are weaker versions of the classical effective disjunction property and the effective interpolation property.\\ The main result of the paper is as follows: Suppose the proof system $EF$ (Extended Frege) has the uniform effective disjunction property, then every sufficiently strong proof system $S$ that corresponds to a theory $T$, which is a theory in the same language as the theory $V_{1}^{1}$, also has the uniform effective disjunction property. Furthermore, if we assume that $EF$ has the uniform effective interpolation property, then the proof system $S$ also has the uniform effective interpolation property.\\ From this, it easily follows that if $EF$ has the uniform effective interpolation property, then for every disjoint $NE$-pair, there exists a set in $E$ that separates this pair. Thus, if $EF$ has the uniform effective interpolation property, it specifically holds that $NE \cap coNE = E$. + Additionally, at the end of the article, the following is proven: Suppose the proof system $EF$ has the uniform effective interpolation property, and let $A_{1}$ and $A_{2}$ be a (not necessarily disjoint) NE-pair such that $A_{1} \cup A_{2} = \mathbb{N}$; then there exists an exponential time algorithm which for every input $n$ (of length $O(\log n)$) finds $i\in\{1,2\}$ such that $n\in A_{i}$. + oai:arXiv.org:2601.02821v1 + math.LO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Martin Maxa + + + DeepFP: Deep-Unfolded Fractional Programming for MIMO Beamforming + https://arxiv.org/abs/2601.02822 + arXiv:2601.02822v1 Announce Type: new +Abstract: This work proposes a mixed learning-based and optimization-based approach to the weighted-sum-rates beamforming problem in a multiple-input multiple-output (MIMO) wireless network. The conventional methods, i.e., the fractional programming (FP) method and the weighted minimum mean square error (WMMSE) algorithm, can be computationally demanding for two reasons: (i) they require inverting a sequence of matrices whose sizes are proportional to the number of antennas; (ii) they require tuning a set of Lagrange multipliers to account for the power constraints. The recently proposed method called the reduced WMMSE addresses the above two issues for a single cell. In contrast, for the multicell case, another recent method called the FastFP eliminates the large matrix inversion and the Lagrange multipliers by using an improved FP technique, but the update stepsize in the FastFP can be difficult to decide. As such, we propose integrating the deep unfolding network into the FastFP for the stepsize optimization. Numerical experiments show that the proposed method is much more efficient than the learning method based on the WMMSE algorithm. + oai:arXiv.org:2601.02822v1 + cs.IT + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jianhang Zhu, Tsung-Hui Chang, Liyao Xiang, Kaiming Shen + + + Collapsed Structured Block Models for Community Detection in Complex Networks + https://arxiv.org/abs/2601.02828 + arXiv:2601.02828v1 Announce Type: new +Abstract: Community detection seeks to recover mesoscopic structure from network data that may be binary, count-valued, signed, directed, weighted, or multilayer. The stochastic block model (SBM) explains such structure by positing a latent partition of nodes and block-specific edge distributions. In Bayesian SBMs, standard MCMC alternates between updating the partition and sampling block parameters, which can hinder mixing and complicate principled comparison across different partitions and numbers of communities. We develop a collapsed Bayesian SBM framework in which block-specific nuisance parameters are analytically integrated out under conjugate priors, so the marginal likelihood p(Y|z) depends only on the partition z and blockwise sufficient statistics. This yields fast local Gibbs/Metropolis updates based on ratios of closed-form integrated likelihoods and provides evidence-based complexity control that discourages gratuitous over-partitioning. We derive exact collapsed marginals for the most common SBM edge types-Beta-Bernoulli (binary), Gamma-Poisson (counts), and Normal-Inverse-Gamma (Gaussian weights)-and we extend collapsing to gap-constrained SBMs via truncated conjugate priors that enforce explicit upper bounds on between-community connectivity. We further show that the same collapsed strategy supports directed SBMs that model reciprocity through dyad states, signed SBMs via categorical block models, and multiplex SBMs where multiple layers contribute additive evidence for a shared partition. Across synthetic benchmarks and real networks (including email communication, hospital contact counts, and citation graphs), collapsed inference produces accurate partitions and interpretable posterior block summaries of within- and between-community interaction strengths while remaining computationally simple and modular. + oai:arXiv.org:2601.02828v1 + math.ST + stat.ME + stat.TH + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Marios Papamichalis, Regina Ruane + + + Varadhan Functions, Variances, and Means on Compact Riemannian Manifolds + https://arxiv.org/abs/2601.02832 + arXiv:2601.02832v1 Announce Type: new +Abstract: Motivated by Varadhan's theorem, we introduce Varadhan functions, variances, and means on compact Riemannian manifolds as smooth approximations to their Fr\'echet counterparts. Given independent and identically distributed samples, we prove uniform laws of large numbers for their empirical versions. Furthermore, we prove central limit theorems for Varadhan functions and variances for each fixed $t\ge0$, and for Varadhan means for each fixed $t>0$. By studying small time asymptotics of gradients and Hessians of Varadhan functions, we build a strong connection to the central limit theorem for Fr\'echet means, without assumptions on the geometry of the cut locus. + oai:arXiv.org:2601.02832v1 + math.PR + math.ST + stat.ME + stat.TH + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yueqi Cao + + + Une br\`eve histoire des perturbations non-hermitiennes de rang un + https://arxiv.org/abs/2601.02834 + arXiv:2601.02834v1 Announce Type: new +Abstract: Les perturbations de faible rang de matrices al\'eatoires ont \'et\'e au c{\oe}ur de nombreux travaux ces vingt derni\`eres ann\'ees. En particulier, les cas non-hermitiens, moins repr\'esent\'es dans la litt\'erature en r\`egle g\'en\'erale, font ici l'objet d'une attention sp\'eciale en raison de leurs applications \`a la physique et \`a l'\'etude des r\'eseaux de neurones. Petit tour d'horizon. + -- + A brief history of non-Hermitian perturbations of rank one: Low-rank perturbations of random matrices have been the focus of active research over the past twenty years. We give an overview of different non-Hermitian models, which are generally less represented in the literature, as well as some of their applications in physics and the study of neural networks. + oai:arXiv.org:2601.02834v1 + math.PR + math-ph + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Gazette SMF 182, 14-26, 2024 + Guillaume Dubach, Jana Reker + + + Quantum isometry groups of log-Laplacians on Cuntz--Krieger algebras + https://arxiv.org/abs/2601.02835 + arXiv:2601.02835v1 Announce Type: new +Abstract: We compute the quantum isometry groups of Cuntz--Krieger algebras endowed with the spectral triples coming from the Ahlfors regular structure of the underlying topological Markov chain. This allows us to exhibit a new family of compact quantum groups, mixing features from quantum automorphism groups of graphs and easy quantum groups. Contrary to the classical isometry groups, whose actions on the Cuntz--Krieger algebras are never ergodic, the quantum isometry group acts ergodically in the case of the Cuntz algebra. This also leads to the construction of a (genuinely quantum) ergodic and faithful action of a compact matrix quantum group on the Cantor space. + oai:arXiv.org:2601.02835v1 + math.OA + math.QA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Amaury Freslon, Dimitris Michail Gerontogiannis, Adam Skalski + + + Data-Driven Modeling of Global Bifurcations and Chaos in a Mechanical System under Delayed and Quantized Control + https://arxiv.org/abs/2601.02838 + arXiv:2601.02838v1 Announce Type: new +Abstract: We illustrate how the recent theory of Spectral Submanifolds (SSM) can capture global bifurcations and complex dynamics in mechanical systems even under delay and spatial discretization. Specifically, we build a parameter-dependent SSM-reduced model that predicts global heteroclinic and local bifurcations in a Furuta pendulum under control with delay, and verify these predictions numerically. Under additional spatial discretization of the digital controller, we also obtain an SSM-reduced model that correctly reproduces a numerically and experimentally observed microchaotic attractor in the system. + oai:arXiv.org:2601.02838v1 + math.DS + nlin.CD + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Giacomo Abbasciano, Bal\'azs Endr\'esz, G\'abor St\'ep\'an, George Haller + + + Large-scale geometry of graphs interpolating between curve graphs and pants graphs + https://arxiv.org/abs/2601.02839 + arXiv:2601.02839v1 Announce Type: new +Abstract: We study two types of graphs interpolating between the curve graph and the pants graph from the viewpoint of large-scale geometry. One was introduced by Erlandsson and Fanoni, and the other by Mahan Mj. These graphs were developed independently in different contexts. In this paper, we provide explicit formulae for computing their quasi-flat ranks. These formulae depend on the genus and the number of boundary components of the underlying surface, as well as the interpolation parameter. We also classify geometries of the interpolating graphs into the hyperbolic, relatively hyperbolic, and thick cases. Our approach relies on the theory of twist-free graphs of multicurves, which is developed by Vokes and Russel. + oai:arXiv.org:2601.02839v1 + math.GT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Erika Kuno, Rin Kuramochi, Kento Sakai + + + The Sequence Reconstruction of Permutations under Hamming Metric with Small Errors + https://arxiv.org/abs/2601.02844 + arXiv:2601.02844v1 Announce Type: new +Abstract: The sequence reconstruction problem asks for the recovery of a sequence from multiple noisy copies, where each copy may contain up to $r$ errors. In the case of permutations on \(n\) letters under the Hamming metric, this problem is closely related to the parameter $N(n,r)$, the maximum intersection size of two Hamming balls of radius $r$. While previous work has resolved \(N(n,r)\) for small radii (\(r \leq 4\)) and established asymptotic bounds for larger \(r\), we present new exact formulas for \(r \in \{5,6,7\}\) using group action techniques. In addition, we develop a formula for \(N(n,r)\) based on the irreducible characters of the symmetric group \(S_n\), along with an algorithm that enables computation of \(N(n,r)\) for larger parameters, including cases such as \(N(43,8)\) and \(N(24,14)\). + oai:arXiv.org:2601.02844v1 + math.GR + cs.IT + math.CO + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + A. Abdollahi, J. Bagherian, H. Eskandari, F. Jafari, M. Khatami, F. Parvaresh + + + Stability and error estimates of a linear and partitioned finite element method approximating nonlinear fluid-structure interactions + https://arxiv.org/abs/2601.02847 + arXiv:2601.02847v1 Announce Type: new +Abstract: We propose and analyze a linear and partitioned finite element method for fluid-shell interactions under the arbitrary Lagrangian-Eulerian (ALE) framework. We adopt the P1-bubble/P1/P1 elements for the fluid velocity, pressure, and structure velocity, respectively. We show the stability and error estimates of the scheme without assuming infinitesimal structural deformation nor neglecting fluid convection effects. The theoretical convergence rate is further corroborated by numerical experiments. + oai:arXiv.org:2601.02847v1 + math.NA + cs.NA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Bangwei She, Tian Tian, Karel Tuma + + + Constructing $\lambda$-Angenent curve by flow method + https://arxiv.org/abs/2601.02853 + arXiv:2601.02853v1 Announce Type: new +Abstract: Using a modified curve shortening flow, we construct $\lambda$-Angenent curve, which was first constructed by the shooting method. + oai:arXiv.org:2601.02853v1 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Pak Tung Ho + + + Context-aware Privacy Bounds for Linear Queries + https://arxiv.org/abs/2601.02855 + arXiv:2601.02855v1 Announce Type: new +Abstract: Linear queries, as the basis of broad analysis tasks, are often released through privacy mechanisms based on differential privacy (DP), the most popular framework for privacy protection. However, DP adopts a context-free definition that operates independently of the data-generating distribution. In this paper, we revisit the privacy analysis of the Laplace mechanism through the lens of pointwise maximal leakage (PML). We demonstrate that the distribution-agnostic definition of the DP framework often mandates excessive noise. To address this, we incorporate an assumption about the prior distribution by lower-bounding the probability of any single record belonging to any specific class. With this assumption, we derive a tight, context-aware leakage bound for general linear queries, and prove that our derived bound is strictly tighter than the standard DP guarantee and converges to the DP guarantee as this probability lower bound approaches zero. Numerical evaluations demonstrate that by exploiting this prior knowledge, the required noise scale can be reduced while maintaining privacy guarantees. + oai:arXiv.org:2601.02855v1 + cs.IT + cs.CR + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Heng Zhao (KTH Royal Institute of Technology), Sara Saeidian (KTH Royal Institute of Technology, Inria Saclay), Tobias J. Oechtering (KTH Royal Institute of Technology) + + + Higher order H{\"o}lder approximation by solutions of second order elliptic equations + https://arxiv.org/abs/2601.02859 + arXiv:2601.02859v1 Announce Type: new +Abstract: For a given second order elliptic operation $\mathcal{L}$ in a domain $\Omega\subset{\mathbb{R}}^\mathbf{N}$, $\mathbf{N}\ $, and a compact set $\mathbf{K}\subset\Omega$, order $\mathbf{N}$-$2$-Ahlfors-David regular, we define the space $\mathcal{H}^{\mathbf{r}+\omega}_{\mathcal{L}}(\mathbf{K})$ of continuous functions $f(x),\, x\in\mathbf{K}$, admitting, for any $\delta>0$, a local approximation in the $\delta $-neighborhood of any point $x\in\mathbf{K}$, with $\delta^{\mathbf{r}}\omega(\delta)$-error estimate, by solutions of the equation $\mathcal{L} u=0$. For such functions, we prove the existence of a global approximation $v_\delta$ on $\mathbf{K}$ with the same order of error estimate, by a solution of the same equation in a $\delta$-neighborhood of $\mathbf{K}$. A number of properties of these functions $v_\delta$ and their derivatives are established. + oai:arXiv.org:2601.02859v1 + math.AP + math.CA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Grigori Rozenblum, Nikolay Shirokov + + + Morse index of min-max stationary integral varifolds + https://arxiv.org/abs/2601.02860 + arXiv:2601.02860v1 Announce Type: new +Abstract: We prove an upper bound for the Morse index of min-max stationary integral varifolds realizing the $d$-dimensional $p$-width of a closed Riemannian manifold. + oai:arXiv.org:2601.02860v1 + math.DG + math.AP + math.MG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Mitchell Gaudet, Talant Talipov + + + Global H\"{o}lder Solvability of parabolic equations on domains with capacity density conditions + https://arxiv.org/abs/2601.02863 + arXiv:2601.02863v1 Announce Type: new +Abstract: We investigate the Cauchy-Dirichlet problem for linear parabolic equations in divergence form. Under mild assumptions on the source term and the domain, we prove the existence of globally H\"{o}lder continuous solutions. Notably, our results accommodate data exhibiting singularities nearly as critical as the inverse square of the distance from the boundary. + oai:arXiv.org:2601.02863v1 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Takanobu Hara + + + New biorthogonal sequences generated by index integrals of the weight functions + https://arxiv.org/abs/2601.02866 + arXiv:2601.02866v1 Announce Type: new +Abstract: We exhibit new biorthogonal sequences generated by index integrals of the squares of the modified Bessel functions and gamma functions. The composition orthogonality, involving differential operators is employed. Generalized Wilson polynomials are introduced. Some properties are investigated. + oai:arXiv.org:2601.02866v1 + math.CA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/publicdomain/zero/1.0/ + Semyon Yakubovich + + + The W-Operator: A Volterra Fractional Time Operator with Non-Bernstein Symbol + https://arxiv.org/abs/2601.02876 + arXiv:2601.02876v1 Announce Type: new +Abstract: We introduce a new two-parameter fractional time operator with Volterra structure, denoted by the W-operator, defined through a generalized Laplace symbol. The operator preserves the Caputo-type high-frequency behavior while allowing a controlled modification of the low-frequency regime through an additional parameter, leading to regularized memory effects. We develop a complete symbolic and Volterra theory, including explicit Prabhakar-type kernels, a left-inverse Volterra integral, and a fractional fundamental theorem of calculus. We show that the natural factorization of the Laplace symbol does not fit the classical Bernstein product mechanism and that the symbol is not a Bernstein function in general. Despite this non-Bernstein character, we establish well-posedness of abstract fractional Cauchy problems with sectorial generators by resolvent estimates and Laplace inversion, yielding a W-resolvent family with temporal regularity and smoothing properties. As an illustration, we apply the theory to a W-fractional diffusion model and discuss the influence of the modulation parameter on the relaxation of spectral modes. + oai:arXiv.org:2601.02876v1 + math.AP + math.FA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mohamed Wakrim + + + Constructing Cospectral Vertices Through Orbits of Subgraphs + https://arxiv.org/abs/2601.02892 + arXiv:2601.02892v1 Announce Type: new +Abstract: A constructive method is given for obtaining cospectral vertices in undirected graphs, along with an operation that preserves this construction. We prove that the construction yields cospectral vertices, as well as strongly cospectral vertices under additional conditions. Furthermore, we generalize cospectral vertices to the case of the graph Laplacian and provide an analogous construction. + oai:arXiv.org:2601.02892v1 + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Onur Ege Erden, Fatihcan M. Atay + + + Rational-Kernel Fractional Evolution Equations with Almost Sectorial Operators: A Resolvent Framework Unifying ABC and W Dynamics + https://arxiv.org/abs/2601.02894 + arXiv:2601.02894v1 Announce Type: new +Abstract: We study fractional evolution equations driven by rational-kernel time operators with non-singular memory, including the Atangana-Baleanu-Caputo operator and a generalized W-operator. These operators are characterized by Laplace symbols that do not necessarily belong to the classical Bernstein class. The analysis is carried out in the framework of almost sectorial operators, which allows resolvent estimates beyond standard analytic semigroup theory. Existence, uniqueness, and temporal regularity of mild solutions are established by Laplace transform techniques and contour integration, leading to the construction of associated resolvent families. A unified resolvent framework is developed, enabling a precise comparison between ABC and W dynamics and clarifying the influence of rational memory kernels on decay and smoothing properties. Several examples, including fractional diffusion-type equations, illustrate the abstract theory and highlight the impact of non-singular memory on long-time behavior. + oai:arXiv.org:2601.02894v1 + math.AP + math.FA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mohamed Wakrim + + + Homotopical algebra of Lie-Rinehart pairs + https://arxiv.org/abs/2601.02895 + arXiv:2601.02895v1 Announce Type: new +Abstract: Dwyer-Kan localization at pairs of quasi-isomorphisms of the category of dg Lie-Rinehart pairs $(A,M)$, where $A$ is a semi-free cdga over a field $k$ of characteristic zero and $M$ a cell complex in $A$-modules, is shown to be equivalent to that of strong homotopy Lie-Rinehart (SH LR) pairs satisfying the same cofibrancy condition. Latter is a category of fibrant objects. We introduce cofibrations of SH LR pairs, construct factorizations, and prove lifting properties. Applying them, we show uniqueness up to homotopy of certain BV-type resolutions. Restricting to dg LR pairs whose underlying cdga is of finite type, and using a different (co)fibrancy condition, we show that the functor $(A,M)\mapsto A$ is a Cartesian fibration with presentable fibers. The two (co)fibrancy conditions yield equivalent $\infty$-categories under Dwyer-Kan localization. + oai:arXiv.org:2601.02895v1 + math.AT + math.AG + math.CT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Damjan Pi\v{s}talo + + + Relating Checkpoint Update Probabilities to Momentum Parameters in Single-Loop Variance Reduction Methods + https://arxiv.org/abs/2601.02899 + arXiv:2601.02899v1 Announce Type: new +Abstract: Variance reduction (VR) is a crucial tool for solving finite-sum optimization problems, including the composite general convex setting, which is the focus of this work. On the one hand, denoting the number of component functions as $n$ and the target accuracy as $\epsilon$, some VR methods achieve the near-optimal complexity $\widetilde{\mathcal{O}}\left(n+\sqrt{n}/\sqrt{\epsilon}\right)$, but they all have nested structure and fail to provide convergence guarantee for the iterate sequence itself. On the other hand, single-loop VR methods, being free from the aforementioned disadvantages, have complexity no better than $\mathcal{O}\left(n+n/\sqrt{\epsilon}\right)$ which is the complexity of the deterministic method FISTA, thus leaving a critical gap unaddressed. In this work, we propose the \textit{Harmonia} technique which relates checkpoint update probabilities to momentum parameters in single-loop VR methods. Based on this technique, we further propose to vary the growth rate of the momentum parameter, creating a novel continuous trade-off between acceleration and variance reduction, controlled by the key parameter $\alpha\in[0,1]$. The proposed techniques lead to following favourable consequences. First, several known complexity of quite different algorithms are re-discovered under the proposed unifying algorithmic framework Katyusha-H. Second, under an extra mild condition, Katyusha-H achieves the near-optimal complexity for $\alpha$ belonging to a certain interval, highlighting the effectiveness of the acceleration-variance reduction trade-off. Last, without extra conditions, Katyusha-H achieves the complexity $\widetilde{\mathcal{O}}(n+\sqrt{n}/\sqrt{\epsilon})$ with $\alpha=1$ and proper mini-batch sizes. The proposed idea and techniques may be of general interest beyond the considered problem in this work. + oai:arXiv.org:2601.02899v1 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Hai Liu, Tiande Guo, Congying Han + + + The 2-systole on compact K\"ahler surfaces with positive scalar curvature + https://arxiv.org/abs/2601.02901 + arXiv:2601.02901v1 Announce Type: new +Abstract: We study the 2-systole on compact K\"ahler surfaces of positive scalar curvature. For any such surface $(X,\omega)$, we prove the sharp estimate \(\min_X S(\omega)\cdot\syst_2(\omega)\le12\pi\), with equality if and only if $X=\PP^2$ and $\omega$ is the Fubini--Study metric. Using the classification of positive scalar curvature K\"ahler surfaces by their minimal models, we also determine the optimal constant in each case and describe the corresponding rigid models: $12\pi$ when the minimal model is $\PP^2$, $8\pi$ for Hirzebruch surfaces, and $4\pi$ for non-rational ruled surfaces. In the non-rational ruled case, we also give an independent analytic proof, adapting Stern's level set method to the holomorphic fibration in K\"ahler setting. + oai:arXiv.org:2601.02901v1 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Zehao Sha + + + Inhomogeneous nonlinear Schr\"odinger equations with competing singular nonlinearities + https://arxiv.org/abs/2601.02909 + arXiv:2601.02909v1 Announce Type: new +Abstract: We study nonlinear elliptic equations arising as stationary states of inhomogeneous nonlinear Schr\"odinger equations with competing singular nonlinearities. Working in a weighted Sobolev space that combines the homogeneous Sobolev space with a weighted Lebesgue term, we establish continuous and compact embeddings of Caffarelli--Kohn--Nirenberg type. These embeddings, together with a model that displays a natural scaling, allow us to apply the abstract critical point framework of Mercuri and Perera (2025), yielding a sequence of nonlinear eigenvalues for the associated problem. This scaling property leads to a classification of weighted power-type nonlinearities into subscaled, scaled, and superscaled regimes. Within this variational setting, we obtain broad existence and multiplicity results for equations driven by sums of weighted power nonlinearities, covering superscaled, scaled, and subscaled interactions, both in the subcritical and critical cases. We also provide a nonexistence result as a consequence of a Pohozaev-type identity. Finally, in the radial setting we employ improved radial CKN inequalities to enlarge the admissible embedding ranges. This yields strengthened radial versions of all our main results, including two-dimensional configurations with more singular weights, where no compact embeddings are available in the nonradial case. + oai:arXiv.org:2601.02909v1 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Elisandra Gloss, Kanishka Perera, Bruno Ribeiro + + + Diamond Open Access: The AMR Experiment + https://arxiv.org/abs/2601.02910 + arXiv:2601.02910v1 Announce Type: new +Abstract: Diamond open access journals charge neither readers nor authors. Despite long-standing support for this ideal within mathematics, relatively few such journals exist. This article documents the Association for Mathematical Research's experience building and operating diamond open access journals, focusing on the infrastructure, cost, and editorial practices that make the model viable. It aims to clarify why earlier reform efforts have been difficult to replicate and how a lightweight institutional framework can lower the barrier to adoption. + oai:arXiv.org:2601.02910v1 + math.HO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Alex Kontorovich + + + Characteristic quasi-polynomials of truncated arrangements + https://arxiv.org/abs/2601.02912 + arXiv:2601.02912v1 Announce Type: new +Abstract: Given an (affine) integral arrangement $\mathcal{A}$ in $\mathbb{R}^n$, the reduction of $\mathcal{A}$ modulo an arbitrary positive integer $q$ naturally yields an arrangement $\mathcal{A}_q$ in $\mathbb{Z}_q^n$. Our primary objective is to study the combinatorial aspects of the restriction $\mathcal{A}^{(B,\bm b)}$ to the solution space of $B\bm x=\bm b$, and its reduction $\mathcal{A}_q^{(B,\bm b)}$ modulo $q$. This work generalizes the earlier results of Kamiya, Takemura and Terao, as well as Chen and Wang. + The purpose of this paper is threefold as follows. Firstly, we derive an explicit counting formula for the cardinality of the complement $M\big(\mathcal{A}_q^{(B,\bm b)}\big)$ of $\mathcal{A}_q^{(B,\bm b)}$; and prove that for all positive integers $q>q_0$, this cardinality coincides with a quasi-polynomial $\chi^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},q\big)$ in $q$ with a period $\rho_C$. Secondly, we weaken Chen and Wang's original hypothesis $a \mid b$ to a strictly more general condition $\gcd(a,\rho_C)\mid \gcd(b,\rho_C)$, and introduce the concept of combinatorial equivalence for positive integers. Within this framework, we establish three unified comparison relations: between the unsigned coefficients of $\chi^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},a\big)$ and $\chi^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},b\big)$; between the unsigned coefficients of distinct constituents of $\chi^{\text{quasi}}\big(\mathcal{A}^{(B,\bm b)},q\big)$; and between the cardinalities of $M\big(\mathcal{A}_q^{(B,\bm b)}\big)$ and $M\big(\mathcal{A}_{pq}^{(B,\bm b)}\big)$. Thirdly, using our method, we revisit the enumerative aspects of group colorings and nowhere-zero nonhomogeneous form flows from the early work of Forge, Zaslavsky and Kochol. + oai:arXiv.org:2601.02912v1 + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Ying Cao, Houshan Fu + + + The compositional inverses of some permutation polynomials of the form $x+\gamma\operatorname{Tr}_q^{q^2}(h(x))$ + https://arxiv.org/abs/2601.02919 + arXiv:2601.02919v1 Announce Type: new +Abstract: Recently, Jiang et al. \cite{JIANG2025102522} obtained several classes of Permutation Polynomial of the form $x+\gamma\operatorname{Tr}_q^{q^2}(h(x))$ over finite fields $\mathbb{F}_{q^2},q=2^n$. In this paper, we find the compositional inverse of six classes of permutation polynomials of this form. + oai:arXiv.org:2601.02919v1 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Rajesh P. Singh, Dinesh Kumar, Jitendra Prakash + + + Ramaswami Type translation formulae for the polylogarithm functions + https://arxiv.org/abs/2601.02921 + arXiv:2601.02921v1 Announce Type: new +Abstract: In 1934, Ramaswami proved a number of curious translation formulae satisfied by the Riemann zeta function. Such translation formulae, in turn give the meromorphic extension of the Riemann zeta function. In 1954, Apostol extended those identities to establish a family of such similar translation formulae. In this article, we establish many such Ramaswami and Apostol type translation formulae for the Dirichlet series defining the polylogarithm functions. This extended set up has many interesting applications, for example, it allows us to also find some (seemingly new) recurrence relations between the Bernoulli numbers, and use them to deduce some congruence properties of the tangent numbers. + oai:arXiv.org:2601.02921v1 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Pawan Singh Mehta, Biswajyoti Saha + + + Consistency of square bracket partition relation + https://arxiv.org/abs/2601.02923 + arXiv:2601.02923v1 Announce Type: new +Abstract: Characteristic earlier results were of the form CON$(2^{\aleph_0} \to [\lambda]^2_{n, 2})$, with $2^{\aleph_0} $ an ex-large cardinal, in the best case the first weakly Mahlo cardinal. Characteristic new results are CON$((2^{\aleph_0} = \aleph_m) + \aleph_l \to [\aleph_k]^2_{n, 2})$, for suitable $k < l < m$. So we improve in three respects: the continuum may be small (e.g. not a weakly Mahlo), we use no large cardinal, and the cardinals $\lambda$ involved are $ < 2^{\aleph_0}$ after the forcing. + oai:arXiv.org:2601.02923v1 + math.LO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Saharon Shelah + + + With what probability does an inscribed triangle contain a given point? + https://arxiv.org/abs/2601.02929 + arXiv:2601.02929v1 Announce Type: new +Abstract: Three points uniformly selected on the unit circle form a triangle containing a point $X$ at distance $r \in [0; 1]$ from its center with probability $P(r) = \frac{1}{4} - \frac{3}{2 \pi^2}\textrm{Li}_2(r^2)$, where $\textrm{Li}_2$ is the dilogarithm function (Jeremy Tan Jie Rui, 2018). In this paper we present an alternative proof of this fact. We also discuss a couple of other geometric probability problems where the dilogarithm function arises. + oai:arXiv.org:2601.02929v1 + math.PR + math.MG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Abdulamin Ismailov + + + Dimension-decaying diffusion processes as the scaling limit of condensing zero-range processes + https://arxiv.org/abs/2601.02935 + arXiv:2601.02935v1 Announce Type: new +Abstract: In this article, we prove that, on the diffusive time scale, condensing zero-range processes converge to a dimension-decaying diffusion process on the simplex \[ \Sigma = \{(x_1,\dots,x_S) : x_i \ge 0,\; \sum_{i\in S} x_i = 1\}, \] where $S$ is a finite set. This limiting diffusion has the distinctive feature of being absorbed at the boundary of the simplex. More precisely, once the process reaches a face \[ \Sigma_A = \{(x_1,\dots,x_S) : x_i \ge 0,\; \sum_{i\in A} x_i = 1\}, \qquad A \subset S, \] it remains confined to this set and evolves in the corresponding lower-dimensional simplex according to a new diffusion whose parameters depend on the subset $A$. This mechanism repeats itself, leading to successive reductions of the dimension, until one of the vertices of the simplex is reached in finite time. At that point, the process becomes permanently trapped. + The proof relies on a method to extend the domain of the associated martingale problem, which may be of independent interest and useful in other contexts. + oai:arXiv.org:2601.02935v1 + math.PR + cond-mat.stat-mech + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Johel Beltr\'an, Kyuhyeon Choi, Claudio Landim + + + On a theorem Dan Rudolph: Part II: Amenable groups + https://arxiv.org/abs/2601.02939 + arXiv:2601.02939v1 Announce Type: new +Abstract: We prove an analog of Rudolph's theorem for actions of countable amenable groups, which asserts that among invariant measures with entropy at least c on the $G$-shift $(\Lambda^G,\sigma)$, a typical measure has entropy $c$ and is Bernoulli. We also address a relative version of this theorem. + oai:arXiv.org:2601.02939v1 + math.DS + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Tomasz Downarowicz, Jean-Paul Thouvenot, Benjamin Weiss + + + Rational stable homotopy type of equivariant projective spaces and Grassmannians + https://arxiv.org/abs/2601.02940 + arXiv:2601.02940v1 Announce Type: new +Abstract: We prove explicit rational stable splittings of equivariant complex projective spaces $\mathbb{C}P(V)$ and Grassmannians $Gr_n(V)$, for complex representations $V$. When $V$ is a sum of one-dimensional representations, both $\mathbb{C}P(V)$ and $Gr_n(V)$ are rationally a wedge of representation spheres. For general finite groups $G$ and $V$ a sum of irreducible representations which are not necessarily one-dimensional, we show that $\mathbb{C}P(V)$ splits rationally as a wedge of Thom spaces over irreducible factors in $V$. For $Gr_n(V)$, the factors in the corresponding rational splitting are a smash product of Thom spaces over lower Grassmannians on irreducible factors in $V$. + oai:arXiv.org:2601.02940v1 + math.AT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Samik Basu, Vanny Doem, Chandal Nahak + + + Hopfield neural networks as port-Hamiltonian and gradient systems + https://arxiv.org/abs/2601.02951 + arXiv:2601.02951v1 Announce Type: new +Abstract: The structure of continuous Hopfield networks is revisited from a system-theoretic point of view. After adopting a novel electrical network interpretation involving nonlinear capacitors, it is shown that Hopfield networks admit a port-Hamiltonian formulation provided an extra passivity condition is satisfied. Subsequently it is shown that any Hopfield network can be represented as a gradient system, with Riemannian metric given by the inverse of the Hessian matrix of the total energy stored in the nonlinear capacitors. On the other hand, the well-known 'energy' function employed by Hopfield turns out to be the dissipation potential of the gradient system, and this potential is shown to satisfy a dissipation inequality that can be used for analysis and interconnection. + oai:arXiv.org:2601.02951v1 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Arjan van der Schaft + + + The left-to-right minima basis of the group algebra of the symmetric group (updated version) + https://arxiv.org/abs/2601.02952 + arXiv:2601.02952v1 Announce Type: new +Abstract: We introduce a new basis of the group algebra of the symmetric group, built using the left-to-right minima sets of permutations. We show that on this basis, the descent algebra acts by triangular operators, thus making it an analogue of a cellular basis. The proof involves Dynkin elements (nested commutators) of the free algebra and their interactions with the $\mathbf B$-basis. + oai:arXiv.org:2601.02952v1 + math.CO + math.RA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Darij Grinberg, Ekaterina A. Vassilieva + + + K-stability of Fano weighted hypersurfaces via plt flags and convex geometry + https://arxiv.org/abs/2601.02974 + arXiv:2601.02974v1 Announce Type: new +Abstract: We develop a framework to study the K-stability of weighted Fano hypersurfaces based on a combination of birational and convex-geometric techniques. As an application, we prove that all quasi-smooth weighted Fano hypersurfaces of index 1 with at most two weights greater than 1 are K-stable. We also construct several examples of K-unstable quasi-smooth weighted Fano hypersurfaces of low indices. To prove these results, we establish lower bounds for stability thresholds using the method of Abban-Zhuang, which reduces the problem to lower-dimensional cases. A key feature of our approach is the use of plt flags that are not necessarily admissible. + oai:arXiv.org:2601.02974v1 + math.AG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Livia Campo, Kento Fujita, Taro Sano, Luca Tasin + + + A Fourth-Order Cut-cell Multigrid Method for Generic Elliptic Equations on Arbitrary Domains + https://arxiv.org/abs/2601.02975 + arXiv:2601.02975v1 Announce Type: new +Abstract: To numerically solve a generic elliptic equation on two-dimensional domains with rectangular Cartesian grids, we propose a cut-cell geometric multigrid method that features (1) general algorithmic steps that apply to all forms of elliptic equations and all types of boundary conditions, (2) the versatility of handling both regular and irregular domains with arbitrarily complex topology and geometry, (3) the fourth-order accuracy even at the presence of ${\cal C}^1$ discontinuities on the domain boundary, and (4) the optimal complexity of $O(h^{-2})$. Test results demonstrate the generality, accuracy, efficiency, robustness, and excellent conditioning of the proposed method. + oai:arXiv.org:2601.02975v1 + math.NA + cs.NA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jiyu Liu, Zhixuan Li, Jiatu Yan, Zhiqi Li, Qinghai Zhang + + + Uniform distribution of saddle connection lengths in all $\mathsf{SL}(2,\mathbb{R})$ orbits + https://arxiv.org/abs/2601.02979 + arXiv:2601.02979v1 Announce Type: new +Abstract: For every flat surface, almost every flat surface in its $\mathsf{SL}(2,\mathbb{R})$ orbit has the following property: the sequence of its saddle connection lengths in non-decreasing order is uniformly distributed in the unit interval. + oai:arXiv.org:2601.02979v1 + math.DS + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.1007/s10711-023-00800-3 + Geometriae Dedicata 217 (2023) + Donald Robertson, Benjamin Dozier + + + Coupling Brownian loop soups and random walk loop soups at all polynomial scales + https://arxiv.org/abs/2601.02992 + arXiv:2601.02992v1 Announce Type: new +Abstract: Lawler and Trujillo Ferreras constructed a well-known coupling between the Brownian loop soups in $\mathbb{R}^2$ and the random walk loop soups on $\mathbb{Z}^2$ (one rescales the random walk loops by $1/N$, their time parametrizations by $1/(2N^2)$, and let $N\to \infty$), which led to numerous applications. It nevertheless only holds for loops with time length at least $N^{\theta-2}$ for $\theta \in(2/3,2)$. In particular, there is no control on mesoscopic loops with time length less than $N^{-4/3}$ (i.e.\ roughly diameter less than $N^{-2/3}$). + In this paper, we find a simple way to remove the restriction $\theta>2/3$, so that such a coupling works for all $\theta\in (0,2)$, i.e. for loops at all polynomial scales. We also establish this coupling in any dimension $d\ge 1$ (i.e. for random walk loop soups on $\mathbb{Z}^d$ and Brownian loop soups on $\mathbb{R}^d$). + oai:arXiv.org:2601.02992v1 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Wei Qian + + + Transducing Linear Decompositions of Tournaments + https://arxiv.org/abs/2601.02999 + arXiv:2601.02999v1 Announce Type: new +Abstract: Boja\'nczyk, Pilipczuk, and Grohe [LICS '18] proved that for graphs of bounded linear clique-width, clique-decompositions of bounded width can be produced by a CMSO transduction. We show that in the case of tournaments, a first-order transduction suffices. This implies that the logics CMSO and existential MSO are equivalent over bounded linear clique-width tournaments. + oai:arXiv.org:2601.02999v1 + math.CO + cs.DM + cs.LO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Colin Geniet, Fatemeh Ghasemi, Mamadou Moustapha Kant\'e + + + Remarks on $d$-independent topological groups + https://arxiv.org/abs/2601.03000 + arXiv:2601.03000v1 Announce Type: new +Abstract: A non-trivial topological group is called \emph{$d$-independent} if for every subgroup of cardinality less than the continuum there exists a countable dense subgroup intersecting it trivially. This notion was introduced by M\'arquez and Tkachenko and has been intensively studied in the metrizable setting. In particular, they proved that a second-countable locally compact abelian group is $d$-independent if and only if it is algebraically an $M$-group, and asked whether the same conclusion holds for all separable locally compact groups. + In this paper we give an affirmative answer to this question. We show that every separable locally compact abelian $M$-group is $d$-independent, thereby removing the metrizability assumption from the result of M\'arquez and Tkachenko. + In addition, we investigate several further aspects of $d$-independence. We study its behaviour under taking powers of topological groups and extend the notion of $d$-independence to the non-abelian setting. Moreover, we prove that every separable connected compact group is $d$-independent, thereby answering another question posed by M\'arquez and Tkachenko. + oai:arXiv.org:2601.03000v1 + math.GR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Zhouxiang Huang, Dekui Peng, Gao Zhang + + + G-BSDEs with time-varying monotonicity condition + https://arxiv.org/abs/2601.03006 + arXiv:2601.03006v1 Announce Type: new +Abstract: In this paper, we study backward stochastic differential equations driven by G-Brownian motion where the generator has time-varying monotonicity with respect to y and Lipsitz property with respect to z. Through the Yosida approximation, we have proved the existence and uniqueness of the solutions to these equations. + oai:arXiv.org:2601.03006v1 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Xue Zhang, Renxing Li + + + A Relaxation Method for Nonsmooth Nonlinear Optimization with Binary Constraints + https://arxiv.org/abs/2601.03008 + arXiv:2601.03008v1 Announce Type: new +Abstract: We study binary optimization problems of the form \( \min_{x\in\{-1,1\}^n} f(Ax-b) \) with possibly nonsmooth loss \(f\). Following the lifted rank-one semidefinite programming (SDP) approach\cite{qian2023matrix}, we develop a majorization-minimization algorithm by using the difference-of-convexity (DC) reformuation for the rank-one constraint and the Moreau envelop for the nonsmooth loss. We provide global complexity guarantees for the proposed \textbf{D}ifference of \textbf{C}onvex \textbf{R}elaxation \textbf{A}lgorithm (DCRA) and show that it produces an approximately feasible binary solution with an explicit bound on the optimality gap. Numerical experiments on synthetic and real datasets confirm that our method achieves superior accuracy and scalability compared with existing approaches. + oai:arXiv.org:2601.03008v1 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Lianghai Xiao, Yitian Qian, Shaohua Pan + + + Mathematical aspects of registration methods in bounded domains + https://arxiv.org/abs/2601.03010 + arXiv:2601.03010v1 Announce Type: new +Abstract: Registration methods in bounded domains have received significant attention in the model reduction literature, as a valuable tool for nonlinear approximation. The aim of this work is to provide a concise yet complete overview of relevant results for registration methods in $n$-dimensional domains, from the perspective of parametric model reduction. We present a thorough analysis of two classes of methods, vector flows and compositional maps: we discuss the enforcement of the bijectivity constraint and we comment on the approximation properties of the two methods, for Lipschitz $n$-dimensional domains. + oai:arXiv.org:2601.03010v1 + math.NA + cs.NA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Angelo Iollo, Jon Labatut, Pierre Mounoud, Tommaso Taddei + + + Adaptive Control of Unknown Linear Switched Systems via Policy Gradient Methods + https://arxiv.org/abs/2601.03016 + arXiv:2601.03016v1 Announce Type: new +Abstract: We consider the policy gradient adaptive control (PGAC) framework, which adaptively updates a control policy in real time, by performing data-based gradient descent steps on the linear quadratic regulator cost. This method has empirically shown to react to changing circumstances, such as model parameters, efficiently. To formalize this observation, we design a PGAC method which stabilizes linear switched systems, where both model parameters and switching time are unknown. We use sliding window data for the policy gradient estimate and show that under a dwell time condition and small dynamics variation, the policy can track the switching dynamics and ensure closed-loop stability. We perform simulations to validate our theoretical results. + oai:arXiv.org:2601.03016v1 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Felix Laurent, Feiran Zhao, Jaap Eising, Florian D\"orfler + + + Periodicity of traces of Hecke operators modulo prime powers + https://arxiv.org/abs/2601.03029 + arXiv:2601.03029v1 Announce Type: new +Abstract: We study traces of Hecke operators on spaces of elliptic cusp forms and Drinfeld cusp forms and show that, modulo any prime power, these traces are periodic in the weight. + oai:arXiv.org:2601.03029v1 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Jonas Bergstr\"om, Sjoerd de Vries + + + On the Hilbert-Chow crepant resolution conjecture + https://arxiv.org/abs/2601.03036 + arXiv:2601.03036v1 Announce Type: new +Abstract: We prove the Hilbert-Chow crepant resolution conjecture in the exceptional curve classes for all projective surfaces and all genera. In particular, this confirms Ruan's cohomological Hilbert-Chow crepant resolution conjecture. The proof exploits Fulton-MacPherson compactifications, reducing the conjecture to the case of the affine plane. As an application, using previous results of the author, we also deduce the families DT/GW correspondence for threefolds $S \times C$ in classes that are zero on the first factor, yielding a wall-crossing proof of the correspondence in this case. Finally, we speculate on the relationship between Hilbert schemes and Fulton-MacPherson compactifications beyond the topics considered in this work. + oai:arXiv.org:2601.03036v1 + math.AG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Denis Nesterov + + + Generalized Toeplitz determinants for Starlike Mappings in Several Complex Variables + https://arxiv.org/abs/2601.03039 + arXiv:2601.03039v1 Announce Type: new +Abstract: This paper establishes sharp bounds for the second and third-order Toeplitz determinants associated with starlike functions $f$ in the unit disk such that $f(z)-z$ has a zero of order $k+1$ at $z=0$. These bounds are further extended to starlike mappings defined on the unit ball in a complex Banach space and on bounded starlike circular domains in $\mathbb{C}^n$. The derived results generalize several known bounds as special cases. + oai:arXiv.org:2601.03039v1 + math.CV + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Surya Giri, S. Sivaprasad Kumar + + + Egorov-Type Semiclassical Limits for Open Quantum Systems with a Bi-Lindblad Structure + https://arxiv.org/abs/2601.03041 + arXiv:2601.03041v1 Announce Type: new +Abstract: This paper develops a bridge between bi-Hamiltonian structures of Poisson-Lie type, contact Hamiltonian dynamics, and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) formalism for quantum open systems. On the classical side, we consider bi-Hamiltonian systems defined by a Poisson pencil with non-trivial invariants. Using an exact symplectic realization, these invariants are lifted and projected onto a contact manifold, yielding a completely integrable contact Hamiltonian system in terms of dissipated quantities and a Jacobi-commutative algebra of observables. On the quantum side, we introduce a class of contact-compatible Lindblad generators: GKSL evolutions whose dissipative part preserves a commutative $C^\ast$-subalgebra generated by the quantizations of the classical dissipated quantities, and whose Hamiltonian part admits an Egorov-type semiclassical limit to the contact dynamics. This construction provides a mathematical mechanism compatible with the semiclassical limit for pure dephasing, compatible with integrability and contact dissipation. An explicit Euler-top-type Poisson-Lie pencil, inspired by deformed Euler top models, is developed as a fully worked-out example illustrating the resulting bi-Lindblad structure and its semiclassical behavior. + oai:arXiv.org:2601.03041v1 + math-ph + math.DG + math.MP + math.SG + quant-ph + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Leonardo Colombo, Asier L\'opez-Gord\'on + + + Classification of reductive homogeneous spaces satisfying strict inequality for Benoist-Kobayashi's $\rho$ functions + https://arxiv.org/abs/2601.03049 + arXiv:2601.03049v1 Announce Type: new +Abstract: Let $G$ be a real reductive Lie group and $H$ a reductive subgroup of $G$. Benoist-Kobayashi studied when $L^2(G/H)$ is a tempered representation of $G$. They introduced the functions $\rho$ on Lie algebras and gave a necessary and sufficient condition for the temperedness of $L^2(G/H)$ in terms of an inequality on $\rho$. In a joint work with Y. Oshima, we considered when $L^2(G/H)$ is equivalent to a unitary subrepresentation of $L^2(G)$ and gave a sufficient condition for this in terms of a strict inequality of $\rho$. In this paper, we will classify the pairs $(\mathfrak{g}, \mathfrak{h})$ with $\mathfrak{g}$ complex reductive and $\mathfrak{h}$ complex semisimple which satisfy that strict inequality of $\rho$. + oai:arXiv.org:2601.03049v1 + math.RT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Kazushi Maeda + + + Existence and concentration of ground state solutions for an exponentially critical Choquard equation involving mixed local-nonlocal operators + https://arxiv.org/abs/2601.03060 + arXiv:2601.03060v1 Announce Type: new +Abstract: We study the Choquard equation involving mixed local and nonlocal operators \[-\varepsilon^{2}\Delta u+\varepsilon^{2s}(-\Delta)^{s}u+V(x)u=\varepsilon^{\mu-2}\left(\frac{1}{|x|^{\mu}}*F(u)\right)f(u)\quad \text{in }\R^{2},\] where \(\varepsilon>0\), \(s\in(0,1)\), \(0<\mu<2\), \(f\) has Trudinger--Moser critical exponential growth, and \(F(t)=\int_{0}^{t}f(\tau)\,d\tau\). By variational methods, combined with the Trudinger--Moser inequality and compactness arguments adapted to the critical growth and the nonlocal interaction term, we prove the existence of ground state solutions and describe their concentration behavior as \(\varepsilon\to0^{+}\). + oai:arXiv.org:2601.03060v1 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Shaoxiong Chen, Min Yang, Zhipeng Yang + + + Similarity-Sensitive Entropy: Induced Kernels and Data-Processing Inequalities + https://arxiv.org/abs/2601.03064 + arXiv:2601.03064v1 Announce Type: new +Abstract: We study an entropy functional $H_K$ that is sensitive to a prescribed similarity structure on a state space. For finite spaces, $H_K$ coincides with the order-1 similarity-sensitive entropy of Leinster and Cobbold. We work in the general measure-theoretic setting of kernelled probability spaces $(\Omega,\mu,K)$ introduced by Leinster and Roff, and develop basic structural properties of $H_K$. + Our main results concern the behavior of $H_K$ under coarse-graining. For a measurable map $f:\Omega\to Y$ and input law $\mu$, we define a law-induced kernel on $Y$ whose pullback minimally dominates $K$, and show that it yields a coarse-graining inequality and a data-processing inequality for $H_K$, for both deterministic maps and general Markov kernels. We also introduce conditional similarity-sensitive entropy and an associated mutual information, and compare their behavior to the classical Shannon case. + oai:arXiv.org:2601.03064v1 + math.PR + cs.IT + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Joseph Samuel Miller + + + Hamiltonian reductions as affine closures of cotangent bundles + https://arxiv.org/abs/2601.03068 + arXiv:2601.03068v1 Announce Type: new +Abstract: For an irreducible non-singular affine $G$-variety $Y$ whose action is $2$-large, we prove that the Hamiltonian reduction $T^*Y/\!\!/\!\!/G$ is a symplectic variety with terminal singularities, isomorphic to the affine closure of $T^*Z_{\text{reg}}$ for $Z:=Y/\!/G$. As applications, we provide a short proof of G. Schwarz's theorem on the graded surjectivity of the push-forward map $\mathcal{D}(Y)^G\rightarrow \mathcal{D}(Z)$, and we establish the surjectivity of the symbol map on $Z$. + oai:arXiv.org:2601.03068v1 + math.AG + math.RT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Baohua Fu, Jie Liu + + + Average gradient localisation for degenerate elliptic equations in the plane + https://arxiv.org/abs/2601.03078 + arXiv:2601.03078v1 Announce Type: new +Abstract: We consider Lipschitz solutions to the possibly highly degenerate elliptic equation $ \dv G(\nabla u)=0 $ in $B_1\subset\R^2 $, for any continuous strictly monotone vector field $ G\colon\R^2\to\R^2$. We show that $u$ is either $C^1$ at $0$, or any blowup limit $v(x)=\lim \frac{u(\delta x)-u(0)}{\delta} $ along a sequence $\delta\to 0$ satisfies $ \nabla v\in \mathcal{D}\cap \mathcal{S} \text{ a.e} $. Here, $ \mathcal{D}$ and $\mathcal{S}$ can be roughly interpreted as the sets where ellipticity degenerates from below and above, that is, the symmetric parts of $ \nabla G$ and $(\nabla G)^{-1}$ have a zero eigenvalue. This is a strong indication in favor of the expected continuity of $H(\nabla u)$ for any continuous $H$ vanishing on $\mathcal{D}\cap \mathcal{S}$. In contrast with previous results in the same spirit, we do not make any assumption on the structure of $G$ besides its continuity and strict monotony. + oai:arXiv.org:2601.03078v1 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Thibault Lacombe + + + A proof of Xin-Zhang's tridiagonal determinant conjecture + https://arxiv.org/abs/2601.03082 + arXiv:2601.03082v1 Announce Type: new +Abstract: We confirm a recent conjecture by Xin and Zhang, which establishes a simple product formula for the characteristic polynomial of an $(n-1) \times (n-1)$ tridiagonal matrix $C$. This characteristic polynomial arises from a recurrence relation that enumerates $n \times n$ nonnegative integer matrices with all row and column sums equal to $t$, also called the Ehrhart polynomial of the $n$th Birkhoff polytope. The proof relies on an unexpected observation: shifting $C$ by a scalar multiple of the identity matrix yields a matrix similar to a lower triangular matrix. In triangular form, the characteristic polynomial reduces to the product of the diagonal entries, leading to the desired closed-form expression. Moreover, we extend this method to broader families of tridiagonal matrices. This provides a new approach for deriving exact enumeration formulas. + oai:arXiv.org:2601.03082v1 + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jiaqiang Hu, Chen Zhang + + + Pretrain Finite Element Method: A Pretraining and Warm-start Framework for PDEs via Physics-Informed Neural Operators + https://arxiv.org/abs/2601.03086 + arXiv:2601.03086v1 Announce Type: new +Abstract: We propose a Pretrained Finite Element Method (PFEM),a physics driven framework that bridges the efficiency of neural operator learning with the accuracy and robustness of classical finite element methods (FEM). PFEM consists of a physics informed pretraining stage and an optional finetuning stage. In the pretraining stage, a neural operator based on the Transolver architecture is trained solely from governing partial differential equations, without relying on labeled solution data. The model operates directly on unstructured point clouds, jointly encoding geometric information, material properties, and boundary conditions, and produces physically consistent initial solutions with extremely high computational efficiency. PDE constraints are enforced through explicit finite element, based differentiation, avoiding the overhead associated with automatic differentiation. In the fine-tuning stage, the pretrained prediction is used as an initial guess for conventional FEM solvers, preserving their accuracy, convergence guarantees, and extrapolation capability while substantially reducing the number of iterations required to reach a prescribed tolerance. PFEM is validated on a broad range of benchmark problems, including linear elasticity and nonlinear hyperelasticity with complex geometries, heterogeneous materials, and arbitrary boundary conditions. Numerical results demonstrate strong generalization in the pretraining stage with relative errors on the order of 1\%, and speedups of up to one order of magnitude in the fine-tuning stage compared to FEM with zero initial guesses. + oai:arXiv.org:2601.03086v1 + math.NA + cs.NA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yizheng Wang, Zhongkai Hao, Mohammad Sadegh Eshaghi, Cosmin Anitescu, Xiaoying Zhuang, Timon Rabczuk, Yinghua Liu + + + Pseudo-differential operators associated with the fractional Hankel-Bessel transform + https://arxiv.org/abs/2601.03091 + arXiv:2601.03091v1 Announce Type: new +Abstract: We introduce and study a new class of pseudo-differential operators associated with a fractional Hankel--Bessel transform. Motivated by the classical Hankel transform and the pseudo-differential operators associated with Bessel operators studied by Pathak and Pandey \cite{PathakPandey1995}, we define a fractional variant by inserting a fractional Fourier-type phase into the Hankel kernel. We then introduce global Shubin-type symbol classes adapted to this transform, derive kernel estimates and integral representations, and establish boundedness results on weighted L^{p}-spaces and on fractional Hankel--Sobolev spaces. This provides a new framework parallel to the classical Hankel pseudo-differential calculus, but in a fractional and global setting. + oai:arXiv.org:2601.03091v1 + math.FA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Durgesh Pasawan + + + Stability of Hyperk\"ahler Flow + https://arxiv.org/abs/2601.03092 + arXiv:2601.03092v1 Announce Type: new +Abstract: In this work, we discuss the stability of Donaldson's flow of surfaces in a hyperk\"ahler 4-manifold. In \cite{WT2}, Wang and Tsai proved a uniqueness theorem and $C^1$ dynamic stability theorem of the mean curvature flow for minimal surface. We extend their results and obtain a similar dynamic stability theorem of the hyperk\"ahler flow. + oai:arXiv.org:2601.03092v1 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Kuan-Hui Lee + + + Submanifolds of almost quaternionic skew-Hermitian manifolds + https://arxiv.org/abs/2601.03094 + arXiv:2601.03094v1 Announce Type: new +Abstract: We investigate several classes of submanifolds of almost quaternionic skew-Hermitian manifolds $(M^{4n}, Q, \omega)$, including almost symplectic, almost complex, almost pseudo-Hermitian and almost quaternionic submanifolds. In the torsion-free case, we realize each type of submanifold considered in the theoretical part by constructing explicit examples of submanifolds of semisimple quaternionic skew-Hermitian symmetric spaces. + oai:arXiv.org:2601.03094v1 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ioannis Chrysikos, Jan Gregorovi\v{c} + + + A Kirchhoff equation with infinite conservation laws + https://arxiv.org/abs/2601.03095 + arXiv:2601.03095v1 Announce Type: new +Abstract: We show here that the quasilinear Kirchhoff-Pokhozhaev equation $$u_{tt}-\big(a\int_{\mathbb{R}^n} |\nabla u |^2 dx + b \big)^{-2} \Delta u = 0,$$ with $a\neq0$, admits conservation laws of all orders. + oai:arXiv.org:2601.03095v1 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Chiara Boiti, Renato Manfrin + + + Spherical Ricci tori with rotational symmetry + https://arxiv.org/abs/2601.03096 + arXiv:2601.03096v1 Announce Type: new +Abstract: In this article, we study $c$-spherical Ricci metrics, that is, Riemannian metrics whose Gaussian curvature $K$ satisfies \begin{equation*} + (K - c)\Delta K - |\nabla K|^2 - 4K(K - c)^2 = 0, \end{equation*} for some $c>0$. We explicitly construct a two-parameter family of such metrics with rotational symmetry and show that infinitely many non-isometric examples can be realized on the same torus. Moreover, we investigate their realization as induced metrics on compact rotational surfaces in $\mathbb{S}^3$, establishing the existence of embedded compact spherical Ricci surfaces by controlling a period function associated with the isometric immersion. + oai:arXiv.org:2601.03096v1 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Iury Domingos, Irene. I. Onnis + + + Point-set models for homotopy coherent coalgebras + https://arxiv.org/abs/2601.03101 + arXiv:2601.03101v1 Announce Type: new +Abstract: We show a first rectification result for homotopy chain coalgebras over a field. On the one hand, we consider the $\infty$-category obtained by localizing differential graded coalgebras over an operad with respect to quasi-isomorphisms; on the other, we give a general definition of an $\infty$-category of coalgebras over an enriched $\infty$-operad. We show by induction over cell attachments that these two $\infty$-categories are in fact equivalent when the operad is cofibrant. This yields explicit point-set models for $E_n$-coalgebras and $E_\infty$-coalgebras in the derived $\infty$-category of chain complexes over a field, and an explicit point-set model for the cellular chains functor with its $E_\infty$-coalgebra structure. After Bachmann--Burklund, this gives a point-set algebraic model for nilpotent $p$-adic homotopy types. + oai:arXiv.org:2601.03101v1 + math.AT + math.CT + math.QA + math.RA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Victor Roca i Lucio, Dan Petersen, Sinan Yalin + + + On the monotonicity of the entropy production in the Landau-Maxwell equation + https://arxiv.org/abs/2601.03107 + arXiv:2601.03107v1 Announce Type: new +Abstract: We study the homogeneous Landau equation with Maxwell molecules and prove that the entropy production is non-increasing provided the directional temperatures are well-distributed and the solution admits a moment of order $\ell$, for some $\ell$ arbitrarily close to $2$. It implies that for an initial condition with finite moment of order $\ell$, the entropy production is guaranteed to be non-increasing after a certain time, that we explicitly compute. This is the first partial answer to a conjecture made by Henry P. McKean in 1966 on the sign of the time-derivatives of the entropy. We also obtain algebraic decay estimates for the entropy production for large time; as well as a short-time estimate without moment assumptions. + oai:arXiv.org:2601.03107v1 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + C\^ome Tabary + + + First passage times for decoupled random walks + https://arxiv.org/abs/2601.03109 + arXiv:2601.03109v1 Announce Type: new +Abstract: Motivated by a connection to the infinite Ginibre point process, decoupled random walks were introduced in a recent article Alsmeyer, Iksanov and Kabluchko (2025). The decoupled random walk is a sequence of independent random variables, in which the $n$th variable has the same distribution as the position at time $n$ of a standard random walk with nonnegative increments. We prove distributional convergence in the Skorokhod space equipped with the $J_1$-topology of the running maxima and the first passage times of decoupled random walks. We show that there exist five different regimes, in which distinct limit theorems arise. Rather different functional limit theorems for the number of visits of decoupled standard random walk to the interval $[0,t]$ as $t\to\infty$ were earlier obtained in the aforementioned paper Alsmeyer, Iksanov and Kabluchko (2025). While the limit processes for the first passage times are inverse extremal-like processes, the limit processes for the number of visits are stationary Gaussian. + oai:arXiv.org:2601.03109v1 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Alexander Iksanov, Zakhar Kabluchko, Vitali Wachtel + + + Dualities for finite abelian groups and applications to coding theory + https://arxiv.org/abs/2601.03126 + arXiv:2601.03126v1 Announce Type: new +Abstract: The choice of an isomorphism, a duality, between a finite abelian group $A$ and its character group allows one to define dual codes of additive codes over $A$. Properties of dualities and dual codes are studied, continuing work of Delsarte from 1973 and more recent work of Dougherty and his collaborators. + oai:arXiv.org:2601.03126v1 + cs.IT + math.GR + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jay A. Wood + + + On derived categories of module categories over multiring categories + https://arxiv.org/abs/2601.03128 + arXiv:2601.03128v1 Announce Type: new +Abstract: Let $\mathcal{A}$ and $\mathcal{B}$ be subcategories of tensor categories $\mathcal{C}$ and $\mathcal{D}$, respectively, both of which are abelian categories with finitely many isomorphism classes of simple objects. We prove that if their derived categories $\mathbf{D}^b(\mathcal{A})$ and $\mathbf{D}^b(\mathcal{B})$ are left triangulated tensor ideals and are equivalent as triangulated $\mathbf{D}^b(\mathcal{C})$-module categories via an equivalence induced by a monoidal triangulated functor $F:\mathbf{D}^b(\mathcal{C})\rightarrow \mathbf{D}^b(\mathcal{D})$, then the original module categories $\mathcal{A}$ and $\mathcal{B}$ are themselves equivalent. We then apply this result to smash product algebras. Furthermore, the localization theory of module categories and triangulated module categories is investigated. + oai:arXiv.org:2601.03128v1 + math.RT + math.CT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jing Yu + + + Lipschitz extension and Lipschitz-free spaces over nets in normed spaces + https://arxiv.org/abs/2601.03131 + arXiv:2601.03131v1 Announce Type: new +Abstract: We consider subsets $S$ of a metric space $M$ such that Lipschitz mappings defined on $S$ can be extended to Lipschitz mappings on $M$, and we show that the union of such subsets has the same property under appropriate geometric conditions. We then derive several consequences to the isomorphic structure and classification of Lipschitz and Lipschitz-free spaces. Our main result is that the Lipschitz-free space $\mathcal{F}(M)$ is isomorphic to its countable $\ell_1$-sum when $M$ is either a net $N_X$ in any Banach space $X$ or the integer grid $\mathbb{Z}_{\ell_1}$ in $\ell_1$. We also prove that the Lipschitz space $\mathrm{Lip}_0(\mathbb{Z}_{\ell_1})$ is isomorphic to $\mathrm{Lip}_0(\ell_1)$ and that $\mathrm{Lip}_0(N_X)$ contains a complemented copy of $\mathrm{Lip}_0(X)$, among other results. This answers questions raised by Albiac, Ansorena, C\'uth and Doucha and Candido, C\'uth and Doucha, respectively, and extends previous results by the same authors as well as H\'ajek and Novotn\'y. + oai:arXiv.org:2601.03131v1 + math.FA + math.MG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ram\'on J. Aliaga, Rub\'en Medina + + + Freely floating cylinder on a 3D fluid governed by the Boussinesq equations in the axisymmetric without swirl case + https://arxiv.org/abs/2601.03133 + arXiv:2601.03133v1 Announce Type: new +Abstract: This paper deals with the interactions of waves governed by a non-linear dispersive Boussinesq type system with the vertical displacement of a cylindrical floating structure in an axisymmetric without swirl situation. The Boussinesq regime is a good approximation of free surface Euler's equations when the non-linear parameter and the shallowness parameter are small. The vertical motion of the floating body is governed by the Newton equation. The full coupled wave-structure interaction problem under consideration is reduced to a boundary problem. The boundary condition satisfied by the discharge is given in terms of the vertical displacement of the floating cylinder. The latter is calculated using an ODE, which requires knowledge of the trace of the surface elevation and its second-time derivative. We use the dispersion in order to exhibit a hidden second order ODE on the trace of the surface elevation. This finally allows us to rewrite the waves-structure interaction problem as a system of non-local conservative PDEs plus bounded radial terms with a dispersive boundary layer, combined with an ODE at the boundary. This is what we call the Augmented formulation. Afterwards we showed that this formulation is well-posed with two different methods. Finally, we study the return to equilibrium situation in the linear regime. In particular, we improved previous results on the explicit time decay. We showed that the center mass of the floating body cannot converge to its equilibrium faster than $\mathcal{O}(t^{-1/2})$ in 2D without viscosity and faster than $\mathcal{O}(t^{-3/2})$ with viscosity. + oai:arXiv.org:2601.03133v1 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Geoffrey Beck, Ewan Contentin, Ludovic Martaud + + + Classifying the Fine Polyhedral Spectrum + https://arxiv.org/abs/2601.03145 + arXiv:2601.03145v1 Announce Type: new +Abstract: In this paper, we examine an analogue of the recently solved spectrum conjecture by Fujita in the setting of Fine polyhedral adjunction theory. We present computational results for lower-dimensional polytopes, which lead to a complete classification of the highest numbers of this Fine spectrum in any dimension. Moreover, we present a classification of the Fine spectrum in dimensions one, two and (almost) three, while providing a framework for general classification results in any dimension. + oai:arXiv.org:2601.03145v1 + math.CO + math.AG + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Sof\'ia Garz\'on Mora, Christian Haase + + + Normalization flow and Poincar\'e-Dulac theory + https://arxiv.org/abs/2601.03147 + arXiv:2601.03147v1 Announce Type: new +Abstract: In this article, we develop a new approach to the Poincar\'e--Dulac normal form theory for a system of differential equations near a singular point. Using the continuous averaging method, we construct a normalization flow that moves a vector field to its normal form. We prove that, in the algebra of formal vector fields (given by power series), the normalization procedure achieves full normalization. When convergence is taken into account, we show that the radius of convergence admits a lower bound of order $1/(1+A\delta)$, with $A>0$, as $\delta \to +\infty$. Based on the methods of this work and on the approaches of \cite{Tres2}, we provide a new proof of the Siegel--Brjuno theorem on the convergence of the normalizing transformation. + oai:arXiv.org:2601.03147v1 + math.DS + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Andrey Chernyshev + + + Vaught's Conjecture and Theories of Partial Order Admitting a Finite Lexicographic Decomposition + https://arxiv.org/abs/2601.03155 + arXiv:2601.03155v1 Announce Type: new +Abstract: A complete theory ${\mathcal T}$ of partial order is an FLD$_1$-theory iff some (equivalently, any) of its models ${\mathbb X}$ admits a finite lexicographic decomposition ${\mathbb X} =\sum _{{\mathbb I}}{\mathbb X} _i$, where ${\mathbb I}$ is a finite partial order and ${\mathbb X} _i$-s are partial orders with a largest element. Then we write $\sum _{{\mathbb I}}{\mathbb X}_i\in {\mathcal D} ({\mathcal T})$ and call $\sum _{{\mathbb I}}{\mathbb X}_i$ a VC-decomposition (resp. a VC$^\sharp$-decomposition} iff ${\mathbb X} _i$ satisfies Vaught's conjecture (VC) (resp. VC$^\sharp$: $I({\mathbb X} _i)\in \{ 1,{\mathfrak{c}}\}$), for each $i\in I$. ${\mathcal T}$ is called actually Vaught's iff for some $\sum _{{\mathbb I}}{\mathbb X}_i\in {\mathcal D} ({\mathcal T})$ there are sentences $\tau _i\in \mathop{\rm Th}\nolimits ({\mathbb X} _i)$, $i\in I$, providing VC. We prove that: (1) VC is true for ${\mathcal T}$ iff ${\mathcal T}$ is large or its atomic model has a VC decomposition; (2) VC is true for each actually Vaught's FLD$_1$ theory; (3) VC$^\sharp$ is true for ${\mathcal T}$, if there is a VC$^\sharp$-decomposition of a model of ${\mathcal T}$. VC is true for the partial orders from the closure $\langle {\mathcal C} ^{\rm reticle}_0\cup {\mathcal C} ^{\rm ba}\rangle _{\Sigma}$, where $\langle {\mathcal C}\rangle _{\Sigma}$ denotes the closure of a class ${\mathcal C}$ under finite lexicographic sums. VC$^{\sharp}$ is true for a large class of partial orders of the form $\sum _{{\mathbb I}}(\dot{\bigcup}_{j<n_i}\prod _{k<m_i^j}{\mathbb X} _i^{j,k})_r$, where ${\mathbb X} _i^{j,k}$-s can be linear orders, or Boolean algebras, or belong to a wide class of trees. + oai:arXiv.org:2601.03155v1 + math.LO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Milo\v{s} S. Kurili\'c + + + Stability, convergence, and geometric properties of second-order-in-time space-time discretizations for linear and semilinear wave equations + https://arxiv.org/abs/2601.03160 + arXiv:2601.03160v1 Announce Type: new +Abstract: We revisit second-order-in-time space-time discretizations of the linear and semilinear wave equations by establishing precise equivalences with first-order-in-time formulations. Focusing on schemes using continuous piecewise-polynomial trial functions in time, we analyze their stability, convergence, and geometric properties. We consider first a weak space-time formulation with test functions projected onto discontinuous polynomials of one degree lower in time, showing that it is equivalent to the scheme proposed in [French, Peterson 1996] in the linear case, and extended in [Karakashian, Makridakis 2005] to the semilinear case. In particular, this equivalence shows that this method conserves energy at mesh nodes but is not symplectic. We then introduce two symplectic variants, obtained through Gauss-Legendre and Gauss-Lobatto quadratures in time, and show that they correspond to specific Runge-Kutta time integrators. These connections clarify the geometric structure of the space-time methods considered. + oai:arXiv.org:2601.03160v1 + math.NA + cs.NA + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-sa/4.0/ + Matteo Ferrari, Ilaria Perugia, Enrico Zampa + + + Forward self-similar solutions to the 2D Navier--Stokes equations + https://arxiv.org/abs/2601.03161 + arXiv:2601.03161v1 Announce Type: new +Abstract: We construct self-similar solutions to the 2D Navier--Stokes equations evolving from arbitrarily large $-1$--homogeneous initial data and present numerical evidence for their non-uniqueness. + oai:arXiv.org:2601.03161v1 + math.AP + physics.flu-dyn + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Dallas Albritton, Julien Guillod, Mikhail Korobkov, Xiao Ren + + + On the Euclidean duals of the cyclic codes generated by cyclotomic polynomials + https://arxiv.org/abs/2601.03165 + arXiv:2601.03165v1 Announce Type: new +Abstract: In this article, we determine the minimum distance of the Euclidean dual of the cyclic code $\mathcal{C}_n$ generated by the $n$th cyclotomic polynomial $Q_n(x)$ over $\mathbb{F}_q$, for every positive integer $n$ co-prime to $q$. In particular, we prove that the minimum distance of $\mathcal{C}_{n}^{\perp}$ is a function of $n$, namely $2^{\omega(n)}$. This was precisely the conjecture posed by us in \cite{BHAGAT2025}. + oai:arXiv.org:2601.03165v1 + cs.IT + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Anuj Kumar Bhagat, Ritumoni Sarma + + + Valuations on polyhedra and topological arrangements + https://arxiv.org/abs/2601.03176 + arXiv:2601.03176v1 Announce Type: new +Abstract: We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was mostly concerned with variations of ground fields/rings, over which the vertices of polytopes are defined, we consider more general collections of defining hyperplanes. No algebraic structures are imposed on these collections. This setting allows us to uncover a close relationship between scissors congruence problems on the one hand and finite hyperplane arrangements on the other hand: there are many parallel results in these fields, for which the parallelism seems to have gone unnoticed. In particular, certain properties of the Varchenko--Gelfand algebras for arrangements translate to properties of polytope rings or valuations. Studying these properties is possible in a general topological setting, that is, in the context of the so-called topological arrangements, where hyperplanes do not have to be straight and may even have nontrivial topology. + oai:arXiv.org:2601.03176v1 + math.CO + math.AT + math.GT + math.KT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Askold Khovanskii, Valentina Kiritchenko, Vladlen Timorin + + + Deformations of the connected sum of Gorenstein algebras + https://arxiv.org/abs/2601.03179 + arXiv:2601.03179v1 Announce Type: new +Abstract: We prove that the Gorenstein locus of the Hilbert scheme of points on $\mathbb A^n$ is non-reduced for $n>9$; we construct examples of non-reduced points that come from apolar algebras of the sum of general cubics. As a corollary, we get a non-reducedness result for the cactus scheme. We generalise the Bia{\l}ynicki-Birula decomposition to abstract deformation functors, providing a new method of studying deformation theory. Our construction gives us fractal structures on the nested Hilbert scheme. + oai:arXiv.org:2601.03179v1 + math.AG + math.AC + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Piotr Oszer + + + Strongly finitary metric monads are too strong + https://arxiv.org/abs/2601.03180 + arXiv:2601.03180v1 Announce Type: new +Abstract: Varieties of quantitative algebras are fully described by their free-algebra monads on the category Met of metric spaces. For a longer time it has been an open problem whether the resulting enriched monads are precisely the strongly finitary ones (determined by their values on finite discrete spaces). We present a counter-example: the variety of algebras on two close binary operations yields a monad which is not strongly finitary. A full characterization of free-algebra monads of varieties is: they are the semi-strongly finitary monads, i.e., weighted colimits of strongly finitary monads (in the category of finitary monads). + oai:arXiv.org:2601.03180v1 + math.CT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Jiri Adamek + + + Subjective-Objective Median-based Importance Technique (SOMIT) to Aid Multi-Criteria Renewable Energy Evaluation + https://arxiv.org/abs/2601.03182 + arXiv:2601.03182v1 Announce Type: new +Abstract: Accelerating the renewable energy transition requires informed decision-making that accounts for the diverse financial, technical, environmental, and social trade-offs across different renewable energy technologies. A critical step in this multi-criteria decision-making (MCDM) process is the determination of appropriate criteria weights. However, deriving these weights often solely involves either subjective assessment from decision-makers or objective weighting methods, each of which has limitations in terms of cognitive burden, potential bias, and insufficient contextual relevance. This study proposes the subjective-objective median-based importance technique (SOMIT), a novel hybrid approach for determining criteria weights in MCDM. By tailoring SOMIT to renewable energy evaluation, the method directly supports applied energy system planning, policy analysis, and technology prioritization under carbon neutrality goals. The practical utility of SOMIT is demonstrated through two MCDM case studies on renewable energy decision-making in India and Saudi Arabia. Using the derived weights from SOMIT, the TOPSIS method ranks the renewable energy alternatives, with solar power achieving the highest performance scores in both cases. The main contributions of this work are five-fold: 1) the proposed SOMIT reduces the number of required subjective comparisons from the conventional quadratic order to a linear order; 2) SOMIT is more robust to outliers in the alternatives-criteria matrix (ACM); 3) SOMIT balances subjective expert knowledge with objective data-driven insights, thereby mitigating bias; 4) SOMIT is inherently modular, allowing both its individual parts and the complete approach to be seamlessly coupled with a wide range of MCDM methods commonly applied in energy systems and policy analysis; 5) a dedicated Python library, pysomit, is developed for SOMIT. + oai:arXiv.org:2601.03182v1 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by-nc-nd/4.0/ + 10.1016/j.apenergy.2025.126872 + Applied Energy, 402 (2025) 126872 + Ding Ding, Yang Li, Poh Ling Neo, Zhiyuan Wang, Chongwu Xia + + + Flat simplices and kissing polytopes + https://arxiv.org/abs/2601.03183 + arXiv:2601.03183v1 Announce Type: new +Abstract: We consider how flat a lattice simplex contained in the hypercube $[0,k]^d$ can be. This question is related to the notion of kissing polytopes: two lattice polytopes contained in the hypercube $[0,k]^d$ are kissing when they are disjoint but their distance is as small as possible. We show that the smallest possible distance of a lattice point $P$ contained in the cube $[0,k]^3$ to a lattice triangle in the same cube that does not contain $P$ is $$ \frac{1}{\sqrt{3k^4-4k^3+4k^2-2k+1}} $$ when $k$ is at least $2$. We also improve the known lower bounds on the distance of kissing polytopes for $d$ at least $4$ and $k$ at least $2$. + oai:arXiv.org:2601.03183v1 + math.MG + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Antoine Deza, Lionel Pournin + + + Subprincipal Controlled Quasimodes and Spectral Instability + https://arxiv.org/abs/2601.03188 + arXiv:2601.03188v1 Announce Type: new +Abstract: Here we explore, in a series of articles, semiclassical quasimodes u(h,b), approximative solutions P(h)u(h,b)\sim 0, depending on $0<h<1$, and on b, the subprincipal symbol. We study a pseudodifferential operator with transversal intersections of bicharacteristics, where the principal symbol has double multiplicity, $p=dp=0$, in a small neigborhood $\Omega$. Because of this fact, we instead study the subprincipal symbol b, and we can conclude that we get transport equations depending on b where sign changes for the imaginary part of b give approximative solutions with small support. These modes are used to estimate spectral instability, or the pseudospectrum. We also investigate the possibility that we can factorize the model operator as $P(h)=h^2P_1P_2,$ in this way actually annihilating the subprincipal symbol, thus there is no condition for the imaginary part of b. In a follow-up article, we examine different cases for more complex operators with tangential intersections of bicharacteristics, thereby generalizing the findings here. + oai:arXiv.org:2601.03188v1 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/publicdomain/zero/1.0/ + Pelle Brooke Borgeke + + + HOMFLY parabolic restriction, defect skein theory and the Turaev coproduct + https://arxiv.org/abs/2601.03196 + arXiv:2601.03196v1 Announce Type: new +Abstract: We define a HOMFLY version of the category $\text{Rep}_q\text{P}$ of quantum representations of a parabolic subgroup $\text{P}\subseteq\text{GL}_{m+n}$ of block triangular matrices. Alongside this category, we construct functors that interpolate the usual restriction functors between $\text{GL}_{m+n}$, $\text{P}$ and the subgroup $\text{L}\subseteq\text{GL}_{m+n}$ of block-diagonal matrices, yielding a universal version of the formalism of parabolic restriction. Based on this formalism, we construct central algebras and centred bimodules which serve as algebraic ingredients for defining a skein theory on $3$-manifolds with surface and line defects. We recover the Turaev coproduct on the HOMFLY skein algebra as a particular instance of this theory. In particular, this coproduct is compatible with the cutting and gluing of surfaces. + oai:arXiv.org:2601.03196v1 + math.QA + math.AT + math.CT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Juan Ram\'on G\'omez Garc\'ia + + + A short proof of a bound on the size of finite irreducible semigroups of rational matrices + https://arxiv.org/abs/2601.03206 + arXiv:2601.03206v1 Announce Type: new +Abstract: I give a short proof of a recent result due to Kiefer and Ryzhikov showing that a finite irreducible semigroup of $n\times n$ matrices has cardinality at most $3^{n^2}$. + oai:arXiv.org:2601.03206v1 + math.GR + cs.FL + math.RT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Benjamin Steinberg + + + Signature invariants of monomial ideals + https://arxiv.org/abs/2601.03208 + arXiv:2601.03208v1 Announce Type: new +Abstract: Let $I$ be a monomial ideal of a polynomial ring $R=K[x_1,\ldots,x_n]$ over a field $K$ and let ${\rm sgn}(I)$ be its signature ideal. If $I$ is not a principal ideal, we show that the depth of $R/I$ is the depth of $R/{\rm sgn}(I)$, and the regularity of $R/{\rm sgn}(I)$ is at most the regularity of $R/I$. For ideals of height at least $2$, we show that the height and the associated primes of $I$ and its signature ${\rm sgn}(I)$ are the same, and we show that $I$ is Cohen--Macaulay (resp. Gorenstein) if and only if ${\rm sgn}(I)$ is Cohen--Macaulay (resp. Gorenstein), and furthermore we show that the v-number of ${\rm sgn}(I)$ is at most the v-number of $I$. We give an algorithm to compute the signature of a monomial ideal using \textit{Macaulay}$2$, and an algorithm to examine given families of monomial ideal by computing their signature ideals and determining which of these are either Cohen--Macaulay or Gorenstein. + oai:arXiv.org:2601.03208v1 + math.AC + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jovanny Ibarguen, Carlos E. Valencia, Rafael H. Villarreal + + + Values of ternary quadratic forms at integers and the Berry-Tabor conjecture for 3-tori + https://arxiv.org/abs/2601.03209 + arXiv:2601.03209v1 Announce Type: new +Abstract: Berry and Tabor conjectured in 1977 that spectra of generic integrable quantum systems have the same local statistics as a Poisson point process. We verify their conjecture in the case of the two-point spectral density for a quantum particle in a three-dimensional box, subject to a Diophantine condition on the domain's proportions. A permissible choice of width, height and depth is for example $1,2^{1/3},2^{-1/3}$. This extends previous work of Eskin, Margulis and Mozes (Annals of Math., 2005) in dimension two, where the problem reduces to the quantitative Oppenheim conjecture for quadratic forms of signature $(2,2)$. The difficulty in three and higher dimensions is that we need to consider the distribution of indefinite forms in shrinking rather than fixed intervals, which we are able to resolve for special diagonal forms of signature $(3,3)$ in various scalings, including a rate of convergence. A key step of our approach is to represent the relevant counting problem as an average of a theta function on $\mathrm{SL}(2,\mathbb{Z})^3\backslash\mathrm{SL}(2,\mathbb{R})^3$ over an expanding family of one-parameter unipotent orbits. The asymptotic behaviour of these unipotent averages follows from Ratner's measure classification theorem and subtle escape of mass estimates. + oai:arXiv.org:2601.03209v1 + math.NT + math-ph + math.DS + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Wooyeon Kim, Jens Marklof, Matthew Welsh + + + Lattice coverings and homogeneous covering congruences + https://arxiv.org/abs/2601.03212 + arXiv:2601.03212v1 Announce Type: new +Abstract: We consider the problem of covering $\mathbb{Z}^2$ with a finite number of sublattices of finite index, satisfying a simple minimality or non-degeneracy condition. We show how this problem may be viewed as a projective (or homogeneous) version of the well-known problem of covering systems of congruences. We give a construction of minimal coverings which produces many, but not all, minimal coverings, and determine all minimal coverings with at most $8$ sublattices. + oai:arXiv.org:2601.03212v1 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + J. E. Cremona, P. Koymans + + + Generalized affine buildings for semisimple algebraic groups over real closed fields + https://arxiv.org/abs/2601.03226 + arXiv:2601.03226v1 Announce Type: new +Abstract: We use real algebraic geometry to construct an affine $\Lambda$-building $B$ associated to the $\mathbb{F}$-points of a semisimple algebraic group, where $\mathbb{F}$ is a valued real closed field. We characterize the spherical building at infinity and the local building at a base point. We compute stabilizers of various subsets of $B$ and obtain group decompositions. + oai:arXiv.org:2601.03226v1 + math.GR + math.GT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Raphael Appenzeller + + + Algorithmic randomness in harmonic analysis + https://arxiv.org/abs/2601.03239 + arXiv:2601.03239v1 Announce Type: new +Abstract: Within the last fifteen years, a program of establishing relationships between algorithmic randomness and almost-everywhere theorems in analysis and ergodic theory has developed. In harmonic analysis, Franklin, McNicholl, and Rute characterized Schnorr randomness using an effective version of Carleson's Theorem. We show here that, for computable $1<p<\infty$, the reals at which the Fourier series of a weakly computable vector in $L^p[-\pi,\pi]$ converges are precisely the Martin-L\"{o}f random reals. Furthermore, we show that radial limits of the Poisson integral of an $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Schnorr random reals and that radial limits of the Poisson integral of a weakly $L^1(\mathbb{R})$-computable function coincide with the values of the function at exactly the Martin-L\"{o}f random reals. + oai:arXiv.org:2601.03239v1 + math.LO + cs.LO + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Johanna N. Y. Franklin, Lucas E. Rodriguez, Diego A. Rojas + + + On the Capacity Region of Individual Key Rates in Vector Linear Secure Aggregation + https://arxiv.org/abs/2601.03241 + arXiv:2601.03241v1 Announce Type: new +Abstract: We provide new insights into an open problem recently posed by Yuan-Sun [ISIT 2025], concerning the minimum individual key rate required in the vector linear secure aggregation problem. Consider a distributed system with $K$ users, where each user $k\in [K]$ holds a data stream $W_k$ and an individual key $Z_k$. A server aims to compute a linear function $\mathbf{F}[W_1;\ldots;W_K]$ without learning any information about another linear function $\mathbf{G}[W_1;\ldots;W_K]$, where $[W_1;\ldots;W_K]$ denotes the row stack of $W_1,\ldots,W_K$. The open problem is to determine the minimum required length of $Z_k$, denoted as $R_k$, $k\in [K]$. In this paper, we characterize a new achievable region for the rate tuple $(R_1,\ldots,R_K)$. The region is polyhedral, with vertices characterized by a binary rate assignment $(R_1,\ldots,R_K) = (\mathbf{1}(1 \in \mathcal{I}),\ldots,\mathbf{1}(K\in \mathcal{I}))$, where $\mathcal{I}\subseteq [K]$ satisfies the \textit{rank-increment condition}: $\mathrm{rank}\left(\bigl[\mathbf{F}_{\mathcal{I}};\mathbf{G}_{\mathcal{I}}\bigr]\right) =\mathrm{rank}\bigl(\mathbf{F}_{\mathcal{I}}\bigr)+N$. Here, $\mathbf{F}_\mathcal{I}$ and $\mathbf{G}_\mathcal{I}$ are the submatrices formed by the columns indexed by $\mathcal{I}$. Our results uncover the novel fact that it is not necessary for every user to hold a key, thereby strictly enlarging the best-known achievable region in the literature. Furthermore, we provide a converse analysis to demonstrate its optimality when minimizing the number of users that hold keys. + oai:arXiv.org:2601.03241v1 + cs.IT + cs.CR + cs.NI + eess.SP + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Lei Hu, Sennur Ulukus + + + Sets of Lengths of Integer-Valued Polynomials on Prime Ideals of Principal Ideal Domains + https://arxiv.org/abs/2601.03246 + arXiv:2601.03246v1 Announce Type: new +Abstract: Let $D$ be a principal ideal domain with infinite spectrum such that for every nonzero prime ideal $M$ of $D$, the residue field $D/M$ is finite. Let $K$ be the quotient field of $D$. We investigate sets of lengths in the ring of integer-valued polynomials on $M$, $\text{Int}(M, D) = \{f \in K[x] ~ \vert ~ f(M) \subseteq D\}$. For every multiset of integers $1 < z_1 \leq z_2 \leq \cdots \leq z_n$, we explicitly construct an element of $\text{Int}(M, D)$ with exactly $n$ essentially different factorizations into irreducible elements of $\text{Int}(M, D)$ whose lengths are $z_1, z_2, \ldots, z_n$. Furthermore, we show that $\text{Int}(M, D)$ is not a transfer Krull domain. These results spark off the study of sets of lengths in the rings $\text{Int}(S, D) \neq \text{Int}(D)$, where $S$ is an infinite subset of $D$. + oai:arXiv.org:2601.03246v1 + math.AC + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Zaituni Kansiime, Sholastica Luambano, Sarah Nakato, Hadijah Nalule, Yvette Ndayikunda + + + Nonlinear Spectral Modeling and Control of Soft-Robotic Muscles from Data + https://arxiv.org/abs/2601.03247 + arXiv:2601.03247v1 Announce Type: new +Abstract: Artificial muscles are essential for compliant musculoskeletal robotics but complicate control due to nonlinear multiphysics dynamics. Hydraulically amplified electrostatic (HASEL) actuators, a class of soft artificial muscles, offer high performance but exhibit memory effects and hysteresis. Here we present a data-driven reduction and control strategy grounded in spectral submanifold (SSM) theory. In the adiabatic regime, where inputs vary slowly relative to intrinsic transients, trajectories rapidly converge to a low-dimensional slow manifold. We learn an explicit input-to-output map on this manifold from forced-response trajectories alone, avoiding decay experiments that can trigger hysteresis. We deploy the SSM-based model for real-time control of an antagonistic HASEL-clutch joint. This approach yields a substantial reduction in tracking error compared to feedback-only and feedforward-only baselines under identical settings. This record-and-control workflow enables rapid characterization and high-performance control of soft muscles and muscle-driven joints without detailed physics-based modeling. + oai:arXiv.org:2601.03247v1 + math.DS + cs.CE + cs.RO + cs.SY + eess.SY + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Leonardo Bettini, Amirhossein Kazemipour, Robert K. Katzschmann, George Haller + + + Quantum polylogarithms + https://arxiv.org/abs/2601.00472 + arXiv:2601.00472v1 Announce Type: cross +Abstract: Multiple polylogarithms are periods of variations of mixed Tate motives. Conjecturally, they deliver all such periods. We introduce deformations of multiple polylogarithms depending on a complex parameter h. We call them quantum polylogarithms. Their asymptotic expansion as h goes to 0 recovers multiple polylogarithms. The quantum dilogarithm was studied by Barnes in the XIX century. Its exponent appears in many areas of Mathematics and Physics. Quantum polylogarithms satisfy a holonomic systems of modular difference equations with coefficients in variations of mixed Hodge-Tate structures of motivic origin. If h is a rational number, the quantum polylogarithms can be expressed via multiple polylogarithms. Otherwise quantum polylogarithms are not periods of variations of mixed motives, i.e. they can not be given by integrals of rational differential forms on algebraic varieties. Instead, quantum polylogarithms are integrals of differential forms built from both rational functions and exponentials of rational functions. We call them rational exponential integrals. We suggest that quantum polylogarithms reflect a very general phenomenon: Periods of variations of mixed motives should have quantum deformations. + oai:arXiv.org:2601.00472v1 + math.AG + math-ph + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Alexander B. Goncharov + + + Effect of Electric Charge on Biotherapeutic Transport, Binding and Absorption: A Computational Study + https://arxiv.org/abs/2601.00505 + arXiv:2601.00505v1 Announce Type: cross +Abstract: This study explores the effects of electric charge on the dynamics of drug transport and absorption in subcutaneous injections of monoclonal antibodies (mAbs). We develop a novel mathematical and computational model, based on the Nernst-Planck equations and porous media flow theory, to investigate the complex interactions between mAbs and charged species in subcutaneous tissue. The model enables us to study short-term transport dynamics and long-term binding and absorption for two mAbs with different electric properties. We examine the influence of buffer pH, body mass index, injection depth, and formulation concentration on drug distribution and compare our numerical results with experimental data from the literature. + oai:arXiv.org:2601.00505v1 + cs.CE + cs.NA + math.NA + physics.flu-dyn + physics.med-ph + q-bio.BM + Wed, 07 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Mario de Lucio, Pavlos P. Vlachos, Hector Gomez + + + Color-kinematics duality from an algebra of superforms + https://arxiv.org/abs/2601.02478 + arXiv:2601.02478v1 Announce Type: cross +Abstract: Color-kinematics duality states that the kinematic numerators of the cubic tree-level Yang-Mills scattering amplitudes obey the same symmetry properties that the color factors obey due to the Jacobi identity. We present a novel strategy for deriving this duality, based on the differential forms on a superspace. This space of superforms carries a generalization of a Batalin-Vilkovisky (BV) algebra (BV$^{\square}$ algebra). We show that the homotopy algebra of color-stripped Yang-Mills theory is obtained as a quotient of this space in which a subspace, which is an ideal `up to homotopy', is modded out. This algebra is a subsector of a BV$_{\infty}^{\square}$ algebra. Deriving the latter would provide a first-principle proof of color-kinematics duality from field theory. + oai:arXiv.org:2601.02478v1 + hep-th + math-ph + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Roberto Bonezzi, Christoph Chiaffrino, Olaf Hohm, Maria Foteini Kallimani + + + Exact critical-temperature bounds for two-dimensional Ising models + https://arxiv.org/abs/2601.02502 + arXiv:2601.02502v1 Announce Type: cross +Abstract: We derive exact critical-temperature bounds for the classical ferromagnetic Ising model on two-dimensional periodic tessellations of the plane. For any such tessellation or lattice, the critical temperature is bounded from a above by a universal number that is solely determined by the largest coordination number on the lattice. Crucially, these bounds are tight in some cases such as the Honeycomb, Square, and Triangular lattices. We prove the bounds using the Feynman--Kac--Ward formalism, confirm their validity for a selection of over two hundred lattices, and construct a two-dimensional lattice with 24-coordinated sites and record high critical temperature. + oai:arXiv.org:2601.02502v1 + cond-mat.stat-mech + cond-mat.mes-hall + math-ph + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Davidson Noby Joseph, Igor Boettcher + + + Compressed Qubit Noise Spectroscopy: Piecewise-Linear Modeling and Rademacher Measurements + https://arxiv.org/abs/2601.02516 + arXiv:2601.02516v1 Announce Type: cross +Abstract: Random pulse sequences are a powerful method for qubit noise spectroscopy, enabling efficient reconstruction of sparse noise spectra. Here, we advance this method in two complementary directions. First, we extend the method using a regularizer based on the total generalized variation (TGV) norm, in order to reconstruct a larger class of noise spectra, namely piecewise-linear noise spectra, which more realistically model many physical systems. We show through numerical simulations that the new method resolves finer spectral features, while maintaining an order-of-magnitude speedup over conventional approaches to noise spectroscopy. Second, we simplify the experimental implementation of the method, by introducing Rademacher measurements for reconstructing sparse noise spectra. These measurements use pseudorandom pulse sequences that can be generated in real time from a short random seed, reducing experimental complexity without compromising reconstruction accuracy. Together, these developments broaden the reach of random pulse sequences for accurate and efficient noise characterization in realistic quantum systems. + oai:arXiv.org:2601.02516v1 + quant-ph + cs.IT + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Kaixin Huang, Demitry Farfurnik, Dror Baron, Yi-Kai Liu + + + A novel finite-sample testing procedure for composite null hypotheses via pointwise rejection + https://arxiv.org/abs/2601.02529 + arXiv:2601.02529v1 Announce Type: cross +Abstract: We propose a novel finite-sample procedure for testing composite null hypotheses. Traditional likelihood ratio tests based on asymptotic $\chi^2$ approximations often exhibit substantial bias in small samples. Our procedure rejects the composite null hypothesis $H_0: \theta \in \Theta_0$ if the simple null hypothesis $H_0: \theta = \theta_t$ is rejected for every $\theta_t$ in the null region $\Theta_0$, using an inflated significance level. We derive formulas that determine this inflated level so that the overall test approximately maintains the desired significance level even with small samples. Whereas the traditional likelihood ratio test applies when the null region is defined solely by equality constraints--that is, when it forms a manifold without boundary--the proposed approach extends to null hypotheses defined by both equality and inequality constraints. In addition, it accommodates null hypotheses expressed as unions of several component regions and can be applied to models involving nuisance parameters. Through several examples featuring nonstandard composite null hypotheses, we demonstrate numerically that the proposed test achieves accurate inference, exhibiting only a small gap between the actual and nominal significance levels for both small and large samples. + oai:arXiv.org:2601.02529v1 + stat.ME + math.ST + stat.TH + Wed, 07 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Joonha Park, Ming Wang + + + Normalized Conditional Mutual Information Surrogate Loss for Deep Neural Classifiers + https://arxiv.org/abs/2601.02543 + arXiv:2601.02543v1 Announce Type: cross +Abstract: In this paper, we propose a novel information theoretic surrogate loss; normalized conditional mutual information (NCMI); as a drop in alternative to the de facto cross-entropy (CE) for training deep neural network (DNN) based classifiers. We first observe that the model's NCMI is inversely proportional to its accuracy. Building on this insight, we introduce an alternating algorithm to efficiently minimize the NCMI. Across image recognition and whole-slide imaging (WSI) subtyping benchmarks, NCMI-trained models surpass state of the art losses by substantial margins at a computational cost comparable to that of CE. Notably, on ImageNet, NCMI yields a 2.77% top-1 accuracy improvement with ResNet-50 comparing to the CE; on CAMELYON-17, replacing CE with NCMI improves the macro-F1 by 8.6% over the strongest baseline. Gains are consistent across various architectures and batch sizes, suggesting that NCMI is a practical and competitive alternative to CE. + oai:arXiv.org:2601.02543v1 + cs.LG + cs.AI + cs.CV + cs.IT + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Linfeng Ye, Zhixiang Chi, Konstantinos N. Plataniotis, En-hui Yang + + + Quantum-Enhanced Neural Contextual Bandit Algorithms + https://arxiv.org/abs/2601.02870 + arXiv:2601.02870v1 Announce Type: cross +Abstract: Stochastic contextual bandits are fundamental for sequential decision-making but pose significant challenges for existing neural network-based algorithms, particularly when scaling to quantum neural networks (QNNs) due to issues such as massive over-parameterization, computational instability, and the barren plateau phenomenon. This paper introduces the Quantum Neural Tangent Kernel-Upper Confidence Bound (QNTK-UCB) algorithm, a novel algorithm that leverages the Quantum Neural Tangent Kernel (QNTK) to address these limitations. + By freezing the QNN at a random initialization and utilizing its static QNTK as a kernel for ridge regression, QNTK-UCB bypasses the unstable training dynamics inherent in explicit parameterized quantum circuit training while fully exploiting the unique quantum inductive bias. For a time horizon $T$ and $K$ actions, our theoretical analysis reveals a significantly improved parameter scaling of $\Omega((TK)^3)$ for QNTK-UCB, a substantial reduction compared to $\Omega((TK)^8)$ required by classical NeuralUCB algorithms for similar regret guarantees. Empirical evaluations on non-linear synthetic benchmarks and quantum-native variational quantum eigensolver tasks demonstrate QNTK-UCB's superior sample efficiency in low-data regimes. This work highlights how the inherent properties of QNTK provide implicit regularization and a sharper spectral decay, paving the way for achieving ``quantum advantage'' in online learning. + oai:arXiv.org:2601.02870v1 + cs.LG + cs.IT + math.IT + quant-ph + Wed, 07 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Yuqi Huang, Vincent Y. F Tan, Sharu Theresa Jose + + + Data-driven Reduction of Transfer Operators for Particle Clustering Dynamics + https://arxiv.org/abs/2601.02932 + arXiv:2601.02932v1 Announce Type: cross +Abstract: We develop an operator-based framework to coarse-grain interacting particle systems that exhibit clustering dynamics. Starting from the particle-based transfer operator, we first construct a sequence of reduced representations: the operator is projected onto concentrations and then further reduced by representing the concentration dynamics on a geometric low-dimensional manifold and an adapted finite-state discretization. The resulting coarse-grained transfer operator is finally estimated from dynamical simulation data by inferring the transition probabilities between the Markov states. Applied to systems with multichromatic and Morse interaction potentials, the reduced model reproduces key features of the clustering process, including transitions between cluster configurations and the emergence of metastable states. Spectral analysis and transition-path analysis of the estimated operator reveal implied time scales and dominant transition pathways, providing an interpretable and efficient description of particle-clustering dynamics. + oai:arXiv.org:2601.02932v1 + cond-mat.stat-mech + math.DS + physics.comp-ph + Wed, 07 Jan 2026 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Nathalie Wehlitz, Grigorios A. Pavliotis, Christof Sch\"utte, Stefanie Winkelmann + + + Fast Surrogate Models for Adaptive Aircraft Trajectory Prediction in En route Airspace + https://arxiv.org/abs/2601.03075 + arXiv:2601.03075v1 Announce Type: cross +Abstract: Trajectory prediction (TP) is crucial for ensuring safety and efficiency in modern air traffic management systems. It is, for example, a core component of conflict detection and resolution tools, arrival sequencing algorithms, capacity planning, as well as several future concepts. However, TP accuracy within operational systems is hampered by a range of epistemic uncertainties such as the mass and performance settings of aircraft and the effect of meteorological conditions on aircraft performance. It can also require considerable computational resources. + This paper proposes a method for adaptive TP that has two components: first, a fast surrogate TP model based on linear state space models (LSSM)s with an execution time that was 6.7 times lower on average than an implementation of the Base of Aircraft Data (BADA) in Python. It is demonstrated that such models can effectively emulate the BADA aircraft performance model, which is based on the numerical solution of a partial differential equation (PDE), and that the LSSMs can be fitted to trajectories in a dataset of historic flight data. Secondly, the paper proposes an algorithm to assimilate radar observations using particle filtering to adaptively refine TP accuracy. Comparison with baselines using BADA and Kalman filtering demonstrate that the proposed framework improves system identification and state estimation for both climb and descent phases, with 46.3% and 64.7% better estimates for time to top of climb and bottom of descent compared to the best performing benchmark model. In particular, the particle filtering approach provides the flexibility to capture non-linear performance effects including the CAS-Mach transition. + oai:arXiv.org:2601.03075v1 + cs.CE + math.DS + Wed, 07 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Nick Pepper, Marc Thomas, Zack Xuereb Conti + + + Dual-quaternion learning control for autonomous vehicle trajectory tracking with safety guarantees + https://arxiv.org/abs/2601.03097 + arXiv:2601.03097v1 Announce Type: cross +Abstract: We propose a learning-based trajectory tracking controller for autonomous robotic platforms whose motion can be described kinematically on $\mathrm{SE}(3)$. The controller is formulated in the dual quaternion framework and operates at the velocity level, assuming direct command of angular and linear velocities, as is standard in many aerial vehicles and omnidirectional mobile robots. Gaussian Process (GP) regression is integrated into a geometric feedback law to learn and compensate online for unknown, state-dependent disturbances and modeling imperfections affecting both attitude and position, while preserving the algebraic structure and coupling properties inherent to rigid-body motion. + The proposed approach does not rely on explicit parametric models of the unknown effects, making it well-suited for robotic systems subject to sensor-induced disturbances, unmodeled actuation couplings, and environmental uncertainties. A Lyapunov-based analysis establishes probabilistic ultimate boundedness of the pose tracking error under bounded GP uncertainty, providing formal stability guarantees for the learning-based controller. + Simulation results demonstrate accurate and smooth trajectory tracking in the presence of realistic, localized disturbances, including correlated rotational and translational effects arising from magnetometer perturbations. These results illustrate the potential of combining geometric modeling and probabilistic learning to achieve robust, data-efficient pose control for autonomous robotic systems. + oai:arXiv.org:2601.03097v1 + cs.RO + cs.SY + eess.SY + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Omayra Yago Nieto, Alexandre Anahory Simoes, Juan I. Giribet, Leonardo Colombo + + + Finite Memory Belief Approximation for Optimal Control in Partially Observable Markov Decision Processes + https://arxiv.org/abs/2601.03132 + arXiv:2601.03132v1 Announce Type: cross +Abstract: We study finite memory belief approximation for partially observable (PO) stochastic optimal control (SOC) problems. While belief states are sufficient for SOC in partially observable Markov decision processes (POMDPs), they are generally infinite-dimensional and impractical. We interpret truncated input-output (IO) histories as inducing a belief approximation and develop a metric-based theory that directly relates information loss to control performance. Using the Wasserstein metric, we derive policy-conditional performance bounds that quantify value degradation induced by finite memory along typical closed-loop trajectories. Our analysis proceeds via a fixed-policy comparison: we evaluate two cost functionals under the same closed-loop execution and isolate the effect of replacing the true belief by its finite memory approximation inside the belief-level cost. For linear quadratic Gaussian (LQG) systems, we provide closed-form belief mismatch evaluation and empirically validate the predicted mechanism, demonstrating that belief mismatch decays approximately exponentially with memory length and that the induced performance mismatch scales accordingly. Together, these results provide a metric-aware characterization of what finite memory belief approximation can and cannot achieve in PO settings. + oai:arXiv.org:2601.03132v1 + eess.SY + cs.IT + cs.LG + cs.SY + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Mintae Kim + + + Higher-Dimensional Anyons via Higher Cohomotopy + https://arxiv.org/abs/2601.03150 + arXiv:2601.03150v1 Announce Type: cross +Abstract: We highlight that integer Heisenberg groups at level 2 underlie topological quantum phenomena: their group algebras coincide with the algebras of quantum observables of abelian anyons in fractional quantum Hall (FQH) systems on closed surfaces. Decades ago, these groups were shown to arise as the fundamental groups of the space of maps from the surface to the 2-sphere -- which has recently been understood as reflecting an effective FQH flux quantization in 2-Cohomotopy. Here we streamline and generalize this theorem using the homotopy theory of H-groups, showing that for $k \in \{1,2,4\}$, the non-torsion part of $\pi_1 \mathrm{Map}\big({(S^{2k-1})^2, S^{2k}}\big)$ is an integer Heisenberg group of level 2, where we identify this level with 2 divided by the Hopf invariant of the generator of $\pi_{4k-1}(S^{2k})$. This result implies the existence of higher-dimensional analogs of FQH anyons in the cohomotopical completion of 11D supergravity ("Hypothesis H"). + oai:arXiv.org:2601.03150v1 + cond-mat.str-el + hep-th + math-ph + math.AT + math.MP + quant-ph + Wed, 07 Jan 2026 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Sadok Kallel, Hisham Sati, Urs Schreiber + + + On the Bezrukavnikov-Kaledin quantization of symplectic varieties in characteristic $p$ + https://arxiv.org/abs/2011.08259 + arXiv:2011.08259v2 Announce Type: replace +Abstract: We prove that after inverting the Planck constant $h$ the Bezrukavnikov-Kaledin quantization $(X, \mathcal{O}_h)$ of symplectic variety $X$ in characteristic $p$ is Morita equivalent to a certain central reduction of the algebra of differential operators on $X$. + oai:arXiv.org:2011.08259v2 + math.AG + math.RT + math.SG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + 10.1112/S0010437X23007601 + Compositio Mathematica, Volume 160, Issue 2, February 2024, pp. 411 - 450 + Ekaterina Bogdanova, Vadim Vologodsky + + + Equivariant Chevalley, Giambelli, and Monk Formulae for the Peterson Variety + https://arxiv.org/abs/2111.15663 + arXiv:2111.15663v2 Announce Type: replace +Abstract: We present a formula for the Poincar\'e dual in the flag manifold of the equivariant fundamental class of any regular nilpotent or regular semisimple Hessenberg variety as a polynomial in terms of certain Chern classes. We then develop a type-independent proof of the Giambelli formula for the Peterson variety, and use this formula to compute the intersection multiplicity of a Peterson variety with an opposite Schubert variety corresponding to a Coxeter word. Finally, we develop an equivariant Chevalley formula for the cap product of a divisor class with a fundamental class, and a dual Monk rule, for the Peterson variety. + oai:arXiv.org:2111.15663v2 + math.AG + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Rebecca Goldin, Rahul Singh + + + Stability of three-dimensional stochastic Navier-Stokes equation with Markov switching + https://arxiv.org/abs/2203.15971 + arXiv:2203.15971v2 Announce Type: replace +Abstract: A right continuous Markov chain is introduced in the noise terms of the three-dimensional stochastic Navier-Stokes equation, and we call such stochastic system as stochastic Navier-Stokes equation with Markov switching. In the present article, we study the $p$-th moment exponential stability and the almost surely exponential stability of the solution to the equation. + oai:arXiv.org:2203.15971v2 + math.PR + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Po-Han Hsu + + + A Host--Kra ${\mathbf F}_2^\omega$-system of order $5$ that is not Abramov of order $5$, and non-measurability of the inverse theorem for the $U^6({\mathbf F}_2^n)$ norm + https://arxiv.org/abs/2303.04853 + arXiv:2303.04853v4 Announce Type: replace +Abstract: It was conjectured by Bergelson, Tao, and Ziegler \cite{btz} that every Host--Kra $\F_p^\omega$-system of order $k$ is an Abramov system of order $k$. This conjecture has been verified for $k \leq p+1$. In this paper we show that the conjecture fails when $k=5, p=2$. We in fact establish a stronger (combinatorial) statement, in that we produce a bounded function $f: \F_2^n \to \C$ of large Gowers norm $\|f\|_{U^6(\F_2^n)}$ which (as per the inverse theorem for that norm) correlates with a non-classical quintic phase polynomial $e(P)$, but with the property that all such phase polynomials $e(P)$ are ``non-measurable'' in the sense that they cannot be well approximated by functions of a bounded number of random translates of $f$. A simpler version of our construction can also be used to answer a question of Candela, Gonz\'alez-S\'anchez, and Szegedy \cite{CGSS}. + oai:arXiv.org:2303.04853v4 + math.DS + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Asgar Jamneshan, Or Shalom, Terence Tao + + + Global Koszul duality + https://arxiv.org/abs/2304.08409 + arXiv:2304.08409v4 Announce Type: replace +Abstract: We construct a monoidal model structure on the category of all curved coalgebras and show that it is Quillen equivalent, via the extended bar-cobar adjunction, to another model structure we construct on the category of curved algebras. When the coalgebras under consideration are conilpotent and the algebras are dg, i.e. uncurved, this corresponds to the ordinary dg Koszul duality of Positselski and Keller-Lef\`evre. As an application we construct global noncommutative moduli spaces for flat connections on vector bundles, holomorphic structures on almost complex vector bundles, dg modules over a dg algebra, objects in a dg category, and others. + oai:arXiv.org:2304.08409v4 + math.CT + math.AG + math.AT + math.KT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Matt Booth, Andrey Lazarev + + + An $\mathfrak{sl}_2$ action on link homology of T(2,k) torus links + https://arxiv.org/abs/2307.01910 + arXiv:2307.01910v2 Announce Type: replace +Abstract: We determine an $\mathfrak{sl}_2$ module structure on the equivariant Khovanov-Rozansky homology of (2,k)-torus links following the framework defined in arXiv:2306.10729. + oai:arXiv.org:2307.01910v2 + math.GT + math.QA + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Felix Roz + + + The Beauville-Voisin conjecture for double EPW sextics + https://arxiv.org/abs/2307.15240 + arXiv:2307.15240v2 Announce Type: replace +Abstract: We prove that the Beauville-Voisin conjecture is true for any double EPW sextic, i.e. the subalgebra of the Chow ring generated by divisors and Chern classes of the tangent bundle injects into cohomology. + oai:arXiv.org:2307.15240v2 + math.AG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Kyoto J. of Math. 65 no. 4 (2025), 715-736 + Robert Laterveer + + + Doubly-weighted zero-sum constants + https://arxiv.org/abs/2311.00090 + arXiv:2311.00090v3 Announce Type: replace +Abstract: Let $A,B\subseteq\mathbb Z_n$ be given and $S=(x_1,\ldots, x_k)$ be a sequence in $\mathbb Z_n$. We say that $S$ is an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ and $b_1,\ldots,b_k\in B$ such that $a_1x_1+\cdots+a_kx_k=0$ and $b_1a_1+\cdots+b_ka_k=0$. We show that if $S$ has length $2n-1$, then $S$ has an $(A,B)$-weighted zero-sum subsequence of length $n$. The constant $E_{A,B}$ is defined to be the smallest positive integer $k$ such that every sequence of length $k$ in $\mathbb Z_n$ has an $(A,B)$-weighted zero-sum subsequence of length $n$. A sequence in $\mathbb Z_n$ of length $E_{A,B}-1$ which does not have any $(A,B)$-weighted zero-sum subsequence of length $n$ is called an $E$-extremal sequence for $(A,B)$. We determine the constant $E_{A,B}$ and characterize the $E$-extremal sequences for some pairs $(A,B)$. We also study the related constants $C_{A,B}$ and $D_{A,B}$ which are defined in the article. + oai:arXiv.org:2311.00090v3 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Krishnendu Paul, Shameek Paul + + + On the kernels of the pro-$p$ outer Galois representations associated to once-punctured CM elliptic curves + https://arxiv.org/abs/2312.04196 + arXiv:2312.04196v3 Announce Type: replace +Abstract: In this paper, we compare a certain field arising from the pro-$p$ outer Galois representation associated to a once-punctured CM elliptic curve over an imaginary quadratic field $K$ with the maximal pro-$p$ Galois extension of the mod-$p$ ray class field $K(p)$ of $K$ unramified outside $p$. We prove that these two fields coincide for every prime $p$ which satisfies certain assumptions, assuming an analogue of the Deligne-Ihara conjecture. This may be regarded as an analogue of a result of Sharifi on the kernel of the pro-$p$ outer Galois representation associated to the projective line minus three points. + oai:arXiv.org:2312.04196v3 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Shun Ishii + + + Decision Making under Costly Sequential Information Acquisition: the Paradigm of Reversible and Irreversible Decisions + https://arxiv.org/abs/2401.00569 + arXiv:2401.00569v4 Announce Type: replace +Abstract: Decision making in modern stochastic systems, including e-commerce platforms, financial markets and healthcare systems, has evolved into a multifaceted process that combines information acquisition and adaptive information sources. This paper initiates a study on such integrated settings, where these elements are not only fundamental but, also, interact in a complex and stochastically intertwined manner. + We introduce a relatively simple model, which, however, captures the involved novel elements. A decision maker (DM) may choose between an established product $A$ of known value and a new product $B$ whose value is unknown. In parallel, the DM observes signals about the unknown value of product $B$ and can, also, opt to exchange it for product $A$ if $B$ is initially chosen. Mathematically, the model gives rise to sequential optimal stopping problems with distinct informational regimes (before and after buying product $B$), differentiated by the initial, coarser signal and the subsequent, more accurate one. We analyze in detail the underlying problems using predominantly viscosity solution techniques, departing from the existing literature on information acquisition which is based on traditional optimal stopping arguments. + More broadly, the modeling approach introduced herein offers a novel framework for developing more complex interactions among decisions, information sources and information costs in stochastic environments, through a sequence of nested obstacle problems. + oai:arXiv.org:2401.00569v4 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Renyuan Xu, Thaleia Zariphopoulou, Luhao Zhang + + + Quasi-ergodic theorems for Feynman-Kac semigroups and large deviation for additive functionals + https://arxiv.org/abs/2401.17997 + arXiv:2401.17997v2 Announce Type: replace +Abstract: We study the long-time behavior of an additive functional that takes into account the jumps of a symmetric Markov process. This process is assumed to be observed through a biased observation scheme that includes the survival to events of extinction and the Feynman-Kac weight by another similar additive functional. Under conditioning for the convergence to a quasi-stationary distribution and for two-sided estimates of the Feynmac-Kac semigroup to be obtained, we shall discuss general assumptions on the symmetric Markov process. For the law of additive functionals, we will prove a quasi-ergodic theorem, namely a conditional version of the ergodic theorem and a conditional functional weak law of large numbers. As an application, we also establish a large deviation principle for the mean ratio of additive functionals. + oai:arXiv.org:2401.17997v2 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Daehong Kim, Takara Tagawa, Aur\'elien Velleret + + + On the permutation automorphisms of binary cubic codes + https://arxiv.org/abs/2402.10667 + arXiv:2402.10667v3 Announce Type: replace +Abstract: A binary linear code whose permutation automorphism group has a fixed point free permutation of order $3$ is called a binary cubic code. The scope of this paper is to investigate the structural properties of binary cubic codes. Let $C$ be a binary cubic $[n,k]$ code. In this paper, we prove that if $n\geq 30$ and $C$ has permutation automorphism group of order three, then $k\geq 6$. Additionally, we show that if $n < 30$ and $k\leq 4$, then the permutation automorphism group of $C$ has order greater than three. Moreover, along the way, we provide some results on the structure of the higher dimensional cubic codes. In particular, we present some results concerning the structure of the putative extremal self-dual $[72,36,16]$ code under the assumption that it is cubic. + oai:arXiv.org:2402.10667v3 + cs.IT + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Murat Altunbulak, Fatma Altunbulak Aksu, Roghayeh Hafezieh, \.Ipek Tuvay + + + A Method For Bounding Tail Probabilities + https://arxiv.org/abs/2402.13662 + arXiv:2402.13662v3 Announce Type: replace +Abstract: We present a method for upper and lower bounding the right and the left tail probabilities of continuous random variables (RVs). For the right tail probability of RV $X$ with probability density function $f (x)$, this method requires first setting a continuous, positive, and strictly decreasing function $g (x)$ such that $-f (x)/g' (x)$ is a decreasing and increasing function, $\forall x>x_0$, which results in upper and lower bounds, respectively, given in the form $-f (x) g (x)/g' (x)$, $\forall x>x_0$, where $x_0$ is some point. Similarly, for the upper and lower bounds on the left tail probability of $X$, this method requires first setting a continuous, positive, and strictly increasing function $g (x)$ such that $f (x)/g' (x)$ is an increasing and decreasing function, $\forall x<x_0$, which results in upper and lower bounds, respectively, given in the form $f (x) g (x)/g' (x)$, $\forall x<x_0$. We provide some examples of good candidates for the function $g (x)$. We also establish connections between the new bounds and Markov's inequality and Chernoff's bound. In addition, we provide an iterative method for obtaining ever tighter lower and upper bounds, under certain conditions. As an application, we use the proposed method to derive a novel closed-form asymptotic expression of the converse bound on the capacity of the additive white Gaussian noise (AWGN) channel in the finite-blocklength regime, which is tighter than the closed-form asymptotic expression by Polyanskiy-Poor-Verd\'u. Finally, we provide numerical examples where we show the tightness of the bounds obtained by the proposed method. + oai:arXiv.org:2402.13662v3 + math.PR + cs.IT + math.IT + math.ST + stat.ML + stat.TH + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.1109/ACCESS.2026.3650974 + IEEE Access, 2026 + Nikola Zlatanov + + + A Context for Manifold Calculus + https://arxiv.org/abs/2403.03321 + arXiv:2403.03321v5 Announce Type: replace +Abstract: We develop Weiss's manifold calculus in the setting of $\infty$-categories, where we allow the target $\infty$-category to be any $\infty$-category with small limits. We will establish the connection between polynomial functors, Kan extensions, and Weiss sheaves, and will classify homogeneous functors. We will also generalize Weiss and Boavida de Brito's theorem to functors taking values in arbitrary $\infty$-categories with small limits. + oai:arXiv.org:2403.03321v5 + math.AT + math.CT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Kensuke Arakawa + + + Convergence of Decentralized Stochastic Subgradient-based Methods for Nonsmooth Nonconvex functions + https://arxiv.org/abs/2403.11565 + arXiv:2403.11565v4 Announce Type: replace +Abstract: In this paper, we focus on the decentralized stochastic subgradient-based methods in minimizing nonsmooth nonconvex functions without Clarke regularity, especially in the decentralized training of nonsmooth neural networks. We propose a general framework that unifies various decentralized subgradient-based methods, such as decentralized stochastic subgradient descent (DSGD), DSGD with gradient-tracking technique (DSGD-T), and DSGD with momentum (DSGD-M). To establish the convergence properties of our proposed framework, we relate the discrete iterates to the trajectories of a continuous-time differential inclusion, which is assumed to have a coercive Lyapunov function with a stable set $\mathcal{A}$. We prove the asymptotic convergence of the iterates to the stable set $\mathcal{A}$ with sufficiently small and diminishing step-sizes. These results provide first convergence guarantees for some well-recognized of decentralized stochastic subgradient-based methods without Clarke regularity of the objective function. Preliminary numerical experiments demonstrate that our proposed framework yields highly efficient decentralized stochastic subgradient-based methods with convergence guarantees in the training of nonsmooth neural networks. + oai:arXiv.org:2403.11565v4 + math.OC + cs.LG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Siyuan Zhang, Nachuan Xiao, Xin Liu + + + The global Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups I: Coarse expansions of the relative trace formulae + https://arxiv.org/abs/2404.07342 + arXiv:2404.07342v3 Announce Type: replace +Abstract: This is the first of a series of three papers where we prove the Gan--Gross--Prasad conjecture for Fourier--Jacobi periods on unitary groups and an Ichino--Ikeda type refinement. Our strategy is based on the comparison of relative trace formulae formulated by Liu. The goal of this first paper is to introduce the relative trace formulae and establish the coarse expansions. + oai:arXiv.org:2404.07342v3 + math.RT + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Paul Boisseau, Weixiao Lu, Hang Xue + + + Real plane separating (M-2)-curves of degree d and totally real pencils of degree d-3 + https://arxiv.org/abs/2404.09671 + arXiv:2404.09671v4 Announce Type: replace +Abstract: It is well known that a non-singular real plane projective curve of degree five with five connected components is separating if and only if its ovals are in non-convex position. In this article, this property is set into a different context and generalised to all real plane separating (M-2)-curves. + oai:arXiv.org:2404.09671v4 + math.AG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-sa/4.0/ + \'Epijournal de G\'eometrie Alg\'ebrique, Volume 10 (2026), Article no. 1 + Matilde Manzaroli + + + Master equations with indefinite nonlinearities + https://arxiv.org/abs/2405.02091 + arXiv:2405.02091v2 Announce Type: replace +Abstract: In this paper, we consider the following indefinite fully fractional heat equation involving the master operator \begin{equation} (\partial_t -\Delta)^{s} u(x,t) = x_1u^p(x,t)\ \ \mbox{in}\ \R^n\times\R , \end{equation} where $s\in(0,1)$, and $-\infty < p < \infty$. Under mild conditions, we prove that there is no positive bounded solutions. To this end, we first show that the solutions are strictly increasing along $x_1$ direction by employing the direct method of moving planes. Then by constructing an unbounded sub-solution, we derive the nonexistence of bounded solutions. + To circumvent the difficulties caused by the fully fractional master operator, we introduced some new ideas and novel approaches that, as we believe, will become useful tool in studying a variety of other fractional elliptic and parabolic problems. + oai:arXiv.org:2405.02091v2 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Wenxiong Chen, Yahong Guo + + + Common Noise by Random Measures: Constructing Mean-Field Equilibria for Competitive Investment and Hedging + https://arxiv.org/abs/2408.01175 + arXiv:2408.01175v2 Announce Type: replace +Abstract: We construct Nash-equilibria in mean-field portfolio games of optimal investment and hedging under relative performance concerns with exponential (CARA) utility preferences. Common noise dynamics are modeled by integer-valued random measures, for instance Poisson random measures, in addition to Brownian motions. Agents differ in individual risk aversions, competition weights, and initial capital endowments, while their contingent claim liabilities depend on both common and idiosyncratic risk factors. Mean-field equilibria are characterized by solutions to McKean-Vlasov backward stochastic differential equations with jumps, for which we prove existence and uniqueness of solutions, without assuming mean field interaction to be small. Moreover, we show how the equilibrium can be constructed from the optimal strategy of a single-agent optimization problem (without mean-field interaction) via an appropriate projection. Using successive changes of measure, our derivation provides a decomposition of the equilibrium strategy into three components with clear interpretations. Finally, we show how a limiting mean-field game of quadratic (instead of utility-based) hedging with relative performance concerns arises for vanishing risk aversion. + oai:arXiv.org:2408.01175v2 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Dirk Becherer, Stefanie Hesse + + + Monodromy and vanishing cycles for complete intersection curves + https://arxiv.org/abs/2408.06479 + arXiv:2408.06479v3 Announce Type: replace +Abstract: We compute the topological monodromy of every family of complete intersection curves. Like in the case of plane curves previously treated by the second-named author, we find the answer is given by the $r$-spin mapping class group associated to the maximal root of the adjoint line bundle. Our main innovation is a suite of tools for studying the monodromy of sections of a tensor product of very ample line bundles in terms of the monodromy of sections of the factors, allowing for an induction on (multi-)degree. + oai:arXiv.org:2408.06479v3 + math.AG + math.GT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ishan Banerjee, Nick Salter + + + Instability of Legendrian knottedness, and non-regular Lagrangian concordances of knots + https://arxiv.org/abs/2409.00290 + arXiv:2409.00290v2 Announce Type: replace +Abstract: We show that the family of smoothly non-isotopic Legendrian pretzel knots from the work of Cornwell-Ng-Sivek that all have the same Legendrian invariants as the standard unknot have front-spuns that are Legendrian isotopic to the front-spun of the unknot. Besides that, we construct the first examples of Lagrangian concordances between Legendrian knots that are not regular, and hence not decomposable. Finally, we show that the relation of Lagrangian concordance between Legendrian knots is not anti-symmetric, and hence does not define a partial order. The latter two results are based upon a new type of flexibility for Lagrangian concordances with stabilised Legendrian ends. + oai:arXiv.org:2409.00290v2 + math.SG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Georgios Dimitroglou Rizell, Roman Golovko + + + A combination theorem for hierarchically quasiconvex subgroups, and application to geometric subgroups of mapping class groups + https://arxiv.org/abs/2409.03602 + arXiv:2409.03602v2 Announce Type: replace +Abstract: We provide sufficient conditions for two subgroups of a hierarchically hyperbolic group to generate an amalgamated free product over their intersection. The result applies in particular to certain geometric subgroups of mapping class groups of finite-type surfaces, that is, those subgroups coming from the embeddings of closed subsurfaces. In the second half of the paper, we study under which hypotheses our amalgamation procedure preserves several notions of convexity, such as hierarchical quasiconvexity (as introduced by Behrstock, Hagen, and Sisto) and strong quasiconvexity (every quasigeodesic with endpoints on the subset lies in a uniform neighbourhood). This answers a question of Russell, Spriano, and Tran. + oai:arXiv.org:2409.03602v2 + math.GR + math.GT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Giorgio Mangioni + + + Approximability of deep equilibria + https://arxiv.org/abs/2409.06064 + arXiv:2409.06064v4 Announce Type: replace +Abstract: We introduce a structural framework for computations involving floating-point operations.Informed by real-valued logic, we introduce deep computations (ultracomputations) and deep iterates, formalizing the ideas of "asymptotic limit" of computations and compositional iterates, respectively. + As an application of this framework, we prove the existence of deep equilibria, which hitherto have been found only empirically (yielding remarkable memory savings in deep learning). Our proof of existence of deep equilibria is based on the concept of idempotent ultrafilter from combinatorics and inspired by the notion of indiscernibility from model theory. + We study and characterize deep computations (and hence deep equilibria) that are bona fide computable, i.e., uniformly approximable by a priori given computable primitive real-valued functions. Informed by model theory of real-valued structures, as well as Cp-theory from topology, we use a classical result of Grothendieck to characterize computability of deep computations in terms of continuous extendibility. + Our framework does not impose a priori uniform/global bounds on real-valued quantities; therefore, our structures yield non-compact types spaces. Such type spaces require a more nuanced topological treatment than compact ones arising in model theory of [0,1]-valued structures. + oai:arXiv.org:2409.06064v4 + math.LO + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Samson Alva, Eduardo Due\~nez, Jose Iovino, Claire Walton + + + The leftmost particle of branching subordinators + https://arxiv.org/abs/2409.16617 + arXiv:2409.16617v2 Announce Type: replace +Abstract: We define a family of continuous-time branching particle systems on the non-negative real line, called branching subordinators, where particles move as independent subordinators. Each particle can also split (at possibly infinite rate) into several children (possibly infinitely many) whose positions relative to the position of the parent are random. These particle systems are in the continuity of branching L\'evy processes introduced by Bertoin and Mallein [Ann. Probab. 47(3): 1619-1652 (2019)]. We pay a particular attention to the asymptotic behavior of the leftmost particle of branching subordinators. It turns out that, under some assumptions, the rate of growth of the position of the leftmost particle is of order $t^{\gamma}$ where $\gamma \in (0,1)$ depends explicitly on the parameters. This sub-linear growth is significantly different from the classical linear growth observed for regular branching random walks with non-negative displacements. + oai:arXiv.org:2409.16617v2 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Alexis Kagan, Gr\'egoire V\'echambre + + + A Machine Learning Model for Solving Lane-Emden Equation using Legendre Wavelet Neural Network + https://arxiv.org/abs/2410.05409 + arXiv:2410.05409v2 Announce Type: replace +Abstract: As we know differential equations are very useful for electrical engineers to solve a variety of problems like: voltage across a capacitor, input versus output voltage, etc. Therefore, the goal of this paper is to find the solutions of non-linear differential equations based on the Lane Emden equation of second order using the Legendre wavelet neural network (LWNN) method. Here all the considered equations are singular initial value problems. To manage the singularity challenge, we have employed an artificial neural network method. This approach utilizes a neural network of a single layer, where the hidden layer is omitted by enlarging the input using Legendre wavelets functions. We have applied a feed-forward neural network method to the proposed problem along with the principle of error backpropagation. The effectiveness of the Legendre wavelet Neural Network method is validated through Lane Emden equations.. + oai:arXiv.org:2410.05409v2 + math.NA + cs.NA + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Vijay Kumar Patel, Abhishekh, Dileep Kumar, Nitin Kumar + + + Stability of reverse isoperimetric inequalities in the plane: area, Cheeger, and inradius + https://arxiv.org/abs/2410.06096 + arXiv:2410.06096v2 Announce Type: replace +Abstract: In this paper, we present sharp stability results for various reverse isoperimetric problems in $\mathbb R^2$. Specifically, we prove the stability of the reverse isoperimetric inequality for $\lambda$-convex bodies -- convex bodies with the property that each of their boundary points $p$ supports a ball of radius $1/\lambda$ so that the body lies inside the ball in a neighborhood of $p$. For convex bodies with smooth boundaries, $\lambda$-convexity is equivalent to having the curvature of the boundary bounded below by $\lambda > 0$. Additionally, within this class of convex bodies, we establish stability for the reverse inradius inequality and the reverse Cheeger inequality. Even without its stability version, the sharp reverse Cheeger inequality is new in dimension $2$. + oai:arXiv.org:2410.06096v2 + math.DG + math.MG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Kostiantyn Drach, Kateryna Tatarko + + + Mapping class group orbit closures for Deroin-Tholozan representations + https://arxiv.org/abs/2411.10269 + arXiv:2411.10269v2 Announce Type: replace +Abstract: We prove that infinite mapping class group orbits are dense in the character variety of Deroin-Tholozan representations. In other words, the action is minimal except for finite orbits. Our arguments rely on the symplectic structure of the character variety, emphasizing this geometric perspective over its algebraic properties. + oai:arXiv.org:2411.10269v2 + math.DS + math.GT + math.SG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.3934/jmd.2025019 + Journal of Modern Dynamics, 2025, 21: 805-850 + Yohann Bouilly, Gianluca Faraco, Arnaud Maret + + + Information geometric regularization of unidimensional pressureless Euler equations yields global strong solutions + https://arxiv.org/abs/2411.15121 + arXiv:2411.15121v2 Announce Type: replace +Abstract: Partial differential equations describing compressible fluids are prone to the formation of shock singularities, arising from faster upstream fluid particles catching up to slower, downstream ones. In geometric terms, this causes the deformation map to leave the manifold of diffeomorphisms. Information geometric regularization addresses this issue by changing the manifold geometry to make it geodesically complete. Empirical evidence suggests that this results in smooth solutions without adding artificial viscosity. This work makes a first step towards understanding this phenomenon rigorously, in the setting of the unidimensional pressureless Euler equations. It shows that their information geometric regularization has smooth global solutions. By establishing $\Gamma$-convergence of its variational description, it proves convergence of these solutions to entropy solutions of the nominal problem, in the limit of vanishing regularization parameter. A consequence of these results is that manifolds of unidimensional diffeomorphisms with information geometric regularization are geodesically complete. + oai:arXiv.org:2411.15121v2 + math.AP + math.DG + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ruijia Cao, Florian Sch\"afer + + + Dg-separable dg-extensions + https://arxiv.org/abs/2412.06526 + arXiv:2412.06526v2 Announce Type: replace +Abstract: We define and characterise completely dg-separable dg-extensions $\varphi:(A,d_A)\rightarrow (B,d_B)$. We completely characterise the case of graded commutative dg-division algebras in characteristic different from $2$. We prove that for a dg-separable extension a short exact sequence of dg-modules over $(B,d_B)$ splits if and only if the restriction to $(A,d_A)$ splits. + oai:arXiv.org:2412.06526v2 + math.RA + math.RT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Alexander Zimmermann + + + Degree Realization by Bipartite Multigraphs + https://arxiv.org/abs/2501.15515 + arXiv:2501.15515v5 Announce Type: replace +Abstract: The problem of realizing a given degree sequence by a multigraph can be thought of as a relaxation of the classical degree realization problem (where the realizing graph is simple). This paper concerns the case where the realizing multigraph is required to be bipartite. + The problem of characterizing sequences that can be realized by a bipartite graph has two variants. In the simpler one, termed BDR$^P$, the partition of the sequence into two sides is given as part of the input. A complete characterization for realizability in this variant was given by Gale and Ryser over sixty years ago. However, the variant where the partition is not given, termed BDR, is still open. + For bipartite multigraph realizations, there are also two variants. For BDR$^P$, where the partition is given as part of the input, a characterization was known for determining whether there is a multigraph realization whose underlying graph is bipartite, such that the maximum number of copies of an edge is at most $r$. We present a characterization for determining if there is a bipartite multigraph realization such that the total number of excess edges is at most $t$. We show that optimizing these two measures may lead to different realizations, and that optimizing by one measure may increase the other substantially. As for the variant BDR, where the partition is not given, we show that determining whether a given (single) sequence admits a bipartite multigraph realization is NP-hard. Moreover, we show that this hardness result extends to any graph family which is a sub-family of bipartite graphs and a super-family of paths. On the positive side, we provide an algorithm that computes optimal realizations for the case where the number of balanced partitions is polynomial, and present sufficient conditions for the existence of bipartite multigraph realizations that depend only on the largest degree of the sequence. + oai:arXiv.org:2501.15515v5 + math.CO + cs.DM + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.46298/dmtcs.15158 + Discrete Mathematics & Theoretical Computer Science 28:2 #7, 2026 + Amotz Bar-Noy, Toni Bohnlein, David Peleg, Dror Rawitz + + + Regularity of solutions for fully fractional parabolic equations + https://arxiv.org/abs/2502.07530 + arXiv:2502.07530v2 Announce Type: replace +Abstract: In this paper, we study the fully fractional heat equation involving the master operator: $$ (\partial_t -\Delta)^{s} u(x,t) = f(x,t)\ \ \mbox{in}\ \mathbb{R}^n\times\mathbb{R} , $$ where $s\in(0,1)$ and $f(x,t) \geq 0$. + First we derive H\"{o}lder and Schauder estimates for nonnegative solutions of this equation. Due to the {\em nonlocality} of the master operator, existing results (cf. \cite{ST}) rely on global bounds of the solutions $u$ to control their higher local norms. However, such results are inadequate for blow-up and rescaling analysis aimed at obtaining a priori estimates for solutions to {\em nonlocal } equations on unbounded domains, as the global norms of the rescaled functions may diverge. + This limitation raises to a natural and challenging question: + {\em Can local bounds of solutions replace global bounds to control their higher local norms?} + Here, we provide an affirmative answer to this question for nonnegative solutions. To achieve this, we introduced several new ideas and novel techniques. One of the key innovations is to use a {\em directional perturbation average} to derive an important estimate for the fully fractional heat kernel, as stated in Lemma \ref{key0}. We believe this estimate, along with other new techniques introduced here, will serve as powerful tools in regularity estimates for a wide range of nonlocal equations. + Building on this breakthrough, we employ the blow-up and rescaling arguments to establish a priori estimates for solutions to a broader class of nonlocal equations in unbounded domains, such as $$(\partial_t -\Delta)^{s} u(x,t) = b(x,t) |\nabla_x u (x,t)|^q + f(x, u(x,t))\ \ \mbox{in}\ \ \mathbb{R}^n\times\mathbb{R}.$$ Under appropriate conditions, we prove that all nonnegative solutions, along with their spatial gradients, are uniformly bounded. + oai:arXiv.org:2502.07530v2 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Wenxiong Chen, Yahong Guo, Congming Li + + + Lie algebras of quotient groups + https://arxiv.org/abs/2502.10260 + arXiv:2502.10260v2 Announce Type: replace +Abstract: We give conditions on a diffeological group $G$ and a normal subgroup $H$ under which the quotient group $G/H$ differentiates to a Lie algebra for which $\operatorname{Lie}(G/H) \cong \operatorname{Lie}(G)/\operatorname{Lie}(H)$. Our Lie functor is instantiated by the tangent structure on elastic diffeological spaces introduced by Blohmann. The requisite conditions on $G$ and $H$ hold, for example, when $G$ is a convenient infinite-dimensional Lie group and $H$ is countable, or when $G$ is finite-dimensional and $H$ is arbitrary. To recognize that convenient infinite-dimensional manifolds are elastic diffeological spaces, we give a characterization of convenience in terms of the diffeological tangent functor: a separated and bornological locally convex topological vector space $E$ is convenient if and only if the natural map $E \times E \to TE$ is an isomorphism of diffeological spaces. As an application, we integrate some classically non-integrable Banach-Lie algebras to diffeological groups. + oai:arXiv.org:2502.10260v2 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-sa/4.0/ + David Miyamoto + + + The shift-homological spectrum and parametrising kernels of rank functions + https://arxiv.org/abs/2502.11939 + arXiv:2502.11939v2 Announce Type: replace +Abstract: For any compactly generated triangulated category we introduce two topological spaces, the shift-spectrum and the shift-homological spectrum. We use them to parametrise a family of thick subcategories of the compact objects, which we call radical. These spaces can be viewed as non-monoidal analogues of the Balmer and homological spectra arising in tensor-triangular geometry: we prove that for monogenic tensor-triangulated categories the Balmer spectrum is a subspace of the shift-spectrum. To construct these analogues we utilise quotients of the module category, rather than the lattice theoretic methods which have been adopted in other approaches. We characterise radical thick subcategories and show in certain cases, such as the perfect derived categories of tame hereditary algebras or monogenic tensor-triangulated categories, that every thick subcategory is radical. We establish a close relationship between the shift-homological spectrum and the set of irreducible integral rank functions, and provide necessary and sufficient conditions for every radical thick subcategory to be given by an intersection of kernels of rank functions. In order to facilitate these results, we prove that both spaces we introduce may equivalently be described in terms of the Ziegler spectrum. + oai:arXiv.org:2502.11939v2 + math.CT + math.AT + math.RT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.1112/jlms.70337 + J. Lond. Math. Soc. (2), 112(6):58pp, Id/No e70337, 2025 + Isaac Bird, Jordan Williamson, Alexandra Zvonareva + + + A Note on the Phragmen-Lindelof Theorem + https://arxiv.org/abs/2502.13282 + arXiv:2502.13282v3 Announce Type: replace +Abstract: We provide a generalization of the Phragm\'en-Lindel\"of principal of Rademacher with the aim of correcting, or at least provide a pathway to correcting, several errors appearing in the literature. + oai:arXiv.org:2502.13282v3 + math.NT + math.CV + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Andrew Fiori + + + Global law of conjugate kernel random matrices with heavy-tailed weights + https://arxiv.org/abs/2502.18428 + arXiv:2502.18428v2 Announce Type: replace +Abstract: We study the asymptotic spectral distribution of the conjugate kernel random matrix $YY^\top$, where $Y= f(WX)$ arises from a two-layer neural network model. We consider the setting where $W$ and $X$ are random rectangular matrices with i.i.d.\ entries, where the entries of $W$ follow a heavy-tailed distribution, while those of $X$ have light tails. Our assumptions on $W$ include a broad class of heavy-tailed distributions, such as symmetric $\alpha$-stable laws with $\alpha \in ]0,2[$ and sparse matrices with $\mathcal{O}(1)$ nonzero entries per row. The activation function $f$, applied entrywise, is bounded, smooth, odd, and nonlinear. We compute the limiting eigenvalue distribution of $YY^\top$ through its moments and show that heavy-tailed weights induce strong correlations between the entries of $Y$, resulting in richer and fundamentally different spectral behavior compared to the light-tailed case. + oai:arXiv.org:2502.18428v2 + math.PR + cs.LG + stat.ML + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Alice Guionnet, Vanessa Piccolo + + + Characterizations of Tilt-Stable Local Minimizers of a Class of Matrix Optimization Problems + https://arxiv.org/abs/2503.03217 + arXiv:2503.03217v3 Announce Type: replace +Abstract: Tilt stability plays a pivotal role in understanding how local solutions of an optimization problem respond to small, targeted perturbations of the objective. Although quadratic bundles are a powerful tool for capturing second-order variational behavior, their characterization remains incomplete beyond well-known polyhedral and certain specialized nonpolyhedral settings. To help bridge this gap, we propose a new point-based criterion for tilt stability in prox-regular, subdifferentially continuous functions by exploiting the notion of minimal quadratic bundles. Furthermore, we derive an explicit formula for the minimal quadratic bundle associated with a broad class of general spectral functions, thus providing a practical and unifying framework that significantly extends existing results and offers broader applicability in matrix optimization problems. + oai:arXiv.org:2503.03217v3 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Chao Ding, Ebrahim Sarabi, Shiwei Wang + + + On action rate admissibility criteria + https://arxiv.org/abs/2503.03491 + arXiv:2503.03491v5 Announce Type: replace +Abstract: We formulate new admissibility criteria for initial value problems motivated by the least action principle. These are applied to a two-dimensional Riemann initial value problem for the isentropic compressible Euler fluid flow. It is shown that the criterion prefers the 2-shock solution to solutions obtained by convex integration by Chiodaroli and Kreml or to the hybrid solutions recently constructed by Markfelder and Pellhammer. + oai:arXiv.org:2503.03491v5 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.1007/s00033-025-02705-5 + Heiko Gimperlein, Michael Grinfeld, Robin J. Knops, Marshall Slemrod + + + The large sieve for square moduli, revisited + https://arxiv.org/abs/2503.18009 + arXiv:2503.18009v4 Announce Type: replace +Abstract: We revisit the large sieve for square moduli and obtain conditional improvements under hypotheses on higher additive energies of modular square roots. + oai:arXiv.org:2503.18009v4 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Stephan Baier + + + Secant varieties of Segre-Veronese varieties $\mathbb{P}^m\times\mathbb{P}^n$ embedded by $\mathcal{O}(1,2)$ are non-defective for $n\gg m^3$, $m\geq3$ + https://arxiv.org/abs/2503.21972 + arXiv:2503.21972v2 Announce Type: replace +Abstract: We prove that for any $m\geq3$, $n\gg m^3$, all secant varieties of the Segre-Veronese variety $\mathbb{P}^m\times\mathbb{P}^n$ have the expected dimension. This was already proved by Abo and Brambilla in the subabundant case, hence we focus on the superabundant case. We generalize an approach due to Brambilla and Ottaviani into a construction we call the inductant. With this, the proof of non-defectivity reduces to checking a finite collection of base cases, which we verify using a computer-assisted proof. + oai:arXiv.org:2503.21972v2 + math.AG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + 10.1016/j.jsc.2025.102546 + J. Symbolic Comput. 135 (2026) 102546 + Mat\v{e}j Dole\v{z}\'alek, Nikhil Ken + + + Bounded domains on Kobayashi hyperbolic manifolds covering compact complex manifolds + https://arxiv.org/abs/2504.03935 + arXiv:2504.03935v2 Announce Type: replace +Abstract: The lack of a uniformization theorem in several complex variables leads to a desire to classify all of the simply connected domains. We use established computational methods and a localization technique to generalize a recently-published classification. In particular, we show that if a domain with $C^{1,1}$ boundary on a Kobayashi hyperbolic complex manifold contains a totally real boundary point and covers a compact manifold, then its universal cover must be the Euclidean ball. + oai:arXiv.org:2504.03935v2 + math.CV + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Nicholas Newsome + + + Hanf numbers for poset games + https://arxiv.org/abs/2504.07317 + arXiv:2504.07317v3 Announce Type: replace +Abstract: Given two well partial orders $(P;\leq_P)$ and $(T;\leq_T)$, each with a minimum element, we study the following question: which player has a winning strategy for Chomp on the poset $(P\times T;\leq_{P\times T})$? Here, $(P\times T;\leq_{P\times T})$ denotes the poset obtained as the Cartesian product of $P$ and $T$, equipped with the corresponding lexicographic order. The answer to this very natural question depends strongly on the specific choice of $(P;\leq_P)$ and $(T;\leq_T)$. For this reason, we restrict our attention to classes of posets given by powers of a fixed poset: $\{(P^\sigma;\leq_{P^\sigma})\mid \sigma\in\mathrm{Ord}\}$. A fundamental fact about these classes of structures is that, if the second player does not have a winning strategy for all the posets in $\{(P^\sigma;\leq_{P^\sigma})\mid \sigma\in\mathrm{Ord}\}$, there exists an ordinal $\xi$ such that the second player has a winning strategy on $(P^{\xi};\leq_{P^{\xi}})$ but not on $(P^{\gamma};\leq_{P^{\gamma}})$ for all $\gamma\geq\xi+1$. Determining the corresponding ordinal for this Hanf number-style property constitutes the main objective of this work. + Inspired by results of Garc\'ia-Marco and Knauer, we focus on classes of posets with a purely algebraic definition. These posets arise from submonoids (with respect to the natural sum, or Hessenberg sum) of ordinals of the form $\omega^\sigma$ and are generated by sets of ordinals. In the process, we provide a test to determine whether a finite set $\Gamma$ of ordinals indeed yields well partial orders, and, using set-theoretic techniques, we establish an upper bound for the ordinal $\xi$: if $\Gamma\subset\omega$, then $\xi<\omega_1$, and otherwise $\xi<|\bigcup\Gamma|^+$. + oai:arXiv.org:2504.07317v3 + math.CO + math.LO + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Fabi\'an Rivero Herrera + + + AdGT: Decentralized Gradient Tracking with Tuning-free Per-Agent Stepsize + https://arxiv.org/abs/2504.15196 + arXiv:2504.15196v4 Announce Type: replace +Abstract: In decentralized optimization, the choice of stepsize plays a critical role in algorithm performance. A common approach is to use a shared stepsize across all agents to ensure convergence. However, selecting an optimal stepsize often requires careful tuning, which can be time-consuming and may lead to slow convergence, especially when there is significant variation in the smoothness (L-smoothness) of local objective functions across agents. Individually tuning stepsizes per agent is also impractical, particularly in large-scale networks. To address these limitations, we propose AdGT, an adaptive gradient tracking method that enables each agent to adjust its stepsize based on the smoothness of its local objective. We prove that AdGT achieves linear convergence to the global optimal solution. Through numerical experiments, we compare AdGT with fixed-stepsize gradient tracking methods and demonstrate its superior performance. Additionally, we compare AdGT with adaptive gradient descent (AdGD) in a centralized setting and observe that fully adaptive stepsizes offer greater benefits in decentralized networks than in centralized ones. + oai:arXiv.org:2504.15196v4 + math.OC + cs.SY + eess.SY + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Diyako Ghaderyan, Stefan Werner + + + Vertex evaluation of multiplex graphs using Forman Curvature + https://arxiv.org/abs/2504.17286 + arXiv:2504.17286v2 Announce Type: replace +Abstract: The identification of vertices that play a central role in network analysis is a fundamental challenge. Although traditional centrality measures have been extensively employed for this purpose, the increasing complexity of modern networks necessitates the use of sophisticated metrics. The concept of Forman curvature has recently garnered significant attention as a promising approach. We define the Forman curvature for multiplex graphs, which are a category of complex networks characterized by multiple layers of connections between nodes. We then prove the key properties of the Forman curvature in the context of multiplex graphs and show its usefulness in identifying vertices occupying central positions within these networks. Moreover, through a series of comparative experiments with traditional graph features and graph kernels, we demonstrate that the Forman curvature can function as an effective metric for classifying the overall structure of networks. + oai:arXiv.org:2504.17286v2 + math.CO + cs.DM + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Taiki Yamada + + + On Minimal generating sets of splitting field, Cluster towers and Multiple transitivity of Galois groups + https://arxiv.org/abs/2505.00672 + arXiv:2505.00672v3 Announce Type: replace +Abstract: A natural generating set for a Galois extension regarded as the splitting field of an irreducible polynomial is introduced and investigated here. Minimal generating sets arising in this context throw many surprises compared to the analogous concept in linear algebra: they can be of different cardinalities. In fact we establish that for a certain family of polynomials over the rationals, we have minimal generating sets of all cardinalities in a certain range and that these are the only possible cardinalities for minimal generating set for such a polynomial. We also study how minimal generating sets behave under multiple transitivity of the Galois group and consequently prove the existence of polynomials with all minimal generating sets of uniformly same cardinality. We also connect minimal generating sets with the concept of root cluster tower of an irreducible polynomial introduced by the second author and Krithika in [8]. + oai:arXiv.org:2505.00672v3 + math.NT + math.AC + math.CO + math.GR + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Shubham Jaiswal, P Vanchinathan + + + Minimal Simplicial Degree $d$ Maps from Genus $g$ Surfaces to the Torus + https://arxiv.org/abs/2505.02386 + arXiv:2505.02386v2 Announce Type: replace +Abstract: The degree of a map between orientable manifolds is a fundamental concept in topology, offering deep insights into the structure of the manifolds and the nature of the corresponding maps. This concept has been extensively studied, particularly in the context of simplicial maps between orientable triangulable spaces. In 1982, Gromov proved that if degree $d$ maps exist from a genus $g$ orientable surface to a genus $h$ orientable surface for every $d \in \mathbb{Z}$, then $h$ must be 0 or 1. + Recently, degree $d$ self-maps on spheres, particularly on genus 0 surfaces, have been investigated. In this paper, we focus on the unique minimal 7-vertex triangulation of the torus. We construct simplicial degree $d$ maps from a triangulation of a genus $g$ surface to the 7-vertex triangulation of the torus for $g \geq 1$. Our construction of degree $d$ maps is minimal for every $d$ when $g = 1,2$. If $g \geq 3$, then our construction remains minimal for $|d| \geq 2g - 1$. + We believe that this concept will be highly useful in combinatorial topology, as it leads to several intriguing open research problems. In the final section, we propose some of these open research problems. + oai:arXiv.org:2505.02386v2 + math.CO + math.GT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Biplab Basak, Ayushi Trivedi + + + Bounded cohomology, quotient extensions, and hierarchical hyperbolicity + https://arxiv.org/abs/2505.20462 + arXiv:2505.20462v3 Announce Type: replace +Abstract: We call a central extension bounded if its Euler class is represented by a bounded cocycle. We prove that a bounded central extension of a hierarchically hyperbolic group (HHG) is still a HHG; conversely if a central extension is a HHG, then the extension is bounded, and under a further mild assumption the quotient is commensurable to a HHG. Motivated by questions on hierarchical hyperbolicity of quotients of mapping class groups, we therefore consider the general problem of determining when a quotient of a bounded central extension is still bounded, which we prove to be equivalent to an extendability problem for quasihomomorphisms. Finally, we show that quotients of the 4-strands braid group by suitable powers of a pseudo-Anosov are HHG, and in fact bounded central extensions of some HHG. We also speculate on how to extend the previous result to all mapping class groups. + oai:arXiv.org:2505.20462v3 + math.GR + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Francesco Fournier-Facio, Giorgio Mangioni, Alessandro Sisto + + + Frostman and Fourier characterisations of fractal dimensions + https://arxiv.org/abs/2505.21217 + arXiv:2505.21217v2 Announce Type: replace +Abstract: We examine Frostman-type characterisations and other extremal measure criteria for a range of fractal dimensions of sets. In particular we derive properties of the less familiar modified lower box dimension and upper correlation dimension. We also express a number of fractal dimensions in terms of Fourier properties of measures. + oai:arXiv.org:2505.21217v2 + math.MG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Kenneth J. Falconer, Shuqin Zhang + + + On Differential and Boomerang Properties of a Class of Binomials over Finite Fields of Odd Characteristic + https://arxiv.org/abs/2506.11486 + arXiv:2506.11486v2 Announce Type: replace +Abstract: In this paper, we investigate the differential and boomerang properties of a class of binomial $F_{r,u}(x) = x^r(1 + u\chi(x))$ over the finite field $\mathbb{F}_{p^n}$, where $r = \frac{p^n+1}{4}$, $p^n \equiv 3 \pmod{4}$, and $\chi(x) = x^{\frac{p^n -1}{2}}$ is the quadratic character in $\mathbb{F}_{p^n}$. We show that $F_{r,\pm1}$ is locally-PN with boomerang uniformity $0$ when $p^n \equiv 3 \pmod{8}$. To the best of our knowledge, it is the second known non-PN function class with boomerang uniformity $0$, and the first such example over odd characteristic fields with $p > 3$. Moreover, we show that $F_{r,\pm1}$ is locally-APN with boomerang uniformity at most $2$ when $p^n \equiv 7 \pmod{8}$. We also provide complete classifications of the differential and boomerang spectra of $F_{r,\pm1}$. Furthermore, we thoroughly investigate the differential uniformity of $F_{r,u}$ for $u\in \mathbb{F}_{p^n}^* \setminus \{\pm1\}$. + oai:arXiv.org:2506.11486v2 + cs.IT + cs.CR + math.IT + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Namhun Koo, Soonhak Kwon + + + Enumerating log rational curves on some toric varieties + https://arxiv.org/abs/2506.13975 + arXiv:2506.13975v3 Announce Type: replace +Abstract: The genus 0, fixed-domain log Gromov-Witten invariants of a smooth, projective toric variety X enumerate maps from a general pointed rational curve to a smooth, projective toric variety passing through the maximal number of general points and with prescribed multiplicities along the toric boundary. We determine these invariants completely for the projective bundle X=P_{P^r}(O^s+O(-a)), proving a conjecture of Cela--Iribar L\'opez. A different conjecture when X is the blow-up of P^r at r points is disproven. Whereas the conjectures were predicted using tropical methods, we give direct intersection-theoretic calculations on moduli spaces of "naive log quasimaps." + oai:arXiv.org:2506.13975v3 + math.AG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Carl Lian, Naufil Sakran + + + Maximal transitivity of the cactus group on standard Young tableaux + https://arxiv.org/abs/2506.16561 + arXiv:2506.16561v2 Announce Type: replace +Abstract: The action of the cactus group $C_n$ on Young tableaux of a given shape $\lambda$ goes back to Berenstein and Kirillov and arises naturally in the study of crystal bases and quantum integrable systems. We show that this action is $2$-transitive on standard Young tableaux of the shape $\lambda$ if and only if $\lambda$ is not self-transpose and not a single hook. Moreover, we show that in these cases, the image of the cactus group in the permutation group of standard Young tableaux is either the whole permutation group or the alternating group, and prove that both cases are possible for infinitely many $\lambda$ (though the alternating group is more frequent). As an application, this implies that the Galois group of solutions to the Bethe ansatz in the Gaudin model attached to the Lie group $GL_d$ is, in many cases, at least the alternating group. This also extends the results of Sottile and White on the multiple transitivity of the Galois group of Schubert calculus problems in Grassmannians to many new cases. + oai:arXiv.org:2506.16561v2 + math.CO + math.AG + math.QA + math.RT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Sophia Liao, Leonid Rybnikov + + + Non-unique equilibrium measures and freezing phase transitions for matrix cocycles for negative $t$ + https://arxiv.org/abs/2507.01148 + arXiv:2507.01148v2 Announce Type: replace +Abstract: We consider a one-step matrix cocycle generated by a pair of non-negative parabolic matrices and study the equilibrium measures for $t\log \|\mathcal A\|$ as $t$ runs over the reals. We show that there is a freezing first order phase transition at some parameter value $t_c$ so that for $t<t_{c}$ the equilibrium measure is non-unique and supported on the two fixed points, while for $t>t_c$, the equilibrium measure is unique, non-atomic and fully supported. The phase transition closely resembles the classical Hofbauer example. In particular, our example shows that there may be non-unique equilibrium measures for negative $t$ even if the cocycle is strongly irreducible and proximal. + oai:arXiv.org:2507.01148v2 + math.DS + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Reza Mohammadpour, Anthony Quas + + + Matrix Fej\'er-Riesz type theorem for a union of an interval and a point + https://arxiv.org/abs/2507.01357 + arXiv:2507.01357v2 Announce Type: replace +Abstract: The matrix Fej\'er-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line. In the previous work of the second-named author this was extended to the characterization on arbitrary closed semialgebraic sets $K$ in $\mathbb{R}$ by using matrix quadratic modules from real algebraic geometry. In the compact case there is a denominator-free characterization, while in the non-compact case denominators are needed except when $K$ is the whole line, an unbounded interval, a union of two unbounded intervals, and it was conjectured also when $K$ is a union of an unbounded interval and a point or a union of two unbounded intervals and a point. In this paper, we confirm this conjecture by solving the truncated matrix-valued moment problem (TMMP) on a union of a bounded interval and a point. The presented technique for solving the corresponding TMMP can potentially be used to determine degree bounds in the positivity certificates for matrix polynomials on compact sets $K$. + oai:arXiv.org:2507.01357v2 + math.FA + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Shengding Sun, Alja\v{z} Zalar + + + Bounded diameter monochromatic component covers + https://arxiv.org/abs/2507.05842 + arXiv:2507.05842v2 Announce Type: replace +Abstract: Ryser conjectured that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic trees. Motivated by a question of Austin in analysis, Mili\'cevi\'c predicted something stronger -- that every $r$-edge-coloured complete graph can be covered by $r-1$ monochromatic trees \emph{of bounded diameter}. Here we show that the two conjectures are equivalent. As immediate corollaries we obtain new results about Mili\'cevi\'c's Conjecture, most notably that it is true for $r=5$. We also obtain several new cases of a generalization of Mili\'cevi\'c's Conjecture to non-complete graphs due to DeBiasio-Kamel-McCourt-Sheats. + oai:arXiv.org:2507.05842v2 + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/publicdomain/zero/1.0/ + Alexey Pokrovskiy + + + Classifying Nakayama algebras with a braid group action on $\tau$-exceptional sequences + https://arxiv.org/abs/2507.07608 + arXiv:2507.07608v2 Announce Type: replace +Abstract: We characterise those basic and connected Nakayama algebras $\Lambda$ for which the mutation of $\tau$-exceptional sequences respects the braid group relations. We show that this is the case if and only if $\Lambda$ is hereditary or all indecomposable projective $\Lambda$-modules have length at least $|\Lambda|$. + oai:arXiv.org:2507.07608v2 + math.RT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Maximilian Kaipel, H{\aa}vard Utne Terland + + + Constructions of binary self-orthogonal singly-even minimal linear codes violating the Aschikhmin-Barg condition with few weights + https://arxiv.org/abs/2507.12240 + arXiv:2507.12240v3 Announce Type: replace +Abstract: We first establish a simple yet powerful necessary and sufficient condition for a binary linear code to be SO, leading to a complete characterization of singly-even codes in this family. We further derive necessary and sufficient conditions on Boolean and vectorial Boolean functions for generating such codes via a standard construction method. Building on this foundation, we propose three general frameworks for constructing binary SO singly-even minimal non-AB linear codes with few weights. The first two approaches are based on designing Boolean and vectorial Boolean functions that simultaneously satisfy multiple conditions. The third method generates new SO codes from existing ones. As a result, we obtain many infinite classes of binary self-orthogonal singly-even minimal linear codes violating the AB condition with few weights and fully determined weight distributions. Particularly, numerical results show that some duals of our codes are optimal or near-optimal. + oai:arXiv.org:2507.12240v3 + cs.IT + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Kangquan Li, Hao Chen, Wengang Jin, Longjiang Qu + + + Donaldson-Thomas invariants of $[\mathbb C^4/\mathbb Z_r]$ + https://arxiv.org/abs/2507.21582 + arXiv:2507.21582v2 Announce Type: replace +Abstract: We compute the zero-dimensional Donaldson-Thomas invariants of the quotient stack $[\mathbb{C}^4/\mathbb{Z}_r]$, confirming a conjecture of Cao-Kool-Monavari. Our main theorem is established through an orbifold analogue of Cao-Zhao-Zhou's degeneration formula combined with the zero-dimensional Donaldson-Thomas invariants for $\mathcal{A}_{r-1}\times\mathbb{C}^2$ and an explicit determination of orientations of Hilbert schemes of points on $[\mathbb{C}^4/\mathbb{Z}_r]$. + oai:arXiv.org:2507.21582v2 + math.AG + hep-th + math-ph + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Xiaolong Liu + + + Strong Feller Regularisation of 1-d Nonlinear Transport by Reflected Ornstein-Uhlenbeck Noise + https://arxiv.org/abs/2508.01355 + arXiv:2508.01355v2 Announce Type: replace +Abstract: We consider equations of nonlinear transport on the circle with regular self interactions appearing in aggregation models and deterministic mean field dynamics. We introduce a random perturbation of such systems through a stochastic orientation preserving flow, which is given as an integrated infinite dimensional periodic Ornstein- Uhlenbeck process with reflection. As our main result we show that the induced stochastic dynamics yields a measure valued Markov process on a class of regular measures. Moreover, we show that this process is strong Feller in the corresponding topology. This is interpreted as a qualitative regularisation by noise phenomenon. + oai:arXiv.org:2508.01355v2 + math.PR + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-sa/4.0/ + Max-K. von Renesse, Feng-Yu Wang, Alexander Wei{\ss} + + + Joint distribution of Hecke eigenforms on $\mathbb{H}^3$ + https://arxiv.org/abs/2508.06331 + arXiv:2508.06331v2 Announce Type: replace +Abstract: We prove a joint value equidistribution statement for Hecke-Maa{\ss} cusp forms on the hyperbolic three-space $\mathbb{H}^3$. This supports the conjectural statistical independence of orthogonal cusp forms. + oai:arXiv.org:2508.06331v2 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Didier Lesesvre, Luca Marchesini, Nicole Raulf + + + K-promotion on m-packed labelings of posets + https://arxiv.org/abs/2508.09305 + arXiv:2508.09305v2 Announce Type: replace +Abstract: Schutzenberger's promotion operator, pro, is a fundamental map in dynamical algebraic combinatorics. At first, its action was mainly considered on standard Young tableaux. But pro was subsequently shown to have interesting properties when applied to natural labelings of other posets. Pechenik defined a K-theoretic version of promotion, pro_K, on m-packed labelings of tableaux. The operator pro was then extended to increasing labelings of other posets. The purpose of the current work is to show that the original action of pro_K on m-packed labelings yields interesting results when applied to partially ordered sets in general, and to rooted trees in particular. We show that under certain conditions, the sizes of the orbits and order of pro_K exhibit nice divisibility properties. We also completely determine, for certain values of m, the orbit sizes for the action on various types of rooted trees such as extended stars, combs, zippers, and a type of three-leaved tree. + oai:arXiv.org:2508.09305v2 + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Jamie Kimble (Michigan State University), Bruce E. Sagan (Michigan State University), Avery St. Dizier (Michigan State University) + + + On a Grassmann odd analogue of Carrollian Manifolds + https://arxiv.org/abs/2508.14240 + arXiv:2508.14240v3 Announce Type: replace +Abstract: We define a Grassmann odd analogue of a Carrollian manifold as a supermanifold of dimension $n|1$ with an even degenerate metric such that the kernel is generated by a non-singular odd vector field that is a supersymmetry generator. Alongside other results, we establish that the reduced manifold is a pseudo-Riemannian manifold, and show that compatible affine connections always exist, albeit they must carry torsion. As a physically relevant example, we examine an In\"on\"u--Wigner contraction of the supertranslation algebra on standard superspace $\mathbb{R}^{4|4}$. + oai:arXiv.org:2508.14240v3 + math.DG + gr-qc + math-ph + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Andrew James Bruce + + + Mutations of quivers with 2-cycles + https://arxiv.org/abs/2508.15022 + arXiv:2508.15022v2 Announce Type: replace +Abstract: We develop a mutation theory for quivers with oriented 2-cycles using a structure called a homotopy, defined as a normal subgroupoid of the quiver's fundamental groupoid. This framework extends Fomin-Zelevinsky mutations of 2-acyclic quivers and yields involutive mutations that preserve the fundamental groupoid quotient by the homotopy. It generalizes orbit mutations arising from quiver coverings and allows for infinite mutation sequences even when orbit mutations are obstructed. We further construct quivers with homotopies from triangulations of marked surfaces with colored punctures, and prove that flips correspond to mutations, extending the Fomin-Shapiro-Thurston model to the setting with 2-cycles. + oai:arXiv.org:2508.15022v2 + math.CO + math.RT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Fang Li, Siyang Liu, Lang Mou, Jie Pan + + + Symmetric Poisson geometry, totally geodesic foliations and Jacobi-Jordan algebras + https://arxiv.org/abs/2508.15890 + arXiv:2508.15890v3 Announce Type: replace +Abstract: We introduce symmetric Poisson structures as pairs consisting of a symmetric bivector field and a torsion-free connection satisfying an integrability condition analogous to that in usual Poisson geometry. Equivalent conditions in Poisson geometry have inequivalent analogues in symmetric Poisson geometry and we distinguish between symmetric and strong symmetric Poisson structures. We prove that symmetric Poisson structures correspond to locally geodesically invariant distributions together with a characteristic metric, whereas strong symmetric Poisson structures correspond to totally geodesic foliations together with a leaf metric and a leaf connection. We introduce, using the Patterson-Walker metric, a dynamics on the cotangent bundle and show its connection to symmetric Poisson geometry, the parallel transport equation and the Newtonian equation for conservative systems. Finally, we prove that linear symmetric Poisson structures are in correspondence with Jacobi-Jordan algebras, whereas strong symmetric correspond to those that are moreover associative. + oai:arXiv.org:2508.15890v3 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Filip Mou\v{c}ka, Roberto Rubio + + + Minimal ${A}_{\infty}$-algebras of endomorphisms: The case of $d\mathbb{Z}$-cluster tilting objects + https://arxiv.org/abs/2508.18852 + arXiv:2508.18852v2 Announce Type: replace +Abstract: The Derived Auslander--Iyama Corresponence, a recent result of the authors, provides a classification up to quasi-isomorphism of the derived endomorphism algebras of basic $d\mathbb{Z}$-cluster tilting objects in $\operatorname{Hom}$-finite algebraic triangulated categories in terms of a small amount of algebraic data. In this note we highlight the role of minimal $A_\infty$-algebra structures in the proof of this result, as well as the crucial role of the enhanced $A_\infty$-obstruction theory developed by the second-named author. + oai:arXiv.org:2508.18852v2 + math.RT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Gustavo Jasso, Fernando Muro + + + On discrete Sobolev inequalities for nonconforming finite elements under a semi-regular mesh condition + https://arxiv.org/abs/2509.00505 + arXiv:2509.00505v2 Announce Type: replace +Abstract: We derive a discrete $ L^q-L^p$ Sobolev inequality tailored for the Crouzeix--Raviart and discontinuous Crouzeix--Raviart finite element spaces on anisotropic meshes in both two and three dimensions. Subject to a semi-regular mesh condition, this discrete Sobolev inequality is applicable to all pairs $(q,p)$ that align with the local Sobolev embedding, including scenarios where $q \leq p$. Importantly, the constant is influenced solely by the domain and the semi-regular parameter, ensuring robustness against variations in aspect ratios and interior angles of the mesh. The proof employs an anisotropy-sensitive trace inequality that leverages the element height, a two-step affine/Piola mapping approach, the stability of the Raviart--Thomas interpolation, and a discrete integration-by-parts identity augmented with weighted jump/trace terms on faces. This Sobolev inequality serves as a mesh-robust foundation for the stability and error analysis of nonconforming and discontinuous Galerkin methods on highly anisotropic meshes. + oai:arXiv.org:2509.00505v2 + math.NA + cs.NA + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Hiroki Ishizaka + + + Lipschitz-free spaces and Bossard's reduction argument + https://arxiv.org/abs/2509.00722 + arXiv:2509.00722v2 Announce Type: replace +Abstract: We set up a descriptive set-theoretic framework to study Lipschitz-free spaces and use the reduction argument of Bossard to prove several results. We prove two universality results: if a separable Banach space is isomorphically universal for the class of Lipschitz-free spaces over the countable complete discrete metric spaces then it is isomorphically universal for the class of separable Banach spaces, and if a complete separable metric space is Lipschitz universal for the same class of metric spaces then it is Lipschitz universal for all separable metric spaces. We also show that there exist countable complete discrete metric spaces whose Lipschitz-free spaces fail the bounded approximation property and are thus not isomorphic to any dual Banach space. Finally, we calculate the descriptive complexity of the classes of separable Banach spaces and separable Lipschitz-free spaces having the approximation property. + oai:arXiv.org:2509.00722v2 + math.FA + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Richard J. Smith + + + Global existence of the irrotational Euler-Nordstr\"om equations with a positive cosmological constant: The gravitational field equation + https://arxiv.org/abs/2509.02023 + arXiv:2509.02023v4 Announce Type: replace +Abstract: Our aim is to establish the global existence of classical solutions to the nonlinear irrotational Euler--Nordstr\"om system, which incorporates a linear equation of state and a cosmological constant. In this setting, gravitation is described by a single scalar field satisfying a specific semilinear wave equation. We restrict attention to spatially periodic perturbations of the background metric and therefore study this equation on the three-dimensional torus $\mathbb{T}^3$, working within the Sobolev spaces $H^m(\mathbb{T}^3)$. + We begin by analysing the Nordstr\"om equation in isolation, with a source term generated by an irrotational fluid obeying a linear equation of state. This separation is motivated by the fact that such a fluid produces a source term containing a nonlinear contribution of fractional order. + To obtain a global solution for the gravitational field, the fractional-order nonlinearity $(1+u)^\mu$, with $\mu\in\mathbb{R}$, must remain smooth throughout the evolution. This condition, in turn, requires that $u$ remain small for all time. We ensure this by introducing a suitably chosen energy functional. We also prove that, asymptotically, the solutions tend to a constant. + oai:arXiv.org:2509.02023v4 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Uwe Brauer, Lavi Karp + + + Concentration Inequalities for Branching Random Walk + https://arxiv.org/abs/2509.05860 + arXiv:2509.05860v2 Announce Type: replace +Abstract: While classical concentration inequalities are typically restricted to two special cases -- independence and martingale difference sequences -- we extend concentration inequalities to a much broader class of stochastic processes by relaxing these foundational conditions. %\vspace{0.2\baselineskip} Specifically, heuristically and in the language of calculus, while independence and the martingale difference property correspond to \[ \displaystyle \frac { \partial y } {\partial t}= \text{constant}, + \quad \displaystyle \frac { \partial y } {\partial t} = 0 \] respectively, %\vspace{0.3\baselineskip} we relax these conditions to %\[ \left| \frac { \partial^2 y } {\partial u_i \, \partial t} \right| \le L, \] %thereby allowing the drift $\displaystyle\frac { \partial y } {\partial t}$ to vary with past state $u_i$. \vspace{0.3\baselineskip} a general setting that requires only the existence of a drift $\displaystyle\frac { \partial y } {\partial t}$ which is allowed to vary with the past state. \vspace{0.3\baselineskip} Furthermore, concentration inequalities are established for branching random walks. + oai:arXiv.org:2509.05860v2 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Changqing Liu + + + Quantization of bounded symplectic domains associated with compact Lie groups + https://arxiv.org/abs/2509.05931 + arXiv:2509.05931v2 Announce Type: replace +Abstract: We present a systematic quantization scheme for bounded symplectic domains of the form $D \times G \subset T^\ast G$, where $D \subset \mathfrak{g}^\ast$ is a bounded region defined by algebraic inequalities and $G$ is a compact Lie group with Lie algebra $\mathfrak{g}$. The finiteness of the symplectic volume implies that quantization yields a finite-dimensional Hilbert space, with observables represented by Hermitian matrices, for which we provide an explicit realization. Boundary effects necessitate modifications of the standard von Neumann and Dirac conditions, which usually underlie the correspondence principle. Physically, the compact group $G$ plays the role of momentum space, while $\mathfrak{g}^\ast$ corresponds to the (noncommutative) position space of a particle. The assumption of compact momentum space has profound physical consequences, including the supertunneling phenomenon and the emergence of a maximal fermion density. + oai:arXiv.org:2509.05931v2 + math-ph + hep-th + math.MP + math.QA + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Alexey A. Sharapov + + + K-theoretic Hikita conjecture for quiver gauge theories + https://arxiv.org/abs/2509.06226 + arXiv:2509.06226v2 Announce Type: replace +Abstract: We study variants of Hikita conjecture for Nakajima quiver varieties and corresponding Coulomb branches. First, we derive the equivariant version of the conjecture from the non-equivariant one for a set of gauge theories. Second, we suggest a variant of the conjecture, with K-theoretic Coulomb branches involved. We show that this version follows from the usual (homological) one for a set of theories. We apply this result to prove the conjecture in finite ADE types. In the course of the proof, we show that appropriate completions of K-theoretic and homological (quantized) Coulomb branches are isomorphic. + oai:arXiv.org:2509.06226v2 + math.RT + math-ph + math.KT + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ilya Dumanski, Vasily Krylov + + + $HS$-tensional maps and $HM$-tensional maps + https://arxiv.org/abs/2509.08564 + arXiv:2509.08564v2 Announce Type: replace +Abstract: Let $\psi: (M,g)\longrightarrow (N,h)$ be a smooth map between Riemannian manifolds. The tension field of $\psi$ can be regarded as a map from $(M,g)$ into the Riemannian vector bundle $\psi^{-1}TN$, equipped with the Sasaki metric $G_{S}$. In this paper, we study certain aspects of two types of maps: those whose tension fields are harmonic maps (called $HM$-tensional maps) and those whose tension fields are harmonic sections (called $HS$-tensional maps). + oai:arXiv.org:2509.08564v2 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Bouazza Kacimi, Ahmed Mohammed Cherif, Mustafa \"Ozkan + + + Augmentations, reduced ideal point gluings and compact type degenerations of curves + https://arxiv.org/abs/2509.12429 + arXiv:2509.12429v2 Announce Type: replace +Abstract: In this note we demonstrate some unexpected properties that simple gluings of the simplest derived categories may have. We consider two special cases: the first is an augmented curve, i.e., the gluing of the derived categories of a point and a curve with the gluing bimodule given by the structure sheaf of the curve; the second is an ideal point gluing of curves, i.e., the gluing of the derived categories of two curves with the gluing bimodule given by the ideal sheaf of a point in the product of the curves. We construct unexpected exceptional objects contained in these categories and discuss their orthogonal complements. + We also show that the simplest example of compact type degeneration of curves, a flat family of curves with a smooth general fiber and a 1-nodal reducible central fiber, gives rise to a smooth and proper family of triangulated categories with the general fiber an augmented curve and the central fiber the orthogonal complement of the exotic exceptional object in the ideal point gluing of curves, called the reduced ideal point gluing of curves. + oai:arXiv.org:2509.12429v2 + math.AG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Valery Alexeev, Alexander Kuznetsov + + + The Complexity of Arc-Connectedness Relation in the Plane + https://arxiv.org/abs/2509.24596 + arXiv:2509.24596v2 Announce Type: replace +Abstract: In this paper, we show that the arc-connectedness equivalence relation on a Polish subspace of the real plane is an essentially hyperfinite Borel equivalence relation. This result provides the optimal upper bound for such a Borel equivalence relation. + oai:arXiv.org:2509.24596v2 + math.LO + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yusuf Uyar + + + Impact of memory on clustering in spontaneous particle aggregation + https://arxiv.org/abs/2510.15335 + arXiv:2510.15335v2 Announce Type: replace +Abstract: The effect of short-term and long-term memory on spontaneous aggregation of organisms is investigated using a stochastic agent-based model. Each individual modulates the amplitude of its random motion according to the perceived local density of neighbors. Memory is introduced via a chain of $K$~internal variables that allow agents to retain information about previously encountered densities. The parameter $K$ controls the effective length of memory. A formal mean-field limit yields a macroscopic Fokker--Planck equation, which provides a continuum description of the system in the large-population limit. Steady states of this equation are characterized to interpret the emergence and morphology of clusters. Systematic stochastic simulations in one- and two-dimensional spatial domains reveal that short- or moderate-term memory promotes coarsening, resulting in a smaller number of larger clusters, whereas long-term memory inhibits aggregation and increases the proportion of isolated individuals. Statistical analysis demonstrates that extended memory reduces the agents' responsiveness to environmental stimuli, explaining the transition from aggregation to dispersion as $K$ increases. These findings identify memory as a key factor controlling the collective organization of self-driven agents and provide a bridge between individual-level dynamics and emergent spatial patterns. + oai:arXiv.org:2510.15335v2 + math.DS + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Radek Erban, Jan Haskovec + + + A Linear Representation for Functions on Finite Sets + https://arxiv.org/abs/2510.20167 + arXiv:2510.20167v5 Announce Type: replace +Abstract: We demonstrate that any function $f$ from a finite set $Y$ to itself can be represented linearly. Specifically, we prove the existence of an injective map $j$ from $Y$ into a modular ring $\mathbb{Z}/m\mathbb{Z}$ and a constant $a \in \mathbb{Z}/m\mathbb{Z}$ such that $j(f(y)) = a \cdot j(y)$ in $\mathbb{Z}/m\mathbb{Z}$ holds for all $y \in Y$. This result is established by analyzing the algebraic properties of the adjugate of the characteristic matrix associated with the function's digraph. The proof is constructive, providing a method for finding the embedding $j$, the modulus $m$, and the linear multiplier $a$. + oai:arXiv.org:2510.20167v5 + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Roman Bacik + + + Heterochromatic two-arm probabilities for metric graph Gaussian free fields + https://arxiv.org/abs/2510.20492 + arXiv:2510.20492v2 Announce Type: replace +Abstract: For the Gaussian free field on the metric graph of $\mathbb{Z}^d$ ($d\ge 3$), we consider the heterochromatic two-arm probability, i.e., the probability that two points $v$ and $v'$ are contained in distinct clusters of opposite signs with diameter at least $N$. For all $d\ge 3$ except the critical dimension $d_c=6$, we prove that this probability is asymptotically proportional to $N^{-[(\frac{d}{2}+1)\land 4]}$. Furthermore, we prove that conditioned on this two-arm event, the volume growth of each involved cluster is comparable to that of a typical (unconditioned) cluster; precisely, each cluster has a volume of order $M^{(\frac{d}{2}+1)\land 4}$ within a box of size $M$. + oai:arXiv.org:2510.20492v2 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Zhenhao Cai, Jian Ding + + + Separation and cut edge in macroscopic clusters for metric graph Gaussian free fields + https://arxiv.org/abs/2510.20516 + arXiv:2510.20516v2 Announce Type: replace +Abstract: We prove that for the Gaussian free field (GFF) on the metric graph of $\mathbb{Z}^d$ (for all $d\ge 3$ except the critical dimension $d_c=6$), with uniformly positive probability there exist two distinct sign clusters of diameter at least $cN$ within a box of size $N$ such that their graph distance is less than $N^{-[(d-2)\vee (2d-8)]}$. This phenomenon contrasts sharply with the two-dimensional case, where the distance between two macroscopic clusters is typically on the order of their diameters, following from the basic property of the scaling limit ``conformal loop ensembles'' $\mathrm{CLE}_4$ (Sheffield-Werner'2001). + As a byproduct, we derive that the number of pivotal edges for the one-arm event (i.e., the sign cluster containing the origin has diameter at least $N$) is typically of order $N^{(\frac{d}{2}-1)\land 2}$. This immediately implies that for the incipient infinite cluster (IIC) of the metric graph GFF, the dimension of cut edges (i.e., edges whose removal disconnects the IIC) equals $(\frac{d}{2}-1)\land 2$. Translated in the language of critical loop soups (whose clusters, by the isomorphism theorem, have the same distribution as GFF sign clusters), this leads to the analogous estimates where the counterpart of a pivotal edge is a pivotal loop at scale $1$. This result hints at the new and possibly surprising idea that already in dimension $3$, microscopic loops (even those at scale $1$) play a crucial role in the construction of macroscopic loop clusters. + oai:arXiv.org:2510.20516v2 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Zhenhao Cai, Jian Ding + + + On the gap between cluster dimensions of loop soups on $\mathbb{R}^3$ and the metric graph of $\mathbb{Z}^3$ + https://arxiv.org/abs/2510.20526 + arXiv:2510.20526v2 Announce Type: replace +Abstract: The question of understanding the scaling limit of metric graph critical loop soup clusters and its relation to loop soups in the continuum appears to be one of the subtle cases that reveal interesting new scenarios about scaling limits, with a mixture of macroscopic and microscopic randomness. In the present paper, we show that in three dimensions, scaling limits of the metric graph clusters are strictly larger than the clusters of the limiting continuum Brownian loop soup. We actually show that the upper box counting dimension of the latter clusters is strictly smaller than $5/2$, while that of the former is $5/2$. + oai:arXiv.org:2510.20526v2 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Zhenhao Cai, Jian Ding + + + The power of trees + https://arxiv.org/abs/2510.26419 + arXiv:2510.26419v2 Announce Type: replace +Abstract: We give two consistent constructions of trees $T$ whose finite power $T^{n+1}$ is sharply different from $T^n$: + 1. An $\aleph_1$-tree $T$ whose interval topology $X_T$ is perfectly normal, but $(X_T)^2$ is not even countably metacompact. + 2. For an inaccessible $\kappa$ and a positive integer $n$, a $\kappa$-tree such that all of its $n$-derived trees are Souslin and all of its $(n+1)$-derived trees are special. + oai:arXiv.org:2510.26419v2 + math.LO + math.GN + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ari Meir Brodsky, Assaf Rinot, Shira Yadai + + + Arithmetic Properties of Several Generalized-Constant Sequences, with Implications for $\Gamma^{\left(n\right)}\left(1\right)$ + https://arxiv.org/abs/2511.01849 + arXiv:2511.01849v4 Announce Type: replace +Abstract: Neither the Euler-Mascheroni constant, $\gamma=0.577215...$, nor the Euler-Gompertz constant, $\delta=0.596347...$, is currently known to be irrational. However, it has been proved that at least one of them is transcendental. The two constants are related by a well-known equation of Hardy, equivalent to $\gamma+\delta/e=\mathrm{Ein}(1)$, which recently has been generalized to $\gamma^{(n)}+\delta^{(n)}/e=\eta^{(n)}$, $n\ge0$ for sequences of constants $\gamma^{(n)}$, $\delta^{(n)}$, and $\eta^{(n)}$ (given respectively by raw, conditional, and partial moments of the Gumbel(0,1) probability distribution). Investigating the $\gamma^{(n)}$ through recurrence relations (where $\gamma^{(0)}=1$ and $\gamma^{(1)}=\gamma$), we find that at least one of the pair {$\gamma,\gamma^{(2)}$} and -- conditional on a realistic conjecture verified for $2\leq n\leq26$ -- at least two of each set {$\gamma,\gamma^{(n)},\gamma^{(n+1)},\ldots,\gamma^{(2n)}$} are transcendental, implying that the $\gamma^{(n)}$ are transcendental infinitely often (with analogous results for the sequence $\Gamma^{(n)}(1)=\left(-1\right)^{n}\gamma^{\left(n\right)}$). We then show, via a theorem of Shidlovskii, that the $\eta^{(n)}$ are algebraically independent, and therefore transcendental, for all $n\ge0$, implying that at least one of each pair, {$\gamma^{(n)},\delta^{(n)}/e$} and {$\gamma^{(n)},\delta^{(n)}$}, and at least two of the triple {$\gamma^{(n)},\delta^{(n)}/e,\delta^{(n)}$}, are transcendental for all $n\ge1$. Further analysis of the $\gamma^{(n)}$ and $\eta^{(n)}$ reveals that the values $\delta^{(n)}/e$ are transcendental infinitely often with positive asymptotic density. Finally, we provide parallel results for the sequences $\tilde{\delta}^{(n)}$ and $\tilde{\eta}^{(n)}$ satisfying the "non-alternating analogue" equation $\gamma^{(n)}+\tilde{\delta}^{(n)}/e=\tilde{\eta}^{(n)}$. + oai:arXiv.org:2511.01849v4 + math.NT + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Michael R. Powers + + + Symmetric Iterations with Countable and $<\kappa$-Support: A Framework for Choiceless ZF Extensions + https://arxiv.org/abs/2511.07866 + arXiv:2511.07866v4 Announce Type: replace +Abstract: We present a unified framework for symmetric iterations with countable and, more generally, $<\kappa$-support. Set-length iterations are handled uniformly, and, when the template is first-order definable over a G\"odel-Bernays set theory with Global Choice ground, the same scheme yields class-length iterations. Limit stages with $\mathrm{cf}(\lambda)\ge\kappa$ are treated by direct limits; limits with $\mathrm{cf}(\lambda)<\kappa$ are presented as inverse limits via trees of conditions and tuple-stabilizer filters. The induced limit filters are normal and $\kappa$-complete, which ensures closure of hereditarily symmetric names and preservation of ZF; under a $\kappa$-Baire (strategic-closure) hypothesis we obtain $DC_{<\kappa}$, and under a Localization hypothesis we obtain $DC_\kappa$. For countable support we give an $\omega_1$-length construction that adds reals and refutes AC while preserving ZF+DC, and we show that mixed products (e.g., Cohen with Random) fit naturally via stable pushforwards and restrictions. For singular $\kappa$, we prove the case $\mathrm{cf}(\kappa)=\omega$ in full using block-partition stabilizers and trees; for arbitrary singular $\kappa$ we introduce game-guided fusion of length $\mathrm{cf}(\kappa)$ and a tree-fusion master condition, yielding singular-limit completeness, preservation of $DC_{<\kappa}$, no collapse of $\kappa$, and no new bounded subsets of $\kappa$. The resulting toolkit provides reusable patterns for constructing choiceless inner models that retain targeted fragments of Dependent Choice. + oai:arXiv.org:2511.07866v4 + math.LO + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Frank Gilson + + + Completeness conditions for spacetimes with low-regularity metrics + https://arxiv.org/abs/2511.07867 + arXiv:2511.07867v2 Announce Type: replace +Abstract: We extend Beem's three completeness notions -- finite compactness, timelike Cauchy completeness, and Condition A -- originally defined for spacetimes, to Lorentzian length spaces and study their relationships. We prove that finite compactness implies timelike Cauchy completeness and that timelike Cauchy completeness implies Condition A for globally hyperbolic Lorentzian length spaces. Furthermore, for globally hyperbolic $C^{1}$-spacetimes, we establish the equivalence of the three conditions assuming the causally non-branching and non-intertwining conditions, which in fact imply the continuity of the causal exponential map. These results can be regarded as a Hopf-Rinow type theorem for low-regularity Lorentzian geometry. The appendix presents examples of $C^{1}$-spacetimes -- where geodesic uniqueness may fail -- in which causal geodesics nevertheless behave well, illustrating the scope of our results. + oai:arXiv.org:2511.07867v2 + math.DG + gr-qc + math-ph + math.MG + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Keita Takahashi + + + A Stochastic Genetic Interacting Particle Method for Reaction-Diffusion-Advection Equations + https://arxiv.org/abs/2511.12275 + arXiv:2511.12275v2 Announce Type: replace +Abstract: We develop and analyze a stochastic genetic interacting particle method (SGIP) for reaction-diffusion-advection (RDA) equations. The SGIP method employs operator splitting to approximate the advection-diffusion and reaction processes, treating the former using particle drift-diffusion and the latter via exact or implicit integration of reaction dynamics over bins, where particle density is estimated using a histogram. A key innovation is the incorporation of adaptive resampling to close the loop of particle and density field description of solutions, mimicking the selection mechanism in genetics. Resampling is also crucial for maintaining long-term stability by redistributing particles in accordance with the evolving density field. We provide a comprehensive error analysis and establish convergence bounds under appropriate regularity assumptions. Numerical experiments in one to three space dimensions demonstrate the method's effectiveness across various reaction types (Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP), cubic, Arrhenius) and flow configurations (shear, cellular, cat's eye, Arnold-Beltrami-Childress (ABC) flows), showing excellent agreement with the finite difference method (FDM) while offering computational advantages for complex flow geometries and higher-dimensional problems. + oai:arXiv.org:2511.12275v2 + math.NA + cs.NA + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Boyi Hu, Zhongjian Wang, Jack Xin, Zhiwen Zhang + + + Geometric integrators for adiabatically closed simple thermodynamic systems + https://arxiv.org/abs/2511.14154 + arXiv:2511.14154v2 Announce Type: replace +Abstract: A variational formulation for non-equilibrium thermodynamics was developed by Gay-Balmaz and Yoshimura. In a recent article, the first two authors of the present paper introduced partially cosymplectic structures as a geometric framework for thermodynamic systems, recovering the evolution equations obtained variationally. In this paper, we develop a discrete variational principle for adiabatically closed simple thermodynamic systems, which can be utilised to construct numerical integrators for the dynamics of such systems. The effectiveness of our method is illustrated with several examples. + oai:arXiv.org:2511.14154v2 + math-ph + cs.NA + math.MP + math.NA + physics.class-ph + physics.comp-ph + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jaime Bajo, Manuel de Le\'on, Asier L\'opez-Gord\'on + + + Weak optimal transport with moment constraints: constraint qualification, dual attainment and entropic regularization + https://arxiv.org/abs/2511.16211 + arXiv:2511.16211v2 Announce Type: replace +Abstract: We consider weak optimal problems (possibly entropically penalized) incorporating both soft and hard (including the case of the martingale condition) moment constraints. Even in the special case of the martingale optimal transport problem, existence of Lagrange multipliers corresponding to the martingale constraint is notoriously hard (and may fail unless some specific additional assumptions are made). We identify a condition of qualification of the hard moment constraints (which in the martingale case is implied by well-known conditions in the literature) under which general dual attainment results are established. We also analyze the convergence of entropically regularized schemes combined with penalization of the moment constraint and illustrate our theoretical findings by numerically solving in dimension one, the Brenier-Strassen problem of Gozlan and Juillet and a family of problems which interpolates between monotone transport and left-curtain martingale coupling of Beiglb\"{o}ck and Juillet. + oai:arXiv.org:2511.16211v2 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Guillaume Carlier, Hugo Malamut, Maxime Sylvestre + + + Constancy of an Infinite Cyclotomic Product via Ramanujan Sums + https://arxiv.org/abs/2511.16975 + arXiv:2511.16975v2 Announce Type: replace +Abstract: We show that the infinite product defined by \[ P(z) = -\prod_{n=1}^{\infty} (\Phi_n(z))^{-1/n}, \] where \( \Phi_n(z) \) is the \( n \)-th cyclotomic polynomial, is constant inside the unit disk. The proof translates a result of Ramanujan on Ramanujan sums, equivalent to the prime number theorem, to the setting of infinite products. We also show that similar identities proved by Ramanujan lead to additional results on infinite cyclotomic products. + oai:arXiv.org:2511.16975v2 + math.NT + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Hartosh Singh Bal + + + Boundary regularity and a priori estimates for fractional equations on unbounded domains + https://arxiv.org/abs/2511.17325 + arXiv:2511.17325v2 Announce Type: replace +Abstract: In this paper, we study the boundary H\"older regularity for solutions to the fractional Dirichlet problem in unbounded domains with boundary \begin{equation*} + \begin{cases} + (-\Delta)^s u(x) = g(x),&\text{in } \Omega, + u(x)=0, &\text{in } \Omega^c. + \end{cases} + \end{equation*} Existing results rely on the global $L^{\infty}$ norm of solutions to control their boundary $C^s$ norm, which is insufficient for blow-up and rescaling analysis to obtain a priori estimates in unbounded domains. To overcome this limitation, we first derive a local version of boundary H\"older regularity for nonnegative solutions in which we replace the global $L^{\infty}$ norm by only a local $L^{\infty}$ norm. Then as an important application, we establish a priori estimates for nonnegative solutions to a family of nonlinear equations on unbounded domains with boundaries. + oai:arXiv.org:2511.17325v2 + math.AP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yahong Guo, Congming Li, Yugao Ouyang + + + A Trust-region Funnel Algorithm for Grey-Box Optimisation + https://arxiv.org/abs/2511.18998 + arXiv:2511.18998v2 Announce Type: replace +Abstract: Grey-box optimisation, where some parts of an optimisation problem are represented by explicit algebraic (glass-box) models while others are treated as black-box models lacking analytic derivatives, remains a challenge in process systems engineering. Trust-region (TR) methods provide a robust framework for grey-box problems by combining accurate glass-box derivatives with local reduced models (RMs) for black-box components. However, existing TR approaches often involve complex multi-layered formulations requiring extensive parameter tuning, or lack open-source implementations. Motivated by the recent advances in funnel-based convergence theory for nonlinear optimisation and the TR filter method, we propose a novel TR funnel algorithm for grey-box optimisation that replaces the filter acceptance criterion with a generalisable uni-dimensional funnel, maintaining a monotonically non-increasing upper bound on approximation error of the local black-box RMs. A global convergence proof to a first-order critical point is established. The algorithm, implemented in an open-source Pyomo framework, supports multiple RM forms and globalisation strategies (filter or funnel). Benchmark tests on seven numerical and engineering problems show that the TR funnel algorithm achieves comparable and often improved performance relative to the classical TR filter method. The TR funnel method thus provides a simpler, and extensible alternative for large-scale grey-box optimisation. + oai:arXiv.org:2511.18998v2 + math.OC + cs.NA + math.NA + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Gul Hameed, Tao Chen, Antonio del Rio Chanona, Lorenz T. Biegler, Michael Short + + + Diophantine approximation with primes from short intervals + https://arxiv.org/abs/2512.02174 + arXiv:2512.02174v2 Announce Type: replace +Abstract: In this paper, we establish hybrid results on Diophantine approximation with primes from short intervals. In particular, we prove the following result in a slightly modified form: If $\alpha$ is an irrational number having a continued fraction expansion with bounded terms (in particular, if $\alpha$ is a quadratic irrational), then the number of primes $p$ in the interval $(X-Y,X]$ satisfying $||p\alpha||<\delta$ is asymptotically equal to $2\delta Y/\log X$, provided that $X\ge 10$, $X^{2/3+\varepsilon}\le Y\le X/2$ and $X^{\varepsilon}\max\left\{X^{1/4}Y^{-1/2},X^{2/3}Y^{-1}\right\}\le \delta\le 1/2$. + oai:arXiv.org:2512.02174v2 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Stephan Baier, Sayantan Roy + + + More on the sum-product problem for integers with few prime factors + https://arxiv.org/abs/2512.04931 + arXiv:2512.04931v2 Announce Type: replace +Abstract: We show that if $A\subset \mathbb{Z}$ is a finite set of integers in which every integer is divisible by $O(1)$ many primes then \[\max(\lvert A+A\rvert,\lvert AA\rvert) \geq \lvert A\rvert^{12/7-o(1)}\] and, for any $m\geq 2$, \[\max(\lvert mA\rvert, \lvert A^{(m)}\rvert) \geq \lvert A\rvert^{\frac{2}{3}m+\frac{1}{3}-o(1)}.\] Finally, we show that if $A\subset \mathbb{Q}$ is a finite set of rationals in which the numerator and denominator of every $x\in A$ is divisible by $O(1)$ many primes then $\lvert A+AA\rvert \geq \lvert A\rvert^{2-o(1)}$. + oai:arXiv.org:2512.04931v2 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Rishika Agrawal, Thomas F. Bloom, Giorgis Petridis + + + Singularity of the loops within a cable-graph loop-soup conditioned by its occupation time + https://arxiv.org/abs/2512.05086 + arXiv:2512.05086v2 Announce Type: replace +Abstract: In this note, we show the following feature of the relation between Brownian loop-soups on cable-graphs and their total occupation time-field $\Lambda$: When conditioned on $\Lambda$, the conditional law of individual loops becomes singular with respect to that of unconditioned loops. The idea of the proof is to see that some type of fast points on the curve $\Lambda$ impose an exceptional behaviour of all the loops when they go through these points. + oai:arXiv.org:2512.05086v2 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Arthur Dremaux + + + Critical behaviour of the fully packed loop-$O(n)$ model on planar triangulations + https://arxiv.org/abs/2512.05867 + arXiv:2512.05867v2 Announce Type: replace +Abstract: We study the fully packed loop-$O(n)$ model on planar triangulations. This model is also bijectively equivalent to the Fortuin--Kasteleyn model of planar maps with parameter $q\in (0,4)$ at its self-dual point. These have been traditionally studied using either techniques from analytic combinatorics (based in particular on the gasket decomposition of Borot, Bouttier and Guitter arXiv:1106.0153) or probabilistic arguments (based on Sheffield's hamburger-cheeseburger bijection arXiv:1108.2241). In this paper we establish a dictionary relating quantities of interest in both approaches. This has several consequences. First, we derive an exact expression for the partition function of the fully packed loop-$O(n)$ model on triangulations, as a function of the outer boundary length. This confirms predictions by Gaudin and Kostov. In particular, this model exhibits critical behaviour, in the sense that the partition function exhibits a power-law decay characteristic of the critical regime at this self-dual point. Finally, we derive precise asymptotics for geometric features of the FK model of planar maps when $0 < q <4$, such as the exact tail behaviour of the perimeters of clusters and loops. This sharpens previous results of arXiv:1502.00450 and arXiv:1502.00546. A key step is to use the above dictionary and the probabilistic results to justify rigorously an ansatz commonly assumed in the analytic combinatorics literature. + oai:arXiv.org:2512.05867v2 + math.PR + math-ph + math.CO + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Nathana\"el Berestycki, William Da Silva + + + Low-degree mod 2 cohomology of classifying spaces of $G_2$-gauge groups + https://arxiv.org/abs/2512.12195 + arXiv:2512.12195v2 Announce Type: replace +Abstract: Let $G$ be a simply connected compact simple Lie group and let $\mathcal{G}_k$ denote the gauge group of a principal $G$--bundle over $S^4$ with second Chern class $k\in \pi_4(BG)\cong \mathbb Z$. For $G=G_2$, the $p$--local homotopy types of the gauge groups have been completely classified by Kishimoto--Theriault--Tsutaya and Kameko in terms of the order of the fundamental Samelson product $\langle i_3,1\rangle\in [\Sigma^3G_2,G_2]$. + In this paper, we begin a complementary study of the mod $2$ cohomology of the classifying spaces $B\mathcal{G}_k(G_2)$. Our goal is to understand the structure of $H^*(B\mathcal{G}_k;\mathbb{F}_2)$ as an unstable module over the mod~$2$ Steenrod algebra in a low range of degrees. Using the evaluation fibration \[ \Omega_0^3 G_2 \longrightarrow B\mathcal{G}_k \xrightarrow{\;\mathrm{ev}\;} BG_2 \] together with Serre and Eilenberg--Moore spectral sequences, we study the Serre spectral sequence \[ H^s(BG_2;H^t(\Omega^3_0G_2)) \Longrightarrow H^{s+t}(B\mathcal{G}_k) \] in total degree $\le 10$. A careful analysis of the homotopy groups of $G_2$ shows that \[ H^j(\Omega^3_0G_2;\mathbb{F}_2)=0\quad\text{for }1\le j\le 4, \qquad H^5(\Omega^3_0G_2;\mathbb{F}_2)\neq 0, \] so the first positive-degree generator of the fibre cohomology occurs in degree $5$. As a consequence, there is a distinguished class \[ u_5\in H^5(\Omega^3_0G_2;\mathbb{F}_2) \] whose only possible Serre differential in total degree $\le 10$ is a $d_6$--differential \[ d_6(u_5) = \epsilon(k)\,x_6 \] from $u_5$ to the degree-$6$ generator $x_6\in H^6(BG_2;\mathbb{F}_2)$, for a scalar $\epsilon(k)\in\mathbb{F}_2$ encoding the low-degree effect of the bundle class. In addition, $2$--locally we prove that $\epsilon(k)$ is $4$--periodic in $k$ (i.e. it depends only on $k\bmod 4$) and that $\epsilon(k)=0$ for all $k\equiv 0\pmod 4$. + oai:arXiv.org:2512.12195v2 + math.AT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Dang Vo Phuc + + + Determining subgroups via stationary measures + https://arxiv.org/abs/2512.12966 + arXiv:2512.12966v2 Announce Type: replace +Abstract: In this paper, we consider random walks on the isometry groups of general metric spaces. Under some mild conditions, we show that if two non-elementary random walks on a discrete subgroup of the isometry group have non-singular stationary measures, then subgroups generated by the random walks are commensurable. This result in particular applies to separable Gromov hyperbolic spaces and Teichm\"uller spaces. As a specific application, we prove singularity between stationary measures associated to random walks on different fiber subgroups of the fundamental group of a hyperbolic 3-manifold fibering over the circle. + oai:arXiv.org:2512.12966v2 + math.GT + math.DS + math.GR + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Dongryul M. Kim, Andrew Zimmer + + + Asymptotics of the graph Laplace operator near an isolated singularity + https://arxiv.org/abs/2512.13314 + arXiv:2512.13314v3 Announce Type: replace +Abstract: In this paper, we investigate asymptotics of the continuous graph Laplace operator on a smooth Riemannian manifold $(M,g)$ admitting an isolated singularity $x$. We show that if the curvature function $\kappa$ doesn't grow too fast near $x$, then the graph Laplace operator at $x$ converges to the weighted Laplace-Beltrami operator as the bandwidth $t\downarrow 0.$ On the other hand, we also prove that if one locally modifies a given Riemannian metric across $x$ by a non-constant \textit{purely angular }conformal factor, then $\kappa$ grows too fast and the graph Laplace operator behaves like $O(\frac{1}{\sqrt{t}})$ near $x$, as $t\downarrow 0$, given a mild condition on the angular conformal factor. We provide the Taylor expansion of the graph Laplace operator as $t\downarrow 0$ in specific cases. Numerical simulations at the end illustrate our results. + oai:arXiv.org:2512.13314v3 + math.DG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Susovan Pal + + + Continuized Nesterov Acceleration for Non-Convex Optimization + https://arxiv.org/abs/2512.16533 + arXiv:2512.16533v2 Announce Type: replace +Abstract: In convex optimization, continuous-time counterparts have been a fruitful tool for analyzing momentum algorithms. Fewer such examples are available when the function to minimize is non-convex. In several cases, discrepancies arise between the existing discrete-time results, namely those obtained for momentum algorithms, and their continuous-time counterparts, with the latter typically yielding stronger guarantees. We argue that the continuized framework (Even et al., 2021), mixing continuous and discrete components, can tighten the gap between known continuous and discrete results. This framework relies on computations akin to standard Lyapunov analyses, from which are deduced convergence bounds for an algorithm that can be written as a Nesterov momentum algorithm with stochastic parameters. In this work, we extend the range of applicability of the continuized framework, e.g. by allowing it to handle non-smooth Lyapunov functions. We then strengthen its trajectory-wise guarantees for linear convergence rate, deriving finite time bounds with high probability and asymptotic almost sure bounds. We apply this framework to the non-convex class of strongly quasar convex functions. Adapting continuous-time results that have weaker discrete equivalents to the continuized method, we improve by a constant factor the known convergence rate, and relax the existing assumptions on the set of minimizers. + oai:arXiv.org:2512.16533v2 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Julien Hermant, Jean-Fran\c{c}ois Aujol, Charles Dossal, Lorick Huang, Aude Rondepierre + + + Confusions and Erasures of Error-Bounded Block Decoders with Finite Blocklength + https://arxiv.org/abs/2512.16665 + arXiv:2512.16665v2 Announce Type: replace +Abstract: This paper investigates two distinct types of block errors - undetected errors (confusions) and erasures - in additive white Gaussian noise (AWGN) channels with error-bounded block decoders operating in the finite blocklength (FBL) regime. While block error rate (BLER) is a common metric, it does not distinguish between confusions and erasures, which can have significantly different impacts in cross-layer protocol design, despite upper-layer protocols universally assuming physical (PHY) errors manifest as packet erasures rather than undetected corruptions - an assumption lacking rigorous PHY-layer validation. We present a systematic analysis of confusions and erasures under BLER-constrained maximum likelihood (ML) decoding. Through sphere-packing analysis, we provide analytical bounds for both block confusion and erasure probabilities, and derive the sensitivities of these bounds to blocklength and signal-to-noise ratio (SNR). To the best of our knowledge, this is the first study on this topic in the FBL regime. Our findings provide theoretical validation for the block erasure channel abstraction commonly assumed in medium access control (MAC) and network layer protocols, confirming that, for practical FBL codes, block confusions are negligible compared to block erasures, especially at large blocklengths and high SNR. + oai:arXiv.org:2512.16665v2 + cs.IT + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Bin Han, Yao Zhu, Rafael F. Schaefer, Giuseppe Caire, Anke Schmeink, H. Vincent Poor, Hans D. Schotten + + + Local Topological Constraints on Berry Curvature in Spin--Orbit Coupled BECs + https://arxiv.org/abs/2512.19282 + arXiv:2512.19282v3 Announce Type: replace +Abstract: We establish a local topological obstruction to flattening Berry curvature in spin-orbit-coupled Bose-Einstein condensates (SOC BECs), valid even when the global Chern number vanishes. For a generic two-component SOC BEC, the extended parameter space \(M=T^{2}_{\mathrm{BZ}}\times S^{1}_{\phi_{+}}\times S^{1}_{\phi_{-}}\) carries a natural metric connection \(\nabla^{C}\) whose torsion 3-form encodes the synthetic gauge fields. Under the physically relevant assumption of constant Berry curvatures, the harmonic part of this torsion defines a mixed cohomology class $$ [\omega]\in\bigl(H^{2}(T^{2}_{\mathrm{BZ}})\otimes H^{1}(S^{1}_{\phi_{+}})\bigr)\oplus\bigl(H^{2}(T^{2}_{\mathrm{BZ}})\otimes H^{1}(S^{1}_{\phi_{-}})\bigr), $$ whose mixed tensor rank equals one. Using a general geometric bound for metric connections with totally skew torsion on product manifolds, we show that the obstruction kernel $\mathcal{K}$ vanishes, yielding the sharp inequality $\dim\mathfrak{hol}^{\mathrm{off}}(\nabla^{C})\geq 1$. This forces at least one off-diagonal curvature operator, preventing complete gauging-away of Berry phases even when the total Chern number is zero. This provides the first cohomological lower bound certifying locally irremovable curvature in SOC BECs beyond the Chern-number paradigm. + oai:arXiv.org:2512.19282v3 + math.DG + math.AT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Alexander Pigazzini, Magdalena Toda + + + A Cartesian Cut-Cell Two-Fluid Method for Two-Phase Diffusion Problems + https://arxiv.org/abs/2512.19407 + arXiv:2512.19407v2 Announce Type: replace +Abstract: We present a Cartesian cut-cell finite-volume method for sharp-interface two-phase diffusion problems in static geometries. The formulation follows a two-fluid approach: independent diffusion equations are discretized in each phase on a fixed Cartesian grid, while the phases are coupled through embedded interface conditions enforcing continuity of diffusive flux and a general jump law. Cut cells are treated by integrating the governing equations over phase-restricted control volumes and surfaces, yielding discrete divergence and gradient operators that are locally conservative within each phase. Interface coupling is achieved by introducing a small set of interfacial unknowns per cut cell on the embedded boundary; the resulting algebraic system involves only bulk and interfacial averages. A key feature of the method is the use of a reduced set of geometric information based solely on low-order moments (trimmed volumes, apertures and interface measures/centroids), allowing robust implementation without constructing explicitly cut-cell polytopes. The method supports steady (Poisson) and unsteady (diffusion) regimes and incorporates Dirichlet, Neumann, Robin boundary conditions and general jumps. We validate the scheme on one-, two- and three-dimensional single-phase and two-phase benchmarks, including curved embedded boundaries, Robin conditions and strong property/jump contrasts. The results demonstrate a superlinear convergence behavior, sharp enforcement of interfacial laws and excellent conservation properties. Extensions to moving interfaces and Stefan-type free-boundary problems are natural perspectives of this framework. + oai:arXiv.org:2512.19407v2 + math.NA + cs.NA + physics.comp-ph + physics.flu-dyn + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Louis Libat, Can Sel\c{c}uk, Eric Ch\'enier, Vincent Le Chenadec + + + Graph Sensitivity under Join and Decomposition + https://arxiv.org/abs/2512.19915 + arXiv:2512.19915v2 Announce Type: replace +Abstract: The sensitivity, $\sigma(G)$, of a finite undirected simple graph $G$ is the smallest maximum degree of an induced subgraph on more than the maximum number of independent vertices. Call an indexed family of graphs $G_n$ with maximum degree $\Delta(G_n) \to \infty$ as $n \to \infty$ sensitive if $\sigma(G_n) \to \infty$, and insensitive otherwise. We describe sensitivity under the join operation and decomposition into stable blocks and construct sensitive and insensitive, primarily non-regular, graph families. We determine the sensitivity explicitly for numerous singly- and doubly-indexed graph families, including certain generalized joins - e.g., complete multipartite graphs and some generalized windmill graphs; general rooted products; and families of corona graphs. + oai:arXiv.org:2512.19915v2 + math.CO + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Cathy Kriloff, Jacob Tolman + + + Regularization methods for solving hierarchical variational inequalities with complexity guarantees + https://arxiv.org/abs/2512.20772 + arXiv:2512.20772v2 Announce Type: replace +Abstract: We consider hierarchical variational inequality problems, or more generally, variational inequalities defined over the set of zeros of a monotone operator. This framework includes convex optimization over equilibrium constraints and equilibrium selection problems. In a real Hilbert space setting, we combine a Tikhonov regularization and a proximal penalization to develop a flexible double-loop method for which we prove asymptotic convergence and provide rate statements in terms of gap functions. Our method is flexible, and effectively accommodates a large class of structured operator splitting formulations for which fixed-point encodings are available. Finally, we validate our findings numerically on various examples. + oai:arXiv.org:2512.20772v2 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Daniel Cortild, Meggie Marschner, Mathias Staudigl + + + Trisections and Lefschetz fibrations with $(-n)$-sections + https://arxiv.org/abs/2512.21001 + arXiv:2512.21001v2 Announce Type: replace +Abstract: Castro and Ozbagci constructed a trisection of a closed 4-manifold admitting a Lefschetz fibration with a $(-1)$-section such that the corresponding trisection diagram can be explicitly constructed from a monodromy of the Lefschetz fibration. In this paper, for a closed 4-manifold $X$ admitting an achiral Lefschetz fibration with a $(-n)$-section, we construct a trisection of $X \# n\mathbb{C}P^2$ if $n$ is positive and $X \# (-n)\overline{\mathbb{C}P^2}$ if $n$ is negative such that the corresponding trisection diagram can be explicitly constructed from a monodromy of the Lefschetz fibration. We also construct a trisection of the fiber sum of two achiral Lefschetz fibrations with $n$- and $(-n)$-sections such that the corresponding trisection diagram can be explicitly constructed from monodromies of the Lefschetz fibrations. + oai:arXiv.org:2512.21001v2 + math.GT + math.SG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Tsukasa Isoshima, Reo Yabuguchi + + + Non-finite generatedness of the congruences defined by tropical varieties + https://arxiv.org/abs/2512.21565 + arXiv:2512.21565v2 Announce Type: replace +Abstract: In tropical geometry, there are several important classes of ideals and congruences such as tropical ideals, bend congruences, and the congruences of the form $\mathbf E(Z)$. Although they are analogues of the concept of ideals of rings, it is not well known whether they are finitely generated. In this paper, we study whether the congruences of the form $\mathbf E(Z)$ are finitely generated. In particular, we show that when $Z$ is the support of a tropical variety, $\mathbf E(Z)$ is not finitely generated except for a few specific cases. In addition, we give an explicit minimal generating set of $\mathbf E(|L|)$ for the tropical standard line $L$. + oai:arXiv.org:2512.21565v2 + math.AC + math.AG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Takaaki Ito + + + MSO logic of the real order with the set quantifier ranging over Borel sets + https://arxiv.org/abs/2512.23003 + arXiv:2512.23003v3 Announce Type: replace +Abstract: A celebrated 1969 theorem of Michael Rabin is that the MSO theory of the real order where the monadic quantifier is allowed only to range over the sets of rational numbers, is decidable. In 1975 Saharon Shelah proved that if the monadic quantifier is allowed to range over all subsets of the reals, the resulting MSO theory is undecidable. He conjectured that when we allow the monadic quantifier to range over the Borel subsets of the reals, the resulting MSO theory is decidable. We confirm this conjecture. Namely, the MSO theory of the real order where the set quantifier is allowed to range over the Borel sets, is decidable. + oai:arXiv.org:2512.23003v3 + math.LO + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Mirna D\v{z}amonja + + + Structure preservation and emergent dissipation in stochastic wave equations with transport noise + https://arxiv.org/abs/2512.23309 + arXiv:2512.23309v2 Announce Type: replace +Abstract: We study nonlinear wave equations perturbed by transport noise acting either on the displacement or on the velocity. Such noise models random advection and, under suitable scaling of space covariance, may generate an effective dissipative term. We establish well-posedness in both cases and analyse the associated scaling limits. When the noise acts on the displacement, the system preserves its original structure and converges to the deterministic nonlinear wave equation, whereas if it acts on the velocity, the rescaled dynamics produce an additional Laplacian damping term, leading to a stochastic derivation of a Westervelt-type acoustic model. + oai:arXiv.org:2512.23309v2 + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Chang Liu, Dejun Luo + + + Rational Angle Bisection Problem in Higher Dimensional Spaces and Incenters of Simplices over Fields + https://arxiv.org/abs/2512.24660 + arXiv:2512.24660v2 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 $\mathbb R$: on the coordinate plane, for which rational numbers $a$ and $b$ are the slopes of the angle bisectors between two lines with slopes $a$ and $b$ rational? First, we give a few characterizations of when the angle bisectors between two lines with direction vectors in $k^n$ have direction vectors in $k^n.$ To find solutions to the problem in the case when $k = \mathbb Q,$ we also give a formula for the integral solutions of $x_1{}^2+\dots +x_n{}^2 = dx_{n+1}{}^2,$ which is a generalization of the negative Pell's equation $x^2-dy^2 = -1,$ where $d$ is a square-free positive integer. Second, by applying the above characterizations, we give a necessary and sufficient condition for the incenter of a given $n$-simplex with $k$-rational vertices to be $k$-rational. On the coordinate plane, we prove that every triangle with $k$-rational vertices and incenter can be obtained by scaling a triangle with $k$-rational side lengths and area, which is a generalization of a Heronian triangle. We also state certain fundamental properties of a few centers of a given triangle with $k$-rational vertices. + oai:arXiv.org:2512.24660v2 + math.NT + math.MG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Takashi Hirotsu + + + The Lagrangian and symplectic structures of the Kuramoto oscillator model + https://arxiv.org/abs/2601.00113 + arXiv:2601.00113v2 Announce Type: replace +Abstract: Despite being under intense scrutiny for 50 years, the Kuramoto oscillator model has remained a quintessential representative of non-equilibrium phase transitions. One of the reasons for its enduring relevance is the apparent lack of an optimization formulation, due to the fact that (superficially), the equations of motion seem to not be compatible with a Lagrangian structure. We show that, as a mean-field classical (twisted) spin model on $S^2$, the Kuramoto model can be described variationaly. Based on this result perturbation analysis around (unstable) Kuramoto equilibria are shown to be equivalent to low-energy fluctuations of mean-field Heisenberg spin models. Intriguingly, off-plane perturbations around these equilibria configurations turn out to be described by a semiclassical Gaudin model, pointing to the fact that oscillator synchronization maps to the spin pairing mechanism investigated by Richardson and subsequently by others. + oai:arXiv.org:2601.00113v2 + math-ph + math.CA + math.DS + math.MP + math.SG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Sherwin Kouchekian, Razvan Teodorescu + + + Bilinear forms with Kloosterman fractions and applications + https://arxiv.org/abs/2601.00292 + arXiv:2601.00292v2 Announce Type: replace +Abstract: We establish improved bounds for bilinear forms with Kloosterman fractions of the form ${\sum\sum}_{m,n} \alpha_m \beta_n e(a\overline{m}/(bn))$ with $M<m\le 2M$, $N < n \le 2N$ and $(m,n)=1$. Our approach works directly with arbitrary coefficient sequences $(\alpha_m), (\beta_n) \in \mathbb{C}$, avoiding the temporary restriction to squarefree support used in prior work. While this requires handling additional arithmetic complexity, it yields strictly stronger bounds that improve upon the estimates of Duke, Friedlander, and Iwaniec \cite{DFI} and Bettin-Chandee \cite{BC}; in the balanced case $M \approx N$, the new saving over the trivial bound is $1/12$%, compared to $1/48$ in \cite{DFI} . As an application, we prove a generalized asymptotic formula for the twisted second moment of the Riemann zeta-function with Dirichlet polynomials of length $T^{1/2+\delta}$ for $\delta = 1/46$, extending beyond the previously limiting $\theta = 1/2$ barrier established by Bettin, Chandee, and Radziwi{\l}{\l} \cite{BCR}. We also establish bounds for related Hermitian sums involving Sali\'{e}-type exponential phases and develop techniques for more general bilinear forms with Kloosterman fractions. + oai:arXiv.org:2601.00292v2 + math.NT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Anji Dong, Nicolas Robles, Dirk Zeindler + + + Hadamard-type formulas for real eigenvalues of canonically symplectic operators + https://arxiv.org/abs/2601.00520 + arXiv:2601.00520v2 Announce Type: replace +Abstract: We give first-order asymptotic expansions for the resolvent and Hadamard-type formulas for the eigenvalue curves of one-parameter families of canonically symplectic operators. We allow for parameter dependence in the boundary conditions, bounded perturbations and trace operators associated with each off-diagonal operator, and give formulas for derivatives of eigenvalue curves emanating from the discrete eigenvalue of the unperturbed operator in terms of Maslov crossing forms. We derive the Hadamard-type formulas using two different methods: via a symplectic resolvent difference formula and asymptotic expansions of the resolvent, and using Lyapunov-Schmidt reduction and the implicit function theorem. The latter approach facilitates derivative formulas when the eigenvalue curves are viewed as functions of the spectral parameter. We apply our abstract results to derive a spectral index theorem for the linearised operator associated with a standing wave in the nonlinear Schr\"odinger equation on a compact star graph. + oai:arXiv.org:2601.00520v2 + math.SP + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mitchell Curran, Selim Sukhtaiev + + + Study of Composition Operators in Certain Functional Spaces + https://arxiv.org/abs/2601.00801 + arXiv:2601.00801v2 Announce Type: replace +Abstract: In this thesis we study three problems. The first is the superposition of the operators and their proprities, such as boundedness,continuity,regularity and the inequalities of the norms of the composition of functions in some functional spaces. The second is to generalize some results of the composition of more than two functions, and the third is to give a generalization of Peetre's theorem. + oai:arXiv.org:2601.00801v2 + math.FA + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Mahdi Tahar Brahimi + + + The Maximum of the Volume of a Part of a Cevian Simplex + https://arxiv.org/abs/2601.00876 + arXiv:2601.00876v2 Announce Type: replace +Abstract: The cevians passing through a point in a simplex create a cevian simplex, which is divided by these cevians into smaller simplices. We consider the problem about the maximum of the ratio of the sum of the volumes of some of these smaller simplices by the volume of the reference simplex. The special case of tetrahedron is given as an example. + oai:arXiv.org:2601.00876v2 + math.GM + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Zamina Guliyeva, Yagub Aliyev + + + On the universal curve with unordered marked points in positive characteristic + https://arxiv.org/abs/2601.01336 + arXiv:2601.01336v2 Announce Type: replace +Abstract: We study the relative pro-$\ell$ and continuous relative completions of the algebraic fundamental groups of universal curves over the moduli stack of curves with unordered marked points in positive characteristic. Using specialization and homotopy exact sequences, we compare the ordered and unordered settings and prove that the natural projection from the relative completion of the universal curve over the unordered moduli stack admits no section in positive characteristic. This yields a non-splitting result for the corresponding projection on algebraic fundamental groups. The present paper is a sequel to our earlier work in characteristic zero. + oai:arXiv.org:2601.01336v2 + math.AG + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ma Luo, Tatsunari Watanabe + + + Iterating PP-packages without Choice: A Cohen symmetric seed and a localization framework + https://arxiv.org/abs/2601.01855 + arXiv:2601.01855v2 Announce Type: replace +Abstract: The Partition Principle $\mathsf{PP}$ asserts that whenever there is a surjection $A\twoheadrightarrow B$, there is an injection $B\hookrightarrow A$. Russell conjectured in 1906 that $\mathsf{PP}$ is equivalent to the Axiom of Choice $\mathsf{AC}$; while $\mathsf{AC}\Rightarrow \mathsf{PP}$ is immediate, the converse has remained open. We show that $\mathsf{PP}$ does not imply $\mathsf{AC}$ by constructing a transitive model of $\mathsf{ZF}+\mathsf{DC}+\mathsf{PP}+\neg\mathsf{AC}$. + Starting from a Cohen symmetric model $\mathcal{N}$ of $\mathrm{Add}(\omega,\omega_1)$ built with a countable-support symmetry filter, we fix parameters $S:=A^\omega$ and $T:=\mathcal{P}(S)$ and perform a class-length countable-support symmetric iteration. At successor stages we use orbit-symmetrized packages that split targeted surjections, yielding $\mathsf{PP}\!\restriction T$ and $\mathsf{AC}_{\mathsf{WO}}$, while preserving $\mathsf{DC}$ and ensuring that $A$ remains non-well-orderable. A diagonal-cancellation/diagonal-lift infrastructure supplies a proper $\omega_1$-complete normal filter at limit stages. Finally, Ryan--Smith localization shows that under $\mathsf{SVC}^+(T)$, $\mathsf{PP}$ is equivalent to $\mathsf{PP}\!\restriction T \wedge \mathsf{AC}_{\mathsf{WO}}$, so the final model satisfies $\mathsf{PP}$ but not $\mathsf{AC}$. + oai:arXiv.org:2601.01855v2 + math.LO + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Frank Gilson + + + A Perturbed DCA for Computing d-Stationary Points of Nonsmooth DC Programs + https://arxiv.org/abs/2601.02084 + arXiv:2601.02084v2 Announce Type: replace +Abstract: This paper introduces an efficient perturbed difference-of-convex algorithm (pDCA) for computing d-stationary points of an important class of structured nonsmooth difference-of-convex problems. Compared to the principal algorithms introduced in [J.-S. Pang, M. Razaviyayn, and A. Alvarado, Math. Oper. Res. 42(1):95--118 (2017)], which may require solving several subproblems for a one-step update, pDCA only requires solving a single subproblem. Therefore, the computational cost of pDCA for one-step update is comparable to the widely used difference-of-convex algorithm (DCA) introduced in [D. T. Pham and H. A. Le Thi, Acta Math. Vietnam. 22(1):289--355 (1997)] for computing a critical point. Importantly, under practical assumptions, we prove that every accumulation point of the sequence generated by pDCA is a d-stationary point almost surely. Numerical experiment results on several important examples of nonsmooth DC programs demonstrate the efficiency of pDCA for computing d-stationary points. + oai:arXiv.org:2601.02084v2 + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Zhangcheng Feng, Yancheng Yuan + + + Square integrability of regular representations on reductive homogeneous spaces + https://arxiv.org/abs/2601.02188 + arXiv:2601.02188v2 Announce Type: replace +Abstract: Let $G$ be a real reductive Lie group and $H$ a reductive subgroup of $G$. Benoist-Kobayashi studied when $L^2(G/H)$ is a tempered representation of $G$ and in particular they gave a necessary and sufficient condition for the temperedness in terms of certain functions on Lie algebras. In this paper, we consider when $L^2(G/H)$ is equivalent to a unitary subrepresentation of $L^2(G)$ and we will give a sufficient condition for this in terms of functions introduced by Benoist-Kobayashi. As a corollary, we prove the non-existence of discrete series for homogeneous spaces $G/H$ satisfying certain conditions. + oai:arXiv.org:2601.02188v2 + math.RT + Wed, 07 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Kazushi Maeda, Yoshiki Oshima + + + At the Intersection of Deep Sequential Model Framework and State-space Model Framework: Study on Option Pricing + https://arxiv.org/abs/2012.07784 + arXiv:2012.07784v2 Announce Type: replace-cross +Abstract: Inference and forecast problems of the nonlinear dynamical system have arisen in a variety of contexts. Reservoir computing and deep sequential models, on the one hand, have demonstrated efficient, robust, and superior performance in modeling simple and chaotic dynamical systems. However, their innate deterministic feature has partially detracted their robustness to noisy system, and their inability to offer uncertainty measurement has also been an insufficiency of the framework. On the other hand, the traditional state-space model framework is robust to noise. It also carries measured uncertainty, forming a just-right complement to the reservoir computing and deep sequential model framework. We propose the unscented reservoir smoother, a model that unifies both deep sequential and state-space models to achieve both frameworks' superiorities. Evaluated in the option pricing setting on top of noisy datasets, URS strikes highly competitive forecasting accuracy, especially those of longer-term, and uncertainty measurement. Further extensions and implications on URS are also discussed to generalize a full integration of both frameworks. + oai:arXiv.org:2012.07784v2 + stat.ML + cs.LG + math.DS + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ziyang Ding, Sayan Mukherjee + + + Integrable systems on multiplicative quiver varieties from cyclic quivers + https://arxiv.org/abs/2108.02496 + arXiv:2108.02496v3 Announce Type: replace-cross +Abstract: We consider a class of complex manifolds constructed as multiplicative quiver varieties associated with a cyclic quiver extended by an arbitrary number of arrows starting at a new vertex. Such varieties admit a Poisson structure, which is obtained by quasi-Hamiltonian reduction. We construct several families of Poisson subalgebras inside the coordinate ring of these spaces, which we use to obtain degenerately integrable systems. We also extend the Poisson centre of these algebras to maximal abelian Poisson algebras, hence defining Liouville integrable systems. By considering a suitable set of local coordinates on the multiplicative quiver varieties, we can derive the local Poisson structure explicitly. This allows us to interpret the integrable systems that we have constructed as new generalisations of the spin Ruijsenaars-Schneider system with several types of spin variables. + oai:arXiv.org:2108.02496v3 + nlin.SI + math-ph + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.1088/1751-8121/ada64e + J. Phys. A: Math. Theor. 58 (2025), 045202 + Maxime Fairon + + + Using prior information to boost power in correlation structure support recovery + https://arxiv.org/abs/2111.11278 + arXiv:2111.11278v2 Announce Type: replace-cross +Abstract: Hypothesis testing of structure in correlation and covariance matrices is of broad interest in many application areas. In high dimensions and/or small to moderate sample sizes, high error rates in testing is a substantial concern. This article focuses on increasing power through a frequentist assisted by Bayes (FAB) procedure. This FAB approach boosts power by including prior information on the correlation parameters. In particular, we suppose there is one of two sources of prior information: (i) a prior dataset that is distinct from the current data but related enough that it may contain valuable information about the correlation structure in the current data; and (ii) knowledge about a tendency for the correlations in different parameters to be similar so that it is appropriate to consider a hierarchical model. When the prior information is relevant, the proposed FAB approach can have significant gains in power. A divide-and-conquer algorithm is developed to reduce computational complexity in massive testing dimensions. We show improvements in power for detecting correlated gene pairs in genomic studies while maintaining control of Type I error or false discover rate (FDR). + oai:arXiv.org:2111.11278v2 + stat.ME + math.ST + stat.TH + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ziyang Ding, David Dunson + + + MAST: Model-Agnostic Sparsified Training + https://arxiv.org/abs/2311.16086 + arXiv:2311.16086v2 Announce Type: replace-cross +Abstract: We introduce a novel optimization problem formulation that departs from the conventional way of minimizing machine learning model loss as a black-box function. Unlike traditional formulations, the proposed approach explicitly incorporates an initially pre-trained model and random sketch operators, allowing for sparsification of both the model and gradient during training. We establish the insightful properties of the proposed objective function and highlight its connections to the standard formulation. Furthermore, we present several variants of the Stochastic Gradient Descent (SGD) method adapted to the new problem formulation, including SGD with general sampling, a distributed version, and SGD with variance reduction techniques. We achieve tighter convergence rates and relax assumptions, bridging the gap between theoretical principles and practical applications, covering several important techniques such as Dropout and Sparse training. This work presents promising opportunities to enhance the theoretical understanding of model training through a sparsification-aware optimization approach. + oai:arXiv.org:2311.16086v2 + cs.LG + cs.AI + cs.DC + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Yury Demidovich, Grigory Malinovsky, Egor Shulgin, Peter Richt\'arik + + + On the Structure of Wave Functions in Complex Chern-Simons Theory + https://arxiv.org/abs/2312.00624 + arXiv:2312.00624v2 Announce Type: replace-cross +Abstract: We study the structure of wave functions in complex Chern-Simons theory on the complement of a hyperbolic knot, emphasizing the similarities with the topological string/spectral theory correspondence. We first conjecture a hidden integrality structure in the holomorphic blocks and show that this structure guarantees the cancellation of potential singularities in the full non-perturbative wave function at rational values of the coupling constant. We then develop various techniques to determine the wave function at such rational points. Finally, we illustrate our conjectures and obtain explicit results in the examples of the figure-eight and three-twist knots. In the case of the figure-eight knot, we also perform a direct evaluation of the state integral in the rational case and observe that the resulting wave function has the features of the ground state for a quantum mirror curve. + oai:arXiv.org:2312.00624v2 + hep-th + math-ph + math.GT + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + 10.3842/SIGMA.2026.002 + SIGMA 22 (2026), 002, 45 pages + Marcos Mari\~no, Claudia Rella + + + Learning mirror maps in policy mirror descent + https://arxiv.org/abs/2402.05187 + arXiv:2402.05187v3 Announce Type: replace-cross +Abstract: Policy Mirror Descent (PMD) is a popular framework in reinforcement learning, serving as a unifying perspective that encompasses numerous algorithms. These algorithms are derived through the selection of a mirror map and enjoy finite-time convergence guarantees. Despite its popularity, the exploration of PMD's full potential is limited, with the majority of research focusing on a particular mirror map -- namely, the negative entropy -- which gives rise to the renowned Natural Policy Gradient (NPG) method. It remains uncertain from existing theoretical studies whether the choice of mirror map significantly influences PMD's efficacy. In our work, we conduct empirical investigations to show that the conventional mirror map choice (NPG) often yields less-than-optimal outcomes across several standard benchmark environments. Using evolutionary strategies, we identify more efficient mirror maps that enhance the performance of PMD. We first focus on a tabular environment, i.e. Grid-World, where we relate existing theoretical bounds with the performance of PMD for a few standard mirror maps and the learned one. We then show that it is possible to learn a mirror map that outperforms the negative entropy in more complex environments, such as the MinAtar suite. Additionally, we demonstrate that the learned mirror maps generalize effectively to different tasks by testing each map across various other environments. + oai:arXiv.org:2402.05187v3 + stat.ML + cs.LG + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Carlo Alfano, Sebastian Towers, Silvia Sapora, Chris Lu, Patrick Rebeschini + + + Semiparametric fiducial inference for Cox models + https://arxiv.org/abs/2404.18779 + arXiv:2404.18779v2 Announce Type: replace-cross +Abstract: R. A. Fisher introduced the concept of fiducial as a potential replacement for the Bayesian posterior distribution in the 1930s. During the past century, fiducial approaches have been explored in various parametric and nonparametric settings. However, to the best of our knowledge, no fiducial inference has been developed in the realm of semiparametric statistics. In this paper, we propose a novel fiducial approach for semiparametric models. To streamline our presentation, we use the Cox proportional hazards model, which is the most popular model for the analysis of survival data, as a running example. Other models and extensions are also discussed. In our experiments, we find our method to perform well especially in situations when the maximum likelihood estimator fails. + oai:arXiv.org:2404.18779v2 + stat.ME + math.ST + stat.CO + stat.TH + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yifan Cui, Jan Hannig, Paul Edlefsen + + + A split-step Christov method for approximating rational PDE solutions + https://arxiv.org/abs/2407.04013 + arXiv:2407.04013v3 Announce Type: replace-cross +Abstract: Rational solutions of partial differential equations (PDEs) are notoriously difficult to approximate via spectral Fourier methods due to their algebraically slow decay rate. In this work we discuss approximating rational PDE solutions in a basis of orthogonal functions known as the Fourier series, allowing for the computation of its spectrum via the fast Fourier transform. Spectral differentiation matrices are derived. Several explicit fourth-order split-step integrators are derived and their performance compared. As an application, rogue wave solutions in a family of nonlinear Schr\"odinger equations are explored. Perturbing the constant background is found to generate rogue wave-like structures. The effects of higher-order dispersion and generalized nonlinearities are also examined. + oai:arXiv.org:2407.04013v3 + nlin.PS + cs.NA + math.NA + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Journal of Computational Physics, 2025, 114544 + Justin T. Cole, Troy I. Johnson + + + Bell-CHSH inequality and unitary transformations in Quantum Field Theory + https://arxiv.org/abs/2412.03840 + arXiv:2412.03840v3 Announce Type: replace-cross +Abstract: Unitary transformations are employed to enhance the violations of the Bell-CHSH inequality in relativistic Quantum Field Theory. The case of the scalar field in $1+1$ Minkowski space-time is scrutinized by relying on the Tomita-Takesaki modular theory. The example of the bounded Hermitian operator $sign(\varphi(f))$, where $\varphi(f)$ stands for the smeared scalar field, is worked out. It is shown that unitary deformations enable for violations of the Bell-CHSH inequality. The setup is generalized to the Proca vector field by means of its equivalence with the scalar theory. + oai:arXiv.org:2412.03840v3 + quant-ph + hep-th + math-ph + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + D. O. R. Azevedo, F. M. Guedes, M. S. Guimaraes, I. Roditi, S. P. Sorella, A. F. Vieira + + + Robust Quantum Control for Bragg Pulse Design in Atom Interferometry + https://arxiv.org/abs/2502.04618 + arXiv:2502.04618v3 Announce Type: replace-cross +Abstract: We formulate a robust optimal control algorithm to synthesize minimum energy pulses that can transfer a cold atom system into various momentum states. The algorithm uses adaptive linearization of the evolution operator and sequential quadratic programming to iterate the control towards a minimum energy pulse that achieves optimal target state fidelity. Robustness to parameter variation is achieved using Legendre polynomial approximation over the domain of variation. The method is applied to optimize the Bragg beamsplitting operation in ultra-cold atom interferometry. Even in the presence of 10-40% variability in the initial momentum dispersion of the atomic cloud and the intensity of the optical pulse, the algorithm reliably converges to a control protocol that robustly achieves unprecedented momentum levels with high fidelity for a single-frequency multi-photon Bragg diffraction scheme (e.g. $|\pm 40\hbar k\rangle$). We show the advantages of our method by comparison to stochastic optimization using sampled parameter values, provide detailed sensitivity analyses, and performance of the designed pulses is verified in laboratory experiments. + oai:arXiv.org:2502.04618v3 + quant-ph + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Luke S. Baker, Andre Luiz P. de Lima, Andrew Harter, Ceren Uzun, Liam P. Keeley, Jr-Shin Li, Anatoly Zlotnik, Michael J. Martin, Malcolm G. Boshier + + + Shift orbifolds, decompactification limits, and lattices + https://arxiv.org/abs/2502.18453 + arXiv:2502.18453v2 Announce Type: replace-cross +Abstract: We describe the general shift orbifold of a Narain CFT and use this to investigate decompactification limits in the heterotic Narain moduli space. We also comment on higher rank theories and describe some applications to the CFT based on the Leech lattice and its shift orbifolds. + oai:arXiv.org:2502.18453v2 + hep-th + math.RT + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Dan Israel, Ilarion V. Melnikov, Yann Proto + + + Active operator learning with predictive uncertainty quantification for partial differential equations + https://arxiv.org/abs/2503.03178 + arXiv:2503.03178v2 Announce Type: replace-cross +Abstract: With the increased prevalence of neural operators being used to provide rapid solutions to partial differential equations (PDEs), understanding the accuracy of model predictions and the associated error levels is necessary for deploying reliable surrogate models in scientific applications. Existing uncertainty quantification (UQ) frameworks employ ensembles or Bayesian methods, which can incur substantial computational costs during both training and inference. We propose a lightweight predictive UQ method tailored for Deep operator networks (DeepONets) that also generalizes to other operator networks. Numerical experiments on linear and nonlinear PDEs demonstrate that the framework's uncertainty estimates are unbiased and provide accurate out-of-distribution uncertainty predictions with a sufficiently large training dataset. Our framework provides fast inference and uncertainty estimates that can efficiently drive outer-loop analyses that would be prohibitively expensive with conventional solvers. We demonstrate how predictive uncertainties can be used in the context of Bayesian optimization and active learning problems to yield improvements in accuracy and data-efficiency for outer-loop optimization procedures. In the active learning setup, we extend the framework to Fourier Neural Operators (FNO) and describe a generalized method for other operator networks. To enable real-time deployment, we introduce an inference strategy based on precomputed trunk outputs and a sparse placement matrix, reducing evaluation time by more than a factor of five. Our method provides a practical route to uncertainty-aware operator learning in time-sensitive settings. + oai:arXiv.org:2503.03178v2 + cs.LG + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Nick Winovich, Mitchell Daneker, Lu Lu, Guang Lin + + + Solving the Paint Shop Problem with Flexible Management of Multi-Lane Buffers Using Reinforcement Learning and Action Masking + https://arxiv.org/abs/2504.02644 + arXiv:2504.02644v2 Announce Type: replace-cross +Abstract: In the paint shop problem, an unordered incoming sequence of cars assigned to different colors has to be reshuffled with the objective of minimizing the number of color changes. To reshuffle the incoming sequence, manufacturers can employ a first-in-first-out multi-lane buffer system allowing store and retrieve operations. So far, prior studies primarily focused on simple decision heuristics like greedy or simplified problem variants that do not allow full flexibility when performing store and retrieve operations. In this study, we propose a reinforcement learning approach to minimize color changes for the flexible problem variant, where store and retrieve operations can be performed in an arbitrary order. After proving that greedy retrieval is optimal, we incorporate this finding into the model using action masking. Our evaluation, based on 170 problem instances with 2-8 buffer lanes and 5-15 colors, shows that our approach reduces color changes compared to existing methods by considerable margins depending on the problem size. Furthermore, we demonstrate the robustness of our approach towards different buffer sizes and imbalanced color distributions. + oai:arXiv.org:2504.02644v2 + cs.LG + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + 10.1016/j.ejor.2025.12.017 + Mirko Stappert, Bernhard Lutz, Janis Brammer, Dirk Neumann + + + Constant-Factor Algorithms for Revenue Management with Consecutive Stays + https://arxiv.org/abs/2506.00909 + arXiv:2506.00909v2 Announce Type: replace-cross +Abstract: We study network revenue management problems motivated by applications such as railway ticket sales and hotel room bookings. Requests, each requiring a resource for a consecutive stay, arrive sequentially with known arrival probabilities. We investigate two scenarios: the accept-or-reject scenario, where a request can be fulfilled by assigning any available resource; and the BAM-based scenario, which generalizes the former by incorporating customer preferences through the basic attraction model (BAM), allowing the platform to offer an assortment of available resources from which the customer may choose. We develop polynomial-time policies and evaluate their performance using approximation ratios, defined as the ratio between the expected revenue of our policy and that of the optimal online algorithm. When each arrival has a fixed request type (e.g., the interval of the stay is fixed), we establish constant-factor guarantees: a ratio of 1 - 1/e for the accept-or-reject scenario and 0.25 for the BAM-based scenario. We further extend these results to the case where the request type is random (e.g., the interval of the stay is random). In this setting, the approximation ratios incur an additional multiplicative factor of 1 - 1/e, resulting in guarantees of at least 0.399 for the accept-or-reject scenario and 0.156 for the BAM-based scenario. These constant-factor guarantees stand in sharp contrast to the prior nonconstant competitive ratios that are benchmarked against the offline optimum. + oai:arXiv.org:2506.00909v2 + econ.TH + cs.DS + math.OC + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Ming Hu, Tongwen Wu + + + TTrace: Lightweight Error Checking and Diagnosis for Distributed Training + https://arxiv.org/abs/2506.09280 + arXiv:2506.09280v2 Announce Type: replace-cross +Abstract: Distributed training is essential for scaling the training of large neural network models, such as large language models (LLMs), across thousands of GPUs. However, the complexity of distributed training programs makes them particularly prone to silent bugs, which do not produce explicit error signals but lead to incorrect training outcomes. Effectively detecting and localizing such silent bugs in distributed training is challenging. Common debugging practices based on monitoring training loss or gradient norm curves are indirect, inefficient, and provide no way to localize bugs. To address those challenges, we design and implement TTrace, the first systematic differential testing system for detecting and localizing silent bugs in distributed training. TTrace aligns intermediate tensors from distributed training with those from a trusted reference implementation. To properly compare the floating-point values in the corresponding tensors, we propose a novel mathematical analysis that provides a guideline for setting tolerances, enabling TTrace to distinguish bug-induced errors from numerical errors. Experimental results demonstrate that TTrace effectively detects 11 existing bugs and 3 new bugs in the widely used Megatron-LM framework, while requiring fewer than 10 lines of code changes. TTrace is effective in various training recipes, including low-precision recipes involving BF16 and FP8. Notably, a popular open-source training framework has already adopted the method proposed by TTrace in its development workflow. + oai:arXiv.org:2506.09280v2 + cs.DC + cs.LG + cs.NA + math.NA + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Haitian Jiang, Shaowei Zhu, Zhen Zhang, Zhenyu Song, Xinwei Fu, Zhen Jia, Yida Wang, Jinyang Li + + + Constructive l2-Discrepancy Minimization with Additive Deviations + https://arxiv.org/abs/2508.21423 + arXiv:2508.21423v3 Announce Type: replace-cross +Abstract: The \emph{signed series} problem in the $\ell_2$ norm asks, given set of vectors $v_1,\ldots,v_n\in \mathbf{R}^d$ having at most unit $\ell_2$ norm, does there always exist a series $(\varepsilon_i)_{i\in [n]}$ of $\pm 1$ signs such that for all $i\in [n]$, $\max_{i\in [n]} \|\sum_{j=1}^i \varepsilon_i v_i\|_2 = O(\sqrt{d})$. A result of Banaszczyk [2012, \emph{Rand. Struct. Alg.}] states that there exist signs $\varepsilon_i\in \{-1,1\},\; i\in [n]$ such that $\max_{i\in [n]} \|\sum_{j=1}^i \varepsilon_i v_i\|_2 = O(\sqrt{d+\log n})$. The best constructive bound known so far is of $O(\sqrt{d\log n})$, by Bansal and Garg [2017, \emph{STOC.}, 2019, \emph{SIAM J. Comput.}]. We give a polynomial-time randomized algorithm to find signs $x(i) \in \{-1,1\},\; i\in [n]$ such that \[ \max_{i\in [n]} \|\sum_{j=1}^i x(i)v_i\|_2 = O(\sqrt{d + \log^2 n}) = O(\sqrt{d}+\log n).\] By the constructive reduction of Harvey and Samadi [\emph{COLT}, 2014], this also yields a constructive bound of $O(\sqrt{d}+\log n)$ for the Steinitz problem in the $\ell_2$-norm. Thus, we algorithmically achieve Banaszczyk's bounds for both problems when $d \geq \log^2n$, which also matches the conjectured bounds. Our algorithm is based on the framework on Bansal and Garg, together with a new analysis involving $(i)$ additional linear and spectral orthogonality constraints during the construction of the covariance matrix of the random walk steps, which allow us to control the quadratic variation in the linear as well as the quadratic components of the discrepancy increment vector, alongwith $(ii)$ a ``Freedman-like" version of the Hanson-Wright concentration inequality, for filtration-dependent sums of subgaussian chaoses. + oai:arXiv.org:2508.21423v3 + cs.DM + cs.DS + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Kunal Dutta + + + Toward a Complexity Classification of High-Temperature Bosons: Computational Tractability and Power-Law Clustering + https://arxiv.org/abs/2509.25572 + arXiv:2509.25572v2 Announce Type: replace-cross +Abstract: Determining when quantum many-body systems admit simple, efficiently simulable structure is a central problem. High-temperature thermal states are a natural candidate for such simplicity, yet for bosons, the unbounded local Hilbert space and energy invalidate the usual expectation that large $T$ guarantees tractability. Here we investigate the resulting complexity boundary for interacting lattice bosons and show that the repulsive Bose--Hubbard class lies on the ``simple'' side. For a family with long-range hopping decaying as $r^{-\alpha}$, we prove convergence of a controlled cluster expansion, which implies (above an explicit temperature threshold) an efficient classical algorithm to approximate the partition function and a rigorous power-law clustering bound for connected correlations. More broadly, our results provide a first step toward charting complexity boundaries for high-temperature bosons and suggest the repulsive Bose--Hubbard class as a natural candidate cusp. + oai:arXiv.org:2509.25572v2 + quant-ph + cond-mat.stat-mech + math-ph + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Xin-Hai Tong, Tomotaka Kuwahara + + + A new super integrable hierarchy and a generalized super-AKNS hierarchy + https://arxiv.org/abs/2509.25995 + arXiv:2509.25995v2 Announce Type: replace-cross +Abstract: In this paper, we investigate a non-isospectral problem on the loop algebra of the Lie superalgebra osp(1,6), and construct an super integrable system using the supertrace identity. The resulting super integrable system can be reduced to the super-AKNS hierarchy under certain conditions. By reconsidering a new (2 + 1)-dimensional non-isospectral problem with spectral matrices satisfying these conditions, we obtain a (2+1)-dimensional generalization of the superAKNS hierarchy. + oai:arXiv.org:2509.25995v2 + nlin.SI + math.RA + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Yanhui Bi, Bo Yuan, Yuqi Ruan, Tao Zhang + + + Exact State Evolution and Energy Spectrum in Solvable Bosonic Models + https://arxiv.org/abs/2510.20046 + arXiv:2510.20046v2 Announce Type: replace-cross +Abstract: Solvable bosonic models provide a fundamental framework for describing light propagation in nonlinear media, including optical down-conversion processes that generate squeezed states of light and their higher-order generalizations. In quantum optics a central objective is to determine the time evolution of a given initial state. Exact analytic solution to the state-evolution problem is presented, applicable to a broad class of solvable bosonic models and arbitrary initial states. Moreover, the characteristic equation governing the energy spectrum is derived and the eigenstates are found in the form of continued fractions and as the principal minors of the associated Jacobi matrix. The results provide a solid analytical framework for discussion of exactly solvable bosonic models. + oai:arXiv.org:2510.20046v2 + quant-ph + cond-mat.other + math-ph + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.1016/j.aop.2026.170342 + Valery Shchesnovich + + + Cut-free Deductive System for Continuous Intuitionistic Logic + https://arxiv.org/abs/2510.26849 + arXiv:2510.26849v2 Announce Type: replace-cross +Abstract: We introduce and develop propositional continuous intuitionistic logic and propositional continuous affine logic via complete algebraic semantics. Our approach centres on AC-algebras, which are algebras $USC(\mathcal{L})$ of sup-preserving functions from $[0,1]$ to an integral commutative residuated complete lattice $\mathcal{L}$ (in the intuitionistic case, $\mathcal{L}$ is a locale). We give an algebraic axiomatisation of AC-algebras in the language of continuous logic and prove, using the Macneille completion, that every Archimedean model embeds into some AC-algebra. We also show that (i) $USC(\mathcal{L})$ satisfies $v \dot + v = 2v$ exactly when $\mathcal{L}$ is a locale, (ii) involutiveness of negation in $USC(\mathcal{L})$ corresponds to that in $\mathcal{L} $, and that (iii) adding those conditions recovers classical continuous logic. For each variant -affine, intuitionistic, involutive, classical -we provide a sequent style deductive system and prove completeness and cut admissibility. This yields the first sequent style formulation of classical continuous logic enjoying cut admissibility. + oai:arXiv.org:2510.26849v2 + cs.LO + math.LO + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Guillaume Geoffroy (UCBL, ICJ, AGL) + + + Source-Optimal Training is Transfer-Suboptimal + https://arxiv.org/abs/2511.08401 + arXiv:2511.08401v4 Announce Type: replace-cross +Abstract: We prove that training a source model optimally for its own task is generically suboptimal when the objective is downstream transfer. We study the source-side optimization problem in L2-SP ridge regression and show a fundamental mismatch between the source-optimal and transfer-optimal source regularization: outside of a measure-zero set, $\tau_0^* \neq \tau_S^*$. We characterize the transfer-optimal source penalty $\tau_0^*$ as a function of task alignment and identify an alignment-dependent reversal: with imperfect alignment ($0<\rho<1$), transfer benefits from stronger source regularization, while in super-aligned regimes ($\rho>1$), transfer benefits from weaker regularization. Additionally, in isotropic settings, the decision of whether transfer helps is independent of the target sample size and noise, depending only on task alignment and source characteristics. We verify the linear predictions in a synthetic ridge regression experiment, and we present experiments on MNIST, CIFAR-10, and 20 Newsgroups as evidence that the source-optimal versus transfer-optimal mismatch persists in standard nonlinear transfer learning pipelines. + oai:arXiv.org:2511.08401v4 + stat.ML + cs.LG + math.ST + stat.TH + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/publicdomain/zero/1.0/ + C. Evans Hedges + + + Boundary-driven quantum systems near the Zeno limit: steady states and long-time behavior + https://arxiv.org/abs/2512.12825 + arXiv:2512.12825v2 Announce Type: replace-cross +Abstract: We study composite open quantum systems with a finite-dimensional state space ${\mathcal H}_A\otimes {\mathcal H}_B$ governed by a Lindblad equation $\rho'(t) = {\mathcal L}_\gamma \rho(t)$ where ${\mathcal L}_\gamma\rho = -i[H,\rho] + \gamma {\mathcal D} \rho$, and ${\mathcal D}$ is a dissipator ${\mathcal D}_A\otimes I$ acting non-trivially only on part $A$ of the system, which can be thought of as the boundary, and $\gamma$ is a parameter. It is known that the dynamics simplifies for large $\gamma$: after a time of order $\gamma^{-1}$, $\rho(t)$ is well approximated for times small compared to $\gamma^2$ by $\pi_A\otimes R(t)$ where $\pi_A$ is a steady state of ${\mathcal D}_A$, and $R(t)$ is a solution of $\frac{{\rm d}}{{\rm d}t}R(t) = {\mathcal L}_{P,\gamma}R(t)$ where ${\mathcal L}_{P,\gamma} R := -i[H_P,R] + \gamma^{-1} {\mathcal D}_P R$ with $H_P$ being a Hamiltonian on ${\mathcal H}_B$ and ${\mathcal D}_P$ being a Lindblad generator over ${\mathcal H}_B$. We prove this assuming only that ${\mathcal D}_A$ is ergodic and gapped. In order to better control the long time behavior, and study the steady states $\bar\rho_\gamma$, we introduce a third Lindblad generator ${\mathcal D}_P^\sharp$ that does not involve $\gamma$, but still closely related to ${\mathcal L}_\gamma$. We show that if ${\mathcal D}_P^\sharp$ is ergodic and gapped, then so is ${\mathcal L}_\gamma$ for all large $\gamma$, and if $\bar\rho_\gamma$ denotes the unique steady state for ${\mathcal L}_\gamma$, then $\lim_{\gamma\to\infty}\bar\rho_\gamma = \pi_A\otimes \bar R$ where $\bar R$ is the unique steady state for ${\mathcal D}_P^\sharp$. We show that there is a convergent expansion $\bar\rho_\gamma = \pi_A\otimes\bar R +\gamma^{-1} \sum_{k=0}^\infty \gamma^{-k} \bar n_k$ where, defining $\bar n_{-1} := \pi_A\otimes\bar R$, ${\mathcal D} \bar n_k = -i[H,\bar n_{k-1}]$ for all $k\geq 0$. + oai:arXiv.org:2512.12825v2 + quant-ph + math-ph + math.MP + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Eric A. Carlen, David A. Huse, Joel L. Lebowitz + + + When Indemnity Insurance Fails: Parametric Coverage under Binding Budget and Risk Constraints + https://arxiv.org/abs/2512.21973 + arXiv:2512.21973v2 Announce Type: replace-cross +Abstract: In high-risk environments, traditional indemnity insurance is often unaffordable or ineffective, despite its well-known optimality under expected utility. We compare excess-of-loss indemnity insurance with parametric insurance within a common mean-variance framework, allowing for fixed costs, heterogeneous premium loadings, and binding budget constraints. We show that, once these realistic frictions are introduced, parametric insurance can yield higher welfare for risk-averse individuals, even under the same utility objective. The welfare advantage arises precisely when indemnity insurance becomes impractical, and disappears once both contracts are unconstrained. Our results help reconcile classical insurance theory with the growing use of parametric risk transfer in high-risk settings. + oai:arXiv.org:2512.21973v2 + econ.GN + math.OC + q-fin.EC + q-fin.RM + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Benjamin Avanzi, Debbie Kusch Falden, Mogens Steffensen + + + Exact inference via quasi-conjugacy in two-parameter Poisson-Dirichlet hidden Markov models + https://arxiv.org/abs/2512.22098 + arXiv:2512.22098v2 Announce Type: replace-cross +Abstract: We introduce a nonparametric model for time-evolving, unobserved probability distributions from discrete-time data consisting of unlabelled partitions. The latent process is a two-parameter Poisson-Dirichlet diffusion, and observations arise via exchangeable sampling. Applications include social and genetic data where only aggregate clustering summaries are observed. To address the intractable likelihood, we develop a tractable inferential framework that avoids label enumeration and direct simulation of the latent state. We exploit a duality between the diffusion and a pure-death process on partitions, together with coagulation operators that encode the effect of new data. These yield closed-form, recursive updates for forward and backward inference. We compute exact posterior distributions of the latent state at arbitrary times and predictive distributions of future or interpolated partitions. This enables online and offline inference and forecasting with full uncertainty quantification, bypassing MCMC and sequential Monte Carlo. Compared to particle filtering, our method achieves higher accuracy, lower variance, and substantial computational gains. We illustrate the methodology with synthetic experiments and a social network application, recovering interpretable patterns in time-varying heterozygosity. + oai:arXiv.org:2512.22098v2 + stat.ME + math.PR + math.ST + q-bio.PE + stat.CO + stat.TH + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Marco Dalla Pria, Matteo Ruggiero, Dario Span\`o + + + A UCB Bandit Algorithm for General ML-Based Estimators + https://arxiv.org/abs/2601.01061 + arXiv:2601.01061v2 Announce Type: replace-cross +Abstract: We present ML-UCB, a generalized upper confidence bound algorithm that integrates arbitrary machine learning models into multi-armed bandit frameworks. A fundamental challenge in deploying sophisticated ML models for sequential decision-making is the lack of tractable concentration inequalities required for principled exploration. We overcome this limitation by directly modeling the learning curve behavior of the underlying estimator. Specifically, assuming the Mean Squared Error decreases as a power law in the number of training samples, we derive a generalized concentration inequality and prove that ML-UCB achieves sublinear regret. This framework enables the principled integration of any ML model whose learning curve can be empirically characterized, eliminating the need for model-specific theoretical analysis. We validate our approach through experiments on a collaborative filtering recommendation system using online matrix factorization with synthetic data designed to simulate a simplified two-tower model, demonstrating substantial improvements over LinUCB + oai:arXiv.org:2601.01061v2 + cs.LG + cs.AI + math.PR + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Yajing Liu, Erkao Bao, Linqi Song + + + A Novel Multiple Imputation Approach For Parameter Estimation in Observation-Driven Time Series Models With Missing Data + https://arxiv.org/abs/2601.01259 + arXiv:2601.01259v2 Announce Type: replace-cross +Abstract: Handling missing data in time series is a complex problem due to the presence of temporal dependence. General-purpose imputation methods, while widely used, often distort key statistical properties of the data, such as variance and dependence structure, leading to biased estimation and misleading inference. These issues become more pronounced in models that explicitly rely on capturing serial dependence, as standard imputation techniques fail to preserve the underlying dynamics. This paper proposes a novel multiple imputation method specifically designed for parameter estimation in observation-driven models (ODM). The approach takes advantage of the iterative nature of the systematic component in ODM to propagate the dependence structure through missing data, minimizing its impact on estimation. Unlike traditional imputation techniques, the proposed method accommodates continuous, discrete, and mixed-type data while preserving key distributional and dependence properties. We evaluate its performance through Monte Carlo simulations in the context of GARMA models, considering time series with up to 70\% missing data. An application to the proportion of stocked energy stored in South Brazil further demonstrates its practical utility. + oai:arXiv.org:2601.01259v2 + stat.ME + math.ST + stat.TH + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Guilherme Pumi, Taiane Schaedler Prass, Douglas Krauthein Verdum + + + Rethinking Secure Semantic Communications in the Age of Generative and Agentic AI: Threats and Opportunities + https://arxiv.org/abs/2601.01791 + arXiv:2601.01791v2 Announce Type: replace-cross +Abstract: Semantic communication (SemCom) improves communication efficiency by transmitting task-relevant information instead of raw bits and is expected to be a key technology for 6G networks. Recent advances in generative AI (GenAI) further enhance SemCom by enabling robust semantic encoding and decoding under limited channel conditions. However, these efficiency gains also introduce new security and privacy vulnerabilities. Due to the broadcast nature of wireless channels, eavesdroppers can also use powerful GenAI-based semantic decoders to recover private information from intercepted signals. Moreover, rapid advances in agentic AI enable eavesdroppers to perform long-term and adaptive inference through the integration of memory, external knowledge, and reasoning capabilities. This allows eavesdroppers to further infer user private behavior and intent beyond the transmitted content. Motivated by these emerging challenges, this paper comprehensively rethinks the security and privacy of SemCom systems in the age of generative and agentic AI. We first present a systematic taxonomy of eavesdropping threat models in SemCom systems. Then, we provide insights into how GenAI and agentic AI can enhance eavesdropping threats. Meanwhile, we also highlight potential opportunities for leveraging GenAI and agentic AI to design privacy-preserving SemCom systems. + oai:arXiv.org:2601.01791v2 + eess.SP + cs.IT + cs.NI + math.IT + Wed, 07 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by/4.0/ + Shunpu Tang, Yuanyuan Jia, Zijiu Yang, Qianqian Yang, Ruichen Zhang, Jun Du, Jihong Park, Zhiguo Shi, Jiming Chen +