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" @@ -7,13733 +7,4779 @@ http://www.rssboard.org/rss-specification en-us - Wed, 21 Jan 2026 05:00:21 +0000 + Fri, 23 Jan 2026 05:00:01 +0000 rss-help@arxiv.org - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 Saturday Sunday - Robustness of the Frank-Wolfe Method under Inexact Oracles and the Cost of Linear Minimization - https://arxiv.org/abs/2601.11548 - arXiv:2601.11548v1 Announce Type: new -Abstract: We investigate the robustness of the Frank-Wolfe method when gradients are computed inexactly and examine the relative computational cost of the linear minimization oracle (LMO) versus projection. For smooth nonconvex functions, we establish a convergence guarantee of order $\mathcal{O}(1/\sqrt{k}+\delta)$ for Frank-Wolfe with a $\delta$--oracle. Our results strengthen previous analyses for convex objectives and show that the oracle errors do not accumulate asymptotically. We further prove that approximate projections cannot be computationally cheaper than accurate LMOs, thus extending to the case of inexact projections. These findings reinforce the robustness and efficiency of the Frank-Wolfe framework. - oai:arXiv.org:2601.11548v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Tao Hu - - - Minimal Perimeter Triangle in Nonconvex Quadrilateral:Generalized Fagnano Problem - https://arxiv.org/abs/2601.11552 - arXiv:2601.11552v1 Announce Type: new -Abstract: In 1775, Fagnano introduced the following geometric optimization problem: inscribe a triangle of minimal perimeter in a given acute-angled triangle. A widely accessible solution is provided by the Hungarian mathematician L. Fejer in 1900. This paper presents a specific generalization of the classical Fagnano problem, which states that given a nonconvex quadrilateral (having one reflex angle and others are acute angles), find a triangle of minimal perimeter with exactly one vertex on each of the sides that do not form reflex angle, and the third vertex lies on either of the sides forming the reflex angle. We provide its geometric solution. Additionally, we establish an upper bound for the classic Fagnano problem, demonstrating that the minimal perimeter of the triangle inscribed in a given acute-angled triangle cannot exceed twice the length of any of its sides. - oai:arXiv.org:2601.11552v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + A new iterative three-point method for solving systems of nonlinear equations + https://arxiv.org/abs/2601.15323 + arXiv:2601.15323v1 Announce Type: new +Abstract: A three-point iterative method for solving scalar non-linear equations was selected and then adapted to solve systems of non-linear equations. Subsequently, by applying Taylor's theorem to functions of $\R^{n}$ in $\R^{n}$, it is shown that the new method also has a sixth order of convergence. It is confirmed that the theoretical order of convergence coincides with the computational order of convergence by the numerical solution of two problems. Finally, its computational efficiency is calculated and subsequently compared with that of other three-point methods of fifth and sixth order convergence that also solve systems of non-linear equations. + oai:arXiv.org:2601.15323v1 + math.GM + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Triloki Nath, Manohar Choudhary + Carlos E. Cadenas R., Yorman J. Mendoza N - A Generalized Waist Problem: Optimality Condition and Algorithm - https://arxiv.org/abs/2601.11554 - arXiv:2601.11554v1 Announce Type: new -Abstract: Many years ago John Tyrell a lecturer at King's college London challenged his Ph.D. students with the following puzzle: show that there is a unique triangle of minimal perimeter with exactly one vertex to lie on one of three given lines, pairwise disjoint and not all parallel in the space. The problem in literature is known as the waist problem, and only convexity rescued in this case. Motivated by this we generalize it by replacing lines with a number of convex sets in the Euclidean space and ask to minimize the sum of distances connecting the sets by means of closed polygonal curve. This generalized problem significantly broadens its geometric and practical scope in view of modern convex analysis. We establish the existence of solutions and prove its uniqueness under the condition that at least one of the convex sets is strictly convex and all are in general position: each set can be separated by convex hull of others. A complete set of necessary and sufficient optimality conditions is derived, and their geometric interpretations are explored to link these conditions with classical principles such as the reflection law of light. To address this problem computationally, we develop a projected subgradient descent method and prove its convergence. Our algorithm is supported by detailed numerical experiments, particularly in cases involving discs and spheres. Additionally, we present a real-world analogy of the problem in the form of inter-island connectivity, illustrating its practical relevance. This work not only advances the theory of geometric optimization but also contributes effective methods and insights applicable to facility location, network design, robotics., computational geometry, and spatial planning. - oai:arXiv.org:2601.11554v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + G\"unter Hellwig (1926-2004) -- in memoriam + https://arxiv.org/abs/2601.15329 + arXiv:2601.15329v1 Announce Type: new +Abstract: G\"unter Hellwig was the author of influential textbooks on PDEs and differential operators of mathematical physics, an enthusiastic and inspiring teacher to generations of engineers, organiser of PDE conferences at Oberwolfach and a pioneer in index theory. + oai:arXiv.org:2601.15329v1 + math.HO + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Triloki Nath, Manohar Choudhary, Ram K. Pandey + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Hubert Kalf - A Generalized $(k,m)$ Heron Problem:Optimality Conditions and Algorithm - https://arxiv.org/abs/2601.11555 - arXiv:2601.11555v1 Announce Type: new -Abstract: This paper presents a new extension of the classical Heron problem, termed the generalized $(k,m)$-Heron problem, which seeks an optimal configuration among $k$ feasible and $m$ target non-empty closed convex sets in $\mathbb{R}^n$. The problem is formulated as finding a point in each set that minimizes the pairwise distances from the points in the $k$-feasible sets to the points in the $m$-target sets. This formulation leads to a convex optimization framework that generalizes several well-known geometric distance problems. Using tools from convex analysis, we establish fundamental results on existence, uniqueness, and first-order optimality conditions through subdifferential calculus and normal cone theory. Building on these insights, a Projected Subgradient Algorithm (PSA) is proposed for numerical solution, and its convergence is rigorously proved under a diminishing step-size rule. Numerical experiments in $\mathbb{R}^2$ and $\mathbb{R}^3$ illustrate the algorithm's stability, geometric accuracy, and computational efficiency. Overall, this work provides a comprehensive analytical and algorithmic framework for multi-set geometric optimization with promising implications for location science, robotics, and computational geometry. - oai:arXiv.org:2601.11555v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + Conjectures on Sums of Consecutive Primes + https://arxiv.org/abs/2601.15346 + arXiv:2601.15346v1 Announce Type: new +Abstract: We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd integer. We first formulate an existence conjecture asserting that, for every prime number $p_n$, there exists at least one odd length $k \ge 3$ such that $S_k(p_n)$ is itself a prime number. An exhaustive computational verification covering the first one million prime numbers revealed no counterexamples. We then propose a strengthened conjecture according to which, for every prime number $p_n$, there exist infinitely many odd lengths $k$ such that $S_k(p_n)$ is prime. This strong version is supported by a probabilistic heuristic showing that the series of the corresponding primality probabilities diverges, suggesting that the phenomenon is not exceptional but recurrent. We also analyze the possible modular obstructions, showing that they are local in nature and cannot persist when the length $k$ varies among odd integers. A Diophantine interpretation of the problem is proposed, together with a conceptual comparison with the generalized Goldbach conjecture. Finally, we discuss the role of the Generalized Riemann Hypothesis (GRH) in controlling the distribution of the sums under consideration. These structural, modular, Diophantine, and probabilistic (heuristic) arguments support both conjectures and formalize heuristic theorems of Cram\'er, GRH, and Hardy--Littlewood type explaining the expected absence of counterexamples. + oai:arXiv.org:2601.15346v1 + math.GM + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Triloki Nath, Manohar Choudhary, Ram K. Pandey + Edwige Tolla - The global well-posedness for master equations of mean field games of controls - https://arxiv.org/abs/2601.11588 - arXiv:2601.11588v1 Announce Type: new -Abstract: In this manuscript, we establish the global well-posedness for master equations of mean field games of controls, where the interaction is through the joint law of the state and control. Our results are proved under two different conditions: the Lasry-Lions monotonicity and the displacement $\lambda$-monotonicity, both considered in their integral forms. We provide a detailed analysis of both the differential and integral versions of these monotonicity conditions for the corresponding nonseparable Hamiltonian and examine their relation. The proof of global well-posedness relies on the propagation of these monotonicity conditions in their integral forms and a priori uniform Lipschitz continuity of the solution with respect to the measure variable. - oai:arXiv.org:2601.11588v1 - math.PR - math.AP - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + Maximal Green Sequences for Cluster Algebras Associated to Closed Orbifolds + https://arxiv.org/abs/2601.15389 + arXiv:2601.15389v1 Announce Type: new +Abstract: It is known that the existence of a maximal green sequence for a quiver associated to surfaces is equivalent to the equality of the cluster algebra and upper cluster algebra generated by the quiver. This paper makes the first steps in investigating this behavior in the generalised case of cluster algebras from orbifolds; determining when such surfaces admit a diagram with a maximal green sequence. Specifically, we will provide a triangulation for the orientable surfaces of genus $n$ with an arbitrary number of orbifold points and arbitrary number of punctures, determine when it has a maximal green sequence, and construct one if it exists. + oai:arXiv.org:2601.15389v1 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Shuhui Liu, Xintian Liu, Chenchen Mou, Defeng Sun + Hin Chung Henry Tsang - Poisson semigroup and the Gruet formula for the heat kernels on spaces of constant curvature - https://arxiv.org/abs/2601.11596 - arXiv:2601.11596v1 Announce Type: new -Abstract: This paper is concerned with the Poisson and heat equations on spaces of constant curvature. More explicitly we provide new methods for obtaining old and new explicit formulas for the Poisson and heat semigroups on the Euclidean, spherical and hyperbolic spaces $\R^n$, $\S^n$ and $\H^n$ . We obtain the Gruet formula for the heat kernels in Euclidean and spherical spaces $\R^n$ and $\S^n$, which are new and we provide a new elementary method to derive the classical Gruet formula Gruet\cite{Gruet} for the kernel of the heat semigroup on the hyperbolic space $\H^n$. - oai:arXiv.org:2601.11596v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Understanding FISTA's weak convergence: A step-by-step introduction to the 2025 milestone + https://arxiv.org/abs/2601.15398 + arXiv:2601.15398v1 Announce Type: new +Abstract: Beck and Teboulle's FISTA for finding the minimizer of the sum of two convex functions is one of the most important algorithms of the past decades. While function value convergence of the iterates was known, the actual convergence of the iterates remained elusive until October 2025 when Jang and Ryu, as well as Bo\c{t}, Fadili, and Nguyen proved weak convergence. + In this paper, we provide a gentle self-contained introduction to the proof of their remarkable result. + oai:arXiv.org:2601.15398v1 + math.OC + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mohamed Vall Ould Moustapha + http://creativecommons.org/licenses/by/4.0/ + Heinz H. Bauschke, Walaa M. Moursi - Boundary Delocalization and Spectral Packets for Dirichlet Eigenfunctions - https://arxiv.org/abs/2601.11605 - arXiv:2601.11605v1 Announce Type: new -Abstract: We establish a boundary delocalization principle for high-frequency Dirichlet eigenfunctions on smooth strictly convex domains. The main result excludes persistent boundary concentration at the level of individual eigenmodes when compared to short spectral packets of sublinear length. Quantitatively, we compare boundary energies of single eigenfunctions to packet sums over frequency windows of size N_k = o(k), without asserting any asymptotic gain in magnitude. The main mode-to-packet estimate relies only on the Rellich identity. For the multi-mode bias exclusion we additionally use the boundary local Weyl law to obtain a packet zero-mean cancellation estimate. This mode-to-packet comparison is independent of eigenvalue monotonicity and is stable under eigenvalue crossings. - oai:arXiv.org:2601.11605v1 - math.SP - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + The Geometry of Rough Path Space + https://arxiv.org/abs/2601.15402 + arXiv:2601.15402v1 Announce Type: new +Abstract: We describe $H^p(V)$, a subset of $p$-rough path space $\Omega_p(V)$ which is a vector space under an addition operation $\boxplus$ and a scalar multiplication $\odot$. We show that the domain of $\boxplus$ can be extended to $\Omega_p(V)\times H^p(V)$, allowing any $p$-rough path $X$ to be additively perturbed by an $H\in H^p(V)$. We prove associativity $(X\boxplus H)\boxplus \tilde H = X\boxplus (H\boxplus \tilde H)$ and trivial kernel $X\boxplus H = X \Leftrightarrow H = 1$, where $1$ is the additive zero in $(H^p(V),\boxplus,\odot)$. Finally, we show that enlarging $H^p(V)$ to almost rough paths $H^{am,p}(V)$ does not enlarge the set of displacements of a given $X$, i.e. $\{X\boxplus H: H\in H^p(V)\}=\{X\boxplus H: H\in H^{am,p}(V)\}$. + oai:arXiv.org:2601.15402v1 + math.CA + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Anton Alexa - - - Concatenated Matrix SVD: Compression Bounds, Incremental Approximation, and Error-Constrained Clustering - https://arxiv.org/abs/2601.11626 - arXiv:2601.11626v1 Announce Type: new -Abstract: Large collections of matrices arise throughout modern machine learning, signal processing, and scientific computing, where they are commonly compressed by concatenation followed by truncated singular value decomposition (SVD). This strategy enables parameter sharing and efficient reconstruction and has been widely adopted across domains ranging from multi-view learning and signal processing to neural network compression. However, it leaves a fundamental question unanswered: which matrices can be safely concatenated and compressed together under explicit reconstruction error constraints? Existing approaches rely on heuristic or architecture-specific grouping and provide no principled guarantees on the resulting SVD approximation error. In the present work, we introduce a theory-driven framework for compression-aware clustering of matrices under SVD compression constraints. Our analysis establishes new spectral bounds for horizontally concatenated matrices, deriving global upper bounds on the optimal rank-$r$ SVD reconstruction error from lower bounds on singular value growth. The first bound follows from Weyl-type monotonicity under blockwise extensions, while the second leverages singular values of incremental residuals to yield tighter, per-block guarantees. We further develop an efficient approximate estimator based on incremental truncated SVD that tracks dominant singular values without forming the full concatenated matrix. Therefore, we propose three clustering algorithms that merge matrices only when their predicted joint SVD compression error remains below a user-specified threshold. The algorithms span a trade-off between speed, provable accuracy, and scalability, enabling compression-aware clustering with explicit error control. Code is available online. - oai:arXiv.org:2601.11626v1 - math.NA - cs.LG - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Maksym Shamrai + Martin Geller, Terry Lyons - Qualitative analysis and numerical investigations of time-fractional Zika virus model arising in population dynamics - https://arxiv.org/abs/2601.11636 - arXiv:2601.11636v1 Announce Type: new -Abstract: Epidemic models play a crucial role in population dynamics, offering valuable insights into disease transmission while aiding in epidemic prediction and control. In this paper, we analyze the mathematical model of the time-fractional Zika virus transmission for human and mosquito populations. The fractional derivative is considered in the Caputo sense of order $\alpha\in(0,1).$ We begin by conducting a qualitative analysis using the stability theory of differential equations. The existence and uniqueness of the solution are established, and the model's stability is examined through Hyers-Ulam stability analysis. Furthermore, an efficient difference scheme utilizing the standard L1 technique is developed to simulate the model and analyze the solution's behavior under key parameters. The resulting nonlinear algebraic system is solved using the Newton-Raphson method. Finally, illustrative examples are presented to validate the theoretical findings. Graphical results indicate that the fractional model provides deeper insights and a better understanding of disease dynamics. These findings aid in controlling the virus through contact precautions and recommended therapies while also helping to predict its future spread. - oai:arXiv.org:2601.11636v1 - math.DS - cs.NA - math.NA - q-bio.PE - Wed, 21 Jan 2026 00:00:00 -0500 + F-Purity of Binomial Edge Ideals + https://arxiv.org/abs/2601.15403 + arXiv:2601.15403v1 Announce Type: new +Abstract: In 2012, K. Matsuda introduced the class of weakly closed graphs and investigated when binomial edge ideals are F-pure. He proved that weakly closed binomial edge ideals are F-pure whenever the base field has positive characteristic. He conjectured that: (i) when the base field has characteristic two, every F-pure binomial edge ideal comes from a weakly closed graph; and (ii) that every binomial edge ideal is F-pure provided that the characteristic of the residue field is sufficiently large. + In this paper, we resolve both of Matsuda's conjectures. We confirm Matsuda's first conjecture, showing that the binomial edge ideal of a graph defines an F-pure quotient in characteristic 2 if and only if the graph is weakly closed. We also show that Matsuda's second conjecture is false in a very strong way by showing that graphs containing asteroidal triples, such as the net, define non-F-pure binomial edge ideals in any positive characteristic. Our results yield a complete classification of F-pure binomial edge ideals of chordal graphs as well as large families of standard graded algebras that are F-injective but neither F-pure nor F-rational in all characteristics. + oai:arXiv.org:2601.15403v1 + math.AC + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Gaurav Saini, Bappa Ghosh, Sunita Chand + Adam LaClair, Jason McCullough - A note on Reeb spaces of some explicit real analytic functions - https://arxiv.org/abs/2601.11648 - arXiv:2601.11648v1 Announce Type: new -Abstract: Reeb spaces of smooth functions are fundamental and strong tools in understanding manifolds via smooth functions with mild critical points. They are defined as the natural spaces of all connected components of level sets. They are also important objects in related studies. Realization of graphs as Reeb spaces of smooth functions of certain nice classes is of such studies. - In this paper, we present Reeb spaces of explicit real analytic functions which are not finite graphs. Related problems were started by Sharko, in 2006, who has studied smooth functions with critical points represented by certain elementary polynomials, and followed by a study of Masumoto and Saeki, which is on smooth functions on closed surfaces under an extended situation, and a study of Michalak, which is on Morse functions on closed manifolds. The author has contributed to this by respecting topologies of level sets, and real algebraic construction. - oai:arXiv.org:2601.11648v1 - math.GM - Wed, 21 Jan 2026 00:00:00 -0500 + Partially Polarized Polar Codes: A New Design for 6G Control Channels + https://arxiv.org/abs/2601.15404 + arXiv:2601.15404v1 Announce Type: new +Abstract: We introduce a new family of polar-like codes, called Partially Polarized Polar (PPP) codes. PPP codes are constructed from conventional polar codes by selectively pruning polarization kernels, thereby modifying the synthesized bit-channel capacities to ensure a guaranteed number of non-frozen bits available early in decoding. These early-access information bits enable more effective early termination, which is particularly valuable for blind decoding in downlink control channels, where user equipment (UE) must process multiple candidates, many of which carry no valid control information. Our results show that PPP codes offer substantial performance gains over conventional polar codes, particularly at larger block lengths where hardware limitations restrict straightforward scaling. Compared with existing methods such as aggregation or segmentation, PPP codes achieve higher efficiency without the need for additional hardware support. Finally, we propose several frozen-bitmap design strategies tailored to PPP codes. + oai:arXiv.org:2601.15404v1 + cs.IT + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Naoki Kitazawa + Arman Fazeli, Mohammad M. Mansour, Ziyuan Zhu, Louay Jalloul - Dirichlet Extremals for Discrete Plateau Problems in GT-Bezier Spaces via PSO - https://arxiv.org/abs/2601.11677 - arXiv:2601.11677v1 Announce Type: new -Abstract: We study a discrete analogue of the parametric Plateau problem in a non-polynomial tensor-product surface spaces generated by the generalized trigonometric (GT)--B\'ezier basis. Boundary interpolation is imposed by prescribing the boundary rows and columns of the control net, while the interior control points are selected by a Dirichlet principle: for each admissible choice of B\'ezier basis shape parameters, we compute the unique Dirichlet-energy extremal within the corresponding GT--B\'ezier patch space, which yields a parameter-dependent symmetric linear system for the interior control net under standard nondegeneracy assumptions. The remaining design freedom is thereby reduced to a four-parameter optimization problem, which we solve by particle swarm optimization. Numerical experiments show that the resulting two-level procedure consistently decreases the Dirichlet energy and, in our tests, often reduces the realized surface area relative to classical Bernstein--B\'ezier Dirichlet patches and representative quasi-harmonic and bending-energy constructions under identical boundary control data. We further adapt the same Dirichlet-extremal methodology to a hybrid tensor-product/bilinear Coons framework, obtaining minimality-biased TB--Coons patches from sparse boundary specifications. - oai:arXiv.org:2601.11677v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 + Cancellation elements in multiplicative lattices + https://arxiv.org/abs/2601.15405 + arXiv:2601.15405v1 Announce Type: new +Abstract: We extend to multiplicative lattices a theorem of Anderson and Roitman characterizing the cancellation ideals of a commutative ring. + oai:arXiv.org:2601.15405v1 + math.AC + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Muhammad Ammad, Md Yushalify Misro, Samia Bibi, Ahmad Ramli + Tiberiu Dumitrescu - Five Circles: Real Analysis Theorems equivalent to Completeness - https://arxiv.org/abs/2601.11681 - arXiv:2601.11681v1 Announce Type: new -Abstract: This is an exposition of the work of O. Riemenschneider about five ''circles'' of implications relating real analysis theorems each equivalent to the Dedekind completeness of the real field. These circles cover five elements of real function theory: convergence, connectedness, differentiability, compactness and integration. - oai:arXiv.org:2601.11681v1 - math.HO - Wed, 21 Jan 2026 00:00:00 -0500 + On the diagonal of low bidegree hypersurfaces + https://arxiv.org/abs/2601.15409 + arXiv:2601.15409v1 Announce Type: new +Abstract: We study the existence of a decomposition of the diagonal for bidegree hypersurfaces in a product of projective spaces. Using a cycle theoretic degeneration technique due to Lange, Pavic and Schreieder, we develop an inductive procedure that allows one to raise the degree and dimension starting from the quadric surface bundle of Hassett, Pirutka and Tschinkel. Furthermore, we are able to raise the dimension without raising the degree in a special case, showing that a very general $(3,2)$ complete intersection in $\mathbb P^4\times \mathbb P^3$ does not admit a decomposition of the diagonal. As a corollary of these theorems, we show that in a certain range, bidegree hypersurfaces which were previously only known to be stably irrational over fields of characteristic zero by results of Moe, Nicaise and Ottem, are not retract rational over fields of characteristic different from two. + oai:arXiv.org:2601.15409v1 + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rafael Cantuba + Morten L\"uders, Elia Fiammengo - Age-Based Scheduling for a Memory-Constrained Quantum Switch - https://arxiv.org/abs/2601.11698 - arXiv:2601.11698v1 Announce Type: new -Abstract: In a time-slotted system, we study the problem of scheduling multipartite entanglement requests in a quantum switch with a finite number of quantum memory registers. Specifically, we consider probabilistic link-level entanglement (LLE) generation for each user, probabilistic entanglement swapping, and one-slot decoherence. To evaluate the performance of the proposed scheduling policies, we introduce a novel age-based metric, coined age of entanglement establishment (AoEE). We consider two families of low-complexity policies for which we obtain closed-form expressions for their corresponding AoEE performance. Optimizing over each family, we obtain two policies. Further, we propose one more low-complexity policy and provide its performance guarantee. Finally, we numerically compare the performance of the proposed policies. - oai:arXiv.org:2601.11698v1 - cs.IT - cs.NI - cs.SY - eess.SY - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + What is... hierarchical hyperbolicity? + https://arxiv.org/abs/2601.15410 + arXiv:2601.15410v1 Announce Type: new +Abstract: This is a very short introduction to hierarchically hyperbolic spaces and groups. It is aimed at non-experts, including anyone who may encounter a group with some similarities to mapping class groups. + oai:arXiv.org:2601.15410v1 + math.GR + math.GT + math.HO + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Stavros Mitrolaris, Subhankar Banerjee, Sennur Ulukus + http://creativecommons.org/licenses/by/4.0/ + Alex Wright - Stability and Accuracy Trade-offs in Statistical Estimation - https://arxiv.org/abs/2601.11701 - arXiv:2601.11701v1 Announce Type: new -Abstract: Algorithmic stability is a central concept in statistics and learning theory that measures how sensitive an algorithm's output is to small changes in the training data. Stability plays a crucial role in understanding generalization, robustness, and replicability, and a variety of stability notions have been proposed in different learning settings. However, while stability entails desirable properties, it is typically not sufficient on its own for statistical learning -- and indeed, it may be at odds with accuracy, since an algorithm that always outputs a constant function is perfectly stable but statistically meaningless. Thus, it is essential to understand the potential statistical cost of stability. In this work, we address this question by adopting a statistical decision-theoretic perspective, treating stability as a constraint in estimation. Focusing on two representative notions-worst-case stability and average-case stability-we first establish general lower bounds on the achievable estimation accuracy under each type of stability constraint. We then develop optimal stable estimators for four canonical estimation problems, including several mean estimation and regression settings. Together, these results characterize the optimal trade-offs between stability and accuracy across these tasks. Our findings formalize the intuition that average-case stability imposes a qualitatively weaker restriction than worst-case stability, and they further reveal that the gap between these two can vary substantially across different estimation problems. - oai:arXiv.org:2601.11701v1 - math.ST - stat.ML - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 + Asymptotic behaviour of coupled random dynamical systems with multiscale aspects + https://arxiv.org/abs/2601.15411 + arXiv:2601.15411v1 Announce Type: new +Abstract: We examine a class of stochastic differential inclusions involving multiscale effects designed to solve a class of generalized variational inequalities. This class of problems contains constrained convex non-smooth optimization problems, constrained saddle-point problems and various equilibrium problems in economics and engineering. In order to respect constraints we adopt a penalty approach, introducing an explicit time-dependency into the evolution system. The resulting dynamics are described in terms of a non-autonomous stochastic evolution equation governed by maximally monotone operators in the drift and perturbed by a Brownian motion. We study the asymptotic behavior, as well as finite time convergence rates in terms of gap functions. The condition we use to prove convergence involves a Legendre transform of the function describing the set C, a condition first used by Attouch and Czarnecki (J. Differ. Equations, Vol. 248, Issue 6, 2010) in the context of deterministic evolution equations. We also establish a large deviations principle showing that individual trajectories exhibit exponential concentration around the solution set. Finally we show how our continuous-time approach relates to penalty-regulated algorithms of forward-backward type after performing a suitable Euler-Maruyama discretisation. + oai:arXiv.org:2601.15411v1 + math.OC + math.DS + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Abhinav Chakraborty, Yuetian Luo, Rina Foygel Barber + D. Russell Luke, Johannes-Carl Schnebel, Mathias Staudigl, Juan Peypouquet, Siqi Qu - Detecting Mutual Excitations in Non-Stationary Hawkes Processes - https://arxiv.org/abs/2601.11717 - arXiv:2601.11717v1 Announce Type: new -Abstract: We consider the problem of learning the network of mutual excitations (i.e., the dependency graph) in a non-stationary, multivariate Hawkes process. We consider a general setting where baseline rates at each node are time-varying and delay kernels are not shift-invariant. Our main results show that if the dependency graph of an $n$-variate Hawkes process is sparse (i.e., it has a maximum degree that is bounded with respect to $n$), our algorithm accurately reconstructs it from data after observing the Hawkes process for $T = \mathrm{polylog}(n)$ time, with high probability. Our algorithm is computationally efficient, and provably succeeds in learning dependencies even if only a subset of time series are observed and event times are not precisely known. - oai:arXiv.org:2601.11717v1 - math.ST - math.PR - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 + Counting point configurations in projective space + https://arxiv.org/abs/2601.15421 + arXiv:2601.15421v1 Announce Type: new +Abstract: We investigate the enumerative geometry of point configurations in projective space. We define "projective configuration counts": these enumerate configurations of points in projective space such that certain specified subsets are in fixed relative positions. The $\mathbb{P}^1$ case recovers cross-ratio degrees, which arise naturally in numerous contexts. We establish two main results. The first is a combinatorial upper bound given by the number of weighted transversals of a bipartite graph. The second is a recursion that relates counts associated to projective spaces of different dimensions, by projecting away from a given point. Key inputs include the Gelfand-MacPherson correspondence, the Jacobi-Trudi and Thom-Porteous formulae, and the notion of surplus from matching theory of bipartite graphs. + oai:arXiv.org:2601.15421v1 + math.AG + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Elchanan Mossel, Anirudh Sridhar + Alex Fink, Navid Nabijou, Rob Silversmith - Asymptotically Optimal Tests for One- and Two-Sample Problems - https://arxiv.org/abs/2601.11727 - arXiv:2601.11727v1 Announce Type: new -Abstract: In this work, we revisit the one- and two-sample testing problems: binary hypothesis testing in which one or both distributions are unknown. For the one-sample test, we provide a more streamlined proof of the asymptotic optimality of Hoeffding's likelihood ratio test, which is equivalent to the threshold test of the relative entropy between the empirical distribution and the nominal distribution. The new proof offers an intuitive interpretation and naturally extends to the two-sample test where we show that a similar form of Hoeffding's test, namely a threshold test of the relative entropy between the two empirical distributions is also asymptotically optimal. A strong converse for the two-sample test is also obtained. - oai:arXiv.org:2601.11727v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + Isotropic meta Kazhdan--Lusztig combinatorics I: Ext-quiver presentation for the Hecke category + https://arxiv.org/abs/2601.15426 + arXiv:2601.15426v1 Announce Type: new +Abstract: We provide an ${\rm Ext}$-quiver and relations presentation for the basic algebra of the anti-spherical Hecke categories of isotropic Grassmannians, $H_{(D_n, A_{n-1})}$, in terms of cup-cap meta Kazhdan--Lusztig combinatorics and Temperley--Lieb diagrammatics. + oai:arXiv.org:2601.15426v1 + math.RT + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Arick Grootveld, Biao Chen, Venkata Gandikota + Ben Mills - Positive energy-momentum theorems for asymptotically AdS spin initial data sets with charge - https://arxiv.org/abs/2601.11728 - arXiv:2601.11728v1 Announce Type: new -Abstract: For complete spin initial data sets with an asymptotically anti--de Sitter end, we introduce a charged energy--momentum defined as a linear functional arising from the Einstein--Maxwell constraints. Under a dominant energy condition adapted to the presence of a negative cosmological constant, we establish positive energy--momentum theorems, showing in particular that this functional is non--negative on a natural real cone. We place particular emphasis on the case where the manifold carries a compact inner boundary. In the time--symmetric setting, this yields a mass--charge inequality for asymptotically hyperbolic manifolds with charge. - oai:arXiv.org:2601.11728v1 + A numerical characterization of Dunkl systems + https://arxiv.org/abs/2601.15430 + arXiv:2601.15430v1 Announce Type: new +Abstract: We give a numerical characterization of weighted hyperplane arrangements arising from Dunkl systems. + oai:arXiv.org:2601.15430v1 math.DG - gr-qc - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 + math.AG + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Simon Raulot + Martin de Borbon, Dmitri Panov - Construction of a Gibbs measure for the zonal Dirac equation - https://arxiv.org/abs/2601.11730 - arXiv:2601.11730v1 Announce Type: new -Abstract: We propose a framework to construct Gibbs measures for the Dirac equation. We consider the Dirac equation on the sphere with a "Hartree-type" nonlinearity. We consider a zonal model, that is the analog of a spherically symmetric model but on the sphere. We build a Gibbs measure for this model. With a compactness argument, we prove the existence of a random variable that is a weak solution to the Dirac equation and whose law is the Gibbs measure at all times. - oai:arXiv.org:2601.11730v1 - math.AP + Discrete log-concavity and threshold phenomena for atomic measures + https://arxiv.org/abs/2601.15444 + arXiv:2601.15444v1 Announce Type: new +Abstract: We investigate threshold phenomena for random polytopes $K_N=\conv\{X_1,\dots,X_N\}$ generated by i.i.d.\ samples from an atomic law $\mu$. We identify and provide a missing justification in the discrete-hypercube threshold argument of Dyer--F\"uredi--McDiarmid, where the supporting half-space estimate is derived via a smooth (gradient/uniqueness) step that can fail at boundary contact points. We then compare threshold-driving mechanisms in the continuous log-concave setting -- through the Cram\'{e}r transform and Tukey's half-space depth -- with their discrete analogues. Within this framework, we establish a sharp threshold for lattice $p$-balls $\mathbb{Z}^n \cap rB_p^n$. Finally, we present structural counterexamples showing that sharp thresholds need not hold in general discrete log-concave settings. + oai:arXiv.org:2601.15444v1 math.PR - Wed, 21 Jan 2026 00:00:00 -0500 + math.MG + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Anne-Sophie de Suzzoni, Cyril Mal\'ez\'e + http://creativecommons.org/licenses/by/4.0/ + Silouanos Brazitikos, Minas Pafis - Multiary gradings - https://arxiv.org/abs/2601.11738 - arXiv:2601.11738v1 Announce Type: new -Abstract: This article develops a comprehensive theory of multiary graded polyadic algebras, extending the classical concept of group-graded algebras to higher-arity structures. We introduce the notion of grading by multiary groups and investigate various compatibility conditions between the arity of algebra operations and grading group operations. Key results include quantization rules connecting arities, classification of graded homomorphisms, and concrete examples including ternary superalgebras and polynomial algebras over $n$-ary matrices. The theory reveals fundamentally new phenomena not present in the binary case, such as the existence of higher power gradings and nontrivial constraints on arity compatibility. - oai:arXiv.org:2601.11738v1 - math.RA - Wed, 21 Jan 2026 00:00:00 -0500 + On certain bilinear sums with modular square roots and applications + https://arxiv.org/abs/2601.15448 + arXiv:2601.15448v1 Announce Type: new +Abstract: We extend bounds on additive energies of modular square roots by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu and apply these results to obtain bounds on certain bilinear exponential sums with modular square roots. From here, we make partial progress on the large sieve for square moduli. + oai:arXiv.org:2601.15448v1 + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Steven Duplij + Stephan Baier - Integrated Optimization of Scheduling and Flexible Charging in Mixed Electric-Diesel Urban Transit Bus Systems - https://arxiv.org/abs/2601.11751 - arXiv:2601.11751v1 Announce Type: new -Abstract: The transition of transit fleets to alternative powertrains offers a potential pathway to reducing the cost of mobility. However, the limited range and long charging durations of battery electric buses (BEBs) introduce significant operational complexities, necessitating innovative scheduling and charging strategies. This study proposes an integrated mixed-integer linear programming model to optimize vehicle scheduling and charging strategies for mixed fleets of BEBs and diesel buses. Unlike existing models, which often assume a fixed BEB fleet size or restrict charging to a single charger type, our approach simultaneously determines the optimal fleet composition, scheduling, and flexible partial charging strategy incorporating both slow and fast chargers at garages and terminal stations. The model minimizes combined fleet purchase and operational costs. A queuing strategy is introduced, departing from traditional first-come, first-served methods by dynamically allocating waiting and charging times based on operational priorities and resource availability, improving overall scheduling efficiency. To overcome computational complexities arising from numerous variables, a column generation framework is developed, facilitating scalable solutions for large-scale transit networks. Numerical experiments using real-world transit data from the Chicago Transit Authority and the Pace suburban bus systems demonstrate the model's effectiveness. Results indicate that while a full transition to alternative powertrains results in a modest cost increase, optimal mixed-fleet configurations can actually reduce total system costs. Furthermore, sensitivity analyses reveal that restricting charging to garages significantly increases fleet size and operational costs, underscoring the potential of distributed opportunistic charging. - oai:arXiv.org:2601.11751v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + Variance bounds in product measures without exponential tails + https://arxiv.org/abs/2601.15450 + arXiv:2601.15450v1 Announce Type: new +Abstract: We establish analogs of Cheeger's inequality for probability measures with heavy tails. As one of the principal applications, suppose $\lambda > 3$ and define the (Pareto) probability measure $\mu_{\lambda}$ on $[1,\infty)$ by $d\mu_{\lambda}(x) = (\lambda - 1) x^{-\lambda}$. Let $\mu_{\lambda}^n$ denote the product measure of $\mu_{\lambda}$ on $\mathbb{R}^n$. Then, for any $1$-Lipschitz function (with respect to the Euclidean distance) $f : \mathbb{R}^n \to \mathbb{R}$, we obtain the variance bound $\operatorname{Var}_{\mu_{\lambda}^n}(f) \le C(\lambda)\, n^{\frac{2}{\lambda - 1}}$, where $C(\lambda)$ is an explicit constant depending only on $\lambda$. This improves upon the existing bound $\operatorname{Var}_{\mu_{\lambda}^n}(f) = O(n)$ derived from the Efron--Stein inequality. Moreover, this bound is asymptotically tight when considering the $1$-Lipschitz function $f(x) = |x|_{\infty}$ corresponding to the $L^{\infty}$ norm. In probabilistic terms, suppose $X_1, \dots, X_n$ are i.i.d.\ random variables with distribution $\mu_{\lambda}$. Then, for any $1$-Lipschitz function $f$, we have $\operatorname{Var}(f(X_1, \dots, X_n)) \le C'(\lambda)\operatorname{Var}(\max\{X_1, \dots, X_n\}) = \Theta\!\left(n^{\frac{2}{\lambda - 1}}\right)$, where $C'(\lambda)$ is another explicit constant depending only on $\lambda$. + oai:arXiv.org:2601.15450v1 + math.PR + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sadjad Bazarnovi, Taner Cokyasar, Omer Verbas, Abolfazl Kouros Mohammadian + Shi Feng - Existence of Decreasing Nambu Solutions to the Rainbow Ladder Gap Equation of QCD by Cone Compression - https://arxiv.org/abs/2601.11752 - arXiv:2601.11752v1 Announce Type: new -Abstract: Studying Nambu solutions of the rainbow-ladder gap equation in QCD at zero temperature and chemical potential, we prove that the mass function emerges continuously from zero as the interaction strength is increased past the critical point for all positive, asymptotically perturbative kernels almost everywhere continuous in $L^1$ using the Krasnosel'skii-Guo Cone Compression Theorem. We prove that the coupled system of equations must have a positive, continuous Nambu solution with decreasing mass function for all current quark masses for a class of models which includes the physical point of a popular model of QCD by using a hybrid Krasnosel'skii-Schauder Fixed Point Theorem. - oai:arXiv.org:2601.11752v1 - math-ph - math.FA - math.MP - nucl-th - Wed, 21 Jan 2026 00:00:00 -0500 + Generalized Ramsey Numbers in the Hypercube + https://arxiv.org/abs/2601.15451 + arXiv:2601.15451v1 Announce Type: new +Abstract: We study the generalized Ramsey numbers $f(Q_n, C_{k}, q)$, that is, the minimum number of colors needed to edge-color the hypercube $Q_n$ so that every copy of the cycle $C_{k}$ has at least $q$ colors. Our main result is that for any integers $k,q$ satisfying $k \geq 6$ and $3 \leq q \leq k/2+1$, we have $f(Q_n, C_{k}, q)= o\left( n^{\frac{k/2-1}{k-q+1}} \right).$ We also prove a few other upper and lower bounds in the special cases $k=4$ and $k=6$. This continues the line of research initiated by Faudree, Gy\'arf\'as, Lesniak, and Schelp and Mubayi and Stading who studied the case $k=q$, and by Conder who considered the case $k=6$ and $q=2$. + oai:arXiv.org:2601.15451v1 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alex Roberts + http://creativecommons.org/licenses/by/4.0/ + Emily Heath, Coy Schwieder, Shira Zerbib - A volume formula for Reuleaux polyhedra - https://arxiv.org/abs/2601.11756 - arXiv:2601.11756v1 Announce Type: new -Abstract: A ball polyhedron is a finite intersection of congruent balls in $\mathbb{R}^3$. These shapes arise in various contexts in discrete and convex geometry. We focus on Reuleaux polyhedra, the subclass of ball polyhedra whose centers and vertices coincide. Building on Bogosel's recent work on the volume of Meissner polyhedra, we derive a formula for the volume of Reuleaux polyhedra in terms of their edges. - oai:arXiv.org:2601.11756v1 - math.MG - Wed, 21 Jan 2026 00:00:00 -0500 + The paper "On the constant in a transference inequality for the vector-valued Fourier transform" revisited + https://arxiv.org/abs/2601.15454 + arXiv:2601.15454v1 Announce Type: new +Abstract: The standard proof of the equivalence of Fourier type on \(\mathbb R^d\) and on the torus \(\mathbb T^d\) is usually stated in terms of an implicit constant which can be expressed in terms of the global minimiser of the functions \[f_r(x)=\sum_{m\in\mathbb{Z}}\left|\frac{\sin(\pi(x+m))}{\pi(x+m)}\right|^{2r},\qquad x\in [0,1], \ r\ge 1.\] The aim of this note is to provide a short proof of a result of the authors which states that each \(f_r\) takes a global minimum at the point \(x = \frac12\). + oai:arXiv.org:2601.15454v1 + math.CA + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ryan Hynd + Dion Gijswijt. Jan van Neerven - Nonautonomous Linear Systems: Exponential Dichotomy and its Applications - https://arxiv.org/abs/2601.11759 - arXiv:2601.11759v1 Announce Type: new -Abstract: The first purpose of this work is to provide a friendly introduction to the theory of nonautonomous linear systems of ordinary differential equations, the property of exponential dichotomy and its corresponding spectral theory. The second purpose of this work is disseminate the linearization results carried out by the authors in a nonautonomous framework. - The actual structure of this work is a consequence of several elective courses (2014, 2016, 2019, 2021 and 2023) carried out by the authors for undergraduate and graduated students at the Department of Mathematics of the Universidad de Chile. The monography assumes a good knowledge of multivariate calculus, linear algebra and ordinary differential equations. - oai:arXiv.org:2601.11759v1 - math.CA - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 + Brauer groups of varieties over local fields of finite characteristic + https://arxiv.org/abs/2601.15461 + arXiv:2601.15461v1 Announce Type: new +Abstract: We show that the non-log version of Kato's ramification filtration on the Brauer group of a separated and finite type regular scheme over a positive characteristic local field coincides with the evaluation filtration. This extends a recent result of Bright-Newton to positive characteristics. Among several applications, we extend some results of Ieronymou, Saito-Sato and Kai to positive characteristics. + oai:arXiv.org:2601.15461v1 + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - \'Alvaro Casta\~neda, Gonzalo Robledo + Amalendu Krishna, Subhadip Majumder - Solving High-Dimensional PDEs Using Linearized Neural Networks - https://arxiv.org/abs/2601.11771 - arXiv:2601.11771v1 Announce Type: new -Abstract: Linearized shallow neural networks that are constructed by fixing the hidden-layer parameters have recently shown strong performance in solving partial differential equations (PDEs). Such models, widely used in the random feature method (RFM) and extreme learning machines (ELM), transform network training into a linear least-squares problem. In this paper, we conduct a numerical study of the variational (Galerkin) and collocation formulations for these linearized networks. Our numerical results reveal that, in the variational formulation, the associated linear systems are severely ill-conditioned, forming the primary computational bottleneck in scaling the neural network size, even when direct solvers are employed. In contrast, collocation methods combined with robust least-squares solvers exhibit better numerical stability and achieve higher accuracy as we increase neuron numbers. This behavior is consistently observed for both ReLU$^k$ and $\tanh$ activations, with $\tanh$ networks exhibiting even worse conditioning. Furthermore, we demonstrate that random sampling of the hidden layer parameters, commonly used in RFM and ELM, is not necessary for achieving high accuracy. For ReLU$^k$ activations, this follows from existing theory and is verified numerically in this paper, while for $\tanh$ activations, we introduce two deterministic schemes that achieve comparable accuracy. - oai:arXiv.org:2601.11771v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 + Determinants of modular Collatz graphs and variants + https://arxiv.org/abs/2601.15463 + arXiv:2601.15463v1 Announce Type: new +Abstract: The determinants of modular Collatz graphs and the modular Conway amusical permutation graph are determined, and some interesting number theoretic properties are described. + oai:arXiv.org:2601.15463v1 + math.NT + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Tong Mao, Jinchao Xu, Xiaofeng Xu + Achilleas Karras, Benne de Weger - On the Narrow 2-Class Field Tower of Some Real Quadratic Number Fields: Lengths Heuristics Follow-Up - https://arxiv.org/abs/2601.11773 - arXiv:2601.11773v1 Announce Type: new -Abstract: In this article we continue the investigation of the length of the narrow $2$-class field tower of real quadratic number fields $\mathrm{k}$ whose discriminants are not a sum of two squares and for which their $2$-class groups are elementary of order $4$. Letting $\mathrm{G}$ equal the Galois group of the second Hilbert narrow $2$-class field over $\mathrm{k}$, and $[\mathrm{G}_i]$ denote the lower central series of $\mathrm{G}$, we give heuristic evidence that the length of the narrow $2$-class field tower of $\mathrm{k}$ is equal to $2$ when $\mathrm{G}/\mathrm{G}_3$ is of type $64.150$ (in the tables of Hall and Senior). We also give the formulation of the relevant unit groups of the narrow Hilbert $2$-class field for these fields. - oai:arXiv.org:2601.11773v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 + Rank-metric codes over arbitrary fields: Bounds and constructions + https://arxiv.org/abs/2601.15464 + arXiv:2601.15464v1 Announce Type: new +Abstract: Rank-metric codes, defined as sets of matrices over a finite field with the rank distance, have gained significant attention due to their applications in network coding and connections to diverse mathematical areas. Initially studied by Delsarte in 1978 and later rediscovered by Gabidulin, these codes have become a central topic in coding theory. This paper surveys the development and mathematical foundations, in particular, regarding bounds and constructions of rank-metric codes, emphasizing their extension beyond finite fields to more general settings. We examine Singleton-like bounds on code parameters, demonstrating their sharpness in finite field cases and contrasting this with contexts where the bounds are not tight. Furthermore, we discuss constructions of Maximum Rank Distance (MRD) codes over fields with cyclic Galois extensions and the relationship between linear rank-metric codes with systems and evasive subspaces. The paper also reviews results for algebraically closed fields and real numbers, previously appearing in the context of topology and measure theory. We conclude by proposing future research directions, including conjectures on MRD code existence and the exploration of rank-metric codes over various field extensions. + oai:arXiv.org:2601.15464v1 + cs.IT + math.CO + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Elliot Benjamin, Mohamed Mahmoud Chems-Eddin + Alessandro Neri, Ferdinando Zullo - Mixed-Integer Reaggregated Hull Reformulation of Special Structured Generalized Linear Disjunctive Programs - https://arxiv.org/abs/2601.11782 - arXiv:2601.11782v1 Announce Type: new -Abstract: Generalized Disjunctive Programming (GDP) provides a powerful framework for combining algebraic constraints with logical disjunctions. To solve these problems, mixed-integer reformulations are required, but traditional reformulation schemes, such as Big-M and Hull, either yield a weak continuous relaxation or result in a bloated model size. Castro and Grossmann showed that scheduling problems can be formulated as GDP by modeling task orderings as disjunctions with algebraic timing constraints. Moreover, in their work, a particular representation of the single-unit scheduling problem, namely using a time-slot concept, can be reformulated as a tight yet compact mixed-integer linear program with notable computational performance. Based on that observation, and focusing on the case where the constraints in disjunctions are linear and share the same coefficients, we connect the characterization of the convex hull of these disjunctive sets by Jeroslow and Blair with Castro and Grossmann's time-slot reaggregation strategy to derive a unified reformulation methodology. We test this reformulation in two problems, single-unit scheduling and two-dimensional strip-packing. We derive new formulations of the general precedence concept of single-unit scheduling and symmetry-breaking formulations of the strip-packing problem, yielding mixed-integer programs with strong theoretical guarantees, particularly compact formulations in terms of continuous variables, and efficient computational performance when solving them with commercial mixed-integer solvers for these problems. - oai:arXiv.org:2601.11782v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + Gorenstein flat preenvelopes and weakly Ding injective covers + https://arxiv.org/abs/2601.15469 + arXiv:2601.15469v1 Announce Type: new +Abstract: We consider a (left) coherent ring R. We prove that if the character module of every Ding injective (left) R-module is Gorenstein flat, then the class of Gorenstein flat (right) R-modules, GF, is preenveloping. We show that this is the case when every injective (left) R-module has finite flat dimension. In particular, GF is preenveloping over any Ding-Chen ring.\\ The proofs use the class of weakly Ding injective (left) R-modules, wDI. We show that, when wDI is closed under extensions, the following statements are equivalent:\\ 1. The character module of every Ding injective left R-module is a Gorenstein flat right R-module.\\ 2. The class of weakly Ding injective left R-modules is closed under direct limits.\\ 3. The class of weakly Ding injective modules is covering.\\ The equivalent statements (1)-(3) imply that GF is preenveloping + oai:arXiv.org:2601.15469v1 + math.KT + math.AC + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1021/acs.iecr.5c03172 - Ind. Eng. Chem. Res. 2026, 65 (1) - Albert Joon Lee, David E. Bernal Neira + Alina Iacob - Projected Stochastic Momentum Methods for Nonlinear Equality-Constrained Optimization for Machine Learning - https://arxiv.org/abs/2601.11795 - arXiv:2601.11795v1 Announce Type: new -Abstract: Two algorithms are proposed, analyzed, and tested for solving continuous optimization problems with nonlinear equality constraints. Each is an extension of a stochastic momentum-based method from the unconstrained setting to the setting of a stochastic Newton-SQP-type algorithm for solving equality-constrained problems. One is an extension of the heavy-ball method and the other is an extension of the Adam optimization method. Convergence guarantees for the algorithms for the constrained setting are provided that are on par with state-of-the-art guarantees for their unconstrained counterparts. A critical feature of each extension is that the momentum terms are implemented with projected gradient estimates, rather than with the gradient estimates themselves. The significant practical effect of this choice is seen in an extensive set of numerical experiments on solving informed supervised machine learning problems. These experiments also show benefits of employing a constrained approach to supervised machine learning rather than a typical regularization-based approach. - oai:arXiv.org:2601.11795v1 + Folklore in Multi-Objective Optimisation + https://arxiv.org/abs/2601.15499 + arXiv:2601.15499v1 Announce Type: new +Abstract: In this paper, we present and prove some results in multi-objective optimisation that are considered folklore. For the most part, proofs for these results exist in special cases, but they are used in more general settings since their proofs can be (largely) transferred. We do this transfer explicitly and try to state the results as generally as possible. In particular, we also aim at providing clean and complete proofs for results where the original papers are not rigorous. + oai:arXiv.org:2601.15499v1 math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Qi Wang, Christian Piermarini, Yunlang Zhu, Frank E. Curtis + Oliver Bachtler - The Noisy Quantitative Group Testing Problem - https://arxiv.org/abs/2601.11797 - arXiv:2601.11797v1 Announce Type: new -Abstract: In this paper, we study the problem of quantitative group testing (QGT) and analyze the performance of three models: the noiseless model, the additive Gaussian noise model, and the noisy Z-channel model. For each model, we analyze two algorithmic approaches: a linear estimator based on correlation scores, and a least squares estimator (LSE). We derive upper bounds on the number of tests required for exact recovery with vanishing error probability, and complement these results with information-theoretic lower bounds. In the additive Gaussian noise setting, our lower and upper bounds match in order. - oai:arXiv.org:2601.11797v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + On orthogonality graphs of Okubo algebras + https://arxiv.org/abs/2601.15501 + arXiv:2601.15501v1 Announce Type: new +Abstract: The orthogonality graph of an Okubo algebra with isotropic norm over an arbitrary field $\mathbb{F}$ is considered. Its connected components are described, and their diameters are computed. It is shown that there exist at most two shortest paths between any pair of vertices, and the conditions under which the shortest path is unique are determined. + oai:arXiv.org:2601.15501v1 + math.RA + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Tenghao Li, Neha Sangwan, Xiaxin Li, Arya Mazumdar + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Danil Pavlinov, Svetlana Zhilina - Distance of Quadratic Algebraic Numbers from the Middle-Third Cantor Set - https://arxiv.org/abs/2601.11799 - arXiv:2601.11799v1 Announce Type: new -Abstract: We study the distance from quadratic irrational numbers to the middle-third Cantor set $C$. Mahler asked whether $C$ contains any irrational algebraic numbers; this remains open even for quadratic irrationals. Rather than assuming an answer to this problem, we obtain uniform lower bounds for the distance from a quadratic irrational $\alpha$ to $C$ in terms of the height $H$ of the minimal polynomial of $\alpha$. - We encode $\alpha$ by its orbit under the map $x \mapsto 3x \bmod 1$ and define the exit time $\operatorname{exit}(\alpha)$ as the first iterate that enters the middle interval $[1/3,2/3]$. Our main unconditional result is a quadratic exit bound $\operatorname{exit}(\alpha) \le A (\log_3 H)^2 + B$ for absolute constants $A,B > 0$, valid for all quadratic irrationals whose orbit stays a fixed small distance away from the coarse Cantor boundaries. As a consequence we obtain a distance lower bound $\operatorname{dist}(\alpha,C) \ge H^{-\kappa \log H}$ for some constant $\kappa > 0$. - On the dynamical side we classify orbits by an $L/M/R$ coding and prove that the total number of visits to the right interval $[2/3,1)$ is $O(\log H)$. A finite case analysis on a bounded portion of the orbit is reduced to checking a finite list of explicit affine inequalities on subintervals of $[0,1]$, which we verify with short computer scripts; all Diophantine and dynamical estimates are proved by hand. - oai:arXiv.org:2601.11799v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 + Stabilizer-Code Channel Transforms Beyond Repetition Codes for Improved Hashing Bounds + https://arxiv.org/abs/2601.15505 + arXiv:2601.15505v1 Announce Type: new +Abstract: The quantum hashing bound guarantees that rates up to $1-H(p_I, p_X, p_Y, p_Z)$ are achievable for memoryless Pauli channels, but it is not generally tight. A known way to improve achievable rates for certain asymmetric Pauli channels is to apply a small inner stabilizer code to a few channel uses, decode, and treat the resulting logical noise as an induced Pauli channel; reapplying the hashing argument to this induced channel can beat the baseline hashing bound. We generalize this induced-channel viewpoint to arbitrary stabilizer codes used purely as channel transforms. Given any $ [\![ n, k ]\!] $ stabilizer generator set, we construct a full symplectic tableau, compute the induced joint distribution of logical Pauli errors and syndromes under the physical Pauli channel, and obtain an achievable rate via a hashing bound with decoder side information. We perform a structured search over small transforms and report instances that improve the baseline hashing bound for a family of Pauli channels with skewed and independent errors studied in prior work. + oai:arXiv.org:2601.15505v1 + cs.IT + math.IT + quant-ph + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Frank Gilson + Tyler Kann, Matthieu R. Bloch, Shrinivas Kudekar, Ruediger Urbanke - On large periodic traveling surface waves in porous media - https://arxiv.org/abs/2601.11800 - arXiv:2601.11800v1 Announce Type: new -Abstract: We study large traveling surface waves within a two-dimensional finite depth, free boundary, homogeneous, incompressible and viscous fluid governed by Darcy's law. The fluid is bound by a gravitational force to a flat rigid bottom and meets an atmosphere of constant pressure at the top with its free surface, where it does not experience any capillarity effects. Additionally, the fluid is subject to a fixed, but arbitrarily selected, forcing data profile with variable amplitude. We use the Riemann mapping to equivalently reformulate the resulting two-dimensional free boundary problem as a single one-dimensional fully nonlinear pseudodifferential equation for a function describing the domain's geometry. By discovering a hidden ellipticity in the reformulated equation, we are able to import a global implicit function theorem to construct a connected set of traveling waves, containing both the quiescent solution and large amplitude members. We find that either solutions continue to exist for arbitrarily large data amplitude or else one of a finite number of meaningful breakdown scenarios must occur. This work stands as the first non perturbative construction of large traveling surface waves in any free boundary viscous fluid without surface tension. - oai:arXiv.org:2601.11800v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Maps on Surfaces as a Structural Framework for Genus-One Virtual Knot Classification + https://arxiv.org/abs/2601.15512 + arXiv:2601.15512v1 Announce Type: new +Abstract: We develop a purely combinatorial framework for the systematic enumeration of knot and link diagrams supported on the thickened torus $T^2\times I$. Using the theory of maps on surfaces, cellular $4$--regular torus projections are encoded by permutation pairs $(\alpha,\sigma)$, and unsensed projection classes are enumerated completely and without duplication via canonical representatives. For a fixed projection, crossing assignments are encoded by bit data, and an immediate Reidemeister~II reduction supported by a bigon face is characterized directly in terms of these bits. The genus-one generalized Kauffman-type bracket is then evaluated as a state sum entirely within the permutation model, without drawing diagrams in a fundamental polygon. + The implementation is validated against published genus-one classifications for $N\le 5$ under explicit comparison conventions, with remaining discrepancies explained at the level of global conventions. Beyond the published range, we compute projection and diagram data for crossing numbers up to $N=8$ and provide a public reference implementation together with machine-readable datasets. Via the standard correspondence between virtual knots and knots in thickened surfaces, this yields a canonical and fully reproducible genus-one framework for virtual knot tabulation. + oai:arXiv.org:2601.15512v1 + math.CO + math.GT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Huy Q. Nguyen, Noah Stevenson + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Alexander Omelchenko - Classification of dynamics for a two person model of planned behavior - https://arxiv.org/abs/2601.11804 - arXiv:2601.11804v1 Announce Type: new -Abstract: We study a dynamical system modeling the Theory of Planned Behavior (TPB) in which each individual's behavioral intention evolves continuously under an ODE driven by internal attitudes, perceived social norms, and perceived behavioral control. Actions occur as discrete threshold events: when intention reaches a fixed threshold it is reset to 0 and produces a transient "nudge" that jumps to 1 and then decays exponentially. This yields a hybrid ODE-threshold system with psychologically interpretable parameters. We derive a partial classification in the general case of n individuals. Focusing on the two-individual case (n=2), we obtain explicit formulas for trajectories between action events and derive bounds for first-action times. In the mixed setting where one individual is intrinsically increasing and the other is not, we identify a scalar invariant, M, measuring the net effect of one period of excitation. We prove that non-positive M is equivalent to a partial-action state (only the intrinsically active individual acts countable infinitely often), while positive M is equivalent to full action (both individuals act countably infinitely often). Finally, we demonstrate numerically that these analytic boundaries partition the parameter space with near-perfect agreement, and we provide exploratory simulations suggesting analogous structures for three individuals. - oai:arXiv.org:2601.11804v1 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 + Colour ratio in Prim's ranking of bipartite graphs + https://arxiv.org/abs/2601.15520 + arXiv:2601.15520v1 Announce Type: new +Abstract: We consider a complete bipartite graph of size $n$ endowed with i.i.d. uniform edge weights and run Prim's Algorithm to obtain a ranking of its vertices. Let $\rho^{(n)}_k$ be the proportion of black vertices among the first $k$ vertices in this ranking. We characterise the limit behaviour of $\rho^{(n)}_k$ as both $n$ and $k$ tend to infinity. Our results show that in general the limit of $\rho^{(n)}_k$, when existing, differs from the overall proportion of the black vertices in the graph. + oai:arXiv.org:2601.15520v1 + math.PR + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Rishi Dadlani, John S. McAlister, Tahra L. Eissa, Nina H. Fefferman + F\'elix Kahane, Minmin Wang - Rank of normal functions and Betti strata - https://arxiv.org/abs/2601.11805 - arXiv:2601.11805v1 Announce Type: new -Abstract: In a recent work of the authors, we proved the generic positivity of the Beilinson-Bloch heights of the Gross-Schoen and Ceresa cycles. The geometric part of the proof was to prove the maximality of the rank of the associated normal function and the Zariski closedness of the Betti strata. In this paper, we generalize these geometric results to an arbitrary family of homologically trivial cycles. More generally, we prove a formula to compute the Betti rank and prove the Zariski closedness of the Betti strata, for any admissible normal function of a variation of Hodge structures of weight $-1$. We also define and prove results about degeneracy loci. In the end, we go back to the arithmetic setting and ask some questions about the rationality of the Betti strata and the torsion loci. - oai:arXiv.org:2601.11805v1 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 + Length minimization of filling pairs on hyperbolic surfaces + https://arxiv.org/abs/2601.15524 + arXiv:2601.15524v1 Announce Type: new +Abstract: A filling pair $(\alpha, \beta)$ of a surface $S_g$ is a pair of simple closed curves in minimal position such that the complement of $\alpha\cup\beta$ in $S_g$ is a disjoint union of topological disks. A filling pair is said to be minimally intersecting if the number of intersections between them, or equivalently, the number of complementary disks, is minimal among all filling pairs of $S_g$. For surfaces of genus $g \geq 3$, minimal filling pairs are well understood, whereas in genus two, such a pair divides the surface into exactly two disks. In this paper, we classify all minimal filling pairs up to the action of the mapping class group in genus two and determine the length of the shortest minimal filling pair. + oai:arXiv.org:2601.15524v1 + math.GT + math.MG + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ziyang Gao, Shou-Wu Zhang + http://creativecommons.org/licenses/by/4.0/ + Ni An, Bhola Nath Saha, Bidyut Sanki - Infinitesimal invariants of mixed Hodge structures II: Log Clemens conjecture and log connectivity - https://arxiv.org/abs/2601.11810 - arXiv:2601.11810v1 Announce Type: new -Abstract: Following previous work, we continue the study of infinitesimal methods in mixed Hodge theory. In the first part, inspired by the deformation theory of curves on Calabi-Yau threefolds, we study deformations of smooth $\mathbb{Q}$-log Calabi-Yau pairs $(X,Y)$. We prove unobstructedness results for these pairs under Fano hypotheses. We define families of infinitesimal Abel-Jacobi maps associated with these deformation problems and show that they control the first-order deformations of smooth curves embedded in the pair. Crucially, for the $\frac{1}{2}$-log Calabi-Yau case, we establish an exact duality between deformations and obstructions, recovering the symmetry found in the absolute Calabi-Yau setting. We apply this framework to the cubic threefold, proposing a relative generalization of the Clemens conjecture regarding the injectivity of the infinitesimal Abel-Jacobi map, and establishing a criterion for its non-vanishing. - In the second part, we define infinitesimal invariants for normal functions using extension classes and the log-Leray filtration. Relying on the theory of generalized Jacobian rings developed by Asakura and Saito, we prove a logarithmic Nori connectivity theorem for the universal family of open hypersurfaces, we also deduce a sharp algebraic criterion for the properness of the Hodge loci for open hypersurfaces, generalizing the proof of Carlson-Green-Griffiths-Harris. - oai:arXiv.org:2601.11810v1 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 + The Frog Model on $\mathbb{Z}$ with Discrete Weibull Lifetimes and Random Parameter $p$ + https://arxiv.org/abs/2601.15526 + arXiv:2601.15526v1 Announce Type: new +Abstract: We study the frog model on $\mathbb{Z}$ with particle wise discrete Weibull lifetimes. Each particle has an i.i.d. survival parameter $\pi\in(0,1)$; conditionally on $\pi=p$, its lifetime $\Xi$ satisfies \[ P(\Xi\ge k\mid \pi=p)=p^{k^{\gamma}},\qquad k\in\mathbb{N}_0,\gamma>0. \] The law of $\pi$ has right edge density \[ f_\pi(u)\sim(1-u)^{\beta-1},L\big((1-u)^{-1}\big)\qquad (u\uparrow 1), \] with $\beta>0$ and $L$ slowly varying; let $\eta$ denote the common law of the i.i.d. initial occupation numbers $\{\eta_x\}_{x\in\mathbb{Z}}$. The survival parameter distribution strictly extends the Beta family, while the lifetime distribution extends the geometric case. We prove a sharp extinction and survival dichotomy with the $\gamma-$dependent threshold \[ \beta_c:=\frac{1}{2\gamma}. \] If $\beta>\beta_c$ and $E(\eta)<\infty$, the process becomes extinct almost surely; if $\beta<\beta_c$ and $P(\eta=0)<1$, it survives with positive probability. At the boundary $\beta=\beta_c$ we provide explicit criteria in terms of $\limsup/\liminf$ of $L(n^{2\gamma})$. The case $\gamma=1$ (geometric lifetimes) recovers the benchmark $\beta_c=\frac{1}{2}$ and the critical refinements previously obtained for random geometric lifetimes. + oai:arXiv.org:2601.15526v1 + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rodolfo Aguilar + http://creativecommons.org/licenses/by/4.0/ + J. H. Ram\'irez Gonz\'alez, Gustavo O. Carvalho, F\'abio P. Machado - The Clemens Conjectures for Cubic Threefolds relative to a Hyperplane - https://arxiv.org/abs/2601.11813 - arXiv:2601.11813v1 Announce Type: new -Abstract: We propose an analogue of the Clemens conjectures for $\frac{1}{2}$-log Calabi-Yau threefolds, specifically for the pair $(X, Y)$ where $X$ is a cubic threefold and $Y$ is a hyperplane section. By exploiting a perfect deformation/obstruction duality specific to the $\frac{1}{2}$-log setting, we formulate conjectures regarding the injectivity of the relative infinitesimal Abel-Jacobi map and the finiteness of rational curves with fixed intersection on $Y$. - oai:arXiv.org:2601.11813v1 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 + Palindromicity of multivariate Eulerian polynomials + https://arxiv.org/abs/2601.15527 + arXiv:2601.15527v1 Announce Type: new +Abstract: We lift to the multivariate Eulerian polynomials the identity implying that univariate Eulerian polynomials are palindromic. As a consequence of this generalization, we obtain nice combinatorial identities that can be directly extracted from this polynomial relation and the bijection between permutations involved in the proof of the identity. + oai:arXiv.org:2601.15527v1 + math.CO + math.GR + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rodolfo Aguilar + http://creativecommons.org/publicdomain/zero/1.0/ + Alejandro Gonz\'alez Nevado - The maximal mean equicontinuous factor via regional mean sensitivity - https://arxiv.org/abs/2601.11814 - arXiv:2601.11814v1 Announce Type: new -Abstract: For actions of amenable groups, mean equicontinuity-a natural relaxation of equicontinuity obtained by averaging metrics along orbits-is well known to yield a maximal mean equicontinuous factor. In 2021, Li and Yu introduced the notion of weak sensitivity in the mean for actions of $\mathbb{Z}$ to gain a deeper understanding of this phenomenon, building on earlier work by Qiu and Zhao. - We demonstrate that this relation is insufficient for actions of non-Abelian groups. To overcome this limitation, we introduce the regional mean sensitive relation, which more precisely captures the dynamical behaviour underlying the maximal mean equicontinuous factor. We discuss its fundamental properties and highlight its advantages in the non-Abelian setting. In particular, we show that mean equicontinuity is equivalent to the nonexistence of non-diagonal regional mean sensitive pairs. For this, we work in the context of actions of $\sigma$-compact and locally compact amenable groups. - oai:arXiv.org:2601.11814v1 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 + Variable Stepsize Distributed Forward-Backward Splitting Methods as Relocated Fixed-Point Iterations + https://arxiv.org/abs/2601.15531 + arXiv:2601.15531v1 Announce Type: new +Abstract: We present a family of distributed forward-backward methods with variable stepsizes to find a solution of structured monotone inclusion problems. The framework is constructed by means of relocated fixed-point iterations, extending the approach introduced in arXiv:2507.07428 to conically averaged operators, thus including iteration operators for methods of forward-backward type devised by graphs. The family of methods we construct preserve the per-iteration computational cost and the convergence properties of their constant stepsize counterparts. Specifically, we show that the resulting methods generate a sequence that converges to a fixed-point of the underlying iteration operator, whose shadow sequences converge to a solution of the problem. Numerical experiments illustrate the behaviour of our framework in structured sparse optimisation problems. + oai:arXiv.org:2601.15531v1 + math.OC + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Till Hauser, Chunlin Liu + http://creativecommons.org/licenses/by/4.0/ + Felipe Atenas, Minh N. Dao, Matthew K. Tam - Bayesian ICA for Causal Discovery - https://arxiv.org/abs/2601.11815 - arXiv:2601.11815v1 Announce Type: new -Abstract: Causal discovery based on Independent Component Analysis (ICA) has achieved remarkable success through the LiNGAM framework, which exploits non-Gaussianity and independence of noise variables to identify causal order. However, classical LiNGAM methods rely on the strong assumption that there exists an ordering under which the noise terms are exactly independent, an assumption that is often violated in the presence of confounding. In this paper, we propose a general information-theoretic framework for causal order estimation that remains applicable under arbitrary confounding. Rather than imposing independence as a hard constraint, we quantify the degree of confounding by the multivariate mutual information among the noise variables. This quantity is decomposed into a sum of mutual information terms along a causal order and is estimated using Bayesian marginal likelihoods. The resulting criterion can be interpreted as Bayesian ICA for causal discovery, where causal order selection is formulated as a model selection problem over permutations. Under standard regularity conditions, we show that the proposed Bayesian mutual information estimator is consistent, with redundancy of order $O(\log n)$. To avoid non-identifiability caused by Gaussian noise, we employ non-Gaussian predictive models, including multivariate $t$ distributions, whose marginal likelihoods can be evaluated via MCMC. The proposed method recovers classical LiNGAM and DirectLiNGAM as limiting cases in the absence of confounding, while providing a principled ranking of causal orders when confounding is present. This establishes a unified, confounding-aware, and information-theoretically grounded extension of ICA-based causal discovery. - oai:arXiv.org:2601.11815v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + The formal theory of tangentads PART II + https://arxiv.org/abs/2601.15534 + arXiv:2601.15534v1 Announce Type: new +Abstract: Tangent category theory is a well-established categorical framework for differential geometry. A long list of fundamental geometric constructions, such as the tangent bundle functor, vector fields, Euclidean spaces, and vector bundles have been successfully generalized and internalized within tangent categories. Over the past decade, the theory has also been extended in several directions, yielding concepts such as tangent monads, tangent fibrations, tangent restriction categories, and reverse tangent categories. It is natural to wonder how these new flavours of the theory interact with the geometric constructions. How does a tangent monad or a tangent fibration lift to the tangent category of differential bundles of a tangent category? What is the correct notion of connections for a tangent restriction category? In previous work, we introduced tangentads, a unifying framework that generalizes many tangent-like notions, and developed a formal theory of vector fields for tangentads. In this paper, we extend this formal theory to three further fundamental constructions. These are differential objects, which generalize Euclidean spaces, differential bundles, which represent vector bundles in tangent category theory, and connections on differential bundles, which are the analogue of Koszul connections. These notions are introduced in the general theory of tangentads via appropriate universal properties. We then extend some of the main results of tangent category theory, including the equivalence between differential objects and differential bundles over the terminal object, and show that connections admit well-defined notions of covariant derivative, curvature, and torsion. Finally, we construct connections using PIE limits and apply our framework to several concrete instances of tangentads. + oai:arXiv.org:2601.15534v1 + math.CT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Joe Suzuki + http://creativecommons.org/licenses/by/4.0/ + Marcello Lanfranchi - Large deviations and the matrix product ansatz - https://arxiv.org/abs/2601.11820 - arXiv:2601.11820v1 Announce Type: new -Abstract: We consider probability measures on $A^N$, the set of sequences of symbols on a finite alphabet $A$ of length $N$, that give a weight to each sequence in terms of a collection of matrices with non-negative entries and having rows and columns labeled by a finite or countable set $B$. We prove for such kind of measures large deviations principles for several empirical measures. Our approach is based on a simultaneous combination of an enlargement of the state space to sequences on $A\times B$ and a spectral conjugation that produces a stochastic matrix, as discussed in \cite{GI1}. As a result we describe the measures as hidden Markov measures and can deduce the large deviations results by contraction from the corresponding ones for the enlarged Markov chain. The measure on the enlarged state space is a Markov bridge. The invariant measures of several non equilibrium models of interacting particle systems can be represented by the so called {\it Matrix Product Ansatz} that corresponds to measures of the type that we consider and with matrices labeled by $B$ that is typically countable infinite. The large deviations behavior is different in the cases with $B$ finite or countable. In the finite case we give a variational formula for both the algebraic and the spatial empirical measures, that can be solved in special cases. For the infinite case, we illustrate the method through an example that is the invariant measure of the boundary driven TASEP model in a special regime. We recover in this way the celebrated results in \cite{Der4,Derr7}, and in particular we obtain a variational representation of the rate function similar to that in \cite{Bryc}. Our approach is general and can in principle be applied to any measure represented by the matrix product ansatz with matrices having positive entries. - oai:arXiv.org:2601.11820v1 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 + Rationality of the trivial lattice rank weighted motivic height zeta function for elliptic surfaces + https://arxiv.org/abs/2601.15543 + arXiv:2601.15543v1 Announce Type: new +Abstract: Let $k$ be a perfect field with $\mathrm{char}(k)\neq 2,3$, set $K=k(t)$, and let $\mathcal{W}_n^{\min}$ be the moduli stack of minimal elliptic curves over $K$ of Faltings height $n$ from the height-moduli framework of Bejleri-Park-Satriano applied to $\overline{\mathcal{M}}_{1,1}\simeq \mathcal{P}(4,6)$. For $[E]\in \mathcal{W}_n^{\min}$, let $S \to \mathbb{P}^1_{k}$ be the associated elliptic surface with section. Motivated by the Shioda-Tate formula, we consider the trivariate motivic height zeta function \[ \mathcal{Z}(u,v;t):= \sum_{n\ge0}\Bigl(\sum_{[E]\in \mathcal{W}_n^{\min}} u^{T(S)}v^{\mathrm{rk}(E/K)}\Bigr)t^n \in K_0(\mathrm{Stck}_k)[u,v][[t]] \] which refines the height series by weighting each height stratum with the trivial lattice rank $T(S)$ and the Mordell--Weil rank $\mathrm{rk}(E/K)$. We prove rationality for the trivial lattice specialization $Z_{\mathrm{Triv}}(u;t)=\mathcal{Z}(u,1;t)$ by giving an explicit finite Euler product. We conjecture irrationality for the N\'eron-Severi $Z_{\mathrm{NS}}(w;t)=\mathcal{Z}(w,w;t)$ and the Mordell-Weil $Z_{\mathrm{MW}}(v;t)=\mathcal{Z}(1,v;t)$ specializations. + oai:arXiv.org:2601.15543v1 + math.AG + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Davide Gabrielli, Federica Iacovissi + Jun-Yong Park - A high-order augmented Lagrangian method with arbitrarily fast convergence - https://arxiv.org/abs/2601.11826 - arXiv:2601.11826v1 Announce Type: new -Abstract: We propose a high-order version of the augmented Lagrangian method for solving convex optimization problems with linear constraints, which achieves arbitrarily fast -- and even superlinear -- convergence rates. First, we analyze the convergence rates of the high-order proximal point method under certain uniform convexity assumptions on the energy functional. We then introduce the high-order augmented Lagrangian method and analyze its convergence by leveraging the convergence results of the high-order proximal point method. Finally, we present applications of the high-order augmented Lagrangian method to various problems arising in the sciences, including data fitting, flow in porous media, and scientific machine learning. - oai:arXiv.org:2601.11826v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + Computability of $\mathcal{G}$-Beroulli Measures and Measures of Maximal Entropy on Coded Shift Spaces + https://arxiv.org/abs/2601.15548 + arXiv:2601.15548v1 Announce Type: new +Abstract: In this paper, we investigate the computability of $\mathcal{G}$-Bernoulli measures, with a particular focus on measures of maximal entropy (MMEs) on coded shift spaces. Coded shifts are natural generalizations of sofic shifts and are defined as the closure of all bi-infinite concatenations of words (generators) drawn from a countable generating set $\mathcal{G}$. We begin by establishing a computability criterion for $\mathcal{G}$-Bernoulli measures which are invariant measures given by assigning probability weights to the generators. We then apply this criterion to the setting in which the concatenation entropy exceeds the residual entropy, showing that in this case the unique measure of maximal entropy $\mu_{\rm max}$ on $X$ is computable, provided the Vere--Jones parameter $\kappa$ of $\mathcal{G}$ is computable, based on having oracle access to the generators and the language of $X$. As a consequence, the unique MME is computable for several well-known classes of shift spaces, including $S$-gap shifts, multiple-gap shifts, and $\beta$-shifts. Moreover, the two ergodic MMEs of the Dyck shift are also computable. Finally, we examine the opposite situation, where the residual entropy exceeds the concatenation entropy and the MME is known to be non-unique in general. We show that even when $\mu_{\rm max}$ is unique and the parameter $\kappa$ is computable, the measure $\mu_{\rm max}$ may still fail to be computable. + oai:arXiv.org:2601.15548v1 + math.DS + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Young-Ju Lee, Jongho Park + Tamara Kucherenko, Marco L\'opez, Christian Wolf - Topological and Purely Topological Alignment Dynamics - https://arxiv.org/abs/2601.11828 - arXiv:2601.11828v1 Announce Type: new -Abstract: We study the Euler Alignment system of collective behavior, equipped with `topological' interaction protocols, which were introduced to the mathematical literature by Shvydkoy and Tadmor. Interactions subject to these protocols may depend on both the Euclidean distance between agents and on the mass distribution between them -- the `topological' component. When the interaction protocol is regular, we prove sufficient conditions for the existence of global-in-time classical solutions, related to the initial nonnegativity of a conserved quantity of the system. The remainder of our results explore the case where the interactions are `purely' topological and the interactions do not depend on the Euclidean distance. We show that in this case, the system decouples into an autonomous velocity equation in mass coordinates together with a scalar conservation law with time-dependent flux determined by the velocity. We analyze the long-time behavior for the dynamics associated to both regular and singular protocols. - oai:arXiv.org:2601.11828v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + On the nilpotent residue non-abelian Hodge correspondence for higher-dimensional quasiprojective varieties + https://arxiv.org/abs/2601.15553 + arXiv:2601.15553v1 Announce Type: new +Abstract: In arXiv:2408.16441, the authors proved that on a projective log smooth variety $(\bar{X}, D)$ there is a continuous bijection between the moduli space $M^{\mathrm{nilp}}_{\mathrm{Dol}}(\bar{X}, D)$ of logarithmic Higgs bundles with nilpotent residues and the moduli space $M^{\mathrm{nilp}}_{\mathrm{DR}}(\bar{X}, D)$ of logarithmic connections with nilpotent residues. In this note, we argue that the map is a homeomorphism. + oai:arXiv.org:2601.15553v1 + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Trevor M. Leslie, Jan Peszek + Quoc-Anh Tran - Fractional Supershifts and their associated Cauchy Evolution problems - https://arxiv.org/abs/2601.11829 - arXiv:2601.11829v1 Announce Type: new -Abstract: In this work, we extend the notion of supershifts and superoscillation sequence to fractional Fock spaces based on Gelfond-Leontiev fractional derivatives. We first introduce the fractional supershifts sequence, and then discuss the associated evolution Cauchy problem with the fractional supershifts as initial condition. - oai:arXiv.org:2601.11829v1 - math.CA - math-ph - math.FA - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 + Primes and almost primes between cubes + https://arxiv.org/abs/2601.15564 + arXiv:2601.15564v1 Announce Type: new +Abstract: In this paper we study the problem of detecting prime numbers between all consecutive cubes. Firstly, we use a large computation to show that there is always a prime between $n^3$ and $(n+1)^3$ for $n^3\leq 1.649\cdot 10^{40}$. In addition, we use this computation and a sieve-theoretic argument to show that there exists a number with at most 2 prime factors (counting multiplicity) between $n^3$ and $(n+1)^3$ for all $n\geq 1$. Our sieving argument uses a logarithmic weighting procedure attributed to Richert, which yields significant numerical improvements over previous approaches. + oai:arXiv.org:2601.15564v1 + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Natanael Alpay + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Daniel R. Johnston, Simon N. Thomas, Jonathan P. Sorenson, Jonathan E. Webster - Global Recovery from Local Data: Interior Nudging for 2D Navier-Stokes equations in a Physical Domain - https://arxiv.org/abs/2601.11831 - arXiv:2601.11831v1 Announce Type: new -Abstract: In many real-world applications of data assimilation (DA), the strategic placement of observers is crucial for effective and efficient forecasting. Motivated by practical constraints in sensor deployment, we show that global recovery of the flow field can be achieved using observations available only in a subregion of the domain, possibly far from the boundary. We focus on the two-dimensional incompressible Navier-Stokes equations posed in a bounded physical domain with Dirichlet boundary conditions. Building on the continuous data assimilation framework of Azouani, Olson, and Titi (2014), we rigorously prove that the assimilated solution converges globally to the true solution under suitable conditions on the nudging parameter, spatial resolution, and the geometry of the observation region, specifically, when the maximum distance from any point in the domain to the observational subregion is bounded by a constant multiple of \( \nu^{1/2} \) (in terms of scaling). Our computational results, conducted via finite element methods over complex geometries, support the theoretical findings and reveal even greater robustness in practice. Specifically, synchronization with the true solution is achieved even when the observational subregion lies farther from the rest of the domain than the theoretical threshold permits. Across all three tested scenarios, the local nudging algorithm performs comparably to full-domain assimilation, reaching global accuracy up to machine precision. Interestingly, observational data near the boundary are found to be largely uninformative. This demonstrates that full observability is not necessary: carefully chosen interior observations, even far from the boundary, can suffice. - oai:arXiv.org:2601.11831v1 - math.NA - cs.NA - nlin.CD - physics.flu-dyn - Wed, 21 Jan 2026 00:00:00 -0500 + Non-universality of ternary quadratic forms over fields containing $\sqrt2$ + https://arxiv.org/abs/2601.15568 + arXiv:2601.15568v1 Announce Type: new +Abstract: We prove Kitaoka's conjecture for all totally real number fields of degree 4 -- namely, there is no positive definite classical quadratic form in three variables which is universal. To achieve this, we study the fields (often without restricting the degree) where 2 is a square, because in this arguably most difficult case, the recent results connecting Kitaoka's conjecture to sums of integral squares do not apply. We also prove some other properties of ternary quadratic forms over fields containing $\sqrt2$, for example in relation to the lifting problems for universal quadratic forms and for criterion sets. + oai:arXiv.org:2601.15568v1 + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Rui Fang, Ali Pakzad + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Kristyna Kramer, Jakub Krasensky - Subspaces of $L^2(\mathbb{R}^n)$ Invariant Under Shifts by a Crystal Group - https://arxiv.org/abs/2601.11839 - arXiv:2601.11839v1 Announce Type: new -Abstract: For a crystal group $\Gamma$ in dimension $n$, a closed subspace $\mathcal{V}$ of $L^2(\mathbb{R}^n)$ is called $\Gamma$--shift invariant if, for every $f\in\mathcal{V}$, the shifts of $f$ by every element of $\Gamma$ also belong to $\mathcal{V}$. The main purpose of this paper is to provide a characterization of the $\Gamma$--shift invariant closed subspaces of $L^2(\mathbb{R}^n)$. - oai:arXiv.org:2601.11839v1 - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 + The second Delannoy category + https://arxiv.org/abs/2601.15574 + arXiv:2601.15574v1 Announce Type: new +Abstract: In recent work, Harman and Snowden constructed a symmetric tensor category associated to an oligomorphic group equipped with a measure. The oligomorphic group $\mathbb{G}$ of order preserving automorphisms of the real line admits exactly four measures. The category $\mathcal{C}$ associated to the first measure is called the (first) Delannoy category; it is semi-simple and pre-Tannakian, with numerous special properties. + In this paper, we study the (non-abelian) category $\mathcal{A}$ associated to the second measure, which we call the second Delannoy category. We construct a new pre-Tannakian category $\mathcal{D}$ together with a fully faithful tensor functor $\Psi \colon \mathcal{A} \to \mathcal{D}$. The category $\mathcal{D}$ is the correct ``abelian version'' of the second Delannoy category. Like $\mathcal{C}$, it has remarkable properties: for instance, it is non-semi-simple, but behaves uniformly in the coefficient field (e.g., it has the same Grothendieck ring and $\mathrm{Ext}^1$ quiver over any field). + Additionally, we completely solve the problem of understanding how $\mathcal{A}$ relates to general pre-Tannakian categories. We show that $\mathcal{A}$ admits exactly two local abelian envelopes: the functor $\Psi$, and a previously constructed functor $\Phi \colon \mathcal{A} \to \mathcal{C}$. This is the first case where the local envelopes of a category have been completely determined, outside of cases where there is at most one envelope. This work opens the door to constructing abelian versions of other oligomorphic tensor categories that do not admit a unique envelope. + oai:arXiv.org:2601.15574v1 + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Tom Potter, Keith Taylor + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Kevin Coulembier, Andrew Snowden - Greedily Constructing Small Quasi-Kernels - https://arxiv.org/abs/2601.11847 - arXiv:2601.11847v1 Announce Type: new -Abstract: In a digraph $D$,a quasi-kernel is an independent set $Q$ such that for every vertex $u$, there is a vertex $v \in Q$ satisfying $\text{dist}(v,u)\leq 2$. In 1974 Chv\'atal and Lov\'asz showed every digraph contains a quasi-kernel. In 1976, P. L. Erd\H{o}s and Sz\'ekely conjectured that every sourceless digraph has a quasi-kernel of order at most $\frac{n}{2}$. Despite significant recent attention by the community the problem remains far from solved, with no bound of the form $(1-\epsilon)n$ known. We introduce a polynomial time algorithm which greedily constructs a small quasi-kernel. Using this algorithm we show that if $D$ is a $\vec{K}_{1,d}$-free digraph, then $D$ has a quasi-kernel of order at most $\frac{(d^2 - 2d + 2)n}{d^2-d+1}$. By refining this argument we prove that for any $D$ with maximum out-degree $3$ this algorithm constructs a quasi-kernel of order at most ${4n}/{7}$. Finally, we consider the problem in digraphs forbidding certain orientation of short cycles as subgraphs, concluding that all orientations $D$ of a graph $G$ with girth at least $7$ have a quasi-kernel of order at most $\frac{(d^2+4)n}{(d+2)^2}$, where $d$ is the maximum out-degree of $D$. - oai:arXiv.org:2601.11847v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + Open problems in K-stability of Fano varieties + https://arxiv.org/abs/2601.15576 + arXiv:2601.15576v1 Announce Type: new +Abstract: In this note, we discuss a number of open problems in K-stability theory. + oai:arXiv.org:2601.15576v1 + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Alexander Clow + Chenyang Xu, Ziquan Zhuang - Weighted fractional ultrahyperbolic diffusion on geometrically deformed domains - https://arxiv.org/abs/2601.11851 - arXiv:2601.11851v1 Announce Type: new -Abstract: Standard fractional models on manifolds often conflate geometric anisotropy with medium heterogeneity. In this Letter, we overcome this rigidity by deriving the fundamental solution for a weighted space-time fractional ultrahyperbolic operator, denoted by $(-\Box_{\phi,\omega})^{\beta}$. Using a novel spectral approach based on the Weighted Fourier Transform, we explicitly \textbf{decouple the medium density from the geometric deformation}. A crucial finding is the emergence of a \textbf{geometry-independent drift mechanism} driven purely by the inhomogeneity of the medium. The Green's function is obtained in closed form via the Fox H-function, providing a unified and computable framework for anomalous transport in complex, structurally deformed media. - oai:arXiv.org:2601.11851v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Global solution curves for first order periodic problems, with applications + https://arxiv.org/abs/2601.15579 + arXiv:2601.15579v1 Announce Type: new +Abstract: Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for periodic problems of first order. The results are applied to a population model with fishing, and to the existence and stability of limit cycles. We also describe in detail our numerical computations of curves of periodic solutions, and of limit cycles. + oai:arXiv.org:2601.15579v1 + math.DS + math.CA + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gustavo Dorrego + http://creativecommons.org/licenses/by/4.0/ + Nonlinear Anal., Real World Appl. 56, Article ID 103161, 18 p. (2020) + Philip Korman, Dieter S. Schmidt - Global weak solutions to the isentropic compressible Navier--Stokes equations with vacuum and unbounded density in a half-plane under Dirichlet boundary conditions - https://arxiv.org/abs/2601.11852 - arXiv:2601.11852v1 Announce Type: new -Abstract: We establish the global existence of a class of weak solutions to the isentropic compressible Navier--Stokes equations in a half-plane with Dirichlet boundary conditions, allowing for vacuum both in the interior and at infinity, under a suitably small initial total energy. The solutions constructed here admit unbounded densities and lie in an intermediate regularity regime between the finite-energy weak solutions of Lions--Feireisl and the framework of Hoff. This result generalizes previous works of Hoff (Comm. Pure Appl. Math. 55 (2002), pp. 1365--1407) and Perepelitsa (Arch. Ration. Mech. Anal. 212 (2014), pp. 709--726) concerning discontinuous solutions by allowing vacuum states and unbounded density. Our analysis relies on the Green function method and new estimates involving the specific structure of the equations and the geometry of the half-plane. To the best of our knowledge, this is the first result concerning global weak solutions within Hoff's framework on an unbounded domain that simultaneously accommodates Dirichlet boundary conditions and far-field vacuum. The intermediate-regularity class developed here may be viewed as a natural extension of Hoff's theory, precisely tailored to overcome the two corresponding obstructions: the lack of global space-time control of the effective viscous flux arising from far-field vacuum and the absence of boundary-induced regularity gains in the no-slip setting. - oai:arXiv.org:2601.11852v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + A Wild Steiner-Lehmus Chase + https://arxiv.org/abs/2601.15591 + arXiv:2601.15591v1 Announce Type: new +Abstract: We present a proof the Steiner-Lehmus equal bisectors theorem by applying the Law of sines in rapid succession to a side-by-side comparison. For nearly two centuries, the quest for a direct proof has sustained interest in proving and reproving this theorem. We suggest that a second driving force may also be at play. + oai:arXiv.org:2601.15591v1 + math.HO + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Shuai Wang, Xin Zhong + http://creativecommons.org/licenses/by/4.0/ + 10.1080/00255572.2025.2594896 + Mathematical Gazette, published online 20 January, 2026 + Eric L. Grinberg, Mehmet Z. Orhon - New examples of twisted Brill-Noether loci II - https://arxiv.org/abs/2601.11855 - arXiv:2601.11855v1 Announce Type: new -Abstract: Our purpose in this paper is to construct new examples of twisted Brill Noether loci on curves of genus g greater than 2 with negative expected dimension. We begin by completing the proof of Butler's conjecture for coherent systems of certain type establishing the birationality, smoothness, and irreducibility of the corresponding loci. We also produce new points on the BN map. - oai:arXiv.org:2601.11855v1 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 + Overpartitions with repeated smallest non-overlined part + https://arxiv.org/abs/2601.15601 + arXiv:2601.15601v1 Announce Type: new +Abstract: Inspired by Andrews' and Bachraoui's work on partitions with repeated smallest part, we extend the concept to overpartitions. + oai:arXiv.org:2601.15601v1 + math.CO + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - L. Brambila-Paz, P. E. Newstead + http://creativecommons.org/licenses/by/4.0/ + Amita Malik, Rishabh Sarma - On the R\'enyi Rate-Distortion-Perception Function and Functional Representations - https://arxiv.org/abs/2601.11862 - arXiv:2601.11862v1 Announce Type: new -Abstract: We extend the Rate-Distortion-Perception (RDP) framework to the R\'enyi information-theoretic regime, utilizing Sibson's $\alpha$-mutual information to characterize the fundamental limits under distortion and perception constraints. For scalar Gaussian sources, we derive closed-form expressions for the R\'enyi RDP function, showing that the perception constraint induces a feasible interval for the reproduction variance. Furthermore, we establish a R\'enyi-generalized version of the Strong Functional Representation Lemma. Our analysis reveals a phase transition in the complexity of optimal functional representations: for $0.5<\alpha < 1$, the coding cost is bounded by the $\alpha$-divergence of order $\alpha+1$, necessitating a codebook with heavy-tailed polynomial decay; conversely, for $\alpha > 1$, the representation collapses to one with finite support, offering new insights into the compression of shared randomness under generalized notions of mutual information. - oai:arXiv.org:2601.11862v1 + On the Nonasymptotic Scaling Guarantee of Hyperparameter Estimation in Inhomogeneous, Weakly-Dependent Complex Network Dynamical Systems + https://arxiv.org/abs/2601.15603 + arXiv:2601.15603v1 Announce Type: new +Abstract: Hierarchical Bayesian models are increasingly used in large, inhomogeneous complex network dynamical systems by modeling parameters as draws from a hyperparameter-governed distribution. However, theoretical guarantees for these estimates as the system size grows have been lacking. A critical concern is that hyperparameter estimation may diverge for larger networks, undermining the model's reliability. Formulating the system's evolution in a measure transport perspective, we propose a theoretical framework for estimating hyperparameters with mean-type observations, which are prevalent in many scientific applications. Our primary contribution is a nonasymptotic bound for the deviation of estimate of hyperparameters in inhomogeneous complex network dynamical systems with respect to network population size, which is established for a general family of optimization algorithms within a fixed observation duration. While we firstly establish a consistency result for systems with independent nodes, our main result extends this guarantee to the more challenging and realistic setting of weakly-dependent nodes. We validate our theoretical findings with numerical experiments on two representative models: a Susceptible-Infected-Susceptible model and a Spiking Neuronal Network model. In both cases, the results confirm that the estimation error decreases as the network population size increases, aligning with our theoretical guarantees. This research proposes the foundational theory to ensure that hierarchical Bayesian methods are statistically consistent for large-scale inhomogeneous systems, filling a gap in this area of theoretical research and justifying their application in practice. + oai:arXiv.org:2601.15603v1 + math.ST cs.IT math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + stat.ML + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Jiahui Wei, Marios Kountouris + http://creativecommons.org/licenses/by/4.0/ + Yi Yu, Yubo Hou, Yinchong Wang, Nan Zhang, Jianfeng Feng, Wenlian Lu - Open book decompositions with page a four-punctured sphere - https://arxiv.org/abs/2601.11871 - arXiv:2601.11871v1 Announce Type: new -Abstract: In this paper, we study contact structures supported by open book decompositions whose pages are four-punctured spheres. The paper is split into two parts. In the first part, we find infinitely many overtwisted, right-veering monodromies on the four-punctured sphere. This is done using the techniques developed by Ito-Kawamuro in the papers arXiv:1112.5874, arXiv:1310.6404. Although most of the monodromies that we show are overtwisted are pseudo-Anosov, we are also able to classify precisely which reducible monodromies on the four-punctured sphere are tight. In the second part of the paper, we reprove part of a result of Lekili arXiv:1008.3529 by classifying which reducible mondromies have non-zero Heegaard Floer invariant. This is done by using the bordered contact invariants of Min-Varvarezos arXiv:2410.05511. - oai:arXiv.org:2601.11871v1 - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 + Barcode entropy and relative symplectic cohomology + https://arxiv.org/abs/2601.15606 + arXiv:2601.15606v1 Announce Type: new +Abstract: In this paper, we study the barcode entropy--the exponential growth rate of the number of not-too-short bars--of the persistence module associated with the relative symplectic cohomology $SH_M(K)$ of a Liouville domain $K$ embedded in a symplectic manifold $M$. Our main result establishes a quantitative link between this Floer-theoretic invariant and the dynamics of the Reeb flow on $\partial K$. More precisely, we show that the barcode entropy of the relative symplectic cohomology $SH_M(K)$ is bounded above by a constant multiple of the topological entropy of the Reeb flow on the boundary of the domain, where the constant depends on the embedding of $K$ into $M$. + oai:arXiv.org:2601.15606v1 + math.SG + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Harahm Park + http://creativecommons.org/licenses/by/4.0/ + Jonghyeon Ahn - Simple, subdirectly irreducible weakly dicomplemented lattices - https://arxiv.org/abs/2601.11873 - arXiv:2601.11873v1 Announce Type: new -Abstract: In this work, we exhibit several subclasses of weakly dicomplemented lattices (WDLs) based on their skeletons and dual skeletons. We investigate normal filters (resp. ideals) and show that the set of normal filters (resp. ideals) forms a complete lattice, which is not a sublattice of the lattice of all filters (ideals). The normal filter (ideal) generated by a subset and the join of two normal filters (resp. ieals) are characterized. We further prove that the lattice of normal filters is isomorphic to the lattice of normal ideals, and that the only class of filters (or ideals) that generate a congruence in WDLs is the class of normal filters. For distributive WDLs, the congruences generated by filters are characterized. Using normal filters, we characterize simple, subdirectly irreducible, and regular WDLs. Moreover, it is shown that the congruences generated by normal filters are permutable, and that regular distributive WDLs are congruence-permutable and verify the congruence extension property (CEP). Finally, we prove that, under certain conditions, the lattice of normal filters is isomorphic to the lattice of filters of the Boolean center of a distributive WDL. It is also established that the lattice of normal filters of a WDL $L$ embeds into the lattice of normal filters of the power $L^{X}$ of $L$. - oai:arXiv.org:2601.11873v1 - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 + Lead distance under a pickoff limit in Major League Baseball: A sequential game model + https://arxiv.org/abs/2601.15608 + arXiv:2601.15608v1 Announce Type: new +Abstract: Major League Baseball (MLB) recently limited pitchers to three pickoff attempts, creating a cat-and-mouse game between pitcher and runner. Each failed attempt adds pressure on the pitcher to avoid using another, and the runner can intensify this pressure by extending their leadoff toward the next base. We model this dynamic as a two-player zero-sum sequential game in which the runner first chooses a lead distance, and then the pitcher chooses whether to attempt a pickoff. We establish optimality characterizations for the game and present variants of value iteration and policy iteration to solve the game. Using lead distance data, we estimate generalized linear mixed-effects models for pickoff and stolen base outcome probabilities given lead distance, context, and player skill. We compute the game-theoretic equilibria under the two-player model, as well as the optimal runner policy under a simplified one-player Markov decision process (MDP) model. In the one-player setting, our results establish an actionable rule of thumb: the Two-Foot Rule, which recommends that a runner increase their lead by two feet after each pickoff attempt. + oai:arXiv.org:2601.15608v1 + math.OC + stat.AP + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Yannick Lea Tenkeu Jeufack, Leonard Kwuida + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Scott Powers, Sivaramakrishnan Ramani, Jacob Hahn, Andrew J. Schaefer - On the eigenvalues of cyclic covers of Paley graphs - https://arxiv.org/abs/2601.11877 - arXiv:2601.11877v1 Announce Type: new -Abstract: We study covering graphs of the Paley graph associated to a finite field of characteristic p in the case where the covering transformation group is cyclic of prime order distinct from p. When the field has q = p elements, we show that the eigenvalues of the adjacency matrix determine the graph isomorphism class among translation invariant covers. When q = p^r > p, we construct examples of cospectral covering graphs that are not isomorphic as graphs. - oai:arXiv.org:2601.11877v1 - math.CO + On the Zeros of the Riemann Zeta Function with Two Ordinate Shifts + https://arxiv.org/abs/2601.15610 + arXiv:2601.15610v1 Announce Type: new +Abstract: We prove that for any fixed real numbers y_1, y_2 not equal to 0, and constant C > 0, there exists a threshold T_* = T_*(y_1, y_2, C) > 0 such that for all T >= T_*, the interval [T, T(1 + epsilon)], with epsilon = exp(-C sqrt(log T)), contains at least one gamma satisfying zeta(1/2 + i gamma) = 0, zeta(1/2 + i (gamma + y_1)) != 0, and zeta(1/2 + i (gamma + y_2)) != 0. + This extends earlier work by Banks (for a single shift y) to two distinct shifts y_1, y_2. Our argument is based on the behavior of zeta and L functions in zero-free regions via Perron's formula. + oai:arXiv.org:2601.15610v1 math.NT - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Natalie Dinin, John A. Lind + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Ali Ebadi - Lowest eigenvalues and higher order elliptic differential operators - https://arxiv.org/abs/2601.11882 - arXiv:2601.11882v1 Announce Type: new -Abstract: Let $(M,g)$ be a closed, smooth Riemannian manifold of dimension $m \geq 1$. It is not difficult to produce an example of an elliptic differential operator on $(M,g)$ that has the property that there exists a sign-changing eigenfunction that is associated with the lowest eigenvalue. Indeed, $\Delta_g^2 + \lambda_2 \Delta_g$ does the job, where $\Delta_g:=div_g \nabla_g$. and where $\lambda_2$ is the second lowest eigenvalue of the operator $-\Delta_g$. The question that remains is how rare are elliptic differential operators whose lowest eigenvalue has this property. In this paper, the author proves that elliptic operators of the form $\Delta_g^2 - div_g(T-\lambda_2 g^{-1}) d$, where $T$ is a negative semi-definite $(2,0)$-tensor field on $M$, and where $g^{-1}$ is the inverse metric tensor, have the property that there exists a sign-changing eigenfunction that is associated with the lowest eigenvalue of the operator. This suggests that there are a lot of fourth-order elliptic operators with the property that there exists a sign-changing eigenfunction that is associated with the lowest eigenvalue of the operator. - oai:arXiv.org:2601.11882v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Degree-choosability of proper conflict-free list coloring of sparse graphs + https://arxiv.org/abs/2601.15611 + arXiv:2601.15611v1 Announce Type: new +Abstract: Given a graph $G$ and a mapping $f:V(G) \to \mathbb{N}$, an $f$-list assignment of $G$ is a function that maps each $v \in V(G)$ to a set of at least $f(v)$ colors. For an $f$-list assignment $L$ of a graph $G$, a proper conflict-free $L$-coloring of $G$ is a proper coloring $\phi$ of $G$ such that for every vertex $v \in V(G)$, $\phi(v) \in L(v)$ and some appears precisely once in the neighborhood of $v$. We say that $G$ is proper conflict-free $f$-choosable if for every $f$-list assignment $L$ of $G$, there exists a proper conflict-free $L$-coloring of $G$. If $G$ is proper conflict-free $f$-choosable and there is a constant $k$ such that $f(v)= d_G(v)+k$ for every vertex $v$ of $G$, then we say $G$ is proper conflict-free $({\rm degree}+k)$-choosable. In this paper, we consider graphs with a bounded maximum average degree. We show that every graph with the maximum average degree less than $\frac{10}{3}$ is proper conflict-free $({\rm degree}+3)$-choosable, and that every graph with the maximum average degree less than $\frac{18}{7}$ is proper conflict-free $({\rm degree}+2)$-choosable. As a result, every planar graph with girth at least $5$ is proper conflict-free $({\rm degree}+3)$-choosable, and every planar graph with girth at least $9$ is proper conflict-free $({\rm degree}+2)$-choosable. + oai:arXiv.org:2601.15611v1 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - David Raske + Masaki Kashima, Riste \v{S}krekovski, Rongxing Xu - Analytic Regularization of a Ramanujan Machine Conjecture - https://arxiv.org/abs/2601.11892 - arXiv:2601.11892v1 Announce Type: new -Abstract: We provide a formal analytic derivation of a continued fraction identity for $-\pi/4$ recently conjectured by the Ramanujan Machine~\cite{Raayoni2021}. By utilizing the contiguous relations of the Gauss hypergeometric function ${}_2F_1(a, b; c; z)$, we establish that the conjectured polynomial architecture is a regularized representation of the transcendental ratio $\mathcal{R}(1/2, 0, 1/2; -1)$. Through an explicit equivalence transformation $\mathcal{T}$ defined by a linear scaling sequence, we map the Gaussian unit-denominator expansion to the conjectured form, thereby recovering the quadratic partial numerators $(n-1)^2$ and linear partial denominators $-(2n-1)$. Convergence is rigorously established via the limit-periodicity of the transformed coefficients, which reside on the Worpitzky boundary $L=1/4$. - oai:arXiv.org:2601.11892v1 - math.GM - Wed, 21 Jan 2026 00:00:00 -0500 + Lucas sequences, Pell's equations, and automorphisms of K3 surfaces + https://arxiv.org/abs/2601.15617 + arXiv:2601.15617v1 Announce Type: new +Abstract: We have the correspondences between Lucas sequences, Pell's equations, and the automorphisms of K3 surfaces with Picard number 2. Using these correspondences, we determine the intersections of some Lucas sequences. + oai:arXiv.org:2601.15617v1 + math.AG + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Chao Wang + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.1007/s11139-025-01257-6 + The Ramanujan Journal (2025) 68:108 + Kwangwoo Lee - A Separable and Asymptotic-Preserving Dynamical Low-Rank Method for the Vlasov--Poisson--Fokker--Planck System - https://arxiv.org/abs/2601.11900 - arXiv:2601.11900v1 Announce Type: new -Abstract: We present a dynamical low-rank (DLR) method for the Vlasov--Poisson--Fokker--Planck (VPFP) system. Our main contributions are two-fold: (i) a conservative spatial discretization of the Fokker--Planck operator that factors into velocity-only and space-only components, enabling efficient low-rank projection, and (ii) a time discretization within the DLR framework that properly handles stiff collisions. We propose both first-order and second-order low-rank IMEX schemes. For the first-order scheme, we prove an asymptotic-preserving (AP) property when the field fluctuation is small. Numerical experiments demonstrate accuracy, robustness, and AP property at modest ranks. - oai:arXiv.org:2601.11900v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 + Existence and uniqueness of $L^1$-solutions to time-fractional nonlinear diffusion equations + https://arxiv.org/abs/2601.15618 + arXiv:2601.15618v1 Announce Type: new +Abstract: We establish the global existence and uniqueness of $L^1$-solutions to the Cauchy problem for time-fractional porous medium type nonlinear diffusion equations. Furthermore, we give the mass conservation law for $L^1$-solutions to time-fractional fast diffusion equations, and prove that the finite-time extinction does not occur for any nonnegative $L^1$-solutions. + oai:arXiv.org:2601.15618v1 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Shiheng Zhang, Jingwei Hu + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mikiya Kametaka, Tatsuki Kawakami - The Inverse Symplectic Eigenvalue Problem of a Graph - https://arxiv.org/abs/2601.11912 - arXiv:2601.11912v1 Announce Type: new -Abstract: Symplectic geometry plays an increasingly important role in mathematics, physics and applications, and naturally gives rise to interesting matrix families and properties. One of these is the notion of symplectic eigenvalues, whose existence for positive definite matrices is known as Williamson's theorem or decomposition. This notion of symplectic eigenvalues gives rise to inverse problems. We introduce the inverse symplectic eigenvalue problem for positive definite matrices described by a labeled graph and solve it for several families of labeled graphs and all labeled graphs of order four. To solve these problems we develop various tools such as the Strong Symplectic Spectral Property (SSSP) and its consequences such as the Supergraph Theorem, the Bifurcation Theorem, and the Matrix Liberation Lemma for symplectic eigenvalues, graph couplings to describe collections of labelings of a graph that produce the same symplectic eigenvalues, and coupled graph zero forcing. We establish numerous results for symplectic positive definite matrices, including a sharp lower bound on the number of nonzero entries of such a matrix (or equivalently, the number of edges in its graph). This lower bound is a consequence of a lower bound on the sum of number of nonzero entries in an irreducible positive definite matrix and its inverse. - oai:arXiv.org:2601.11912v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + Limit behavior of linearly edge-reinforced random walks on the half-line + https://arxiv.org/abs/2601.15627 + arXiv:2601.15627v1 Announce Type: new +Abstract: Motivated by the article [M. Takei, Electron. J. Probab. 26 (2021), article no. 104], we study the limit behavior of linearly edge-reinforced random walks on the half-line $\mathbb{Z}_+$ with reinforcement parameter $\delta>0$, and each edge $\{x,x+1\}$ has the initial weight $x^{\alpha}\ln^{\beta}x$ for $x > 1$ and $1$ for $x = 0, 1$. The aim of this paper is to study the almost sure limit behavior of the walk in the recurrent regime, and extend the results of Takei mentioned above. + oai:arXiv.org:2601.15627v1 + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Himanshu Gupta, Leslie Hogben, Bryan Shader, Tony Wong + Zechun Hu, Renming Song, Li Wang - Rate-Distortion-Classification Representation Theory for Bernoulli Sources - https://arxiv.org/abs/2601.11919 - arXiv:2601.11919v1 Announce Type: new -Abstract: We study task-oriented lossy compression through the lens of rate-distortion-classification (RDC) representations. The source is Bernoulli, the distortion measure is Hamming, and the binary classification variable is coupled to the source via a binary symmetric model. Building on the one-shot common-randomness formulation, we first derive closed-form characterizations of the one-shot RDC and the dual distortion-rate-classification (DRC) tradeoffs. We then use a representation-based viewpoint and characterize the achievable distortion-classification (DC) region induced by a fixed representation by deriving its lower boundary via a linear program. Finally, we study universal encoders that must support a family of DC operating points and derive computable lower and upper bounds on the minimum asymptotic rate required for universality, thereby yielding bounds on the corresponding rate penalty. Numerical examples are provided to illustrate the achievable regions and the resulting universal RDC/DRC curves. - oai:arXiv.org:2601.11919v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + The $V_1$- and $V_2$-polynomials of a long virtual knot + https://arxiv.org/abs/2601.15634 + arXiv:2601.15634v1 Announce Type: new +Abstract: We introduce two polynomial invariants $V_1(K;t)$ and $V_2(K;t)$ of a long virtual knot $K$, which generalize the degree-two finite type invariants $v_{2,1}$ and $v_{2,2}$ of Goussarov, Polyak, and Viro. We establish their fundamental properties and show that any pair of Laurent polynomials can be realized as $(V_1(K;t),V_2(K;t))$ for some long virtual knot $K$. While these polynomials are not finite type invariants of any degree with respect to virtualizations, their first derivatives at $t=1$ define finite type invariants of degree three. As an application, we obtain an explicit Gauss diagram formula for the $\alpha_3$-invariant. + oai:arXiv.org:2601.15634v1 + math.GT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Nam Nguyen, Thinh Nguyen, Bella Bose + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Shin Satoh, Kodai Wada - Phase-IDENT: Identification of Two-phase PDEs with Uncertainty Quantification - https://arxiv.org/abs/2601.11922 - arXiv:2601.11922v1 Announce Type: new -Abstract: We propose a novel method, Phase-IDENT, for identifying partial differential equations (PDEs) from noisy observations of dynamical systems that exhibit phase transitions. Such phenomena are prevalent in fluid dynamics and materials science, where they can be modeled mathematically as functions satisfying different PDEs within distinct regions separated by phase boundaries. Our approach simultaneously identifies the underlying PDEs in each regime and accurately reconstructs the phase boundaries. Furthermore, by incorporating change point detection techniques, we provide uncertainty quantification for the detected boundaries, enhancing the interpretability and robustness of our method. We conduct numerical experiments on a variety of two-phase PDE systems under different noise levels, and the results demonstrate the effectiveness of the proposed approach. - oai:arXiv.org:2601.11922v1 - math.NA - cs.NA - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 + Collaboration versus Specialization in Service Systems with Impatient Customers + https://arxiv.org/abs/2601.15636 + arXiv:2601.15636v1 Announce Type: new +Abstract: We study tandem queueing systems in which servers work more efficiently in teams than on their own and customers are impatient in that they may leave the system while waiting for service. Our goal is to determine the server assignment policy that maximizes the long-run average throughput. We show that when each server is equally skilled at all tasks, the optimal policy has all the servers working together at all times. We also provide a complete characterization of the optimal policy for Markovian systems with two stations and two servers when each server's efficiency may be task dependent. We show that the throughput is maximized under the policy which assigns one server to each station (based on their relative skill at that station) unless station 2 has no work (in which case both servers work at station 1) or the number of customers in the buffer reaches a threshold whose value we characterize (in which case both servers work at station 2). We study how the optimal policy varies with the level of server synergy (including no synergy) and also compare the optimal policy for systems with different customer abandonment rates (including no abandonments). Finally, we investigate the case where the synergy among collaborating servers can be task-dependent and provide numerical results. + oai:arXiv.org:2601.15636v1 + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Edward L. Yang, Roy Y. He + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Bihan Chatterjee, Sigr\'un Andrad\'ottir, Hayriye Ayhan - Exact Redundancy for Symmetric Rate-Distortion - https://arxiv.org/abs/2601.11927 - arXiv:2601.11927v1 Announce Type: new -Abstract: For variable-length coding with an almost-sure distortion constraint, Zhang et al. show that for discrete sources the redundancy is upper bounded by $\log n/n$ and lower bounded (in most cases) by $\log n/(2n)$, ignoring lower order terms. For a uniform source with a distortion measure satisfying certain symmetry conditions, we show that $\log n/(2n)$ is achievable and that this cannot be improved even if one relaxes the distortion constraint to be in expectation rather than with probability one. - oai:arXiv.org:2601.11927v1 + A Class of Subadditive Information Measures and their Applications + https://arxiv.org/abs/2601.15639 + arXiv:2601.15639v1 Announce Type: new +Abstract: We introduce a two-parameter family of discrepancy measures, termed \emph{$(G,f)$-divergences}, obtained by applying a non-decreasing function $G$ to an $f$-divergence $D_f$. Building on Csisz\'ar's formulation of mutual $f$-information, we define a corresponding $(G,f)$-information measure $ +I_{G,f}(X;Y)$. A central theme of the paper is subadditivity over product distributions and product channels. We develop reduction principles showing that, for broad classes of $G$, it suffices to verify divergence subadditivity on binary alphabets. Specializing to the functions $G(x)\in\{x,\log(1+x),-\log(1-x)\}$, we derive tractable sufficient conditions on $f$ that guarantee subadditivity, covering many standard $f$-divergences. Finally, we present applications to finite-blocklength converses for channel coding, bounds in binary hypothesis testing, and an extension of the Shannon--Gallager--Berlekamp sphere-packing exponent framework to subadditive $(G,f)$-divergences. + oai:arXiv.org:2601.15639v1 cs.IT math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Sharang M. Sriramu, Aaron B. Wagner + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Hamidreza Abin, Mahdi Zinati, Amin Gohari, Mohammad Hossein Yassaee, Mohammad Mahdi Mojahedian - Classification of connected proper pairs in the affine transformation group - https://arxiv.org/abs/2601.11933 - arXiv:2601.11933v1 Announce Type: new -Abstract: Let $(L, H)$ be closed subgroups of a locally compact group $G$. The pair $(L, H)$ is said to be proper if the action of $L$ on the homogeneous space $G/H$ is proper. - We give a complete list of connected closed proper pairs in the affine transformation group of $\mathbb{R}^2$. This result extends Kobayashi's classification (1992) of connected closed subgroups of the affine transformation group of $\mathbb{R}^2$acting properly on $\mathbb{R}^2$. - oai:arXiv.org:2601.11933v1 - math.GR - Wed, 21 Jan 2026 00:00:00 -0500 + Generative AI-Empowered Semantic Twin Channel Model for ISAC + https://arxiv.org/abs/2601.15642 + arXiv:2601.15642v1 Announce Type: new +Abstract: Integrated sensing and communication (ISAC) increasingly exposes a gap in today's channel modeling. Efficient statistical models focus on coarse communication-centric metrics, and therefore miss the weak but critical multipath signatures for sensing, whereas deterministic models are computationally inefficient to scale for system-level ISAC evaluation. This gap calls for a unifying abstraction that can couple what the environment means for sensing with how the channel behaves for communication, namely, environmental semantics. This article clarifies the meaning and essentiality of environmental semantics in ISAC channel modeling and establishes how semantics is connected to observable channel structures across multiple semantic levels. Based on this perspective, a semantics-oriented channel modeling principle was advocated, which preserves environmental semantics while abstracting unnecessary detail to balance accuracy and complexity. Then, a generative AI-empowered semantic twin channel model (STCM) was introduced to generate a family of physically plausible channel realizations representative of a semantic condition. Case studies further show semantic consistency under challenging multi-view settings, suggesting a practical path to controllable simulation, dataset generation, and reproducible ISAC benchmarking toward future design and standardization. + oai:arXiv.org:2601.15642v1 + cs.IT + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Shunsuke Miyauchi + Yi Chen, Yatao Hu, Ming Li, Chong Han - The nonlinear estimates on quantum Besov space - https://arxiv.org/abs/2601.11934 - arXiv:2601.11934v1 Announce Type: new -Abstract: The superposition operators have been widely studied in nonlinear analysis, which are essential for the well-posedness theory of nonlinear equations. In this paper, we investigate the boundedness estimates of superposition operators with non-smooth symbols on quantum Besov spaces, which significantly generalize McDonald's results \cite{McNLE} for infinitely differentiable symbols and have rich applications in the well-posedness theory of noncommutative PDEs. As a byproduct, we prove the equivalence of the two descriptions of quantum Besov spaces, resolving the conjecture proposed in \cite[Remark 3.16]{McNLE}. The new ingredients in the proof also involve quantum chain rule and nonlinear interpolation. - oai:arXiv.org:2601.11934v1 - math.FA + Linear stability of the first bifurcation in a tumor growth free boundary problem via local bifurcation structure + https://arxiv.org/abs/2601.15647 + arXiv:2601.15647v1 Announce Type: new +Abstract: In this paper, we consider a 3-dimensional free boundary problem modeling tumor growth with the Robin boundary condition. The system involves a positive parameter $\mu$ which reflects the intensity of tumor aggressiveness. Huang, Zhang and Hu [Nonlinear Anal. Real World Appl. 2017(35), 483-502] have shown that for each $\mu_n$ ($n$ even) in a strictly increasing sequence $\{ \mu_n \}(n\geq 2)$, there exists a stationary bifurcation solution $(\sigma_n(\varepsilon),p_n(\varepsilon),r_n(\varepsilon))$ with $\mu = \mu_n(\varepsilon)$ bifurcating from $\mu_n$. We first derive that the bifurcation curve $(r_2(\varepsilon),\mu_2(\varepsilon))$ exhibits a transcritical bifurcation with $\mu_2'(0)<0$. Moreover, we show that the stationary bifurcation solution $(\sigma_2(\varepsilon),p_2(\varepsilon),r_2(\varepsilon))$ is linearly unstable for small $|\varepsilon|$ under non-radially symmetric perturbations. In contrast to the linear stability of the radially symmetric stationary solution, the lack of explicit expressions for bifurcation solutions adds great difficulty in analyzing their linear stability. The novelty of this paper lies in the use of the bifurcation curve's structure to overcome the above difficulties. Moreover, this linear stability result is not established using the standard method, due to an eight-dimensional generalized kernel at eigenvalue 0 for the linearized operator. + oai:arXiv.org:2601.15647v1 math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Deyu Chen, Guixiang Hong + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Junying Chen, Ruixiang Xing - Small-Error Cascaded Group Testing - https://arxiv.org/abs/2601.11945 - arXiv:2601.11945v1 Announce Type: new -Abstract: Group testing concerns itself with the accurate recovery of a set of "defective" items from a larger population via a series of tests. While most works in this area have considered the classical group testing model, where tests are binary and indicate the presence of at least one defective item in the test, we study the cascaded group testing model. In cascaded group testing, tests admit an ordering, and test outcomes indicate the first defective item in the test under this ordering. Under this model, we establish various achievability bounds for several different recovery criteria using both non-adaptive and adaptive (with "few" stages) test designs. - oai:arXiv.org:2601.11945v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + Iterative Derivations on Central Simple Algebras + https://arxiv.org/abs/2601.15648 + arXiv:2601.15648v1 Announce Type: new +Abstract: We prove that an iterative derivation $\delta_F$ on a field $F$ can be extended to an iterative derivation $\delta_A$ on a central simple $F-$algebra $A$ if the characteristic of $F$ does not divide the exponent of $A$ in the Brauer group of $F.$ For a central simple $F-$algebra with an iterative derivation, we show the existence of a unique (up to isomorphism) Picard-Vessiot splitting field and from the nature its Galois group, we also describe the structure of the central simple algebra in terms of its $\delta_A-$right ideals. + oai:arXiv.org:2601.15648v1 + math.RA + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Daniel McMorrow, Nikhil Karamchandani, Sidharth Jaggi + http://creativecommons.org/licenses/by/4.0/ + Manujith K. Michel, Varadharaj R. Srinivasan - Observer design and boundary output feedback stabilization for semilinear parabolic system over general multidimensional domain - https://arxiv.org/abs/2601.11948 - arXiv:2601.11948v1 Announce Type: new -Abstract: This paper investigates the output feedback stabilization of parabolic equation with Lipschitz nonlinearity over general multidimensional domain using spectral geometry theories. First, a novel nonlinear observer is designed, and the error system is shown to achieve any prescribed decay rate by leveraging the Berezin-Li-Yau inequality from spectral geometry, which also provides effective guidance for sensor placement. Subsequently, a finite-dimensional state feedback controller is proposed, which ensures the quantitative rapid stabilization of the linear part. By integrating this control law with the observer, an efficient boundary output feedback control strategy is developed. The feasibility of the proposed control design is rigorously verified for arbitrary Lipschitz constants, thereby resolving a persistent theoretical challenge. Finally, a numerical case study confirms the effectiveness of the approach. - oai:arXiv.org:2601.11948v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + An index theory for transverse trajectories + https://arxiv.org/abs/2601.15651 + arXiv:2601.15651v1 Announce Type: new +Abstract: In this work, we present an alternative definition of the Le Roux index, which generalizes the Poincar\'e-Hopf index for non-singular planar flows to the broader setting of Brouwer homeomorphisms. This new approach answers a question raised by Le Roux by establishing a connection between the index of a Brouwer homeomorphism and the structure of its transverse foliations, in the sense of Le Calvez. + oai:arXiv.org:2601.15651v1 + math.DS + math.GN + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kai Liu, Hua-Cheng Zhou, Zhong-Jie Han, Xiangyang Peng + Nelson Schuback - Computations of higher elliptic units - https://arxiv.org/abs/2601.11961 - arXiv:2601.11961v1 Announce Type: new -Abstract: In this paper we present a conjecture on the construction of generalised elliptic units above number fields with exactly one complex place. These elliptic units obtained as values of multiple elliptic Gamma functions. These form a collection of multivariate meromorphic functions which were studied in the late 1990s and early 2000s in mathematical physics. Our construction extends the scheme of a recent article by Bergeron, Charollois and Garc\'ia where they constructed conjectural elliptic units above complex cubic fields using the elliptic Gamma function. The elliptic units we construct are expected to generate specific abelian extensions of the base field where they are evaluated, thus giving a conjectural solution to Hilbert's 12th problem for the number fields with exactly one complex place. We provide several examples to support our conjecture in optimal cases for cubic, quartic and quintic fields. - oai:arXiv.org:2601.11961v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 + Construction and Box-counting Dimension of the Edelstein Hidden Variable Fractal Interpolation Function + https://arxiv.org/abs/2601.15658 + arXiv:2601.15658v1 Announce Type: new +Abstract: This paper presents the construction of a hidden variable fractal interpolation function using Edelstein contractions in an iterated function system based on a finite collection of data points. The approach incorporates an iterated function system where variable functions act as vertical scaling factors leading to a generalised vector-valued fractal interpolation function. Furthermore, the paper rigorously examines the smoothness of the constructed function and establishes an upper bound for the box-counting dimension of its graph. + oai:arXiv.org:2601.15658v1 + math.DS + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Pierre L. L. Morain + Aiswarya T, Srijanani Anurag Prasad - A Survey on Spherical Designs: Existence, Numerical Constructions, and Applications - https://arxiv.org/abs/2601.11963 - arXiv:2601.11963v1 Announce Type: new -Abstract: This paper provides a survey of spherical designs and their applications, with a particular emphasis on the perspective of ``numerical analysis''. A set \(X_N\) of \(N\) points on the unit sphere \(\mathbb{S}^d\) is called a \textit{spherical \(t\)-design} if the average value of any polynomial of degree at most \(t\) over \(X_N\) equals its average over the entire sphere. Spherical designs represent one of the most significant topics in the study of point distributions on spheres. They are deeply connected to algebraic combinatorics, discrete geometry, differential geometry, approximation theory, optimization, coding theory, quantum physics, and other fields, which have led to the development of profound and elegant mathematical theories. This article reviews fundamental theoretical results, numerical construction methods, and applied outcomes related to spherical designs. Key topics covered include existence proofs, optimization-based construction techniques, fast computational algorithms, and applications in interpolation, numerical integration, hyperinterpolation, signal and image processing, as well as numerical solutions to partial differential and integral equations. - oai:arXiv.org:2601.11963v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Congpei An, Xiaosheng Zhuang - - - On efficient estimates of the rate of convergence for Markov chains - https://arxiv.org/abs/2601.11973 - arXiv:2601.11973v1 Announce Type: new -Abstract: The paper presents efficient approaches for evaluating convergence rate in total variation for finite and general linear Markov chains. The motivation for studying convergence rate in this metric is its usefulness in various limit theorems. For homogeneous Markov chains the goal is to compare several different methods: (1) the second eigenvalue for the transition matrix method (the method no. 1), (2) the method based on Markov -- Dobrushin's ergodic coefficient, and the new spectral method developed in earlier works, as well as modifications of they both by iterations (the ``other methods''). We answer the question whether or not the ``other methods'' may provide the optimal or close to optimal convergence rate in the case of homogeneous Markov chains. The answer turns out to be positive for appropriate modifications of both ``other methods''. The analogues of these ``other methods'' for the non-homogeneous Markov chains are also presented. The work is theoretical. However, the methods of computing efficient bounds of convergence rates may be in demand in various applied areas. - oai:arXiv.org:2601.11973v1 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexander Veretennikov - - - The linearization approach to the Calder\'on problem revisited: reconstruction via the Born approximation - https://arxiv.org/abs/2601.11975 - arXiv:2601.11975v1 Announce Type: new -Abstract: Linearization techniques are widely used in the analysis and numerical solution of the Calder\'on inverse problem, even if their theoretical basis is not fully understood. In this article, we study the effectiveness of linearization for reconstructing a conductivity from its Dirichlet-to-Neumann (DtN) map, combining rigorous analysis with numerical experiments. In particular, we prove that any DtN map arising from a radial conductivity in the unit ball of $\mathbb{R}^d$ admits an exact representation as a linearized DtN map for a uniquely determined integrable function, the Born approximation. We linearize on a family of background conductivities that includes the constant case, giving a rigorous foundation for linearization-based methods in this framework. We also characterize the Born approximation as a solution of a generalized moment problem. Since this moment problem is formally well-defined even for non-radial conductivities, we use it to develop a numerical algorithm to reconstruct the Born approximation of a general conductivity on the unit disk. We provide numerical experiments to test the resolution and robustness of the Born approximation in different situations. Finally, we show how it can be used as the starting point of an algorithm for reconstructing a conductivity from its DtN map. - oai:arXiv.org:2601.11975v1 - math.NA - cs.NA + Local smoothing estimates for bilinear Fourier integral operators + https://arxiv.org/abs/2601.15667 + arXiv:2601.15667v1 Announce Type: new +Abstract: We formulate a local smoothing conjecture for bilinear Fourier integral operators in every dimension $d \ge 2,$ derived from the celebrated linear case due to Sogge, which we refer to as the \emph{bilinear smoothing conjecture}. We show that the linear local smoothing conjecture implies this bilinear version. As a consequence of our approach and due to the recent progress on the subject, we establish local smoothing estimates for Fourier integral operators in dimension $d=2,$ that is, on $\mathbb{R}^2_x \times \mathbb{R}_t$. Also, a partial progress is presented for the high-dimensional case $d\geq 3.$ In particular, our method allows us to deduce that the bilinear local smoothing conjecture holds for all odd dimensions $d$. + oai:arXiv.org:2601.15667v1 math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Carlos Castro, Fabricio Maci\`a, Crist\'obal Mero\~no, Daniel S\'anchez-Mendoza + http://creativecommons.org/licenses/by/4.0/ + Duv\'an Cardona - The small cancellation flat torus theorem - https://arxiv.org/abs/2601.11991 - arXiv:2601.11991v1 Announce Type: new -Abstract: We establish Flat Torus Theorem type results for groups acting on small cancellation complexes satisfying C(6), C(4)-T(4) and C(3)-T(6) conditions. For C(3)-T(6) complexes the result closely parallels the CAT(0) setting. For C(6) complexes we prove an analogous theorem using a refined notion of flat, exploiting the relationship between C(6) complexes and their duals. In the C(4)-T(4) case we demonstrate that genuine flats do not necessarily exist, providing an explicit example of a C(4)-T(4) complex with an action of $\mathbb{Z}^2$ without invariant flat, and hence not admitting any CAT(0) metric. We introduce the notion of quasi-flats and prove a Flat Torus Theorem for quasi-flats by passing to quadric complexes via quadrization and invoking the Quadric Flat Torus Theorem of Hoda-Munro. - oai:arXiv.org:2601.11991v1 - math.GR - Wed, 21 Jan 2026 00:00:00 -0500 + Geometric wavefront sets of genuine Iwahori-spherical representations + https://arxiv.org/abs/2601.15670 + arXiv:2601.15670v1 Announce Type: new +Abstract: For Iwahori-spherical genuine representations of central covers with positive real Satake parameters, we prove the upper bound inequality for their geometric wavefront sets, formulated for general genuine representations in an earlier work by Gao--Liu--Lo--Shahidi. Meanwhile, we show the equality is attained for covers of type A groups and for some representations of covers of the exceptional groups. We also verify the equality for certain Iwahori-spherical representations occurring in regular unramified principal series; this uses and generalizes the earlier work of Karasiewicz--Okada--Wang on theta representations. Lastly, we determine the leading coefficients in the Harish-Chandra character expansion of a theta representation when its geometric wavefront set is of a special type. + oai:arXiv.org:2601.15670v1 + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Karol Duda + http://creativecommons.org/licenses/by/4.0/ + Fan Gao, Runze Wang - On examples of duals Saito's basis of some inhomogeneous divisors, and application - https://arxiv.org/abs/2601.11992 - arXiv:2601.11992v1 Announce Type: new -Abstract: We investigate a class of non-quasi-homogeneous free divisors in the sense of Saito. These divisors are defined by equations of the form $D:= \{h=0\}$ on $\mathbb{C}^p$, where the polynomial $h$ is specific linear combination of monomials involving the product of coordinates. For this class, we explicitly construct a Saito basis for the module of logarithmic vector fields $Der(logD)$. This construction is then applied to the setting of logarithmic Poisson geometry. Focusing on the example defined by $h=xy+x^{2}y^{2}+x^3y^3$ on the Poisson algebra $(\mathcal{A}=\mathbb{C}[x,y], \{-,-\}_{h})$, where the Poisson bracket is induced by the bivector $\pi = h\partial x\wedge\partial y$. We define the associated Koszul bracket on the module of logarithmic 1-forms. This enables us to prove that $\pi$ endows the sheaf of logarithmic 1-forms $\Omega^{1}(log D )$ with a Lie-Rinehart algebra structure. Furthermore, we introduce and provide explicit descriptions for the resulting cohomology theory, which we term the logarithmic Poisson cohomology $H_{log}^{\bullet} $ of $\{-,-\}_{h}$. As a related and foundational computation, we also calculate the corresponding logarithmic De Rham cohomology $H^{\bullet}_{DR}$ for the divisor $D$ and we make a generalization in dimension 2. - oai:arXiv.org:2601.11992v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 + Arithmetic Properties of Colored Partitions Restricted by Parity of the Parts + https://arxiv.org/abs/2601.15680 + arXiv:2601.15680v1 Announce Type: new +Abstract: Let $a_{r,s}(n)$ denote the number of mutlicolored partitions of $n$, wherein both even parts and odd parts may appear in one of $r$-colors and $s$-colors, respectively, for fixed $r,s\ge 1$. The paper aims to study arithmetic properties satisfied by $a_{r,s}(n)$, using elementary generating function manipulations and classical $q$-series techniques. + oai:arXiv.org:2601.15680v1 + math.CO + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/publicdomain/zero/1.0/ - Kamtila Kari, Joseph Dongho, Prosper Rosaire Mama Assandje, Thomas Bouetou Bouetou + http://creativecommons.org/licenses/by/4.0/ + M. P. Thejitha, James A. Sellers, S. N. Fathima - Conjugacy limits of certain subgroups in $\SL(2,\mathbb{R})\ltimes\mathbb{R}^2$ - https://arxiv.org/abs/2601.11994 - arXiv:2601.11994v1 Announce Type: new -Abstract: We study conjugacy limits of certain of subgroups inside $\SL(2,\R)\ltimes\R^2$. These subgroups have a common feature that any two in the same category are conjugates of each other. - oai:arXiv.org:2601.11994v1 - math.GR - math.DS - math.GN - Wed, 21 Jan 2026 00:00:00 -0500 + Parallelizable Riemannian Alternating Direction Method of Multipliers for Non-convex Pose Graph Optimization + https://arxiv.org/abs/2601.15684 + arXiv:2601.15684v1 Announce Type: new +Abstract: Pose graph optimization (PGO) is fundamental to robot perception and navigation systems, serving as the mathematical backbone for solving simultaneous localization and mapping (SLAM). Existing solvers suffer from polynomial growth in computational complexity with graph size, hindering real-time deployment in large-scale scenarios. In this paper, by duplicating variables and introducing equality constraints, we reformulate the problem and propose a Parallelizable Riemannian Alternating Direction Method of Multipliers (PRADMM) to solve it efficiently. Compared with the state-of-the-art methods that usually exhibit polynomial time complexity growth with graph size, PRADMM enables efficient parallel computation across vertices regardless of graph size. Crucially, all subproblems admit closed-form solutions, ensuring PRADMM maintains exceptionally stable performance. Furthermore, by carefully exploiting the structures of the coefficient matrices in the constraints, we establish the global convergence of PRADMM under mild conditions, enabling larger relaxation step sizes within the interval $(0,2)$. Extensive empirical validation on two synthetic datasets and multiple real-world 3D SLAM benchmarks confirms the superior computational performance of PRADMM. + oai:arXiv.org:2601.15684v1 + math.OC + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Manoj Choudhuri, C. R. E. Raja + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Xin Chen, Chunfeng Cui, Deren Han, Liqun Qi - Microscopic derivation of a one-dimensional lubrication model with roughness - https://arxiv.org/abs/2601.11999 - arXiv:2601.11999v1 Announce Type: new -Abstract: We derive a hydrodynamic model for the motion of inertial particles with a spherical hard core, interacting through lubrication forces and pairwise repulsive forces. The repulsion arises from the assumption that each particle is surrounded by a thin rough layer of reduced permeability. We prove that, as the number of particles tends to infinity (and their size tends to 0), the microscopic dynamics converges to a macroscopic hydrodynamic model in which congestion effects are encoded directly into the macroscopic interaction forces, depending on a local critical density transported by the flow. In particular, we extend the work of Lefebvre-Lepot and Maury where non-inertial particles, submitted to only a lubrication force were considered, and present the convergence proof when inertial effects and roughness are taken into account. - oai:arXiv.org:2601.11999v1 + Global regularity for the Navier-Stokes equations with application to global solvability for the Euler equations + https://arxiv.org/abs/2601.15685 + arXiv:2601.15685v1 Announce Type: new +Abstract: We show that any Leray-Hopf weak solution to the $d$-dimensional Navier-Stokes equations $(d\geq 3)$ with initial values $u_0\in H^{s}(\mathbb R^d)$, $s\geq -1+\frac{d}{2}$, belongs to $L^\infty(0,\infty; H^{s}(\mathbb R^d))$ and thus it is globally regular. For the proof, first, we construct a supercritical space which has very sparse inverse logarithmic weight in the frequency domain, compared to the critical homogeneous Sobolev $\dot{H}^{-1+d/2}$-norm. Then we obtain the energy estimates of high frequency parts of the solution which involve the supercritical norm as a factor of the upper bounds. Finally, we superpose the energy norm of high frequency parts of the solution to get estimates of the critical and subcritical norms independent of the viscosity coefficient for the weak solution via the re-scaling argument. + oai:arXiv.org:2601.15685v1 math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + math-ph + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Aline Lefebvre-Lepot, Muhammed Ali Mehmood, Charlotte Perrin, Ewelina Zatorska + Myong-Hwan Ri - A Multi-Level Deep Framework for Deep Solvers of Partial Differential Equations - https://arxiv.org/abs/2601.12000 - arXiv:2601.12000v1 Announce Type: new -Abstract: In this paper, inspired by the multigrid method, we propose a multi-level deep framework for deep solvers. Overall, it divides the entire training process into different levels of training. At each level of training, an adaptive sampling method proposed in this paper is first employed to obtain new training points, so that these points become increasingly concentrated in computational regions corresponding to high-frequency components. Then, the generalization ability of deep neural networks are utilized to update the PDEs for the next level of training based on the results from all previous levels. Rigorous mathematical proofs and detailed numerical experiments are employed to demonstrate the effectiveness of the proposed method. - oai:arXiv.org:2601.12000v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 + Symbolic Rees algebras of space monomial primes of degree 5 + https://arxiv.org/abs/2601.15692 + arXiv:2601.15692v1 Announce Type: new +Abstract: Let K be a field of characteristic 0. Let P_K(5,103,169) be the defining ideal of the space monomial curve {(t^5,t^{103},t^{169})}. In this paper we shall prove that the symbolic Rees algebra R_s(P_K(5,103,169)) is not Noetherian, that is, is not finitely generated over K. + oai:arXiv.org:2601.15692v1 + math.AC + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yu Yang, Qiaolin He + Kazuhiko Kurano - Structure of ind-pro completions of Noetherian rings - https://arxiv.org/abs/2601.12016 - arXiv:2601.12016v1 Announce Type: new -Abstract: We prove some results on the structure of ind-pro completions of Noetherian rings along flags of prime ideals. In particular, we compute the Krull dimension and deduce the criterion on semilocality in the case of essentially of finite type algebras over a field. We also show that ind-pro completion inherits properties of the base ring such as normality, regularity, local equidimensionality, etc. - oai:arXiv.org:2601.12016v1 - math.AC - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 + Rokhlin dimension for actions of residually compact groups + https://arxiv.org/abs/2601.15694 + arXiv:2601.15694v1 Announce Type: new +Abstract: We introduce the concept of Rokhlin dimension for actions of residually compact groups on C*-algebras, which extends and unifies previous notions for actions of compact groups, residually finite groups and the reals. We then demonstrate that finite nuclear dimension (respectively, absorption of a strongly self-absorbing C*-algebra) is preserved under the formation of crossed products by residually compact group actions with finite Rokhlin dimension (respectively, finite Rokhlin dimension with commuting towers). Furthermore, if second countable residually compact group contains a non-open cocompact closed subgroup, then crossed products arising from actions with finite Rokhlin dimension are stable. Finally, we study the relationship between the tube dimension of a topological dynamical system and the Rokhlin dimension of the induced C*-dynamical system. + oai:arXiv.org:2601.15694v1 + math.OA + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Dmitry Badulin + Xin Cao, Xiaochun Fang, Jianchao Wu - Deformation rigidity of some simple affine VOAs - https://arxiv.org/abs/2601.12017 - arXiv:2601.12017v1 Announce Type: new -Abstract: In this paper, we prove that simple affine vertex operator algebras with positive integral levels admit only trivial first-order deformations. Therefore, the deformation rigidity conjecture of strongly rational vertex operator algebras holds for these cases. We also show that the same holds simple affine vertex operator algebra of $\mathfrak{sl}_2$ at the non-integral admissible level $-4/3$. Therefore, neither $C_2$-cofiniteness nor rationality is a necessary condition for deformation rigidity of VOAs. We conjecture that the same should hold for every simple affine VOA that does not coincide with the corresponding universal affine VOA. - oai:arXiv.org:2601.12017v1 - math.QA - hep-th - Wed, 21 Jan 2026 00:00:00 -0500 + Maximal Fuchsian subgroups of the $d=2$ Bianchi group + https://arxiv.org/abs/2601.15700 + arXiv:2601.15700v1 Announce Type: new +Abstract: Let $\Gamma$ denote the $d = 2$ Bianchi group $\operatorname{PSL}(2,\mathbb{Z}[\sqrt{-2}])$. We give an explicit description of all conjugacy classes of maximal nonelementary Fuchsian subgroups of $\Gamma$ as integral orders of certain indefinite quaternion algebras over $\mathbb{Q}$. Using this description, we also provide the covolumes corresponding to each conjugacy class. As an application, we compute the limit $\lim_{x\to\infty} \frac{\Pi(x)}{x}$ where $\Pi(x)$ counts the number of primitive totally geodesic immersed surfaces in the manifold $\Gamma\backslash\mathbb{H}^3$ with area less than $x$. + oai:arXiv.org:2601.15700v1 + math.NT + math.GT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Andrew R. Linshaw, Fei Qi + http://creativecommons.org/licenses/by/4.0/ + Anthony Lee - On Multilinear Forms for Mod $p$ Representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ - https://arxiv.org/abs/2601.12021 - arXiv:2601.12021v1 Announce Type: new -Abstract: Motivated by the study of trilinear forms for complex representations, we investigate the space of $G$-invariant linear forms on tensor products of irreducible admissible representations of $G = \mathrm{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$. Our main result is a complete vanishing theorem: for any $n \ge 1$ and $n$ infinite-dimensional irreducible admissible representations $\pi_1,\dots,\pi_n$ of $G$, \[ \operatorname{Hom}_G(\pi_1 \otimes \cdots \otimes \pi_n, \mathbb{1}) = 0. \] A refined version holds for $B^+ := \begin{pmatrix} p^{\mathbb{Z}} & \mathbb{Q}_p \\ 0 & 1 \end{pmatrix}$-invariant forms when at least one $\pi_i$ is supersingular. The proof proceeds by a detailed analysis of certain subgroups, reducing the problem from $G$ to $B^+$ and ultimately to the representation theory of $\mathbb{Z}_p$. We also deduce partial extensions of the result to $\mathrm{GL}_2(F)$ for finite extensions $F/\mathbb{Q}_p$. - oai:arXiv.org:2601.12021v1 + On mode transition algebras for $\mathbb{Z}$-graded vertex algebras and applications to bosonic ghosts + https://arxiv.org/abs/2601.15701 + arXiv:2601.15701v1 Announce Type: new +Abstract: We study the mode transition algebras and Zhu algebras in the setting of $\mathbb{Z}$-graded vertex algebras, with particular focus on the Weyl vertex algebra at central charge 2 (also known as bosonic ghosts or the $\beta\gamma$-system). We show that the mode transition algebras of the Weyl vertex algebra at central charge 2 admit unity elements that form a family of strong unities in the sense of Damiolini-Gibney-Krashen. The existence of unities for the mode transition algebra of the Weyl vertex algebra at central charge 2 allows us to explicitly construct all higher level Zhu algebras of the Weyl vertex algebra at central charge 2. We further analyze weak modules of the Weyl vertex algebra at central charge 2 induced from Zhu algebras, proving that every such module is already induced from the level-zero Zhu algebra. We then prove that all indecomposable reducible weight modules induced from a Zhu algebra are not weakly interlocked, and hence not strongly interlocked in the sense of Barron-Batistelli-Orosz Hunziker-Yamskulna. More generally, we show that the property of being weakly interlocked is preserved under the action of an invertible Li's $\mathbf{\Delta}$ operator. As an application, we prove that all indecomposable reducible weight modules of the Weyl vertex algebra at central charge 2 obtained via spectral flow of Zhu-induced modules are likewise not weakly interlocked. These results clarify the role of being weakly interlocked in the modularity properties of bosonic ghost modules previously studied by Ridout-Wood and Allen-Wood. + oai:arXiv.org:2601.15701v1 + math.QA + math-ph + math.MP math.RT - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yikun Fan + Katrina Barron, Justine Fasquel, Florencia Orosz Hunziker, Gaywalee Yamskulna - Generalizing the Fano inequality further - https://arxiv.org/abs/2601.12027 - arXiv:2601.12027v1 Announce Type: new -Abstract: Interactive statistical decision making (ISDM) features algorithm-dependent data generated through interaction. Existing information-theoretic lower bounds in ISDM largely target expected risk, while tail-sensitive objectives are less developed. We generalize the interactive Fano framework of Chen et al. by replacing the hard success event with a randomized one-bit statistic representing an arbitrary bounded transform of the loss. This yields a Bernoulli f-divergence inequality, which we invert to obtain a two-sided interval for the transform, recovering the previous result as a special case. Instantiating the transform with a bounded hinge and using the Rockafellar-Uryasev representation, we derive lower bounds on the prior-predictive (Bayesian) CVaR of bounded losses. For KL divergence with the mixture reference distribution, the bound becomes explicit in terms of mutual information via Pinsker's inequality. - oai:arXiv.org:2601.12027v1 + Generalized Information Inequalities via Submodularity, and Two Combinatorial Problems + https://arxiv.org/abs/2601.15723 + arXiv:2601.15723v1 Announce Type: new +Abstract: It is well known that there is a strong connection between entropy inequalities and submodularity, since the entropy of a collection of random variables is a submodular function. Unifying frameworks for information inequalities arising from submodularity were developed by Madiman and Tetali (2010) and Sason (2022). Madiman and Tetali (2010) established strong and weak fractional inequalities that subsume classical results such as Han's inequality and Shearer's lemma. Sason (2022) introduced a convex-functional framework for generalizing Han's inequality, and derived unified inequalities for submodular and supermodular functions. In this work, we build on these frameworks and make three contributions. First, we establish convex-functional generalizations of the strong and weak Madiman and Tetali inequalities for submodular functions. Second, using a special case of the strong Madiman-Tetali inequality, we derive a new Loomis-Whitney-type projection inequality for finite point sets in $\mathbb{R}^d$, which improves upon the classical Loomis-Whitney bound by incorporating slice-level structural information. Finally, we study an extremal graph theory problem that recovers and extends the previously known results of Sason (2022) and Boucheron et al., employing Shearer's lemma in contrast to the use of Han's inequality in those works. + oai:arXiv.org:2601.15723v1 cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Raghav Bongole, Tobias J. Oechtering, Mikael Skoglund - - - High-Dimensional $p$-Normed Flows - https://arxiv.org/abs/2601.12036 - arXiv:2601.12036v1 Announce Type: new -Abstract: We generalize Tutte's integer flows and the $d$-dimensional Euclidean flows of Mattiolo, Mazzuoccolo, Rajn\'{i}k, and Tabarelli to \emph{$d$-dimensional $p$-normed nowhere-zero flows} and define the corresponding flow index $\phi_{d,p}(G)$ to be the infimum over all real numbers $r$ for which $G$ admits a $d$-dimensional $p$-normed nowhere-zero $r$-flow. For any bridgeless graph $G$ and any $p\ge 1$, we establish general upper bounds, including $\phi_{2,p}(G) \le 3$, $\phi_{3,p}(G) \le 1+\sqrt{2}$, and tight bounds for graphs admitting a $4$-NZF. For graphs with oriented $(k+1)$-cycle $2l$-covers, we show that $\phi_{k,p}(G) = 2$, which implies $\phi_{2,p}(G) = 2$ for graphs admitting a nowhere-zero $3$-flow and $\phi_{3,p}(G) = 2$ for those admitting a nowhere-zero $4$-flow. These results extend classical flow theory to arbitrary norms, provide supporting evidences for Tutte's $5$-flow Conjecture and Jain's $S^2$-Flow Conjecture, and connect combinatorial flows with geometric and topological perspectives. - oai:arXiv.org:2601.12036v1 math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chenxing Li, Jiaao Li, Rong Luo, Bo Su + Gunank Jakhar, Gowtham R. Kurri, Suryajith Chillara, Vinod M. Prabhakaran - Critical partition regular functions for compact spaces - https://arxiv.org/abs/2601.12041 - arXiv:2601.12041v1 Announce Type: new -Abstract: We study ideal-based refinements of sequential compactness arising from the class FinBW(I), consisting of topological spaces in which every sequence admits a convergent subsequence indexed by a set outside a given ideal I. A central theme of this work is the existence of critical ideals whose position in the Katetov order determines the relationship between a fixed class of spaces and the corresponding FinBW(I) classes. Building on earlier results characterizing several classical topological classes via such ideals, we extend this theory to a broader framework based on partition regular functions, which unifies ordinary convergence with other non-classical convergence notions such as IP- and Ramsey-type convergence. Furthermore, we investigate the existence of critical ideals associated with function classes motivated by Mazurkiewicz's theorem on uniformly convergent subsequences. - oai:arXiv.org:2601.12041v1 - math.GN - Wed, 21 Jan 2026 00:00:00 -0500 + Four-dimensional Lorentzian algebraic Ricci solitons + https://arxiv.org/abs/2601.15730 + arXiv:2601.15730v1 Announce Type: new +Abstract: We describe four-dimensional Lorentzian algebraic Ricci solitons. In sharp contrast with the Riemannian situation, any connected and simply connected four-dimensional Lie group admits a left-invariant Lorentz metric which is a Ricci soliton. + oai:arXiv.org:2601.15730v1 + math.DG + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Rafa{\l} Filip\'ow, Ma{\l}gorzata Kowalczuk, Hubert Ksi\k{a}\.zek, Adam Kwela, Grzegorz Ucal - - - Koopman Spectral Computation Beyond The Reflexive Regime: Endpoint Solvability Complexity Index And Type-2 Links - https://arxiv.org/abs/2601.12044 - arXiv:2601.12044v1 Announce Type: new -Abstract: We study the Solvability Complexity Index (SCI) of Koopman operator spectral computation in the information-based framework of towers of algorithms. Given a compact metric space $(\mathcal{X},d)$ with a finite Borel measure $\omega$ on $\mathcal{X}$ and a continuous nonsingular map $F:\mathcal{X}\to \mathcal{X}$, our focus is the Koopman operator $\mathcal{K}_F$ acting on $L^p(\mathcal{X},\omega)$ for $p\in\{1,\infty\}$ for the computational problem \[ \Xi_{\sigma_{\mathrm{ap}}}(F) :=\sigma_{\mathrm{ap}}\!\bigl(\mathcal{K}_F\bigr), \] with input access given by point evaluations of $F\mapsto F(x)$ (and fixed quadrature access to $\omega$). - We clarify how the $L^1$ case can be brought into the same oracle model as the reflexive regime $1<p<\infty$ by proving a uniform finite-dimensional quadrature compatibility, while highlighting the fundamentally different role played by non-separability at $p=\infty$. - Beyond Koopman operators, we also construct a prototype family of decision problems $(\Xi_m)_{m\in\mathbb N}$ realizing prescribed finite tower heights, providing a reusable reduction source for future SCI lower bounds. Finally, we place these results deeper in the broader computational landscape of Type-2/Weihrauch theory. - oai:arXiv.org:2601.12044v1 - math.LO - cs.NA - math.DS - math.NA - math.SP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Christopher Sorg + Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey, Ramon Vazquez-Lorenzo - Geometric realisations of type $\tilde{A}_n$ preprojective algebras in homological mirror symmetry - https://arxiv.org/abs/2601.12045 - arXiv:2601.12045v1 Announce Type: new -Abstract: The type $A_n$-singularity $\mathbb{C}^2/\mathbb{Z}_{n+1}$ can be resolved by hyper-K\"ahler manifolds $X_{\zeta}$ with underlying smooth manifolds diffeomorphic to the resolution of singularities $X_{\text{res}}$, whose hyper-K\"ahler structure depends on a parameter $\zeta\in H_2(X_{\text{res}};\mathbb{R})$. The structure as a complex manifold of each such hyper-K\"ahler manifold is equivalent to the resolution of singularities at the poles and the structure of a Milnor fibre with roots determined by $\zeta$ elsewhere; the symplectic structure is exact along the equator and is deformed by areas depending on $\zeta$ on the exceptional $(-2)$-spheres away from the equator. - We show that removing suitable divisors $D_u$ from a fixed $X_{\zeta}$ varying with $u$ in the underlying upper hemisphere of the $S^2$-family of K\"ahler-structures yields a log Calabi--Yau hyper-K\"ahler family (in particular a family of log Calabi--Yau submanifolds), and that mirror symmetry is satisfied (partly conjectural in one direction) for this family by hyper-K\"ahler rotation, in particular by interchanging the structures over the equator and the pole. We furthermore show homological mirror symmetry after adding the missing divisors, which is related to attaching stops and computing singularity categories of certain Landau--Ginzburg potentials on the $A$-side and $B$-side, respectively. - More concretely: we compute wrapped Fukaya categories and compare them with (previous and new) computations of derived categories of coherent sheaves and derived categories of singularities in algebraic geometry. We show that the relevant categories (with two exceptions) are triangulated equivalent to module categories over the additive and the multiplicative preprojective algebras of type $\tilde{A}_n$, or to deformations of these algebras depending on the parameters $\zeta$. - oai:arXiv.org:2601.12045v1 - math.SG - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 + A sequential linear complementarity problem method for generalized Nash equilibrium problems + https://arxiv.org/abs/2601.15742 + arXiv:2601.15742v1 Announce Type: new +Abstract: We propose a sequential linear complementarity problem (SLCP) method for solving generalized Nash equilibrium problems (GNEPs). By introducing a novel merit function that utilizes the specific structure of GNEPs, we establish global convergence of the method. The conditions guaranteeing global convergence are analogous to those for the classical sequential quadratic programming method with exact Lagrange Hessians, making this a natural and reasonable generalization. Moreover, we provide a detailed analysis of the solvability of the mixed linear complementarity subproblems, which are formulated as affine GNEPs. Sufficient characterizations for the local superlinear convergence are also derived, highlighting the efficiency of the proposed method. Finally, numerical experiments demonstrate the practical performance and effectiveness of the SLCP method in comparison with existing approaches. + oai:arXiv.org:2601.15742v1 + math.OC + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Johan Rydholm + Ruoyu Diao, Yu-Hong Dai, Liwei Zhang - Partition identities associated with $A_r$-Surface singularities - https://arxiv.org/abs/2601.12048 - arXiv:2601.12048v1 Announce Type: new -Abstract: We prove a family of partition identities involving integer partitions in three colors. The conditions imposed on the types of partitions appearing in these identities involve constraints that arise in the Rogers-Ramanujan and Andrews-Gordon identities, as well as in their recent extensions. The identities established in this paper are associated with the $A_r$ surface singularities via the arc HP-series, which provides a measure of singularities of algebraic varieties defined using arc spaces. - oai:arXiv.org:2601.12048v1 - math.AG - math.AC - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + Rankin--Cohen brackets in Representation Theory + https://arxiv.org/abs/2601.15750 + arXiv:2601.15750v1 Announce Type: new +Abstract: The Rankin--Cohen brackets provide a basic example of ``non-elementary" differential symmetry breaking operators. They can be interpreted as bi-differential operators remarkable for reflecting the structure of fusion rules for holomorphic discrete series representations of the Lie group $SL(2,\mathbb R)$ and are intimately connected to classical special polynomials. + In this introductory article, we explore the combinatorial structure of these operators and discuss a general framework for constructing their higher-dimensional analogues from the representation-theoretic perspective on branching problems. The exposition is based on lectures delivered by the authors during the thematic semester ``Representation Theory and Noncommutative Geometry", held in Spring 2025 at the Henri Poincar\'e Institute in Paris. + oai:arXiv.org:2601.15750v1 + math.RT + math.CA + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Pooneh Afsharijoo, Pedro D. Gonz\'alez P\'erez, Hussein Mourtada + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Toshiyuki Kobayashi, Michael Pevzner - Function Computation Over Multiple Access Channels via Hierarchical Constellations - https://arxiv.org/abs/2601.12050 - arXiv:2601.12050v1 Announce Type: new -Abstract: We study function computation over a Gaussian multiple-access channel (MAC), where multiple transmitters aim at computing a function of their values at a common receiver. To this end, we propose a novel coded-modulation framework for over-the-air computation (OAC) based on hierarchical constellation design, which supports reliable computation of multiple function outputs using a single channel use. Moreover, we characterize the achievable computation rate and show that the proposed hierarchical constellations can compute R output functions with decoding error probability epsilon while the gap to the optimal computation rate scales as O(\log_2(1/\epsilon)/K) for independent source symbols, where K denotes the number of transmitters. Consequently, this gap vanishes as the network size grows, and the optimal rate is asymptotically attained. - Furthermore, we introduce a shielding mechanism based on variable-length block coding that mitigates noise-induced error propagation across constellation levels while preserving the superposition structure of the MAC. We show that the shielding technique improves reliability, yielding a gap that scales optimally as O(\log_2\ln{(1/\epsilon)}), regardless of the source distribution. Together, these results identify the regimes in which uncoded or lightly coded OAC is information-theoretically optimal, providing a unified framework for low-latency, channel-agnostic function computation. - oai:arXiv.org:2601.12050v1 + Recursive Flow: A Generative Framework for MIMO Channel Estimation + https://arxiv.org/abs/2601.15767 + arXiv:2601.15767v1 Announce Type: new +Abstract: Channel estimation is a fundamental challenge in massive multiple-input multiple-output systems, where estimation accuracy governs the spectral efficiency and link reliability. In this work, we introduce Recursive Flow (RC-Flow), a novel solver that leverages pre-trained flow matching priors to robustly recover channel state information from noisy, under-determined measurements. Different from conventional open-loop generative models, our approach establishes a closed-loop refinement framework via a serial restart mechanism and anchored trajectory rectification. By synergizing flow-consistent prior directions with data-fidelity proximal projections, the proposed RC-Flow achieves robust channel reconstruction and delivers state-of-the-art performance across diverse noise levels, particularly in noise-dominated scenarios. The framework is further augmented by an adaptive dual-scheduling strategy, offering flexible management of the trade-off between convergence speed and reconstruction accuracy. Theoretically, we analyze the Jacobian spectral radius of the recursive operator to prove its global asymptotic stability. Numerical results demonstrate that RC-Flow reduces inference latency by two orders of magnitude while achieving a 2.7 dB performance gain in low signal-to-noise ratio regimes compared to the score-based baseline. + oai:arXiv.org:2601.15767v1 cs.IT math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Saeed Razavikia, Mohammad Kazemi, Deniz G\"und\"uz, Carlo Fischione + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Zehua Jiang, Fenghao Zhu, Chongwen Huang, Richeng Jin, Zhaohui Yang, Xiaoming Chen, Zhaoyang Zhang, M\'erouane Debbah - Ramanujan polar graphs - https://arxiv.org/abs/2601.12057 - arXiv:2601.12057v1 Announce Type: new -Abstract: Recently, a construction of minimal codes arising from a family of almost Ramanujan graphs was shown. Ramanujan graphs are examples of expander graphs that minimize the second-largest eigenvalue of their adjacency matrix. We call such graphs Ramanujan, since all known non-trivial constructions imply the Ramanujan conjecture on arithmetical functions. In this paper, we prove that some families of tangent graphs of finite classical polar spaces satisfy Ramanujan's condition. If the polarity is unitary, or it is orthogonal and the quadric is over the binary field, the tangent graphs are strongly regular, and we know their spectrum. By direct computation, it is possible to show which families of tangent graphs are Ramanujan. - oai:arXiv.org:2601.12057v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + Stochastically forced compressible Navier-Stokes equations with slip boundary conditions of friction type + https://arxiv.org/abs/2601.15768 + arXiv:2601.15768v1 Announce Type: new +Abstract: We study a mathematical model of a compressible viscous fluid driven by stochastic forces under slip boundary conditions of friction type. We introduce a notion of a weak solution that is analytically and probabilistically consistent with this model. Our main result establishes the existence of such weak solutions under slip boundary conditions on bounded domains with $C^{2+\nu}$-boundary ($\nu>0$). The proof of this result combines an extended version of the four-layer approximation scheme on the torus by Breit/Feireisl/Hofmanov\'{a} (2018) with the convex approximation method for absolute value functions studied by Ne\v{c}asov\'{a}/Ogorzaly/Scherz (2023). + oai:arXiv.org:2601.15768v1 + math.PR + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Valentino Smaldore + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Reo Tsuboya - Magnetic spectral inverse problems on compact Anosov manifolds - https://arxiv.org/abs/2601.12058 - arXiv:2601.12058v1 Announce Type: new -Abstract: In this paper, we establish positive results for two spectral inverse problems in the presence of a magnetic potential. Exploiting the principal wave trace invariants, we first show that on closed Anosov manifolds with simple length spectrum, one can recover an electric and a magnetic (up to a natural gauge) potential from the spectrum of the associated magnetic Schr\"odinger operator. This extends a particular instance of a recent positive result on the spectral inverse problem for the Bochner Laplacian in negative curvature, obtained by M.Ceki\'c and T.Lefeuvre (2023). Similarly, we prove that the spectrum of the magnetic Dirichlet-to-Neumann map (or Steklov operator) determines at the boundary both a magnetic potential, up to gauge, and an electric potential, provided the boundary is Anosov with simple length spectrum. Under this assumption, one can actually show that the magnetic Steklov spectrum determines the full Taylor series at the boundary of any smooth magnetic field and electric potential. As a simple consequence, in this case, both an analytic magnetic field and an analytic electric potential are uniquely determined by their Steklov spectrum. - oai:arXiv.org:2601.12058v1 - math.SP - math-ph - math.DG - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 + Representations of the modular group into the isometries of $\mathrm{SL}_3(\mathbb{R})/\mathrm{SO}(3)$ + https://arxiv.org/abs/2601.15781 + arXiv:2601.15781v1 Announce Type: new +Abstract: We describe a connected component of the space of conjugacy classes of representations of the modular group $\mathrm{PSL}_2(\mathbb{Z})$ into the isometry group of the symmetric space $\mathrm{SL}_3(\mathbb{R})/\mathrm{SO}(3)$. This connected component contains the family of representations constructed by Schwartz via Pappus' theorem, as well as their Anosov deformations studied by Barbot, Lee, and Val\'erio. We show that certain representations in this component (far from the Schwartz representations) are Anosov. + oai:arXiv.org:2601.15781v1 + math.GT + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - David dos Santos Ferreira, Benjamin Florentin + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Joan Porti - Almost coherent rings - https://arxiv.org/abs/2601.12059 - arXiv:2601.12059v1 Announce Type: new -Abstract: Inspired from the work of P. Scholze on the finiteness of \(\mathbf{F}_{p}\)-cohomology groups of proper rigid-analytic varieties over \(p\)-adic fields, Zavyalov recently introduced the notion of almost coherent rings, which plays a key role in the almost ring theory. In this paper, we characterize almost coherent rings in terms of almost flat modules and almost absolutely pure modules, integrating numerous classical results into almost mathematics. Besides, we show that every almost coherent $R$-module is not almost isomorphic to a coherent $R$-module, giving a negative answer to a question proposed in [14,B. Zavyalov, {\it Almost coherent modules and almost coherent sheaves}, Memoirs of the European Mathematical Society 19. Berlin: European Mathematical Society (EMS), 2025]. - oai:arXiv.org:2601.12059v1 - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 + Shuriken Graphs Arising from Clean Graphs of Rings and Their Properties Relative to Base Graphs + https://arxiv.org/abs/2601.15783 + arXiv:2601.15783v1 Announce Type: new +Abstract: Let $R$ be a finite ring with identity. The idempotent graph $I(R)$ is the graph whose vertex set consists of the non-trivial idempotent elements of $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx = 0$. The clean graph $Cl_2(R)$ is a graph whose vertices are of the form $(e, u)$, where $e$ is a nonzero idempotent element and $u$ is a unit of $R$. Two distinct vertices $(e,u)$ and $(f, v)$ are adjacent if and only if $ef = fe = 0$ or $uv = vu = 1$. The shuriken graph operation is an operation that arises from the structure of the clean graph and depends on the structure of the associated idempotent graph. In this paper, we study the graph obtained from the shuriken operation and examine how its properties depend on those of the base graph. In particular, we investigate several graph invariants, including the clique number, chromatic number, independence number, and domination number. Moreover, we analyze topological indices and characterize Eulerian and Hamiltonian properties of the resulting shuriken graphs in terms of the properties of the base graphs. + oai:arXiv.org:2601.15783v1 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xiaolei Zhang + Felicia Servina Djuang, Indah Emilia Wijayanti, Yeni Susanti - Invariant Means on $VN^n(G)$ - https://arxiv.org/abs/2601.12063 - arXiv:2601.12063v1 Announce Type: new -Abstract: Let $G$ be a locally compact group, and $VN^n(G)$ is the dual of the multidimensional Fourier algebra $A^n(G)$. In this article, we define invariant means on $VN^n(G)$ and prove that the set of all invariant means on $VN^n(G)$ is non-empty. Further, we investigated the invariant means on $VN^n(G)$ for discrete and non-discrete cases of $G$. Also, we show that if $H$ is an open subgroup of $G$, then the number of invariant means on $VN^n(H)$ is the same as that of $VN^n(G)$. Finally, we study invariant means on the dual of the algebra $A_0^n(G)$, the closure of Fourier algebra $A^n(G)$ in the cb-multiplier norm. - oai:arXiv.org:2601.12063v1 + Remarks about symmetry-type conditions of conditional bases of Banach spaces + https://arxiv.org/abs/2601.15784 + arXiv:2601.15784v1 Announce Type: new +Abstract: We investigate the existence of equivalent p-norms, 0< p 1, under which conditional symmetric or spreading bases in quasi-Banach spaces become isometric. For spreading bases (which need not be unconditional or even Schauder bases), we develop new techniques involving the geometry of spreading sequences and their associated spreading models. We prove that any spreading basis is automatically seminormalized, M-bounded, and uniformly spreading, which allows the construction of an isometric renorming via its spreading model. For symmetric bases, we show they are necessarily spreading and uniformly symmetric, enabling a direct application of a renorming lemma for uniformly bounded semigroups of operators. Consequently, any quasi-Banach space with a symmetric basis admits a renorming making all permutations isometries, and any spreading basis admits a renorming making all increasing maps isometries. These results extend and unify classical isometric renorming theorems for unconditional, subsymmetric, and symmetric Schauder bases to the conditional, non-Schauder setting. + oai:arXiv.org:2601.15784v1 math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Kanupriya Wadhawan, N. Shravan Kumar - - - Expansion and Bounds for the Bias of Empirical Tail Value-at-Risk - https://arxiv.org/abs/2601.12064 - arXiv:2601.12064v1 Announce Type: new -Abstract: Tail Value-at-Risk (TVaR) is a widely adopted risk measure playing a critically important role in both academic research and industry practice in insurance. In data applications, TVaR is often estimated using the empirical method, owing to its simplicity and nonparametric nature. The empirical TVaR has been explicitly advocated by regulatory authorities as a standard approach for computing TVaR. However, prior literature has pointed out that the empirical TVaR estimator is negatively biased, which can lead to a systemic underestimation of risk in finite-sample applications. This paper aims to deepen the understanding of the bias of the empirical TVaR estimator in two dimensions: its magnitude as well as the key distributional and structural determinants driving the severity of the bias. To this end, we derive a leading-term approximation for the bias based on its asymptotic expansion. The closed-form expression associated with the leading-term approximation enables us to obtain analytical insights into the structural properties governing the bias of the empirical TVaR estimator. To account for the discrepancy between the leading-term approximation and the true bias, we further derive an explicit upper bound for the bias. We validate the proposed bias analysis framework via simulations and demonstrate its practical relevance using real data. - oai:arXiv.org:2601.12064v1 - math.ST - stat.AP - stat.ME - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Nadezhda Gribkova, Jianxi Su, Mengqi Wang + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jos\'e L. Ansorena, Alejandro Marcos - Boojums in Liquid Crystals Around a Colloid - https://arxiv.org/abs/2601.12065 - arXiv:2601.12065v1 Announce Type: new -Abstract: We study the Landau-de Gennes theory in the one constant limit. The bulk domain is the exterior of a spherical colloid. A Rapini-Papoular surface potential is imposed on the colloid surface, supplemented by a homogeneous far-field condition at spatial infinity. Under the axially symmetric ansatz and the Lyuksyutov constraint, we show that energy minimizers exhibit boojum disclinations at the two poles of the colloid. The local structure of these boojum disclinations is also characterized. - oai:arXiv.org:2601.12065v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Efficient Numerical Reconstruction of Wave Equation Sources via Droplet-Induced Asymptotics + https://arxiv.org/abs/2601.15787 + arXiv:2601.15787v1 Announce Type: new +Abstract: In this paper, we develop and numerically implement a novel approach for solving the inverse source problem of the acoustic wave equation in three dimensions. By injecting a small high-contrast droplet into the medium, we exploit the resulting wave field perturbation measured at a single external point over time. The method enables stable source reconstructions where conventional approaches fail due to ill-posedness, with potential applications in medical imaging and non-destructive testing. Key contributions include: + 1. Implementation of a theoretically justified asymptotic expansion, from [33], using the eigensystem of the Newtonian operator, with error analysis for the spectral truncation. + 2. Novel numerical schemes for solving the time-domain Lippmann-Schwinger equation and reconstructing the source via Riesz basis expansions and mollification-based numerical differentiations. + 3. Reconstruction requiring only single-point measurements, overcoming traditional spatial data limitations. + 4. 3D numerical experiments demonstrating accurate source recovery under noise (SNR of the order $1/a$), with error analysis for the droplet size (of the order $a$) and the number of spectral modes $N$. + oai:arXiv.org:2601.15787v1 + math.NA + cs.NA + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yuchen Huang, Yong Yu + Shutong Hou, Mourad Sini, Haibing Wang - Boundary Perturbations of Steklov Eigenvalues - https://arxiv.org/abs/2601.12077 - arXiv:2601.12077v1 Announce Type: new -Abstract: We consider the dependence of non-zero Steklov eigenvalues on smooth perturbations of the domain boundary. We prove that these eigenvalues are generically simple under such boundary perturbations. This result complements our previous work on metric perturbations, thereby establishing generic simplicity Steklov eigenvalues under both fundamental geometric variations. - oai:arXiv.org:2601.12077v1 - math.SP + A half-space Liouville theorem for anisotropic minimal graph with free boundary + https://arxiv.org/abs/2601.15788 + arXiv:2601.15788v1 Announce Type: new +Abstract: In this paper we prove the following Liouville-type theorem: any anisotropic minimal graph with free boundary in the half-space must be flat, provided that the graph function has at most one-sided linear growth. This extends the classical results of Bombieri-De Giorgi-Miranda and Simon to an appropriate free boundary setting. + oai:arXiv.org:2601.15788v1 math.DG - Wed, 21 Jan 2026 00:00:00 -0500 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lihan Wang + Guofang Wang, Wei Wei, Chao Xia, Xuwen Zhang - Sharpness of the Osgood Criterion for the Continuity Equation with Divergence-free Vector Fields - https://arxiv.org/abs/2601.12096 - arXiv:2601.12096v1 Announce Type: new -Abstract: For any modulus of continuity $\omega$ that fails the Osgood condition, we construct a divergence-free velocity field $v \in C_t C^\omega_x$ for which the associated ODE admits at least two distinct flow maps. In other words, non-uniqueness does not occur merely for a single or even finitely many trajectories, but instead on a set of initial conditions $E$ of positive Lebesgue measure. In fact, the set $E$ has full measure inside a cube where the construction is supported. Moreover, we also construct a divergence-free velocity field $v \in C_{t}C^\omega_x$ for which the associated continuity equation admits two distinct solutions $\mu^1$ and $\mu^2$ which are absolutely continuous with respect to Lebesgue measure for almost every time, and start from the same initial datum $\bar \mu \ll \mathscr{L}^{d}$. Our construction introduces two novel ideas: (i) We introduce the notion of "parallelization", where at each time, the velocity field consists of simultaneous motion across multiple nested spatial scales. This differs from most explicit constructions in the literature on mixing or anomalous dissipation, where the velocity on different scales acts at separate times. This is crucial to cover the whole class of non-Osgood moduli of continuity. (ii) Inspired by a recent work of Bru\`e, Colombo and Kumar, we develop a new fixed-point framework that naturally incorporates the parallelization mechanism. This framework allows us to construct anomalous solutions of the continuity equation that belong to $L^1(\mathbb{R}^d)$ a.e. in time. - oai:arXiv.org:2601.12096v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Localization of complementarity eigenvalues + https://arxiv.org/abs/2601.15789 + arXiv:2601.15789v1 Announce Type: new +Abstract: Let A, B be symmetric n x n real matrices with B positive definite and strictly diagonally dominant. We derive two localization sets for the complementarity eigenvalues of (A, B), the tightest one assuming additionally that A is copositive. This extends He-Liu-Shen sets to the case where B is not the identity. Moreover, we compare the computable bounds obtained from these new sets with the extreme classical generalized eigenvalues. + oai:arXiv.org:2601.15789v1 + math.OC + math.SP + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Roberto Colombo, Anuj Kumar + Antonio Sasaki (CMA, PSL), Sophie Demassey (CMA, PSL), Valentina Sessa (CMA, PSL) - Higher integrability of solutions to elliptic equations under additional sign constraints - https://arxiv.org/abs/2601.12100 - arXiv:2601.12100v1 Announce Type: new -Abstract: Solutions to elliptic equations often exhibit higher regularity properties such as \emph{higher integrability}. That is, for instance, a solution $u$ to a system that a priori only satisfies $ u \in W^{1,r}$ is more regular and even in the Sobolev space $W^{1,s}$ for some $s>r$. Under additional constraints of the sign of specific terms such as $(\partial_i u)$ this improvement of regularity can be sharpened further. - In this work, we consider two examples of such higher integrability results: First, we show a version of M\"uller's result on the higher integrability of the determinant for maps $u \in W^{1,n} $ such that $\mathrm{det}(\nabla u) \geq 0$ (or $ \mathrm{det}_-(\nabla u) \in L \log L$). Second, we consider (very weak) solutions to the $p$-Laplace equation that satisfy sign constraints for their partial derivatives, i.e. that $(\partial_i u)_- $ is of higher integrability than $(\partial_i u)_+$. To prove our results, we use the method of Lipschitz truncation; for the second example we further develop a variation of this technique, the \emph{asymmetric} Lipschitz truncation. - oai:arXiv.org:2601.12100v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Superpositions of CARMA processes + https://arxiv.org/abs/2601.15796 + arXiv:2601.15796v1 Announce Type: new +Abstract: We introduce supCARMA processes, defined as superpositions of L\'evy-driven CARMA processes with respect to a L\'evy basis, as a natural extension of the superpositions of Ornstein-Uhlenbeck type processes. We then focus on supCAR$(2)$ processes and show that they can be classified into three distinct types determined by the eigenstructure of the underlying CAR$(2)$ matrix. For each type we provide conditions for existence and derive explicit expressions for the correlation function. The resulting correlation structures may exhibit long-range dependence and can be non-monotone. These features make supCAR$(2)$ processes a flexible class for modeling time series with oscillatory correlations or strong dependence. + oai:arXiv.org:2601.15796v1 + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Stefan Schiffer + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Danijel Grahovac, Magdalena Miki\'c - On the Construction and Correlation Properties of Permutation-Interleaved Zadoff-Chu Sequences - https://arxiv.org/abs/2601.12107 - arXiv:2601.12107v1 Announce Type: new -Abstract: Constant amplitude zero auto-correlation (CAZAC) sequences are widely applied in waveforms for radar and communication systems. Motivated by a recent work [Berggren and Popovi\'c, IEEE Trans. Inf. Theory 70(8), 6068-6075 (2024)], this paper further investigates the approach to generating CAZAC sequences by interleaving Zadoff-Chu (ZC) sequences with permutation polynomials (PPs). We propose one class of high-degree PPs over the integer ring Z N , and utilize them and their inverses to interleave ZC sequences for constructing CAZAC sequences. It is known that a CAZAC sequence can be extended to an equivalence class by five basic opertations. We further show that the obtained CAZAC sequences are not covered by the equivalence classes of ZC sequences and interleaved ZC sequences by quadratic PPs and their inverses, and prove the sufficiency of the conjecture by Berggren and Popovi\'c in the aforementioned work. In addition, we also evaluate the aperiodic auto-correlation of certain ZC sequences from quadratic PPs. - oai:arXiv.org:2601.12107v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + Equivariant linear isometries and infinite little discs operads via transfer systems + https://arxiv.org/abs/2601.15800 + arXiv:2601.15800v1 Announce Type: new +Abstract: In this article, we apply the recently developed theory of transfer systems to study the relationship between $G$-equivariant linear isometries and infinite little discs operads, for a finite group $G$. This framework allows us to reduce involved topological problems to discrete problems regarding the subgroup structure and representation theory of the group $G$. Our main result is an example of this: we classify the $G$-universes $\mathcal{U}$ for which the linear isometries operad $\mathcal{L}(\mathcal{U})$ and the infinite little discs operad $\mathcal{D}(\mathcal{U})$ are homotopically equivalent. To achieve this, we use ideas that originate from the work of Balchin-Barnes-Roitzheim on the combinatorics of transfer systems on a total order. Additionally, the use of transfer systems gives us insight into the algebraic structures that arise from equivariant homotopy theory. Compatible pairs of transfer systems provide rules for when multiplicative transfer maps can be paired with additive transfer maps. In the case that the group $G$ is abelian, we provide conditions for when the pair $(\mathcal{L}(\mathcal{U}),\mathcal{D}(\mathcal{U}))$ defines a maximally compatible pair of transfer systems. As a consequence, we contribute to a recent conjecture about equivariant operad pairs. + oai:arXiv.org:2601.15800v1 + math.AT + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Qin Yuan, Chunlei Li, Xiangyong Zeng + Euan Aitken - One-variable equations over the lamplighter group - https://arxiv.org/abs/2601.12112 - arXiv:2601.12112v1 Announce Type: new -Abstract: We study one-variable equations over the lamplighter group $\MZ_2 \wr \MZ$. While the decidability of arbitrary equations over $L_2$ remains open, we prove that the Diophantine problem for single equations in one variable is decidable. Our approach reduces the problem to a divisibility question for families of parametric Laurent polynomials over $\MZ_2$, whose coefficients depend linearly on an integer parameter. We develop an automaton-theoretic framework to analyze divisibility of such polynomials, exploiting eventual periodicity phenomena arising from polynomial division over finite fields. This yields an explicit decision procedure, which is super-exponential in the worst case. On the other hand, we show that for a generic class of equations, solvability can be decided in nearly quadratic time. These results establish a sharp contrast between worst-case and typical computational behavior and provide new tools for the study of equations over wreath products. - oai:arXiv.org:2601.12112v1 - math.GR - Wed, 21 Jan 2026 00:00:00 -0500 + Optimal stochastic impulse control problem with delay with actions decided at the execution time + https://arxiv.org/abs/2601.15803 + arXiv:2601.15803v1 Announce Type: new +Abstract: In this paper, we consider a class of stochastic impulse control problem when there is a fixed delay $\Delta$ between the decision and execution times. The dynamics of the controlled system between two impulses is an arbitrary adapted stochastic process. Unlike the most existing literature, we consider the problem when the impulse sizes are decided at the execution time in both risk-neutral and risk-sensitive cases. This model fits more, in the real life, for some problems such as the pricing of swing options. The horizon T of the problem can be finite or infinite. In each case we show the existence of an optimal strategy. The main tools we use are the notions of reflected Backward Stochastic Differential Equations (BSDEs for short) and the Snell envelope of processes. + oai:arXiv.org:2601.15803v1 + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexander Ushakov, Yankun Wang + Said Hamad\`ene (LMM), Ibtissam Hdhiri - Hodge decomposition for Kato manifolds - https://arxiv.org/abs/2601.12113 - arXiv:2601.12113v1 Announce Type: new -Abstract: We prove that any Kato manifold satisfies the Hodge decomposition, in the sense that $b_k=\sum_{p+q=k}h^{p, q}$, by relating its cohomology to the corresponding cohomology of its modification data. We give, therefore, more evidence supporting a conjecture of Ornea--Verbitsky stating that compact locally conformally K\"ahler manifolds satisfy the Hodge decomposition. We further study Bott--Chern and Aeppli cohomology of Kato manifolds, showing that in certain degrees they coincide with Dolbeault cohomology. - oai:arXiv.org:2601.12113v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 + New results on Fourier multipliers on $L^p$: a perspective through unimodular symbols + https://arxiv.org/abs/2601.15815 + arXiv:2601.15815v1 Announce Type: new +Abstract: The paper focuses on the behaviour of unimodular Fourier multipliers with exponential growth in the context of weighted $L^p$-spaces. Our main result shows that much of the general theory of multipliers is approachable through the theory of unimodular multipliers. Indeed, we show that a bounded measurable function $m$ is a multiplier on $L^p$ for $1\leq p<\infty$ if and only if $e^{itm}$ is a multiplier on $L^p$ and its multiplier norm admits an exponential bound of the form $e^{c|t|^s}$ for suitable $c>0$ and $0<s<1$. We then apply this principle to obtain new results related to the boundedness of homogeneous rough operators, singular operators along curves and oscillatory integrals. A key ingredient in our study is an extension of the classical Stein's theorem on analytic families of operators that studies the behaviour of the derivative operator when $\theta \to 0$. + oai:arXiv.org:2601.15815v1 + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Giacomo Perri + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Mar\'ia Jes\'us Carro, Alberto Salguero-Alarc\'on - Offline Policy Learning with Weight Clipping and Heaviside Composite Optimization - https://arxiv.org/abs/2601.12117 - arXiv:2601.12117v1 Announce Type: new -Abstract: Offline policy learning aims to use historical data to learn an optimal personalized decision rule. In the standard estimate-then-optimize framework, reweighting-based methods (e.g., inverse propensity weighting or doubly robust estimators) are widely used to produce unbiased estimates of policy values. However, when the propensity scores of some treatments are small, these reweighting-based methods suffer from high variance in policy value estimation, which may mislead the downstream policy optimization and yield a learned policy with inferior value. In this paper, we systematically develop an offline policy learning algorithm based on a weight-clipping estimator that truncates small propensity scores via a clipping threshold chosen to minimize the mean squared error (MSE) in policy value estimation. Focusing on linear policies, we address the bilevel and discontinuous objective induced by weight-clipping-based policy optimization by reformulating the problem as a Heaviside composite optimization problem, which provides a rigorous computational framework. The reformulated policy optimization problem is then solved efficiently using the progressive integer programming method, making practical policy learning tractable. We establish an upper bound for the suboptimality of the proposed algorithm, which reveals how the reduction in MSE of policy value estimation, enabled by our proposed weight-clipping estimator, leads to improved policy learning performance. - oai:arXiv.org:2601.12117v1 - math.OC - cs.LG - Wed, 21 Jan 2026 00:00:00 -0500 + Weakly pancyclic vertices in dense nonbipartite graphs + https://arxiv.org/abs/2601.15822 + arXiv:2601.15822v1 Announce Type: new +Abstract: Let $G$ be a graph of girth $g$ and circumference $c.$ A vertex $v$ of $G$ is called weakly pancyclic if $v$ lies on an $\ell$-cycle for every integer $\ell$ with $g\le \ell\le c.$ We prove that if $G$ is a nonbipartite graph of order $n\ge 5$ and size at least $\left\lfloor(n-1)^2/4\right\rfloor+2,$ then $G$ contains three weakly pancyclic vertices, with one exception. This strengthens a result of Brandt from 1997. We also pose a related problem. + oai:arXiv.org:2601.15822v1 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jingren Liu, Hanzhang Qin, Junyi Liu, Mabel C. Chou, Jong-Shi Pang + Yurui Tang, Xingzhi Zhan - On the Hausdorff Dimension of weighted exactly Approximable Vectors - https://arxiv.org/abs/2601.12121 - arXiv:2601.12121v1 Announce Type: new -Abstract: We show that the Hausdorff dimension of $\boldsymbol w$-weighted $\tau$-exactly approximable vectors in $\mathbb R^d$ coincides with the Hausdorff dimension of $\boldsymbol w$-weighted $\tau$-approximable vectors, generalizing a result of the first named author and De Saxc\'e. - oai:arXiv.org:2601.12121v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 + Reversibility and symmetry of affine toral automorphisms + https://arxiv.org/abs/2601.15827 + arXiv:2601.15827v1 Announce Type: new +Abstract: We study reversibility and strong reversibility of affine automorphisms of the two-torus, written as $f_{A,\bar{a}}(\bar{x})=A\bar{x}+\bar{a} \ (\mathrm{mod}\ \mathbb{Z}^2)$. We derive explicit criteria for the reversibility of such maps in terms of the matrix $A$ and the translation $\bar{a}$. If $1$ is not an eigenvalue of $A$, reversibility of the affine map coincides with reversibility of $A$. When $1$ is an eigenvalue, additional arithmetic obstructions appear. We also provide a simple geometric condition, based on Pick's Theorem, that guarantees the existence of fixed points, along with a description of the dynamics of affine toral automorphisms. We also compute the entropy and characterize when conjugacy classes in the affine group are finite or uncountable. + oai:arXiv.org:2601.15827v1 + math.DS + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Prasuna Bandi, Reynold Fregoli + http://creativecommons.org/licenses/by/4.0/ + Kuntal Banerjee, Anubrato Bhattacharyya, Krishnendu Gongopadhyay, Subhamoy Mondal - Navier slip effects in micropolar thin-film flow: a rigorous derivation of Reynolds-type models - https://arxiv.org/abs/2601.12125 - arXiv:2601.12125v1 Announce Type: new -Abstract: We study the stationary flow of incompressible micropolar fluid in a thin three-dimensional domain under Navier slip boundary condition for the velocity and no-spin condition for microrotation. After rescaling the governing equations, we perform a rigorous asymptotic analysis as the film thickness tends to zero, considering a friction coefficient dependent on the small parameter. According to the scaling of the slip coefficient, we identify three distinct regimes: perfect slip, partial slip, and no-slip. For each regime, we derive the corresponding reduced micropolar system and obtain explicit expressions for the velocity and microrotation fields. This leads to a generalized Reynolds-type equation for the pressure, highlighting the impact of slip effects on the micropolar thin-film flow. - oai:arXiv.org:2601.12125v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Quantum Coherence Spaces Revisited: A von Neumann (Co)Algebraic Approach + https://arxiv.org/abs/2601.15832 + arXiv:2601.15832v1 Announce Type: new +Abstract: We describe a categorical model of MALL (Multiplicative Additive Linear Logic) inspired by the Heisenberg-Schr\"odinger duality of finite-dimensional quantum theory. Proofs of formulas with positive logical polarity correspond to CPTP (completely positive trace-preserving) maps in our model, i.e. the quantum operations in the Schr\"odinger picture, whereas proofs of formulas with negative logical polarity correspond to CPU (completely positive unital) maps, i.e. the quantum operations in the Heisenberg picture. The mathematical development is based on noncommutative geometry and finite-dimensional von Neumann (co)algebras, which can be defined as special kinds of (co)monoid objects internal to the category of finite-dimensional operator spaces. + oai:arXiv.org:2601.15832v1 + math.CT + cs.LO + math.FA + math.OA + quant-ph + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by-nc-nd/4.0/ - Mar\'ia Anguiano, Igor Pa\v{z}anin, Francisco J. Su\'arez-Grau + Thea Li, Vladimir Zamdzhiev - Spectral Analysis of the $D_{\log}^{(\lambda, N)}$ Operators - https://arxiv.org/abs/2601.12133 - arXiv:2601.12133v1 Announce Type: new -Abstract: This paper investigates the recent Connes-Consani-Moscovici $D_{\log}^{(\lambda, N)}$ operators, whose spectra are currently hypothesized to approach the zeros of $\zeta\left(\frac{1}{2} +is\right)$ as $\lambda, N \rightarrow \infty$. It turns out that when considering different standard notions of error, the dissonance between the spectra and Riemann $\zeta$ zeros either appears to or can be proven to be inverse logarithmic in nature, which elegantly fits the distribution of prime numbers. - oai:arXiv.org:2601.12133v1 - math.SP - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 + $C^\ast$-extreme points of unital completely positive maps invariant under group action + https://arxiv.org/abs/2601.15840 + arXiv:2601.15840v1 Announce Type: new +Abstract: In this work, we study a sub-collection of unital completely positive maps from a unital $C^\ast$-algebra $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$, the algebra of bounded linear operators on a Hilbert space $\mathcal{H}$ in the setting of $C^\ast$-convexity. Let $\tau$ be an action of a group $G$ on the $C^\ast$-algebra $\mathcal{A}$ through $C^\ast$-automorphisms. We focus our attention to the set of all unital completely positive maps from $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$, which remain invariant under $\tau$. We denote this collection by the notation $\text{UCP}^{G_\tau} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. This collection forms a $C^\ast$-convex set. We characterize the set of $C^\ast$-extreme points of $\text{UCP}^{G_\tau} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. Further, we conclude the article by proving the Krein--Milman type theorem in the setting of $C^\ast$-convexity for the set $\text{UCP}^{G_\tau} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. + oai:arXiv.org:2601.15840v1 + math.OA + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Dominik \'Sliwi\'nski + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Chaitanya J. Kulkarni - Fractional Semilinear Equations on Hyperbolic Spaces - https://arxiv.org/abs/2601.12140 - arXiv:2601.12140v1 Announce Type: new -Abstract: We study a semilinear equation involving the fractional Laplacian on the hyperbolic space $\mathbb{H}^n$. Unlike in conformally compact Einstein manifolds, the fractional Laplacian on $\mathbb{H}^n$ does not enjoy conformal covariance. By employing Helgason-Fourier analysis, we explicitly derive the Green's function of the fractional Laplacian on $\mathbb{H}^n$ and study its asymptotic behaviors. We then apply a direct method of moving planes to the integral form of the equation, establishing symmetry of solutions and nonexistence of positive solutions in the critical and subcritical cases, respectively. In addition, we develop several maximum principles on hyperbolic space. - oai:arXiv.org:2601.12140v1 + Riemann-Hilbert approach for the nonlocal modified Korteweg-de Vries equation with a step-like oscillating background + https://arxiv.org/abs/2601.15841 + arXiv:2601.15841v1 Announce Type: new +Abstract: This work focuses on the Cauchy problem for the nonlocal modified Korteweg-de Vries equation $$ u_t(x,t)+6u(x,t)u(-x,-t)u_x(x,t)+u_{xxx}(x,t)=0, $$ with the oscillating step-like boundary conditions: $u(x,t)\to 0$ as $x\to-\infty$ and $u(x,t)\backsimeq A\cos(2Bx+8B^3t)$ as $x\to\infty$, where $A,B>0$ are arbitrary constants. The main goal is to develop the Riemann-Hilbert formalism for this problem, paying a particular attention to the case of the ``pure oscillating step'' initial data, that is $u(x,0)=0$ for $x<0$ and $u(x,0)=A\cos(2Bx)$ for $x\geq0$. Also, we derive three new families of two-soliton solutions, which correspond to the values of $A$ and $B$ satisfying $B<\frac{A}{4}$, $B>\frac{A}{4}$, and $B=\frac{A}{4}$. + oai:arXiv.org:2601.15841v1 math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jianxiong Wang + Yan Rybalko - On dihedral invariants of the free associative algebra of rank two - https://arxiv.org/abs/2601.12144 - arXiv:2601.12144v1 Announce Type: new -Abstract: Let $K\langle X_d\rangle$ denote the free associative algebra of rank $d \geq 2$ over a field $K$. By results of Lane (1976) and Kharchenko (1978), the algebra of invariants $K\langle X_d\rangle ^G$ is free for any subgroup $G \leq \GL_d(K)$ and any field $K$. - Koryukin (1984) introduced an additional action of the symmetric group $Sym(n)$ on the homogeneous component of degree $n$ of $K\langle X_d\rangle$, given by permuting the positions of the variables. This endows $K\langle X_d\rangle $ with the structure of a $(K\langle X_d\rangle,\circ)$-$S$-algebra. With respect to this action, Koryukin proved that the invariant algebra $K\langle X_d\rangle ^G$ is finitely generated for every reductive group $G$. - In this paper we study the algebra ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ of invariants under the action of the dihedral group D_{2n} $ on the free associative algebra ${\mathbb C} \langle u,v\rangle$ of rank $2$. We compute the Hilbert series of ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ and construct an explicit set of generators for ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ as a free algebra. Furthermore, we describe a finite generating set for the $S$-algebra ${\mathbb C}\langle u,v\rangle^{D_{2n}}$. - oai:arXiv.org:2601.12144v1 - math.RA - Wed, 21 Jan 2026 00:00:00 -0500 + Weak Centrality: AF-algebras, C(X)-algebras, and group C*-algebras + https://arxiv.org/abs/2601.15843 + arXiv:2601.15843v1 Announce Type: new +Abstract: We first prove that every AF-algebra is weakly central, thereby resolving a question left open by Archbold--Gogi\'c. We then establish a new characterization of weak centrality for unital $C^*$-algebras in terms of $C(X)$-algebras. The paper concludes with an appendix that examines weak centrality in full group $C^*$-algebras and places these examples within the hierarchy of group classes. + oai:arXiv.org:2601.15843v1 + math.OA + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Silvia Boumova, Vesselin Drensky, \c{S}ehmus F{\i}nd{\i}k + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Bharat Talwar, Prahlad Vaidyanathan, Stefan Wagner - A $p$-adic cohomological approach to congruences of meromorphic modular forms - https://arxiv.org/abs/2601.12157 - arXiv:2601.12157v1 Announce Type: new -Abstract: We study congruences relating Fourier coefficients of meromorphic modular forms and Frobenius eigenvalues of elliptic curves corresponding to their poles. We develop a $p$-adic cohomological framework that interprets these congruences via the interaction between the rigid cohomology of modular curves and the crystalline structure of the associated elliptic curves. Using comparison theorems and the Gysin sequence, we relate the Frobenius actions in cohomology to the $U_p$-operator acting on spaces of overconvergent modular forms. Our approach applies uniformly to both modular curves and Shimura curves admitting smooth integral models over $\mathbb{Z}_p$. - oai:arXiv.org:2601.12157v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 + Quantitative Borg-Levinson theorem for the magnetic Sch\"odinger operator with unbounded electrical potential + https://arxiv.org/abs/2601.15847 + arXiv:2601.15847v1 Announce Type: new +Abstract: The first author established in [8] a quantitative Borg-Levinson theorem for the Schr\"odinger operator with unbounded potential. In the present work, we extend the results in [8] to the magnetic Schr\"odinger operator. We discuss both the isotropic and anisotropic cases. We establish H\"older stability inequalities of determining the electrical potential or magnetic field from the corresponding boundary spectral data. + oai:arXiv.org:2601.15847v1 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Paolo Bordignon + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mourad Choulli, Hiroshi Takase - Streaming Operator Inference for Model Reduction of Large-Scale Dynamical Systems - https://arxiv.org/abs/2601.12161 - arXiv:2601.12161v1 Announce Type: new -Abstract: Projection-based model reduction enables efficient simulation of complex dynamical systems by constructing low-dimensional surrogate models from high-dimensional data. The Operator Inference (OpInf) approach learns such reduced surrogate models through a two-step process: constructing a low-dimensional basis via Singular Value Decomposition (SVD) to compress the data, then solving a linear least-squares (LS) problem to infer reduced operators that govern the dynamics in this compressed space, all without access to the underlying code or full model operators, i.e., non-intrusively. Traditional OpInf operates as a batch learning method, where both the SVD and LS steps process all data simultaneously. This poses a barrier to deployment of the approach on large-scale applications where dataset sizes prevent the loading of all data into memory at once. Additionally, the traditional batch approach does not naturally allow model updates using new data acquired during online computation. To address these limitations, we propose Streaming OpInf, which learns reduced models from sequentially arriving data streams. Our approach employs incremental SVD for adaptive basis construction and recursive LS for streaming operator updates, eliminating the need to store complete data sets while enabling online model adaptation. The approach can flexibly combine different choices of streaming algorithms for numerical linear algebra: we systematically explore the impact of these choices both analytically and numerically to identify effective combinations for accurate reduced model learning. Numerical experiments on benchmark problems and a large-scale turbulent channel flow demonstrate that Streaming OpInf achieves accuracy comparable to batch OpInf while reducing memory requirements by over 99% and enabling dimension reductions exceeding 31,000x, resulting in orders-of-magnitude faster predictions. - oai:arXiv.org:2601.12161v1 - math.NA - cs.LG - cs.NA - math.DS - physics.comp-ph - Wed, 21 Jan 2026 00:00:00 -0500 + Quadratic discrepancy estimates for probability measures on the Heisenberg group + https://arxiv.org/abs/2601.15850 + arXiv:2601.15850v1 Announce Type: new +Abstract: We initiate the study of quadratic discrepancy for finite point sets on the Heisenberg group $\mathbb H^n$ with respect to upper Ahlfors regular probability measures. For a natural family of test sets given by left translations and dilations of cylindrically defined neighborhoods, we introduce an $L^2$-discrepancy and establish a Roth-type lower bound depending on the homogeneous dimension of $\mathbb H^n$. + This result extends classical discrepancy estimates from the Euclidean and compact settings to a non-commutative, step-two nilpotent Lie group. It should be viewed as a first step toward the development of a discrepancy theory on the Heisenberg group. + oai:arXiv.org:2601.15850v1 + math.CA + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Tomoki Koike, Prakash Mohan, Marc T. Henry de Frahan, Julie Bessac, Elizabeth Qian + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Luca Brandolini, Alessandro Monguzzi, Matteo Monti - Locating critical points attracted to p-adic attracting cycles - https://arxiv.org/abs/2601.12163 - arXiv:2601.12163v1 Announce Type: new -Abstract: In complex dynamics, a fundamental result of Fatou and Julia asserts that every attracting cycle of a rational map attracts a critical point. The analogous statement fails in non-Archimedean dynamics. For a non-Archimedean rational map, this paper establishes a sharp condition on the multiplier of an attracting cycle ensuring it attracts a critical point. - oai:arXiv.org:2601.12163v1 - math.DS - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 + Practical applications of Set Shaping Theory to Non-Uniform Sequences + https://arxiv.org/abs/2601.15853 + arXiv:2601.15853v1 Announce Type: new +Abstract: Set Shaping Theory (SST) moves beyond the classical fixed-space model by constructing bijective mappings the original sequence set into structured regions of a larger sequence space. These shaped subsets are characterized by a reduced average information content, measured by the product of the empirical entropy and the length, yielding (N +k)H0(f(s)) < NH0(s), which represents the universal coding limit when the source distribution is unknown. The principal experimental difficulty in applying Set Shaping Theory to non-uniform sequences arises from the need to order the sequences of both the original and transformed sets according to their information content. An exact ordering of these sets entails exponential complexity, rendering a direct implementation impractical. In this article, we show that this obstacle can be overcome by performing an approximate but informative ordering that preserves the structural requirements of SST while achieving the shaping gain predicted by the theory. This result extends previous experimental findings obtained for uniformly distributed sequences and demonstrates that the shaping advantage of SST persists for non-uniform sequences. Finally, to ensure full reproducibility, the software implementing the proposed method has been made publicly available on GitHub, enabling independent verification of the results reported in this work + oai:arXiv.org:2601.15853v1 + cs.IT + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Juan Rivera-Letelier + A. Schmidt, A. Vdberg, A. Petit - Fractional Quantum Hall States: Infinite Matrix Product Representation and its Implications - https://arxiv.org/abs/2601.12165 - arXiv:2601.12165v1 Announce Type: new -Abstract: We present a novel matrix product representation of the Laughlin and related fractional quantum Hall wavefunctions based on a rigorous version of the correlators of a chiral quantum field theory. This representation enables the quantitative control of the coefficients of the Laughlin wavefunction times an arbitrary monomial symmetric polynomial when expanded in a Slater determinant or permanent basis. It renders the properties, such as factorization and the renewal structure, inherent in such fractional quantum Hall wavefunctions transparent. We prove bounds on the correlators of the chiral quantum field theory and utilize this representation to demonstrate the exponential decay of connected correlations and a gap in the entanglement spectrum on a thin cylinder. - oai:arXiv.org:2601.12165v1 + Synthetic Differential Jet Bundles are Reduced + https://arxiv.org/abs/2601.15862 + arXiv:2601.15862v1 Announce Type: new +Abstract: We have previously observed that the theory of solutions of partial differential equations, regarded as diffieties inside jet bundles, acquires a powerful comonadic formulation after passage from the category of Fr\'echet smooth manifolds to the Cahiers topos of formal smooth sets (a well-adapted model for Synthetic Differential Geometry). However, the tacit assumption that this passage preserves the projective limits that define infinite jet bundles had remained unproven. Here we provide a detailed proof. + oai:arXiv.org:2601.15862v1 + math.DG math-ph - cond-mat.str-el + math.CT math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Severin Schraven, Simone Warzel - - - Balancing adaptability and predictability: K-revision multistage stochastic programming - https://arxiv.org/abs/2601.12166 - arXiv:2601.12166v1 Announce Type: new -Abstract: A standard assumption in multistage stochastic programming is that decisions are made after observing the uncertainty from the prior stage. The resulting solutions can be difficult to implement in practice, as they leave practitioners ill-prepared for future stages. To provide better foresight, we introduce the K-revision approach. This new framework requires plans to be specified in advance. To maintain flexibility, we allow plans to be revised a maximum of K times as new information becomes available. We analyze the complexity of K-revision problems, showing NP-hardness even in a simple setting. We examine, both theoretically and computationally, the impact of the K-revision approach on the objective compared with classical multistage stochastic programming models and the partially adaptive approach introduced in [1, 2]. We develop two MIP formulations, one directly from our definition and the other based on a combinatorial characterization. We analyze the tightness of these formulations and propose several methods to strengthen them. Computational experiments on synthetic problems and practical applications demonstrate that our approach is both computationally tractable and effective in reaching near-optimal performance while increasing the predictability of the solutions produced. - oai:arXiv.org:2601.12166v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chengwenjian Wang, Alexander S. Estes, Jean-Philippe P. Richard + Grigorios Giotopoulos, Igor Khavkine, Hisham Sati, Urs Schreiber - The P\'olya Web - https://arxiv.org/abs/2601.12172 - arXiv:2601.12172v1 Announce Type: new -Abstract: We introduce the P\'olya Web, a system of coalescing random walks based on the classic P\'olya urn model. This construction serves as an analogue to the web of coalescing random walks studied by T\'oth and Werner (1998), replacing simple symmetric random walks with P\'olya walks as primary constituents. First, we study the general web of up-right oriented coalescing random walks. We investigate its geometric properties and prove that certain indicator random variables satisfy negative association. Notably, the proof involves a non-trivial application of the van den Berg-Kesten-Reimer (BKR) inequality. Based on this property, we derive a strong law for the number of connected components generated by walks starting at the same time. Subsequently, we focus on the specific properties of the P\'olya Web. It is well-known that the normalized coordinates of a single P\'olya Walk converge almost surely to a beta-distributed random variable. We determine the joint distribution of these limiting variables in the coalescing framework. Using these joint densities, we provide exact calculations regarding the almost sure convergence of the number of components. Finally, by applying a local scaling to the P\'olya Web at the edges, we introduce the Yule Web, a web of coalescing Yule processes. We demonstrate that the fundamental properties and results derived for the P\'olya Web can be extended to this limiting case. - oai:arXiv.org:2601.12172v1 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 + Tangle structure trees + https://arxiv.org/abs/2601.15870 + arXiv:2601.15870v1 Announce Type: new +Abstract: We introduce a comprehensive data structure, tangle structure trees, which simultaneously displays all the $\mathcal{F}$-tangles of an abstract separation system for very general obstruction sets $\mathcal{F}$. It simultaneously also displays certificates $\sigma\in\mathcal{F}$ for any non-existence of such tangles, or for the non-extendability of low-order tangles to higher-order ones. + Our theorem can be applied to produce the structures of the classical tree-of-tangles and tangle-tree duality theorems, both for graph tangles and for their known generalizations to more general separation systems. It extends those theorems to obstruction sets $\mathcal{F}$ that need not define profiles (as they must in trees of tangles) or consist of stars of separations (as they must in tangle-tree duality). + Our existence proof for these structure trees is constructive. The construction has been implemented in open-source software available for tangle detection and further analysis. + oai:arXiv.org:2601.15870v1 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - \'Akos Urb\'an + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Hanno von Bergen, Reinhard Diestel - Kato's Ramification filtration via de Rham-Witt complex and applications - https://arxiv.org/abs/2601.12177 - arXiv:2601.12177v1 Announce Type: new -Abstract: Given an $F$-finite regular scheme $X$ of positive characteristic and a simple normal crossing divisor $E$ on $X$, we introduce a filtration on the de Rham-Witt complex $W_m\Omega^\bullet_{X\setminus E}$. When $X$ is the spectrum of a henselian discrete valuation ring $A$ with quotient field $K$, this extends the classical filtration on $W_m(K)$ due to Brylinski. We show that Kato's ramification filtration on $H^q_\et(X \setminus E, {\Q}/{\Z}(q-1))$ for $q \ge 1$ admits an explicit description in terms of the above filtration of the de Rham-Witt complex of $X \setminus E$. When $q =1$, this specializes to the results of Kato and Kerz-Saito. - As applications, we prove refinements of the duality theorem of Jannsen-Saito-Zhao for smooth projective schemes over finite fields and the duality theorem of Zhao for semi-stable schemes over henselian discrete valuation rings of positive characteristic with finiteresidue fields. We also prove a modulus version of the duality theorem of Ekedahl. As another application, we prove Lefschetz theorems for Kato's ramification filtrations for smooth projective varieties over $F$-finite fields. This extends a result of Kerz-Saito for $H^1$ to higher cohomology. Similar results are proven for the Brauer group. - oai:arXiv.org:2601.12177v1 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 + Visibility of Lattice Points across Polynomials + https://arxiv.org/abs/2601.15877 + arXiv:2601.15877v1 Announce Type: new +Abstract: The visibility of lattice points from the origin along a polynomial family of curves constitutes a significant generalization of visibility along straight lines. Following the classical notion, where the density equals 1/2, and its generalization to monomial curves of the form y = a x^b, where the density equals 1/(b+1), we study a family of polynomial curves defined by y = q(a_n x^n + ... + a_1 x), where q is a positive rational number. + We introduce a new criterion based on a polynomial greatest common divisor condition that provides a lower bound on the number of visible lattice points in N^2. Conversely, we derive conditions under which a given lattice point becomes the next visible point along such a polynomial curve. Using the principle of inclusion-exclusion, we also obtain an exact double-sum formula for the number of pairs (a, b) less than or equal to N that are visible with respect to this polynomial family. + Finally, we extend the framework to related problems and pose several open questions concerning gap distributions and quantitative bounds for non-visible points. This work provides a broader theoretical foundation for lattice point visibility beyond linear and monomial settings. + oai:arXiv.org:2601.15877v1 + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Amalendu Krishna, Subhadip Majumder + Chahat Ahuja - On a theorem of Artin and the dimension of the space spanned by the rational valued characters of a group - https://arxiv.org/abs/2601.12185 - arXiv:2601.12185v1 Announce Type: new -Abstract: In this paper, we sharpen a theorem of Artin to show that for a finite group, the dimension of the subspace of class functions spanned by the rational valued characters equals the number of conjugacy classes of cyclic subgroups. - oai:arXiv.org:2601.12185v1 - math.GR - Wed, 21 Jan 2026 00:00:00 -0500 + Recovery of nonlinear material parameters in a quasilinear Lam\'e system + https://arxiv.org/abs/2601.15881 + arXiv:2601.15881v1 Announce Type: new +Abstract: We investigate the inverse problem of determining nonlinear elastic material parameters from boundary stress measurements corresponding to prescribed boundary displacements. The material law is described by a nonlinear, space-independent elastic tensor depending on both the displacement and the strain, and gives rise to a general class of quasilinear Lam\'e systems. We prove the unique and stable recovery of a wide class of space-independent nonlinear elastic tensors, including the identification of two nonlinear isotropic Lam\'e moduli as well as certain anisotropic tensors. The boundary measurements are assumed to be available at a finite number of boundary points and, in the isotropic case, at a single point. Moreover, the measurements are generated by boundary displacements belonging to an explicit class of affine functions. The analysis is based on structural properties of nonlinear Lam\'e systems, including asymptotic expansions of the boundary stress and tensorial calculus. + oai:arXiv.org:2601.15881v1 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Mark L. Lewis + David Johansson, Yavar Kian - Sets of Ramsey-limit points and IP-limit points - https://arxiv.org/abs/2601.12187 - arXiv:2601.12187v1 Announce Type: new -Abstract: Let $X$ be an uncountable Polish space and let $\mathcal{H}$ be the Hindman ideal, that is, the family of all $S\subseteq \omega$ which are not $IP$-sets. For each sequence $x=(x_n)_{n \in \omega}$ taking values in $X$, let $\Lambda_{x}(FS)$ be the set of $IP$-limit points of $x$. Also, let $\Lambda_{x}(\mathcal{H})$ be the set of $\mathcal{H}$-limit points of $x$, that is, the set of ordinary limits of subsequences $(x_n)_{n \in S}$ with $S\notin \mathcal{H}$. After proving that these two notions do not coincide in general, we show that both families of nonempty sets of the type $\Lambda_{x}(FS)$ and of the type $\Lambda_{x}(\mathcal{H})$ are precisely the class of nonempty analytic subsets of $X$. - An analogous result holds also for Ramsey convergence. In the proofs, we use the concept of partition regular functions introduced in J. Symb. Log. (2024) [doi:10.1017/jsl.2024.8], which provide a unified approach to these types of convergence. - oai:arXiv.org:2601.12187v1 - math.GN - math.DS - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 + Directional polynomial frames on spheres + https://arxiv.org/abs/2601.15883 + arXiv:2601.15883v1 Announce Type: new +Abstract: We introduce a general framework for the construction of polynomial frames in $L^2(\mathbb{S}^{d-1})$, $d \geq 3$, where the frame functions are obtained as rotated versions of an initial sequence of polynomials $\Psi^j$, $j\in \mathbb{N}_0$. The rotations involved are discretized using suitable quadrature rules. This framework includes classical constructions such as spherical needlets and directional wavelet systems, and at the same time permits the systematic design of new frames with adjustable spatial localization, directional sensitivity, and computational complexity. We show that a number of frame properties can be characterized in terms of simple, easily verifiable conditions on the Fourier coefficients of the functions $\Psi^j$. Extending an earlier result for zonal systems, we establish sufficient conditions under which the frame functions are optimally localized in space with respect to a spherical uncertainty principle, thus making the corresponding systems a viable tool for position-frequency analyses. To conclude this article, we explicitly discuss examples of well-localized and highly directional polynomial frames. + oai:arXiv.org:2601.15883v1 + math.CA + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Rafa{\l} Filip\'ow, Adam Kwela, Paolo Leonetti + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Marzieh Hasannasab, Larissa Kaldewey, Frederic Schoppert - Optimal Leveraging of Smoothness and Strong Convexity for Peaceman--Rachford Splitting - https://arxiv.org/abs/2601.12190 - arXiv:2601.12190v1 Announce Type: new -Abstract: In this paper, we introduce a simple methodology to leverage strong convexity and smoothness in order to obtain an optimal linear convergence rate for the Peaceman--Rachford splitting (PRS) scheme applied to optimization problems involving two smooth strongly convex functions. The approach consists of adding and subtracting suitable quadratic terms from one function to the other so as to redistribute strong convexity in the primal formulation and smoothness in the dual formulation. This yields an equivalent modified optimization problem in which each term has adjustable levels of strong convexity and smoothness. In this setting, the Peaceman--Rachford splitting method converges linearly to the solution of the modified problem with a convergence rate that can be optimized with respect to the introduced parameters. Upon returning to the original formulation, this procedure gives rise to a modified variant of PRS. The optimal linear rate established in this work is strictly better than the best rates previously available in the general setting. The practical performance of the method is illustrated through an academic example and applications in image processing. - oai:arXiv.org:2601.12190v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + 2-Equivariant 2-Vector bundles and 2K-theories + https://arxiv.org/abs/2601.15893 + arXiv:2601.15893v1 Announce Type: new +Abstract: We construct a theory of 2-vector bundles over a Lie groupoid, with fibers modeled by the bicategory of super algebras, bimodules and intertwiners. We demonstrate that these 2-vector bundles form a symmetric monoidal 2-stack. From this structure, we define the 2K-theory as the Grothendieck group of the internal equivalence classes of the 2-vector bundle over the given Lie groupoid, and we construct the spectra representing this theory. We then extend this framework to the equivariant setting. For any Lie groupoid equipped with an action by a coherent 2-group, we introduce the bicategory of 2-equivariant 2-vector bundles over it. This leads to the definition of 2-equivariant 2K-theory as the Grothendieck group of the internal equivalence classes in the bicategory. Furthermore, we define a higher analogue of orbifold, which generalizes Lie groupoids with a 2-group action, and construct the bicategory of 2-orbifold 2-vector bundles. Finally, we can define the 2-orbifold 2K-theory. + oai:arXiv.org:2601.15893v1 + math.AT + math.DG + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Luis Brice\~no-Arias, Fernando Rold\'an + Zhen Huan - Sobolev inequalities for nonlinear Dirichlet forms - https://arxiv.org/abs/2601.12192 - arXiv:2601.12192v1 Announce Type: new -Abstract: In this short note we show an equivalence between Sobolev type inequalities and so called isocapacitary inequalities in the context of a large class of nonlinear Dirichlet forms, their associated Dirichlet spaces and their associated capacities. - oai:arXiv.org:2601.12192v1 + Convergence to shock profiles for Burgers equation with singular fast-diffusion and boundary effect + https://arxiv.org/abs/2601.15900 + arXiv:2601.15900v1 Announce Type: new +Abstract: In this paper, we study the asymptotic stability of viscous shock profile for the Burgers equation $u_t +f(u)_x = (\frac{u_{x}}{u^{1-m}})_x$ on the half-space $(0,+\infty)$, subject to the boundary conditions $u|_{x=0}=u_->0$ and $u|_{x=+\infty}=0$. Here, the parameter $\frac{1}{2}<m<1$ measures the strength of fast diffusion. A key challenge arises from the pronounced singularity in the diffusivity $\left(\frac{u_x}{u^{1-m}} \right)_x$ at $u=0$ and the boundary layer. We demonstrate that the long-time behavior of $u$ converges to a shifted shock profile $U(x-st-d(t))$, where $d(t)$ is governed by the boundary layer dynamics at $x=0$ and driven by the initial data $u(x,0)$. To overcome the singularity from fast diffusion compounded by the bad effect of boundary layer for wave stability, some new techniques for weighted energy estimates are introduced artfully. + oai:arXiv.org:2601.15900v1 math.AP - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ralph Chill, Burkhard Claus + Xiaowen Li, Ming Mei - Coherent Comparison as Information Cost: A Cost-First Ledger Framework for Discrete Dynamics - https://arxiv.org/abs/2601.12194 - arXiv:2601.12194v1 Announce Type: new -Abstract: We develop an information-theoretic framework for discrete dynamics grounded in a comparison-cost functional on ratios. Given two quantities compared via their ratio \(x=a/b\), we assign a cost \(F(x)\) measuring deviation from equilibrium (\(x=1\)). Requiring coherent composition under multiplicative chaining imposes a d'Alembert functional equation; together with normalization (\(F(1)=0\)) and quadratic calibration at unity, this yields a unique reciprocal cost functional (proved in a companion paper): \[ J(x) = \tfrac{1}{2}\bigl(x + x^{-1}\bigr) - 1. \] This cost exhibits reciprocity \(J(x)=J(x^{-1})\), vanishes only at \(x=1\), and diverges at boundary regimes \(x\to 0^+\) and \(x\to\infty\), excluding ``nothingness'' configurations. Using \(J\) as input, we introduce a discrete ledger as a minimal lossless encoding of recognition events on directed graphs. Under deterministic update semantics and minimality (no intra-tick ordering metadata), we derive atomic ticks (at most one event per tick). Explicit structural assumptions (conservation, no sources/sinks, pairwise locality, quantization in \(\delta\mathbb{Z}\)) force balanced double-entry postings and discrete ledger units. To obtain scalar potentials on graphs with cycles while retaining single-edge impulses per tick, we impose time-aggregated cycle closure (no-arbitrage/clearing over finite windows). Under this hypothesis, cycle closure is equivalent to path-independence, and the cleared cumulative flow admits a unique scalar potential on each connected component (up to additive constant), via a discrete Poincar\'e lemma. On hypercube graphs \(Q_d\), atomicity imposes a \(2^d\)-tick minimal period, with explicit Gray-code realization at \(d=3\). The framework connects ratio-based divergences, conservative graph flows, and discrete potential theory through a coherence-forced cost structure. - oai:arXiv.org:2601.12194v1 + Blind Identification of Channel Codes: A Subspace-Coding Approach + https://arxiv.org/abs/2601.15903 + arXiv:2601.15903v1 Announce Type: new +Abstract: The problem of blind identification of channel codes at a receiver involves identifying a code chosen by a transmitter from a known code-family, by observing the transmitted codewords through the channel. Most existing approaches for code-identification are contingent upon the codes in the family having some special structure, and are often computationally expensive otherwise. Further, rigorous analytical guarantees on the performance of these existing techniques are largely absent. This work presents a new method for code-identification on the binary symmetric channel (BSC), inspired by the framework of subspace codes for operator channels, carefully combining principles of hamming-metric and subspace-metric decoding. We refer to this method as the minimum denoised subspace discrepancy decoder. We present theoretical guarantees for code-identification using this decoder, for bounded-weight errors, and also present a bound on the probability of error when used on the BSC. Simulations demonstrate the improved performance of our decoder for random linear codes beyond existing general-purpose techniques, across most channel conditions and even with a limited number of received vectors. + oai:arXiv.org:2601.15903v1 cs.IT math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sebastian Pardo-Guerra, Megan Simons, Anil Thapa, Jonathan Washburn + Pramod Singh, Prasad Krishnan, Arti Yardi - Bruhat Intervals in the Infinite Symmetric Group are Cohen-Macaulay - https://arxiv.org/abs/2601.12195 - arXiv:2601.12195v1 Announce Type: new -Abstract: We show that the (non-Noetherian) Stanley-Reisner ring of the order complex of certain intervals in the Bruhat order on the infinite symmetric group $S_\infty$ of all auto-bijections of $\mathbb{N}$ is Cohen-Macaulay in the sense of ideals and weak Bourbaki unmixed. This gives an infinite-dimensional version of results due to Edelman, Bj\"{o}rner, and Kind and Kleinschmidt for finite symmetric groups $S_n$. - oai:arXiv.org:2601.12195v1 - math.CO - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 + Metric constructions and fixed point theorems in product spaces + https://arxiv.org/abs/2601.15907 + arXiv:2601.15907v1 Announce Type: new +Abstract: The paper studies a general scheme for constructing metrics on a product of metric spaces by means of a family of continuous convex functions. This construction includes the conventional $p$-metrics and generates metrics that are topologically equivalent to the conventional ones. As an application, we study fixed point and approximate fixed point properties for nonexpansive maps on a product space equipped with the constructed metric. We show that existing fixed point results of this type are consequences of our framework. Examples are provided to illustrate the established results. The construction machinery is also used to study products of length and geodesic spaces. The obtained results encompass existing ones and provide a background for potential studies of fixed point properties on these product spaces. + oai:arXiv.org:2601.15907v1 + math.MG + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Nathaniel Gallup, Leo Gray + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Doan Huu Hieu, Vo Minh Tam, Nguyen Duy Cuong - Higher-Order Approximations of Sojourn Times in M/G/1 Queues via Stein's Method - https://arxiv.org/abs/2601.12197 - arXiv:2601.12197v1 Announce Type: new -Abstract: We study the stationary sojourn time distribution in an M/G/1 queue operating under heavy traffic. It is known that the sojourn time converges to an exponential distribution in the limit. Our focus is on obtaining pre-asymptotic, higher-order approximations that go beyond the classical exponential limit. Using Stein's method, we develop an approach based on higher-order expansions of the generator of the underlying Markov process. The key technical step is to represent higher-order derivatives in terms of lower-order ones and control the resulting error via derivative bounds of the Stein equation. Under suitable moment-matching conditions on the service distribution, we show that the approximation error decays as a high-order power of the slack parameter $\varepsilon=1-\rho$. Error bounds are established in the Zolotarev metric, which further imply bounds on the Wasserstein distance as well as the moments. Our results demonstrate that the accuracy of the exponential approximation can be systematically improved by matching progressively more moments of the service distribution. - oai:arXiv.org:2601.12197v1 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 + On the escape rate for intermittent maps with holes shrinking around the indifferent fixed point + https://arxiv.org/abs/2601.15908 + arXiv:2601.15908v1 Announce Type: new +Abstract: We study non-uniformly expanding maps of the unit interval with a parabolic fixed point at the origin that admit an ergodic absolutely continuous invariant measure, which may be finite or infinite. By introducing a hole defined by an interval containing the parabolic fixed point, we analyze the escape rate of the resulting open system and its asymptotic behavior as the hole shrinks. Our approach relies on the transfer operator associated with the dynamical system and on the relationship between the transfer operators of the original system and its induced version. The results extend to this general framework previous investigations which considered special cases. + oai:arXiv.org:2601.15908v1 + math.DS + math-ph + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Bihan Chatterjee, Siva Theja Maguluri, Debankur Mukherjee - - - A curvature-regularized variational problem with an area constraint - https://arxiv.org/abs/2601.12201 - arXiv:2601.12201v1 Announce Type: new -Abstract: Interlocking interfaces are commonly employed to mitigate relative sliding under shear.Indeed, Their geometry is typically selected on grounds of fabrication convenience rather than analytical optimality. There is no reason to suppose that circular or polygonal profiles minimize localized stress concentration under fixed geometric constraints. We propose a variational model in which the interface is represented by a planar curve $y=f(x)$, and localized stress amplification is quantified by a curvature-sensitive functional \[ J[f] = \int_{-a}^{a} \bigl(1+\gamma \kappa^2\bigr) \sqrt{1+f'(x)^2}\,dx, \] defined on the Sobolev space $W^{2,2}([-a,a])$. The functional is motivated by elasticity-theoretic considerations in which curvature enters the leading-order stress field near a singular interface.Indeed, any profile possessing discontinuous tangents yields a divergent integral, thereby rendering it energetically inadmissible within the Sobolev space $W^{2,2}$. An area constraint $\int_{-a}^{a} f(x)\,dx = A_0$ is imposed to model fixed material volume. Using the direct method of the calculus of variations, we establish the existence of a minimizer and derive the associated Euler--Lagrange equation, a nonlinear fourth-order boundary value problem. - Note, however, that constant-curvature and piecewise-linear profiles fail to satisfy the necessary optimality conditions under the imposed constraint. Indeed, we are thus forced to conclude that analytical optimality necessitates a more complex variation in the local tangent angle - The analysis indicates that commonly employed interlock geometries are not variationally optimal for minimizing localized shear stress within this class of admissible interfaces. - oai:arXiv.org:2601.12201v1 - math.RA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chandrasekhar Gokavarapu (Department of Mathematics, Government College) + Claudio Bonanno, Sharvari Neetin Tikekar - Characterizations of Lorentz Type Sobolev Multiplier Spaces and Their Preduals - https://arxiv.org/abs/2601.12206 - arXiv:2601.12206v1 Announce Type: new -Abstract: We provide several characterizations of Sobolev multiplier spaces of Lorentz type and their preduals. Block decomposition and K\"othe dual of such preduals are discussed. As an application, the boundedness of local Hardy-Littlewood maximal function on these spaces will be justified. - oai:arXiv.org:2601.12206v1 - math.FA - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + A fully diagonalized spectral method on the unit ball + https://arxiv.org/abs/2601.15911 + arXiv:2601.15911v1 Announce Type: new +Abstract: Our main objective in this work is to show how Sobolev orthogonal polynomials emerge as a useful tool within the framework of spectral methods for boundary-value problems. The solution of a boundary-value problem for a stationary Schr\"odinger equation on the unit ball can be studied from a variational perspective. In this variational formulation, a Sobolev inner product naturally arises. As test functions, we consider the linear space of the polynomials satisfying the boundary conditions on the sphere, and a basis of mutually orthogonal polynomials with respect to the Sobolev inner product is provided. The basis of the proposed method is given in terms of spherical harmonics and univariate Sobolev orthogonal polynomials. The connection formula between these Sobolev orthogonal polynomials and the classical orthogonal polynomials on the ball is established. Consequently, the Sobolev Fourier coefficients of a function satisfying the boundary value problem are recursively derived. Finally, one numerical experiment is presented. + oai:arXiv.org:2601.15911v1 + math.NA + cs.NA + math.CA + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Keng Hao Ooi + 10.1007/s11075-026-02315-w + Numerical Algorithms (2026) + Miguel A. Pi\~nar - Time-asymptotic stability of composite waves of degenerate Oleinik shock and rarefaction for non-convex conservation laws with Cattaneo's law - https://arxiv.org/abs/2601.12216 - arXiv:2601.12216v1 Announce Type: new -Abstract: This paper examines the large-time behavior of solutions to a one-dimensional conservation law featuring a non-convex flux and an artificial heat flux term regulated by Cattaneo's law, forming a 2$\times$2 system of hyperbolic equations. Under the conditions of small wave strength and sufficiently small initial perturbations, we demonstrate the time-asymptotic stability of a composite wave that combines a degenerate Oleinik shock and a rarefaction wave. The proof utilizes the Oleinik entropy condition, the a-contraction method with time-dependent shifts, and weighted energy estimates. - oai:arXiv.org:2601.12216v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + The distinguishing number of complete bipartite and crown graphs + https://arxiv.org/abs/2601.15913 + arXiv:2601.15913v1 Announce Type: new +Abstract: The distinguishing number of a permutation group $G\leqslant\Sym(\Omega)$ is the minimum number of colours needed to colour $\Omega$ in such a way that the only colour preserving element of $G$ is the identity. The distinguishing number of a graph is the distinguishing number of its automorphism group (as a permutation group on vertices). We determine the distinguishing number of the complete bipartite graphs $K_{n,n}$ and the crown graphs $K_{n,n}-nK_2$, as well as the distinguishing number of some `large' subgroups of their automorphism groups, that is, the subgroups that are vertex- and edge-transitive and such that the induced action on each bipart is $\Alt(n)$ or $\Sym(n)$. We show that, if $G$ is a `large' group of automorphisms of $K_{n,n}$, then $n-1\leqslant D(G) \leqslant n+1$. Similarly, if $G$ is a `large' group of automorphisms of a crown graph, then $\lceil \sqrt{n-1}\rceil \leqslant D(G)\leqslant \lfloor \sqrt{n}\rfloor+1$. + \smallskip + \textit{Keywords:} complete bipartite graph; crown graph; distinguishing number; symmetric group; alternating group + oai:arXiv.org:2601.15913v1 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Yuxi Hu, Ran Song + Lei Chen, Alice Devillers, Luke Morgan, Friedrich Rober - Interval B-Tensors and Interval Double B-Tensors - https://arxiv.org/abs/2601.12217 - arXiv:2601.12217v1 Announce Type: new -Abstract: This paper systematically investigates the properties and characterization of interval B-tensors and interval double B-tensors. We propose verifiable necessary and sufficient conditions that allow for determining whether an entire interval tensor family belongs to these classes based solely on its extreme point tensors. The study elucidates profound connections between these interval tensors and other structured ones such as interval Z-tensors and P-tensors, while also providing simplified criteria for special cases like circulant structures. Furthermore, under the condition of even order and symmetry, we prove that interval B-tensors (double B-tensors) ensure the property of being an interval P-tensor. This work extends interval matrix theory to tensors, offering new analytical tools for fields such as polynomial optimization and complementarity problems involving uncertainty. - oai:arXiv.org:2601.12217v1 + Progressive Power Homotopy for Non-convex Optimization + https://arxiv.org/abs/2601.15915 + arXiv:2601.15915v1 Announce Type: new +Abstract: We propose a novel first-order method for non-convex optimization of the form $\max_{\bm{w}\in\mathbb{R}^d}\mathbb{E}_{\bm{x}\sim\mathcal{D}}[f_{\bm{w}}(\bm{x})]$, termed Progressive Power Homotopy (Prog-PowerHP). The method applies stochastic gradient ascent to a surrogate objective obtained by first performing a power transformation and then Gaussian smoothing, $F_{N,\sigma}(\bm{\mu}):=\mathbb{E}_{\bm{w}\sim\mathcal{N}(\bm{\mu},\sigma^2I_d),\bm{x}\sim\mathcal{D}}[e^{Nf_w(\bm{x})}]$, while progressively increasing the power parameter $N$ and decreasing the smoothing scale $\sigma$ along the optimization trajectory. We prove that, under mild regularity conditions, Prog-PowerHP converges to a small neighborhood of the global optimum with an iteration complexity scaling nearly as $O(d^2\varepsilon^{-2})$. Empirically, Prog-PowerHP demonstrates clear advantages in phase retrieval when the samples-to-dimension ratio approaches the information-theoretic limit, and in training two-layer neural networks in under-parameterized regimes. These results suggest that Prog-PowerHP is particularly effective for navigating cluttered non-convex landscapes where standard first-order methods struggle. + oai:arXiv.org:2601.15915v1 math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + cs.AI + cs.LG + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Li Ye, Yisheng Song + Chen Xu - Stabilization of arbitrary structures in a three-dimensional doubly degenerate nutrient taxis system - https://arxiv.org/abs/2601.12218 - arXiv:2601.12218v1 Announce Type: new -Abstract: The doubly degenerate nutrient taxis system \begin{equation}\label {0.1} \left\{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\chi \nabla \cdot (u^{\alpha}v\nabla v)+\ell uv,&x\in \Omega,\, t>0,\\ & v_{t}=\Delta v-uv,&x\in \Omega,\, t>0,\\ \end{aligned} \right. \end{equation} is considered under zero-flux boundary conditions in a smoothly bounded domain $\Omega\subset\mathbb{R}^3$ where $\alpha>0,\chi>0$ and $\ell> 0$. By developing a novel class of functional inequalities to address the challenges posed by - the doubly degenerate diffusion mechanism in \eqref{0.1}, it is shown that for $\alpha\in(\frac{3}{2},\frac{19}{12})$, the associated initial-boundary value problem admits a global continuous weak solution for sufficiently regular initial data. Furthermore, in an appropriate topological setting, this solution converges to an equilibrium $(u_\infty, 0)$ as $t\rightarrow \infty$. Notably, the limiting profile $u_{\infty}$ is non-homogeneous when the initial signal concentration $v_0$ is sufficiently small, provided the initial data $u_0$ is not identically constant. - oai:arXiv.org:2601.12218v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Mutation Of Matrices Over Group Rings + https://arxiv.org/abs/2601.15920 + arXiv:2601.15920v1 Announce Type: new +Abstract: We give a precise definition of mutation of skew symmetrizable matrices over group rings and relate it to folding and mutation of quivers with symmetries. These matrices can have non-zero diagonal entries and we explain a mutation rule in some of these cases as well. This new rule comes from a notion of a generalized mutation of an entire quiver or sub-quiver. + oai:arXiv.org:2601.15920v1 + math.CO + math.RA + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - De-Ji-Xiang-Mao, Ai Huang, Yifu Wang + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Dani Kaufman, Carmen Alves Sabin - Persistent Sheaf Laplacian Analysis of Protein Stability and Solubility Changes upon Mutation - https://arxiv.org/abs/2601.12219 - arXiv:2601.12219v1 Announce Type: new -Abstract: Genetic mutations frequently disrupt protein structure, stability, and solubility, acting as primary drivers for a wide spectrum of diseases. Despite the critical importance of these molecular alterations, existing computational models often lack interpretability, and fail to integrate essential physicochemical interaction. To overcome these limitations, we propose SheafLapNet, a unified predictive framework grounded in the mathematical theory of Topological Deep Learning (TDL) and Persistent Sheaf Laplacian (PSL). Unlike standard Topological Data Analysis (TDA) tools such as persistent homology, which are often insensitive to heterogeneous information, PSL explicitly encodes specific physical and chemical information such as partial charges directly into the topological analysis. SheafLapNet synergizes these sheaf-theoretic invariants with advanced protein transformer features and auxiliary physical descriptors to capture intrinsic molecular interactions in a multiscale and mechanistic manner. To validate our framework, we employ rigorous benchmarks for both regression and classification tasks. For stability prediction, we utilize the comprehensive S2648 and S350 datasets. For solubility prediction, we employ the PON-Sol2 dataset, which provides annotations for increased, decreased, or neutral solubility changes. By integrating these multi-perspective features, SheafLapNet achieves state-of-the-art performance across these diverse benchmarks, demonstrating that sheaf-theoretic modeling significantly enhances both interpretability and generalizability in predicting mutation-induced structural and functional changes. - oai:arXiv.org:2601.12219v1 - math.SP - cs.LG - q-bio.QM - Wed, 21 Jan 2026 00:00:00 -0500 + A new proof of unboundedness of Riesz operator in $L^\infty$ and applications to mild ill-posedness in $W^{1,\infty}$ of the Euler type equations + https://arxiv.org/abs/2601.15922 + arXiv:2601.15922v1 Announce Type: new +Abstract: In this paper, we first present a new and simple proof of unboundedness of Riesz operator in $L^\infty$ and then establish the mild ill-posedness in $W^{1,\infty}$ of 3D rotating Euler equations and 2D Euler equations with partial damping. To the best of our knowledge, our work is the first one addressing the ill-posedness issue on the rotating Euler equations in $W^{1,\infty}$ without the vorticity formulation. As a further application, we prove the instability of perturbations for the 2D surface quasi-geostrophic equation and porous medium system in $W^{1,\infty}$. + oai:arXiv.org:2601.15922v1 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yiming Ren, Junjie Wee, Xi Chen, Grace Qian, Guo-Wei Wei + Jinlu Li, Yanghai Yu - Mean-Field Games Under Model Uncertainty - https://arxiv.org/abs/2601.12226 - arXiv:2601.12226v1 Announce Type: new -Abstract: We study discrete-time, finite-state mean-field games (MFGs) under model uncertainty, where agents face ambiguity about the state transition probabilities. Each agent maximizes its expected payoff against the worst-case transitions within an uncertainty set. Unlike in classical MFGs, model uncertainty renders the population distribution flow stochastic. This leads us to consider strategies that depend on both individual states and the realized distribution of the population. Our main results establish the asymptotic relationship between $N$-agent games and MFGs: every MFG equilibrium constitutes an $\varepsilon$-Nash equilibrium for sufficiently large populations, and conversely, limits of $N$-agent equilibria are MFG equilibria. We also prove the existence of equilibria for finite-agent games and construct a solvable mean-field example with closed-form solutions. - oai:arXiv.org:2601.12226v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + Higher-dimensional Heegaard Floer homology and spectral networks + https://arxiv.org/abs/2601.15923 + arXiv:2601.15923v1 Announce Type: new +Abstract: Given a closed surface $C$ and a real exact Lagrangian $\Sigma \subset T^*C$ associated to a spectral curve, we construct a homomorphism $\operatorname{BSk}_\kappa(C)\to\operatorname{Mat}(N^{\kappa},\operatorname{BSk}_\kappa(\Sigma))$ from the braid skein algebra of $C$ to the matrix-valued braid skein algebra of $\Sigma$ using Floer theory and in particular higher-dimensional Heegaard Floer homology (HDHF). We sketch a proof that this map coincides with a hybrid Floer-Morse approach which counts HDHF-type holomorphic curves coupled with certain Morse gradient graphs -- called fold\-ed Morse trees -- using a variant of the adiabatic limit theorems of Fukaya-Oh and Ekholm, which compares holomorphic curves and Morse flow trees. + oai:arXiv.org:2601.15923v1 + math.SG + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zongxia Liang, Zhou Zhou, Yaqi Zhuang, Bin Zou + Ko Honda, Yin Tian, Tianyu Yuan - Classical-Quantum Channel Resolvability Using Matrix Multiplicative Weight Update Algorithm - https://arxiv.org/abs/2601.12230 - arXiv:2601.12230v1 Announce Type: new -Abstract: We study classical-quantum (C-Q) channel resolvability. C-Q channel resolvability has been proved by only random coding in the literature. In our previous study, we proved channel resolvability by deterministic coding, using multiplicative weight update algorithm. We extend this approach to C-Q channels and prove C-Q channel resolvability by deterministic coding, using the matrix multiplicative weight update algorithm. This is the first approach to C-Q channel resolvability using deterministic coding. - oai:arXiv.org:2601.12230v1 + A Remark on Downlink Massive Random Access + https://arxiv.org/abs/2601.15928 + arXiv:2601.15928v1 Announce Type: new +Abstract: In downlink massive random access (DMRA), a base station transmits messages to a typically small subset of active users, selected randomly from a massive number of total users. Explicitly encoding the identities of active users would incur a significant overhead scaling logarithmically with the number of total users. Recently, via a random coding argument, Song, Attiah and Yu have shown that the overhead can be reduced to within some upper bound irrespective of the number of total users. In this remark, recognizing that the code design for DMRA is an instance of covering arrays in combinatorics, we show that there exists deterministic construction of variable-length codes that incur an overhead no greater than $1 + log_2 e$ bits. + oai:arXiv.org:2601.15928v1 cs.IT math.IT - quant-ph - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Koki Takahashi, Shun Watanabe + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yuchen Liao, Wenyi Zhang - An optimal boundary control approach to the Cherrier-Escobar problem - https://arxiv.org/abs/2601.12232 - arXiv:2601.12232v1 Announce Type: new -Abstract: We study an optimal boundary control problem associated to the boundary obstacle problem for the couple conformal Laplacian and conformal Robin operator on n-dimensional compact Riemannian manifolds with boundary and with n\geq 3. When the Cherrier-Escobar invariant of the compact Riemannian manifold with boundary is positive, we show that the optimal controls are equal to their associated optimal states. Moreover, we show that the optimal controls are minimizers of the Cherrier-Escobar functional, and hence induce conformal metrics with zero scalar curvature and constant mean curvature. Furthermore, we show the existence of an optimal control under an Aubin type assumption. For the standard unit ball, we derive a sharp Sobolev trace type inequality and prove that the standard bubbles-namely conformal factor of metrics conformal to the standard one with zero scalar curvature and constant mean curvature -- are the only optimal controls and hence equal to their associated optimal states. - oai:arXiv.org:2601.12232v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + The representations of the Lie superalgebra p(3) in prime characteristic + https://arxiv.org/abs/2601.15932 + arXiv:2601.15932v1 Announce Type: new +Abstract: Let g be the Lie superalgebra p(3) of rank 2 over an algebraically closed field K of characteristic p > 3. We classify all irreducible modules of g, and give the character formulae for irreducible modules. + oai:arXiv.org:2601.15932v1 + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Cheikh Birahim Ndiaye, Abdul-Malik Saiid + http://creativecommons.org/publicdomain/zero/1.0/ + Ye Ren - An alternative construction of the $G_2(2)$-graph - https://arxiv.org/abs/2601.12235 - arXiv:2601.12235v1 Announce Type: new -Abstract: In this note, we give an alternative construction of the $G_2(2)$-graph from a $U_3(2)$-geometry. - oai:arXiv.org:2601.12235v1 + An Efficient Algorithm to Generate all Labeled Triangle-free Graphs with a given Graphical Degree Sequence + https://arxiv.org/abs/2601.15943 + arXiv:2601.15943v1 Announce Type: new +Abstract: We extend our previous algorithm that generates all labeled graphs with a given graphical degree sequence to generate all labeled triangle-free graphs with a given graphical degree sequence. The algorithm uses various pruning techniques to avoid having to first generate all labeled realizations of the input sequence and then testing whether each labeled realization is triangle-free. It can be further extended to generate all labeled bipartite graphs with a given graphical degree sequence by adding a simple test whether each generated triangle-free realization is a bipartite graph. All output graphs are generated in the lexicographical ordering as in the original algorithm. The algorithms can also be easily parallelized. + oai:arXiv.org:2601.15943v1 math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + cs.CC + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/publicdomain/zero/1.0/ - Koichi Inoue + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Kai Wang - Explicit symmetric low-regularity integrators for the semilinear Klein-Gordon equation - https://arxiv.org/abs/2601.12246 - arXiv:2601.12246v1 Announce Type: new -Abstract: This paper is concerned with the design and analysis of symmetric low-regularity integrators for the semilinear Klein-Gordon equation. We first propose a general symmetrization procedure that allows for the systematic construction of symmetric schemes from existing explicit (non-symmetric) integrators. Applying this procedure, we derive two novel schemes. Error analyses show that both integrators achieve their optimal convergence orders in the energy space under significantly relaxed regularity assumptions. Furthermore, the symmetry property ensures that the convergence order of a first-order symmetric scheme improves as the regularity of the exact solution increases. A numerical experiment demonstrates that the proposed second-order symmetric scheme nearly preserves the system energy over extended periods. - oai:arXiv.org:2601.12246v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 + Extreme Score Distributions in Countable-Outcome Round-Robin Tournaments of Equally Strong Players + https://arxiv.org/abs/2601.15950 + arXiv:2601.15950v1 Announce Type: new +Abstract: We consider a general class of round-robin tournament models of equally strong players. In these models, each of the $n$ players competes against every other player exactly once. For each match between two players, the outcome is a value from a countable subset of the unit interval, and the scores of the two players in a match sum to one. The final score of each player is defined as the sum of the scores obtained in matches against all other players. We study the distribution of extreme scores, including the maximum, second maximum, and lower-order extremes. Since the exact distribution is computationally intractable even for small values of $n$, we derive asymptotic results as the number of players $n$ tends to infinity, including limiting distributions, and rates of convergence. + oai:arXiv.org:2601.15950v1 + math.PR + math.ST + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zhirui Shen, Bin Wang + Yaakov Malinovsky - Paley-type matrices and $1$-factorizations of complete graphs - https://arxiv.org/abs/2601.12250 - arXiv:2601.12250v1 Announce Type: new -Abstract: Ball, Ortega--Moreno, and Prodromou asked whether, for every odd prime $p$, one can find a $1$-factor of the complete graph $K_{p+1}$ with some arithmetic restrictions related to quadratic residues. This problem is motivated by $1$-factorizations that are compatible with the sign pattern of certain Paley-type matrices. Recently, Afifurrahman et al. made some partial progress. In this paper, we completely resolve the problem. - oai:arXiv.org:2601.12250v1 - math.CO - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 + Geometry of spherical spin glasses + https://arxiv.org/abs/2601.15966 + arXiv:2601.15966v1 Announce Type: new +Abstract: Spherical spin glasses are canonical models for smooth random functions in high dimensions. In this review, we survey several interrelated lines of research on their geometric structure. We begin with results concerning critical points and their relationship to the Gibbs measure. For the pure models, the measure concentrates on spherical bands around critical points that approximately maximize the energy at a particular radius. Next, we present another approach in which a similar picture is derived for general mixed models. At the core of this approach is a free energy functional computed over bands using multiple orthogonal replicas, satisfying a strong concentration of measure. We discuss several implications of this method for a generalized Thouless-Anderson-Palmer (TAP) approach. Finally, we explain how these geometric insights inform optimization algorithms, and briefly relate them to Smale's 17th problem over the real numbers. + oai:arXiv.org:2601.15966v1 + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chi Hoi Yip, Semin Yoo + Eliran Subag - The generalized Lax conjecture is true for topological reasons related to compactness, convexity and determinantal deformations of increasing products of pointwise approximating linear forms - https://arxiv.org/abs/2601.12267 - arXiv:2601.12267v1 Announce Type: new -Abstract: We develop a topological approach to prove the generalized Lax conjecture using the fact that determinants of sufficiently big symmetric linear pencils are able to express the rigidly convex sets of RZ polynomials of any degree $d$. Monicity of the representation is assessed through a topological argument that allows us to perturbate a sufficiently close linear approximation into a suitable nice determinantal multiple of the initial RZ polynomial with the same rigidly convex set. The perturbation can be smoothly performed. This fact is what will allow us to determine that the multiple obtained respects the initial rigidly convex sets. This argument provides thus a full proof of the generalized Lax conjecture. However, an effective proof providing the representation in nice terms seems far from reachable at this moment. - oai:arXiv.org:2601.12267v1 - math.AG + Iteration complexity of the Difference-of-Convex Algorithm for unconstrained optimization: a simple proof + https://arxiv.org/abs/2601.15970 + arXiv:2601.15970v1 Announce Type: new +Abstract: We propose a simple proof of the worst-case iteration complexity for the Difference of Convex functions Algorithm (DCA) for unconstrained minimization, showing that the global rate of convergence of the norm of the objective function's gradients at the iterates converge to zero like o(1/k). A small example is also provided indicating that the rate cannot be improved. + oai:arXiv.org:2601.15970v1 math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/publicdomain/zero/1.0/ - Alejandro Gonz\'alez Nevado - - - The relative GAGA Theorem and an application to the analytic mapping stacks - https://arxiv.org/abs/2601.12299 - arXiv:2601.12299v1 Announce Type: new -Abstract: We prove a relative GAGA theorem for perfect and pseudo-coherent complexes in non-archimedean analytic geometry, allowing bases given by Fredholm analytic rings, including those associated from affinoid perfectoid spaces. This answers a question raised in \cite{heuer2024padicnonabelianhodgetheory}. As an application, we show that for a proper scheme \(X\) and an Artin stack \(Y\) with suitable conditions, the analytification of the algebraic mapping stack \(\mathrm{Map}(X,Y)\) agrees with the intrinsic analytic mapping stack \(\mathrm{Map}(X^{\mathrm{an}},Y^{\mathrm{an}})\). - oai:arXiv.org:2601.12299v1 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Qixiang Wang + Serge Gratton, Philippe L. Toint - On the Minimum Length of Functional Batch Codes with Small Recovery Sets - https://arxiv.org/abs/2601.12302 - arXiv:2601.12302v1 Announce Type: new -Abstract: Batch codes are of potential use for load balancing and private information retrieval in distributed data storage systems. Recently, a special case of batch codes, termed functional batch codes, was proposed in the literature. In functional batch codes, users can query linear combinations of the information symbols, and not only the information symbols themselves, as is the case for standard batch codes. In this work, we consider linear functional batch codes with the additional property that every query is answered by using only a small number of coded symbols. We derive bounds on the minimum length of such codes, and evaluate the results by numerical computations. - oai:arXiv.org:2601.12302v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + On maximal rank properties for symmetric polynomials in an equigenerated monomial complete intersection + https://arxiv.org/abs/2601.15978 + arXiv:2601.15978v1 Announce Type: new +Abstract: It is well known that a monomial complete intersection has the strong Lefschetz property in characteristic zero. This property is equivalent to the statement that any power of the sum of the variables is a maximal rank element on the complete intersection. In this paper, we investigate what happens when this element is replaced by another symmetric polynomial, in an equigenerated complete intersection. + We answer the question completely for the power sum symmetric polynomial using a grading technique, and for any Schur polynomial in the case of two variables by deriving a closed formula for the determinants of a family of Toeplitz matrices. Further, we obtain partial results in three or more variables for the elementary and the complete homogeneous symmetric polynomials and pose several open questions. + oai:arXiv.org:2601.15978v1 + math.AC + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kristiina Oksner, Henk D. L. Hollmann, Ago-Erik Riet, Vitaly Skachek + Filip Jonsson Kling, Samuel Lundqvist - Level of Faces for Exponential Sequence of Arrangements - https://arxiv.org/abs/2601.12328 - arXiv:2601.12328v1 Announce Type: new -Abstract: In this paper, we introduce the bivariate exponential generating function $F_l(x,y)$ for the number of level-$l$ faces of an exponential sequence of arrangements (ESA), and establish the formula $F_l(x,y)=\big(F_1(x,y)\big)^l$ with a combinatorial interpretation. Its specialization at $x=0$ recovers a result first obtained by Chen et al. [3,4] for certain classic ESAs and later generalized to all ESAs by Southerland et al. [8]. As a byproduct, we obtain that an alternating sum of the number of level-$l$ faces is invariant with respect to the choice of ESA, and is exactly the Stirling number of the second kind. We also extend the binomial-basis expansion theorem [3,4,14] and Stanley's formula on ESAs [9] from characteristic polynomials to Whitney polynomials. - oai:arXiv.org:2601.12328v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + Real-Time Inviscid Fluid Dynamics and Aero-acoustics on a Sphere + https://arxiv.org/abs/2601.15982 + arXiv:2601.15982v1 Announce Type: new +Abstract: Real-time fluid and aeroacoustic simulation on complex surfaces can have interactive applications - from globe-based weather visualizations to immersive computer games with physically accurate wind and sound. However, conventional grid-based solvers struggle with numerical instability near surface singularities, and mesh-based approaches lack a straightforward path to solving partial differential equations (PDEs) with stable, high-order accuracy. + Our model presents a unified framework for real-time inviscid fluid simulation and aeroacoustics on spherical surfaces with embedded obstacles, combining the Closest Point Method (CPM), projection-based Navier-Stokes solvers, and the Ffowcs Williams-Hawkings (FWH) analogy. CPM enables surface PDEs to be solved in a Cartesian embedding without parametrization by restricting computation to a narrow band around the sphere. Each band point is mapped to its nearest surface location, where band operators project results onto the local tangent space. Surface obstacles are modelled with signed distance functions (SDFs), enforcing no-slip velocity constraints and Bernoulli-based pressure adjustments for consistent real-world boundary interactions. Aeroacoustic sources are computed directly from surface pressure force derivatives and mapped to real-time audio via frequency and amplitude modulation with artifact-suppressing hysteresis smoothing. + Our findings from this model simulate the behaviour of inviscid fluid on spherical surfaces while generating sound using the pressure of the fluid flowing on the surface. This approach gives results that offer stability, geometric consistency, and support applications in scientific visualization, virtual reality, and educational tools. + oai:arXiv.org:2601.15982v1 + math.AP + physics.flu-dyn + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yanru Chen, Houshan Fu, Weikang Liang, Suijie Wang + http://creativecommons.org/licenses/by/4.0/ + Madhusraba Sinha, Jan Stratmann - A Complete Proof of the Simon--Lukic Conjecture for Higher-Order Szeg\H{o} Theorems - https://arxiv.org/abs/2601.12332 - arXiv:2601.12332v1 Announce Type: new -Abstract: This paper provides a complete proof of Simon-Lukic conjecture for orthogonal polynomials on the unit circle. For a probability measure $d\mu = w(\theta) \frac{d\theta}{2\pi} + d\mu_s$ with Verblunsky coefficients $\alpha=\{\alpha_n\}_{n=0}^\infty$, distinct singular points $(\theta_k)_{k=1}^{\ell}$, and multiplicities $(m_k)_{k=1}^{\ell}$, we establish the equivalence between the entropy condition \[ \int_0^{2\pi} \prod_{k=1}^{\ell} [1 - \cos(\theta - \theta_k)]^{m_k} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \] and the decomposition condition \[ \exists \beta^{(1)}, \ldots, \beta^{(\ell)} : \alpha = \sum_{k=1}^\ell \beta^{(k)} \,\, \text{with} \,\, (S - e^{-i\theta_k})^{m_k} \beta^{(k)} \in \ell^2, \,\, \beta^{(k)} \in \ell^{2m_k + 2}. \] - The proof synthesizes unitary transformations, discrete Sobolev-type inequalities, higher-order Szeg\H{o} expansions, and a novel algebraic decomposition technique. Our resolution affirms that spectral theory is fundamentally local-global behavior emerges from the superposition of local resonances, each governed by its intrinsic scale. - oai:arXiv.org:2601.12332v1 - math.SP - Wed, 21 Jan 2026 00:00:00 -0500 + Primes and The Field of Values of Characters + https://arxiv.org/abs/2601.15987 + arXiv:2601.15987v1 Announce Type: new +Abstract: Let $p$ be a prime. For $p=2$, the fields of values of the complex irreducible characters of finite groups whose degrees are not divisible by $p$ have been classified; for odd primes $p$, a conjectural classification has been proposed. In this work, we extend this conjecture to characters whose degrees are divisible by arbitrary powers of $p$, and we provide some evidence supporting its validity. + oai:arXiv.org:2601.15987v1 + math.RT + math.GR + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Daxiong Piao + http://creativecommons.org/licenses/by/4.0/ + Nguyen N. Hung, Gabriel Navarro, Pham Huu Tiep + + + Rank of elliptic curves and class groups of real quadratic fields + https://arxiv.org/abs/2601.15988 + arXiv:2601.15988v1 Announce Type: new +Abstract: In this paper, we are going to prove the relation between rank of elliptic curves and the non-triviality of class groups of infinitely many real quadratic fields. + oai:arXiv.org:2601.15988v1 + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 + new + http://creativecommons.org/licenses/by/4.0/ + Kalyan Banerjee - Representation theorems for nonvariational solutions of the Helmholtz equation - https://arxiv.org/abs/2601.12335 - arXiv:2601.12335v1 Announce Type: new -Abstract: We consider a possibly multiply connected bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{\max\{1,m\},\alpha}$ for some $m\in {\mathbb{N}}$, $\alpha\in]0,1[$ and we plan to solve both the Dirichlet and the Neumann problem for the Helmholtz equation in $\Omega$ and in the exterior of $\Omega$ in terms of acoustic layer potentials. Then we turn to prove an integral representation theorem solutions of the Helmholtz equation in terms of a single layer acoustic potential. The main focus of the paper is on $\alpha$-H\"{o}lder continuous solutions which may not have a classical normal derivative at the boundary points of $\Omega$ and that may have an infinite Dirichlet integral around the boundary of $\Omega$\, \textit{i.e.}, case $m=0$. Namely for solutions that do not belong to the classical variational setting. - oai:arXiv.org:2601.12335v1 + A sharp criterion and complete classification of global-in-time solutions and finite time blow-up of solutions to a chemotaxis system in supercritical dimensions + https://arxiv.org/abs/2601.15990 + arXiv:2601.15990v1 Announce Type: new +Abstract: We consider the chemotaxis system with indirect signal production in the whole space, \begin{equation}\label{abst:p}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u\nabla v),\\ 0 = \Delta v + w,\\ w_t = \Delta w + u \end{cases} \end{equation} with emphasis on supercritical dimensions. In contrast to the classical parabolic-elliptic Keller--Segel system, where the analysis can be reduced to a single equation, the above system is essentially parabolic-parabolic and does not admit such a reduction. In this paper, we establish a sharp threshold phenomenon separating global-in-time existence from finite time blow-up in terms of scaling-critical Morrey norms of the initial data. In particular, we prove the existence of singular stationary solutions and show that their Morrey norm values serve as the critical thresholds determining the long-time behavior of solutions. Consequently, we identify new critical exponents at which the long-time behavior of solutions changes. This yields a complete classification of the long-time behavior of solutions, providing the first such results for the essentially parabolic-parabolic chemotaxis system \eqref{abst:p} in supercritical dimensions. + oai:arXiv.org:2601.15990v1 math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - M. Lanza de Cristoforis + Yuri Soga - Asymptotic Behavior of the Principal Eigenvalue Problems with Large Divergence-Free Drifts - https://arxiv.org/abs/2601.12342 - arXiv:2601.12342v1 Announce Type: new -Abstract: In this paper, we consider the following principal eigenvalue problem with a large divergence-free drift: \begin{equation}\label{0.1} -\varepsilon\Delta \phi-2\alpha\nabla m(x)\cdot\nabla \phi+V(x)\phi=\lambda_\alpha \phi\ \,\ \text{in}\, \ H_0^1(\Omega),\tag{0.1} \end{equation} where the domain $\Omega\subset \mathbb{R}^N (N\ge 1)$ is bounded with smooth boundary $\partial\Omega$, the constants $\varepsilon>0$ and $\alpha>0$ are the diffusion and drift coefficients, respectively, and $m(x)\in C^{2}(\bar{\Omega})$, $V (x)\in C^{\gamma}(\bar{\Omega})~(0<\gamma<1)$ are given functions. For a class of divergence-free drifts where $m$ is a harmonic function in $\Omega$ and has no first integral in $H_{0}^{1}(\Omega)$, we prove the convergence of the principal eigenpair $(\lambda_\alpha, \phi)$ for (0.1) as $\alpha\rightarrow+\infty$, which addresses a special case of the open question proposed in [H. Berestycki, F. Hamel and N. Nadirashvili, CMP, 2005]. Moreover, we further investigate the refined limiting profiles of the principal eigenpair $(\lambda_\alpha, \phi)$ for (0.1) as $\alpha\rightarrow+\infty$, which display the visible effects of the large divergence-free drifts on the principal eigenpair $(\lambda_\alpha, \phi)$. - oai:arXiv.org:2601.12342v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Counting Saddle Connections on Hyperelliptic Translation Surfaces with a Slit + https://arxiv.org/abs/2601.15993 + arXiv:2601.15993v1 Announce Type: new +Abstract: We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that it satisfies an $L (\log L)^{d-2}$ growth rate, where $d$ is the complex dimension of the hyperelliptic stratum. The upper bound holds for all translation surfaces in the hyperelliptic stratum while the lower bound holds for almost every surface in the hyperelliptic stratum. The proof of the lower bound uses horocycle renormalization. + oai:arXiv.org:2601.15993v1 + math.DS + math.GT + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yujin Guo, Yuan Lou, Hongfei Zhang + David Aulicino, Howard Masur, Huiping Pan, Weixu Su - Classification of the structures of stable radial solutions for semilinear elliptic equations in $\bf R^N$ - https://arxiv.org/abs/2601.12350 - arXiv:2601.12350v1 Announce Type: new -Abstract: We study the stability of radial solutions of the semilinear elliptic equation $\Delta u +f(u)=0$ in ${\bf R^N}$, where $N \geq 3$ and $f$ is a general superciritical nonlinearity. We give a classification of the solution structures with respect to the stability of radial solutions, and establish criteria for the existence and nonexistence of stable radial solutions in terms of the limits of $f'(u)F(u)$ as $u \to 0$ or $\infty$, where $F(u) = \int^{\infty}_u 1/f(t)dt$. Furthermore, we show the relation between the existence of singular stable solutions and the solution structure. - oai:arXiv.org:2601.12350v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Minimax-optimal Halpern iterations for Lipschitz maps + https://arxiv.org/abs/2601.15996 + arXiv:2601.15996v1 Announce Type: new +Abstract: This paper investigates the minimax-optimality of Halpern fixed-point iterations for Lipschitz maps in general normed spaces. Starting from an a priori bound on the orbit of iterates, we derive non-asymptotic estimates for the fixed-point residuals. These bounds are tight, meaning that they are attained by a suitable Lipschitz map and an associated Halpern sequence. By minimizing these tight bounds we identify the minimax-optimal Halpern scheme. For contractions, the optimal iteration exhibits a transition from an initial Halpern phase to the classical Banach-Picard iteration and, as the Lipschitz constant approaches one, we recover the known convergence rate for nonexpansive maps. For expansive maps, the algorithm is purely Halpern with no Banach-Picard phase; moreover, on bounded domains, the residual estimates converge to the minimal displacement bound. Inspired by the minimax-optimal iteration, we design an adaptive scheme whose residuals are uniformly smaller than the minimax-optimal bounds, and can be significantly sharper in practice. Finally, we extend the analysis by introducing alternative bounds based on the distance to a fixed point, which allow us to handle mappings on unbounded domains; including the case of affine maps for which we also identify the minimax-optimal iteration. + oai:arXiv.org:2601.15996v1 + math.OC + math.ST + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yasuhito Miyamoto, Y\=uki Naito + Mario Bravo, Roberto Cominetti, Jongmin Lee - Time-fractional nonlinear evolution equations with time-dependent constraints - https://arxiv.org/abs/2601.12352 - arXiv:2601.12352v1 Announce Type: new -Abstract: This article is devoted to presenting an abstract theory of time-fractional gradient flow equations for time-dependent convex functionals in real Hilbert spaces. The main results are concerned with the existence of strong solutions to time-fractional abstract evolution equations governed by time-dependent subdifferential operators. To prove these results, Gronwall-type lemmas for nonlinear Volterra integral inequalities and fractional chain-rule formulae are developed. Moreover, the obtained abstract results are applied to the initial-boundary value problem for time-fractional nonlinear parabolic equations on moving domains. - oai:arXiv.org:2601.12352v1 + The Recovery of Semilinear Potentials Satisfying Null Conditions From Scattering Data + https://arxiv.org/abs/2601.15997 + arXiv:2601.15997v1 Announce Type: new +Abstract: We construct oscillatory solutions of fully semilinear wave equations in Minkowski space satisfying a null condition of the form $$\square u:=(-\partial_{x_0}^2 +\sum_{j=1}^n \partial_{x_j}^2 )u= q(x,u)((\partial_{x_0}u)^2-|\nabla_{x'}u|^2),$$ $$x=(x_0,x'), \;\ x'=(x_1,\ldots, x_n) \text{ and } x_0=t \text{ is the time variable,}$$ on an interval $x_0\in [-T,T]$, $T<\infty$ arbitrary, which consist of the superposition of a non-oscillatory background solution and a single phase train of highly oscillatory waves of wave length $h\ll1$ and amplitudes given by powers of $h$; the waves interact with the nonlinearity and we measure the response $u(x_0,x')|_{x_0=T'}$ at a fixed time $x_0=T'<T$. + We show that the coefficient of amplitude $h$ of the oscillatory part of the nonlinear geometric optics expansion of the solution determines the light-ray transform of a vector field associated with $q(x,u)$, which determines $q(x,u)$ uniquely in the maximal region determined by the data. Our methods also work for systems of semilinear wave equations satisfying null conditions, but in this paper we focus on the scalar case. + oai:arXiv.org:2601.15997v1 math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + math-ph + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yoshihito Nakajima + Joel Nathe, Ant\^onio S\'a Barreto - Skew brace extensions, second cohomology and complements - https://arxiv.org/abs/2601.12371 - arXiv:2601.12371v1 Announce Type: new -Abstract: We study extensions and second cohomology of skew left braces via the natural semi-direct products associated with the skew left braces. Let $0 \to I \to E \to H \to 0$ be a skew brace extension and $\Lambda_H$ denote the natural semi-direct products associated with the skew left brace $H$. We establish a group homomorphism from ${\rm H}_{Sb}^2(H, I)$ into ${\rm H}_{Gp}^2(\Lambda_H, I \times I)$, which turns out to be an embedding when $I \le {\rm Soc}(E)$. In particular the Schur multiplier of a skew left braces $H$ embeds into the Schur multiplier of the group $\Lambda_H$. Analog of the Schur-Zassenhaus theorem is established for skew left braces in several specific cases. We introduce a concept called minimal extensions (which stay at the extreme end of split extensions) of skew left braces and derive many fundamental results. Several reduction results for split extensions of finite skew left braces by abelian groups (viewed as trivial left braces) are obtained. - oai:arXiv.org:2601.12371v1 - math.GR - math.QA - Wed, 21 Jan 2026 00:00:00 -0500 + Time-Optimal Switching Surfaces for Triple Integrator under Full Box Constraints + https://arxiv.org/abs/2601.16003 + arXiv:2601.16003v1 Announce Type: new +Abstract: Time-optimal control for triple integrator under full box constraints is a fundamental problem in the field of optimal control, which has been widely applied in the industry. However, scenarios involving asymmetric constraints, non-stationary boundary conditions, and active position constraints pose significant challenges. This paper provides a complete characterization of time-optimal switching surfaces for the problem, leading to novel insights into the geometric and algebraic structure of the optimal control. The active condition of position constraints is derived, which is absent from the literature. An efficient algorithm is proposed, capable of planning time-optimal trajectories under asymmetric full constraints and arbitrary boundary states, with a 100% success rate. Computational time for each trajectory is within approximately 10{\mu}s, achieving a 5-order-of-magnitude reduction compared to optimization-based baselines. + oai:arXiv.org:2601.16003v1 + math.OC + cs.SY + eess.SY + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Nishant Rathee, Manoj K. Yadav + Yunan Wang, Chuxiong Hu, Zhao Jin - On a Modification of the Twistor Space - https://arxiv.org/abs/2601.12372 - arXiv:2601.12372v1 Announce Type: new -Abstract: In the paper we construct a modification $S(M)$ of the twistor space of a K\"ahler scalar flat surface $M$ and study its complex-geometric and metric properties. In particular, we construct complete balanced metrics on $S(M)$ and show that $S(M)$ can not be K\"ahler when $M$ is a compact simple hyperk\"ahler manifold. - oai:arXiv.org:2601.12372v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 + A brief note about p-curvature on graphs + https://arxiv.org/abs/2601.16010 + arXiv:2601.16010v1 Announce Type: new +Abstract: In this paper, we consider Wang's $CD_p(m,K)$ condition on graphs, which depends on the $p$-Laplacian $\Delta_p$ for $p>1$ and is an extension of the classical Bakry-\'Emery $CD(m,K)$ curvature dimension condition. We calculate several examples including paths, cycles and star graphs, and we show that the $p$-curvature is non-negative at some vertices in the case $p\geq 2$, while it approaches to $-\infty$ in the case of $1<p<2$. In addition, we observe that a crucial property of $\Gamma_2$ on Cartesian products does no longer hold for $\Gamma_2^p$ in the case of $p > 2$. As a consequence, an analogous proof that non-negative curvature is preserved under taking Cartesian products is not possible for $p > 2$. + oai:arXiv.org:2601.16010v1 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Anna Fino, Gueo Grantcharov, Alberto Pipitone Federico + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Chunyang Hu - An efficient penalty decomposition algorithm for minimization over sparse symmetric sets - https://arxiv.org/abs/2601.12383 - arXiv:2601.12383v1 Announce Type: new -Abstract: This paper proposes an improved quasi-Newton penalty decomposition algorithm for the minimization of continuously differentiable functions, possibly nonconvex, over sparse symmetric sets. The method solves a sequence of penalty subproblems approximately via a two-block decomposition scheme: the first subproblem admits a closed-form solution without sparsity constraints, while the second subproblem is handled through an efficient sparse projection over the symmetric feasible set. Under a new assumption on the gradient of the objective function, weaker than global Lipschitz continuity from the origin, we establish that accumulation points of the outer iterates are basic feasible and cardinality-constrained Mordukhovich stationarity points. To ensure robustness and efficiency in finite-precision arithmetic, the algorithm incorporates several practical enhancements, including an enhanced line search strategy based on either backtracking or extrapolation, and four inexpensive diagonal Hessian approximations derived from differences of previous iterates and gradients or from eigenvalue-distribution information. Numerical experiments on a diverse benchmark of $30$ synthetic and data-driven test problems, including machine-learning datasets from the UCI repository and sparse symmetric instances with dimensions ranging from $10$ to $500$, demonstrate that the proposed algorithm is competitive with several state-of-the-art methods in terms of efficiency, robustness, and strong stationarity. - oai:arXiv.org:2601.12383v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + Infinite random graphs + https://arxiv.org/abs/2601.16013 + arXiv:2601.16013v1 Announce Type: new +Abstract: We study countable graphs that -- up to isomorphism and with probability one -- arise from a random process, in a similar fashion as the Rado graph. Unlike in the classical case, we do not require that probabilities assigned to pairs of points are all equal. We give examples of such generalized random graphs, and show that the class of graphs under consideration has a two-element basis. + oai:arXiv.org:2601.16013v1 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Ahmad Mousavi, Morteza Kimiaei, Saman Babaie-Kafaki, Vyacheslav Kungurtsev + Ziemowit Kostana, Jaros{\l}aw Swaczyna, Agnieszka Widz - Strong Hollowness in Commutative Rings - https://arxiv.org/abs/2601.12388 - arXiv:2601.12388v1 Announce Type: new -Abstract: In this paper we study strongly hollow ideals and completely strongly hollow ideals in commutative rings without finiteness assumptions. We establish basic structural properties, including maximality phenomena and permanence under quotients and surjective homomorphisms. We obtain several characterizations of completely strongly hollow ideals in terms of extremal ideals avoiding a given ideal, and we show that a strongly hollow ideal which is not contained in the Jacobson radical is necessarily completely strongly hollow. As applications, we derive strong restrictions in integral domains and consequences for principal ideal domains, including a discrete valuation ring criterion. We develop the connection between complete hollowness and complete irreducibility and obtain a correspondence between completely strongly hollow ideals and completely strongly irreducible ideals. Finally, we develop a condition related to greatest common divisors which is equivalent to strongly hollowness under mild finiteness conditions. - oai:arXiv.org:2601.12388v1 - math.AC - math.RA - Wed, 21 Jan 2026 00:00:00 -0500 + The T-tensor of spherically symmetric Finsler metrics + https://arxiv.org/abs/2601.16021 + arXiv:2601.16021v1 Announce Type: new +Abstract: This paper is devoted to the study of the T-tensor associated with a spherically symmetric Finsler metric $F=u\phi(r,s)$ on \(\mathbb{R}^n\). We derive a general expression for the T-tensor in terms of the scalar function \(\phi(r, s)\) and its partial derivatives. Furthermore, we characterize all spherically symmetric Finsler metrics satisfying the so-called T-condition, that is, those for which the T-tensor vanishes. In addition, we obtain the formula for the mean Cartan tensor and demonstrate that all spherically symmetric Finsler metrics of dimension $n \geq 3$, with a non-zero mean Cartan tensor are quasi-C-reducible. + oai:arXiv.org:2601.16021v1 + math.DG + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Amartya Goswami, Joseph Israel Zelezniak + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Salah G. Elgendi - $2$-quasi-perfect Lee codes and abelian Ramanujan graphs: a new construction and relationship - https://arxiv.org/abs/2601.12393 - arXiv:2601.12393v1 Announce Type: new -Abstract: In this paper, we obtain a new explicit family of $2$-quasi-perfect Lee codes of arbitrarily large length. Our construction is based on generating sets of abelian (almost) Ramanujan graphs obtained by Forey, Fres\'{a}n, Kowalski and Wigderson. Also, we develop a relationship between certain abelian Ramanujan graphs and $2$-quasi-perfect Lee codes obtained by Mesnager, Tang and Qi. - oai:arXiv.org:2601.12393v1 + Stacked Intelligent Metasurface-Aided Wave-Domain Signal Processing: From Communications to Sensing and Computing + https://arxiv.org/abs/2601.16030 + arXiv:2601.16030v1 Announce Type: new +Abstract: Neural networks possess incredible capabilities for extracting abstract features from data. Electromagnetic computing harnesses wave propagation to execute computational operations. Metasurfaces, composed of subwavelength meta-atoms, are capable of engineering electromagnetic waves in unprecedented ways. What happens when combining these three cutting-edge technologies? This question has sparked a surge of interest in designing physical neural networks using stacked intelligent metasurface (SIM) technology, with the aim of implementing various computational tasks by directly processing electromagnetic waves. SIMs open up an exciting avenue toward high-speed, massively parallel, and low-power signal processing in the electromagnetic domain. This article provides a comprehensive overview of SIM technology, commencing with its evolutionary development. We subsequently examine its theoretical foundations and existing SIM prototypes in depth. Furthermore, the optimization/training strategies conceived to configure SIMs for achieving the desired functionalities are discussed from two different perspectives. Additionally, we explore the diverse applications of SIM technology across the communication, sensing, and computing domains, presenting experimental evidence that highlights its distinctive advantages in supporting multiple functions within a single device. Finally, we identify critical technical challenges that must be addressed to deploy SIMs in next-generation wireless networks and shed light on promising research directions to unlock their full potential. + oai:arXiv.org:2601.16030v1 cs.IT - math.CO math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Shohei Satake + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Jiancheng An, Chau Yuen, Marco Di Renzo, Mehdi Bennis, Merouane Debbah, Lajos Hanzo - Privacy via Modulation Rotation and Inter-Symbol Interference - https://arxiv.org/abs/2601.12394 - arXiv:2601.12394v1 Announce Type: new -Abstract: Two physical-layer mechanisms for achieving user-side differential privacy in communication systems are proposed. Focusing on binary phase-shift keying (BPSK) modulation, differential privacy (DP) is first studied under a deterministic phase rotation applied on the BPSK modulation at the transmitter, while the receiver is assumed to be unaware of the rotation angle. In this setting, privacy is achieved through an effective reduction in the decision distance, resulting in a controlled increase in the bit error rate (BER) without explicit noise injection. Next, a BPSK transmission scheme with intentionally induced inter-symbol interference (ISI) is studied, where the receiver is likewise unaware of the deterministic timing offset that generates the ISI. Unlike the rotated BPSK scheme, the DP obtained via ISI is shown to depend explicitly on the input data distribution. In particular, numerical results demonstrate that, for a fixed ISI parameter, the privacy loss is maximized when the binary input symbols are equiprobable. While conventional DP mechanisms rely on artificially added noise, often incurring additional energy or communication costs, it is shown that structured modifications, such as modulation rotation or induced ISI inherent to realistic communication channels can itself provide DP guarantees. While the analysis focuses on deterministic transmitter modifications unknown to the receiver, it is noted that real-world devices naturally introduce unintentional rotations or ISI due to hardware nonidealities and implementation errors. These effects can therefore provide a level of privacy without requiring explicit noise injection. Hence, it is possible to avoid deliberately perturbing the data, instead leveraging inherent device imperfections to achieve privacy guarantees with no additional privacy cost. - oai:arXiv.org:2601.12394v1 + RIS-Aided Cooperative ISAC Network for Imaging-Based Low-Altitude Surveillance + https://arxiv.org/abs/2601.16033 + arXiv:2601.16033v1 Announce Type: new +Abstract: The low-altitude economy is integral to the advancement of numerous sectors, necessitating the development of advanced low-altitude surveillance techniques. Nevertheless, conventional methods encounter limitations of high deployment costs and low signal strength. This study proposes a reconfigurable intelligent surface (RIS)-aided cooperative integrated sensing and communication (ISAC) network for low-altitude surveillance. This network employs RISs to reflect ISAC signals into low-altitude space for sensing. To enhance signal strength, we employ active RIS (ARIS) to amplify the signals. Moreover, in order to avoid error propagation and data association in traditional sensing methods, we model low-altitude surveillance as an imaging problem based on compressed sensing theory, which can be solved through the subspace pursuit algorithm. We derive the Cramer-Rao lower bound (CRLB) of the proposed RIS-aided low-altitude imaging system and analyze the impacts of various system parameters on sensing performance, providing guidance for ISAC system configuration. Numerical results show that ARIS outperforms passive RIS under identical power constraints, achieving effective imaging and target detection at altitudes up to 300 meters. + oai:arXiv.org:2601.16033v1 cs.IT math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Morteza Varasteh, Pegah Sharifi + Zhixin Chen, Yixuan Huang, Zhengze Ji, Jie Yang, Shi Jin - Anderson Acceleration for Distributed Constrained Optimization over Time-varying Networks - https://arxiv.org/abs/2601.12398 - arXiv:2601.12398v1 Announce Type: new -Abstract: This paper applies the Anderson Acceleration (AA) technique to accelerate the Fenchel dual gradient method (FDGM) to solve constrained optimization problems over time-varying networks. AA is originally designed for accelerating fixed-point iterations, and its direct application to FDGM faces two challenges: 1) FDGM in time-varying networks cannot be formulated as a standard fixed-point update; 2) even if the network is fixed so that FDGM can be expressed as a fixed-point iteration, the direct application of AA is not distributively implementable. To overcome these challenges, we first rewrite each update of FDGM as inexactly solving several \emph{local} problems where each local problem involves two neighboring nodes only, and then incorporate AA to solve each local problem with higher accuracy, resulting in the Fenchel Dual Gradient Method with Anderson Acceleration (FDGM-AA). To guarantee global convergence of FDGM-AA, we equip it with a newly designed safe-guard scheme. Under mild conditions, our algorithm converges at a rate of \(O(1/\sqrt{k})\) for the primal sequence and \(O(1/k)\) for the dual sequence. The competitive performance of our algorithm is validated through numerical experiments. - oai:arXiv.org:2601.12398v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + Tri-Hybrid Beamforming Design for integrated Sensing and Communications + https://arxiv.org/abs/2601.16036 + arXiv:2601.16036v1 Announce Type: new +Abstract: Tri-hybrid beamforming architectures have been proposed to enable energy-efficient communications systems in extra-largescale antenna arrays using low-cost programmable metasurface antennas. We study the tri-hybrid beamforming design for integrated sensing and communications (ISAC) to improve both communications and sensing performances. Specifically, we formulate a multi-objective optimization problem that balances communications signal-to-noise ratio (SNR) and the sensing power at a target direction, subject to constraints on the total power consumption and physical limitations inherent to the trihybrid beamforming architecture. We develop an efficient iterative algorithm in which the variables are updated in a closed form at each iteration, leading to a low-complexity and fast-execution design. Numerical results show that the tri-hybrid architecture improves spatial gain and energy efficiency, though with reduced beam alignment capability compared to conventional hybrid beamforming architectures. + oai:arXiv.org:2601.16036v1 + cs.IT + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Haijuan Liu, Xuyang Wu + Tianyu Fang, Mengyuan Ma, Markku Juntti, Nhan Thanh Nguyen - BiCoLoR: Communication-Efficient Optimization with Bidirectional Compression and Local Training - https://arxiv.org/abs/2601.12400 - arXiv:2601.12400v1 Announce Type: new -Abstract: Slow and costly communication is often the main bottleneck in distributed optimization, especially in federated learning where it occurs over wireless networks. We introduce BiCoLoR, a communication-efficient optimization algorithm that combines two widely used and effective strategies: local training, which increases computation between communication rounds, and compression, which encodes high-dimensional vectors into short bitstreams. While these mechanisms have been combined before, compression has typically been applied only to uplink (client-to-server) communication, leaving the downlink (server-to-client) side unaddressed. In practice, however, both directions are costly. We propose BiCoLoR, the first algorithm to combine local training with bidirectional compression using arbitrary unbiased compressors. This joint design achieves accelerated complexity guarantees in both convex and strongly convex heterogeneous settings. Empirically, BiCoLoR outperforms existing algorithms and establishes a new standard in communication efficiency. - oai:arXiv.org:2601.12400v1 - math.OC + Risk reversal for least squares estimators under nested convex constraints + https://arxiv.org/abs/2601.16041 + arXiv:2601.16041v1 Announce Type: new +Abstract: In constrained stochastic optimization, one naturally expects that imposing a stricter feasible set does not increase the statistical risk of an estimator defined by projection onto that set. In this paper, we show that this intuition can fail even in canonical settings. + We study the Gaussian sequence model, a deliberately austere test best, where for a compact, convex set $\Theta \subset \mathbb{R}^d$ one observes \[ Y = \theta^\star + \sigma Z, \qquad Z \sim N(0, I_d), \] and seeks to estimate an unknown parameter $\theta^\star \in \Theta$. The natural estimator is the least squares estimator (LSE), which coincides with the Euclidean projection of $Y$ onto $\Theta$. We construct an explicit example exhibiting \emph{risk reversal}: for sufficiently large noise, there exist nested compact convex sets $\Theta_S \subset \Theta_L$ and a parameter $\theta^\star \in \Theta_S$ such that the LSE constrained to $\Theta_S$ has strictly larger risk than the LSE constrained to $\Theta_L$. We further show that this phenomenon can persist at the level of worst-case risk, with the supremum risk over the smaller constraint set exceeding that over the larger one. + We clarify this behavior by contrasting noise regimes. In the vanishing-noise limit, the risk admits a first-order expansion governed by the statistical dimension of the tangent cone at $\theta^\star$, and tighter constraints uniformly reduce risk. In contrast, in the diverging-noise regime, the risk is determined by global geometric interactions between the constraint set and random noise directions. Here, the embedding of $\Theta_S$ within $\Theta_L$ can reverse the risk ordering. + These results reveal a previously unrecognized failure mode of projection-based estimators: in sufficiently noisy settings, tightening a constraint can paradoxically degrade statistical performance. + oai:arXiv.org:2601.16041v1 + math.ST cs.LG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Laurent Condat, Artavazd Maranjyan, Peter Richt\'arik - - - Weak quantum hypergroups from finite index C*-inclusions - https://arxiv.org/abs/2601.12406 - arXiv:2601.12406v1 Announce Type: new -Abstract: We study a finite index inclusion of simple unital C*-algebras and construct a canonical completely positive coproduct on the second relative commutant, thereby endowing it with a natural coalgebra structure. Motivated by this construction, we introduce the notion of a weak quantum hypergroup, a generalization of the quantum hypergroups of Chapovsky and Vainerman. We show that every finite index inclusion gives rise to such a weak quantum hypergroup, and that the corresponding weak quantum hypergroup possesses a Haar integral. In the irreducible case, this structure satisfies the axioms of a quantum hypergroup in the sense of Chapovsky and Vainerman, while in the depth 2 setting our framework yields the associated weak Hopf algebra constructed by Nikshych and Vainerman. These results provide a unified and intrinsically C*-algebraic framework for generalized quantum symmetries associated with finite index inclusions. - oai:arXiv.org:2601.12406v1 - math.OA - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 + math.OC + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Keshab Chandra Bakshi, Debashish Goswami, Biplab Pal - - - Majorization between symplectic spectra of positive semidefinite matrices - https://arxiv.org/abs/2601.12408 - arXiv:2601.12408v1 Announce Type: new -Abstract: Given $2n \times 2n$ real symmetric positive semidefinite matrix $A$ with symplectic kernel, there exists a real $2n \times 2n$ \emph{symplectic matrix} $M$ such that $M^TAM= D \oplus D$, where $D$ is an $n \times n$ non-negative diagonal matrix which is unique up to permutation of its diagonal entries. - The diagonal entries of $D$ are called the \emph{symplectic eigenvalues} or symplectic spectrum of $A$. - In this work, we investigate some majorization and weak supermajorization relations between the symplectic spectra of two positive semidefinite matrices. - More explicitly, suppose $A$ and $B$ are $2n \times 2n$ real symmetric positive semidefinite matrices with symplectic kernels. - We show that if the symplectic spectrum of $A$ is majorized by the symplectic spectrum of $B$, then $A$ lies in the convex hull of the symplectic orbit of $B$. - We also establish that only a weak converse of this statement holds; i.e., if $A$ lies in the convex hull of the symplectic orbit of $B$ then the symplectic spectrum of $A$ is \emph{weakly supermajorized} by the symplectic spectrum of $B$. - Several consequences of our results are also presented. - Our methods make use of well-known connections between the theory of majorization, doubly stochastic, doubly superstochastic, and symplectic matrices. - oai:arXiv.org:2601.12408v1 - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Temjensangba, Hemant K. Mishra, Niloy Paul + Omar Al-Ghattas - Dynamic resource allocation in eukaryotic Resource Balance Analysis - https://arxiv.org/abs/2601.12411 - arXiv:2601.12411v1 Announce Type: new -Abstract: Resource Balance Analysis (RBA) is a framework for predicting steady-state cellular growth under resource constraints. However, classical RBA formulations are static and do not capture the dynamic regulation of biosynthetic resources or macromolecular turnover, which is particularly important in eukaryotic cells. In this work, we propose a dynamic extension of eukaryotic RBA based on an optimal control formulation. Cellular growth is modeled as the result of a time-dependent allocation of translational capacity between metabolic enzymes and macromolecular machinery, aimed at maximizing biomass accumulation over a finite time horizon. Using Pontryagin's Maximum Principle, we characterize optimal allocation strategies and show that steady-state RBA solutions arise as limiting regimes of the dynamic problem. - oai:arXiv.org:2601.12411v1 + A Second-Order Dynamical System for Solving Generalized Inverse Mixed Variational Inequality problems + https://arxiv.org/abs/2601.16043 + arXiv:2601.16043v1 Announce Type: new +Abstract: In this paper, we study a class of generalized inverse mixed variational inequality problems (GIMVIPs). We propose a novel projection-based second-order time-varying dynamical system for solving GIMVIPs. Under the assumptions that the underlying operators are strongly monotone and Lipschitz continuous, we establish the existence and uniqueness of solution trajectories and prove their global exponential convergence to the unique solution of the GIMVIP. Furthermore, a discrete-time realization of the continuous dynamical system is developed, resulting in an inertial projection algorithm. We show that the proposed algorithm achieves linear convergence under suitable choices of parameters. Finally, numerical experiments are presented to illustrate the effectiveness and convergence behavior of the proposed method in solving GIMVIPs. + oai:arXiv.org:2601.16043v1 math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Saeed Sadeghi Arjmand - - - Counterexamples, Constructions, and Nonexistence Results for Optimal Ternary Cyclic Codes - https://arxiv.org/abs/2601.12427 - arXiv:2601.12427v1 Announce Type: new -Abstract: Cyclic codes are an important subclass of linear codes with wide applications in communication systems and data storage systems. In 2013, Ding and Helleseth presented nine open problems on optimal ternary cyclic codes $\mathcal{C}_{(1,e)}$. While the first two and the sixth problems have been fully solved, others remain open. In this paper, we advance the study of the third and fourth open problems by providing the first counterexamples to both and constructing two families of optimal codes under certain conditions, thereby partially solving the third problem. Furthermore, we investigate the cyclic codes $\mathcal{C}_{(1,e)}$ where $e(3^h\pm 1)\equiv\frac{3^m-a}{2}\pmod{3^m-1}$ and $a$ is odd. For $a\equiv 3\pmod{4}$, we present two new families of optimal codes with parameters $[3^m-1,3^m-1-2m,4]$, generalizing known constructions. For $a\equiv 1\pmod{4}$, we obtain several nonexistence results on optimal codes $\mathcal{C}_{(1,e)}$ with the aforementioned parameters revealing the constraints of such codes. - oai:arXiv.org:2601.12427v1 - cs.IT - math.CO - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jingjun Bao, Hanlin Zou + http://creativecommons.org/licenses/by/4.0/ + Nam Van Tran - Localization and interpolation of parabolic $L^p$ Neumann problems - https://arxiv.org/abs/2601.12429 - arXiv:2601.12429v1 Announce Type: new -Abstract: We show a localization estimate for local solutions to the parabolic equation $-\partial_t u+\mbox{div} (A\nabla u)=0$ with zero Neumann data, assuming that the $L^p$ Neumann problem and $L^{p'}$ Dirichlet problem for the adjoint operator are solvable in a Lipschitz cylinder for some $p\in(1,\infty)$. Using this result, we establish the solvability of the Neumann problem in the atomic Hardy space for parabolic operators with bounded, measurable, time-dependent coefficients, and hence obtain the interpolation of solvability of the $L^p$ Neumann problem. - oai:arXiv.org:2601.12429v1 - math.AP - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 + On the Identification of Elliptic Curves That Admit Infinitely Many Twists Satisfying the Birch-Swinnerton-Dyer Conjecture + https://arxiv.org/abs/2601.16044 + arXiv:2601.16044v1 Announce Type: new +Abstract: Recent work of Burungale-Skinner-Tian-Wan established the first infinite families of quadratic twists of non-CM elliptic curves over $\mathbb{Q}$ for which the strong Birch-Swinnerton-Dyer (BSD) conjecture holds. Building on their results, we encode the required hypotheses into an explicit algorithm and apply it to the database of elliptic curves in the $L$-functions and Modular Forms Database (LMFDB), identifying all elliptic curves $E$ of conductor at most $500{,}000$ that admit infinitely many quadratic twists satisfying the strong BSD conjecture. Our computations provide certain numerical evidence for a conjecture of Radziwi{\l}{\l} and Soundararajan predicting Gaussian behavior in the analytic order of the Shafarevich-Tate group, while also observing a systematic positive bias within the BSD-satisfying subfamily. + oai:arXiv.org:2601.16044v1 + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Martin Dindo\v{s}, Linhan Li, Jill Pipher + Barinder S. Banwait, Xiaoyu Huang - Periodic families in the homology of $GL_n(F_2)$ - https://arxiv.org/abs/2601.12431 - arXiv:2601.12431v1 Announce Type: new -Abstract: We construct infinite families of nonzero classes in $H_d(GL_n(F_2);F_2)$ along lines of the form $d =\frac{2}{3}n +$(constant), thereby showing that the known slope $\frac{2}{3}$-stability for these homology groups are optimal. Using the new stability Hopf algebra perspective of Randal-Williams, our computations in addition recover the slope-$\frac{2}{3}$ stability for $GL_n(Z)$ with coefficients in $F_2$, improve that for $Aut(F_n)$ to $\frac{2}{3}$, and demonstrate that those slopes are optimal. Perhaps of independent interest, we also provide a manual for computing stability Hopf algebras over $F_2$. - oai:arXiv.org:2601.12431v1 - math.AT - math.KT - Wed, 21 Jan 2026 00:00:00 -0500 + Fujita exponents on quantum Euclidean spaces + https://arxiv.org/abs/2601.16053 + arXiv:2601.16053v1 Announce Type: new +Abstract: We study the well-posedness of a non-linear heat equation with power nonlinearity with positive initial data on quantum Euclidean spaces. We prove a noncommutative analogue of the classical Fujita theorem by identifying the critical exponent separating finite-time blow-up from global existence for small initial data. Moreover, we establish a fundamental inequality in general semifinite von Neumann algebras that is of independent interest and plays a crucial role in the study of global existence and local well-posedness of solutions of nonlinear equations in noncommutative setting. + oai:arXiv.org:2601.16053v1 + math.AP + math.OA + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kelly Wang + Edward McDonald, Michael Ruzhansky, Serikbol Shaimardan, Kanat Tulenov - Disjoint non-forking amalgamation in stable AECs - https://arxiv.org/abs/2601.12439 - arXiv:2601.12439v1 Announce Type: new -Abstract: The \emph{disjoint amalgamation property} (DAP), which asserts that all spans of a class of models can be amalgamated with minimal intersection, is an important property in the context of abstract elementary classes, with connections to both Grossberg's question and Shelah's categoricity conjecture. We prove that, in a nice AEC $\mathbf{K}$ stable in $\lambda \geq \operatorname{LS}(\mathbf{K})$ with a strong enough independence relation, all high cofinality $\lambda$-limit models are disjoint (non-forking) amalgamation bases. - $\textbf{Theorem.}$ - Let $\mathbf{K}$ be an AEC stable in $\lambda$, where $\mathbf{K}_\lambda$ has AP, JEP, and NMM, and let $\mathbf{K}'$ be some AC where $\mathbf{K}_{(\lambda,\geq\kappa)} \subseteq \mathbf{K}' \subseteq \mathbf{K}_\lambda$. Suppose there is an independence relation on $\mathbf{K}'$ satisfying uniqueness, existence, non-forking amalgamation, $\mathbf{K}_{(\lambda,\geq\kappa)}$-universal continuity* in $\mathbf{K}_\lambda$, and $(\geq \kappa)$-local character. - Assume $M_0, M_1, M_2 \in \mathbf{K}_{(\lambda,\geq\kappa)}$, and that $M_0 \leq_{\mathbf{K}} M_l$ and $a_l \in M_l$ for $l = 1, 2$. Then there exist $N \in \mathbf{K}_{(\lambda,\geq\kappa)}$ and $f_l : M_l \rightarrow N$ fixing $M_0$ for $l = 1, 2$ such that $\operatorname{gtp}(f_l(a_l)/f_{3-l}[M_{3-l}], N)$ does not fork over $M_0$ and $f_1[M_1] \cap f_2[M_2] = M_0$. That is, our independence relation has disjoint non-forking amalgamation. - In particular, every $M_0 \in \mathbf{K}_{(\lambda,\geq\kappa)}$ is a disjoint amalgamation base in $\mathbf{K}_\lambda$. - The hypotheses on the independence relation can be weakened (closer to $\lambda$-non-splitting in $\lambda$-stable AECs) if we are willing to give up the `non-forking' conditions of the amalgamation. - oai:arXiv.org:2601.12439v1 - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 + Fully Functional Weighted Testing for Abrupt and Gradual Location Changes in Functional Time Series + https://arxiv.org/abs/2601.16058 + arXiv:2601.16058v1 Announce Type: new +Abstract: Change point tests for abrupt changes in the mean of functional data, i.e., random elements in infinite-dimensional Hilbert spaces, are either based on dimension reduction techniques, e.g., based on principal components, or directly based on a functional CUSUM (cumulative sum) statistic. The former have often been criticized as not being fully functional and losing too much information. On the other hand, unlike the latter, they take the covariance structure of the data into account by weighting the CUSUM statistics obtained after dimension reduction with the inverse covariance matrix. In this paper, as a middle ground between these two approaches, we propose an alternative statistic that includes the covariance structure with an offset parameter to produce a scale-invariant test procedure and to increase power when the change is not aligned with the first components. We obtain the asymptotic distribution under the null hypothesis for this new test statistic, allowing for time dependence of the data. Furthermore, we introduce versions of all three test statistics for gradual change situations, which have not been previously considered for functional data, and derive their limit distribution. Further results shed light on the asymptotic power behavior for all test statistics under various ground truths for the alternatives. + oai:arXiv.org:2601.16058v1 + math.ST + stat.ME + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jeremy Beard - - - Schr\"odinger Operators, Integral Curvature, and the Euler Characteristic of Riemannian Manifolds - https://arxiv.org/abs/2601.12440 - arXiv:2601.12440v1 Announce Type: new -Abstract: We establish new connections between integral curvature bounds and the Euler characteristic of closed Riemannian manifolds through the perspective of Schr\"odinger-type operators. Central to our approach is the twisted Dirac operator \(\mathcal{D}_{\theta}\), whose index equals \(\chi(M)\). Under integral smallness conditions on the negative part of a potential \(V\) and a Sobolev--Poincar\'e inequality, we show that a suitable scaling of \(\theta\) forces the kernel of \(\mathcal{D}_{t\theta}\) to vanish, thereby implying \(\chi(M)=0\). - Applying this framework to geometrically natural potentials yields several topological consequences. In even dimensions, sufficiently small integral bounds on partial sums of curvature operator eigenvalues force \(\chi(M)\) either to vanish or to have a sign determined by the middle dimension. For four-manifolds, a small \(L^{p}\)-norm of the negative Ricci curvature relative to the diameter guarantees \(\chi(M)\ge 0\). Moreover, when \(\chi(M)\neq 0\) we obtain a Li--Yau type lower bound for the first eigenvalue of the rough Laplacian on \(1\)-forms in terms of the diameter and an integral curvature quantity. Subsequently, we provide an explicit lower bound for the first eigenvalue of the Laplacian on $1$-forms under almost nonnegative curvature conditions, thereby giving an affirmative answer to Yau's Problem 79. - oai:arXiv.org:2601.12440v1 - math.DG - math.SP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Teng Huang, Pan Zhang - - - Distinct permutation dot products - https://arxiv.org/abs/2601.12445 - arXiv:2601.12445v1 Announce Type: new -Abstract: We show that for any two sets of reals numbers $A=\{a_1,\dots,a_n\}$ and $B=\{b_1,\dots,b_n\}$, the sums of the form $\sum_{i=1}^n a_i\,b_{\pi(i)}$ always take on $\Omega(n^{3})$ distinct values, as we range over all permutations $\pi \in S_n$. - An important ingredient is a ``supportive'' version of Hal\'asz's anticoncentration theorem from Littlewood-Offord theory, which may be of independent interest. - oai:arXiv.org:2601.12445v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Cosmin Pohoata + Claudia Kirch, Hedvika Rano\v{s}ov\'a, Martin Wendler - On spaces of embeddings of circles in surfaces - https://arxiv.org/abs/2601.12450 - arXiv:2601.12450v1 Announce Type: new -Abstract: We consider the space of embeddings of finitely many circles that bound disks in non-positively curved surfaces. We index the connected components of this space with finite rooted trees and show that the connected components are classifying spaces of the ``braided" automorphism groups of the associated trees. An intermediate step to proving these results is to construct a strong deformation retract onto the subspace of geometric circles; moreover, this strong deformation retraction is equivariant with respect to transformations of the surface. - oai:arXiv.org:2601.12450v1 + Bivariate topological complexity: a framework for coordinated motion planning + https://arxiv.org/abs/2601.16059 + arXiv:2601.16059v1 Announce Type: new +Abstract: We introduce a bivariate version of topological complexity, $\mathrm{TC}(f,g)$, associated with two continuous maps $f\colon X\to Z$ and $g\colon Y\to Z$. This invariant measures the minimal number of continuous motion planning rules required to coordinate trajectories in $X$ and $Y$ through a shared target space $Z$. It recovers Farber's classical topological complexity when $f=g=\mathrm{id}_X$ and Pave\v{s}i\'c's map-based invariant when one of the maps is the identity. + We develop a structural theory for $\mathrm{TC}(f,g)$, including symmetry, product inequalities, stability properties, and a collaboration principle showing that, when one of the maps is a fibration, the complexity of synchronization is controlled by the other. We also introduce a homotopy-invariant bivariate complexity $\mathrm{TC}_H(f,g)$ of Scott type, defined via homotopic distance, and study its relationship with the strict invariant. + Concrete examples reveal rigidity phenomena with no analogue in the classical case, including strict gaps between $\mathrm{TC}(f,g)$ and $\mathrm{TC}_H(f,g)$ and situations where synchronization becomes impossible. Cohomological estimates provide computable obstructions in both the strict and homotopy-invariant settings. + oai:arXiv.org:2601.16059v1 math.AT - math.GR - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ryan C. Gelnett + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Jose Manuel Garcia Calcines, Jose Antonio Vilches Alarcon - Even Sets and Dual Projective Geometric Codes: A Tale of Cylinders - https://arxiv.org/abs/2601.12451 - arXiv:2601.12451v1 Announce Type: new -Abstract: In this paper, we prove that the smallest even sets in ${\rm PG}(n,q)$, i.e. sets that intersect every line in an even number of points, are cylinders with a hyperoval as base. This fits into a more general study of dual projective geometric codes. Let $q$ be a prime power, and define $\mathcal C_k(n,q)^\perp$ as the kernel of the $k$-space vs. point incidence matrix of ${\rm PG}(n,q)$, seen as a matrix over the prime order subfield of $\mathbb F_q$. Determining the minimum weight of this linear code is still an open problem in general, but has been reduced to the case $k=1$. There is a known construction that constructs small weight codewords of $\mathcal C_1(n,q)^\perp$ from minimum weight codewords of $\mathcal C_1(2,q)^\perp$. We call such codewords cylinder codewords. We pose the conjecture that all minimum weight codewords of $\mathcal C_1(n,q)^\perp$ are cylinder codewords. This conjecture is known to be true if $q$ is prime. We take three steps towards proving that the conjecture is true in general: - (1) We prove that the conjecture is true if $q$ is even. This is equivalent to our classification of the smallest even sets. - (2) We prove that the minimum weight of $\mathcal C_1(n,q)^\perp$ is $q^{n-2}$ times the minimum weight of $\mathcal C_1(2,q)^\perp$, which matches the weight of cylinder codewords. Thus, we completely reduce the problem of determining the minimum weight of $\mathcal C_1(n,q)^\perp$ to the case $n=2$. - (3) We prove that if the conjecture is true for $n=3$, it is true in general. - oai:arXiv.org:2601.12451v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + Continuum limit of hypergraph $p$-Laplacian equations on point clouds + https://arxiv.org/abs/2601.16063 + arXiv:2601.16063v1 Announce Type: new +Abstract: This paper studies a class of $p$-Laplacian equations on point clouds that arise from hypergraph learning in a semi-supervised setting. Under the assumption that the point clouds consist of independent random samples drawn from a bounded domain $\Omega\subset\mathbb{R}^d$, we investigate the asymptotic behavior of the solutions as the number of data points tends to infinity, with the number of labeled points remains fixed. We show, for any $p>d$ in the viscosity solution framework, that the continuum limit is a weighted $p$-Laplacian equation subject to mixed Dirichlet and Neumann boundary conditions. The result provides a new discretization of the $p$-Laplacian on point clouds. + oai:arXiv.org:2601.16063v1 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Sam Adriaensen + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Kehan Shi - Unbounded banded matrices with positive bidiagonal factorization and mixed-type multiple orthogonal polynomials - https://arxiv.org/abs/2601.12453 - arXiv:2601.12453v1 Announce Type: new -Abstract: A spectral Favard theorem is proved for semi-infinite banded matrices admitting a positive bidiagonal factorization, without assuming boundedness of the associated operator, thus covering both the bounded and unbounded settings. The result yields a matrix-valued spectral measure and an explicit spectral representation of the matrix powers in terms of the associated mixed-type multiple orthogonal polynomials. The argument follows the constructive truncation scheme: principal truncations are oscillatory, hence have simple positive spectra, and a suitable choice of initial conditions ensures positivity of the Christoffel coefficients and of the resulting discrete matrix-valued measures supported at the truncation eigenvalues. The main difficulty is the passage to the limit of these discrete measures beyond the bounded case. This is resolved by combining the available Gaussian quadrature structure with a Helly-type compactness argument, leading to a limiting matrix-valued measure and completing the spectral theorem. The role of normality (maximal degree pattern) for the mixed-type families is also addressed. - oai:arXiv.org:2601.12453v1 - math.CA - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 + On the Stable Euclidean Distance Degree of Algebraic Layers + https://arxiv.org/abs/2601.16071 + arXiv:2601.16071v1 Announce Type: new +Abstract: We study the projective geometry of algebraic neural layers, namely families of maps induced by a polynomial activation function, with particular emphasis on the generic Euclidean Distance degree ($\mathrm{gED}$). This invariant is projective in nature and measures the number of optimal approximations of a general point in the ambient space with respect to a general metric. For a fixed architecture (i.e. fixed width and activation polynomial), we prove that the $\mathrm{gED}$ is stably polynomial in the dimensions of the input and output spaces. Moreover, we show that this stable polynomial depends only on the degree of the activation function. + Our approach relies on standard intersection theory on the Nash blow-up, which allows us to express the $\gED$ as an intersection number over products of Grassmannians. Stable polynomiality is deduced via equivariant localization, while the reduction to the monomial case follows from an explicit Schubert calculus computation on Grassmannians. + oai:arXiv.org:2601.16071v1 + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Am\'ilcar Branquinho, Ana Foulqui\'e-Moreno, Manuel Ma\~nas + Giacomo Graziani - Hirzebruch-Riemann-Roch for complex analytic infinity-prestacks - https://arxiv.org/abs/2601.12454 - arXiv:2601.12454v1 Announce Type: new -Abstract: We provide a cocycle-level Hirzebruch-Riemann-Roch (HRR) identity for arbitrary complex analytic infinity-prestacks. We view this work as the natural setting for Toledo and Tong's HRR philosophy and technical machinery. - oai:arXiv.org:2601.12454v1 - math.AG - math.AT - Wed, 21 Jan 2026 00:00:00 -0500 + The Hyperrigidity Conjecture for Spectrahedra + https://arxiv.org/abs/2601.16075 + arXiv:2601.16075v1 Announce Type: new +Abstract: We show that if K is a compact spectrahedron whose set of extreme points is closed, then the operator system of continuous affine functions on K is hyperrigid in the C*-algebra C(ex(K)). + oai:arXiv.org:2601.16075v1 + math.OA + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Cheyne Glass, Thomas Tradler, Mahmoud Zeinalian + Marcel Scherer - Structure theory of set addition with two operations - https://arxiv.org/abs/2601.12457 - arXiv:2601.12457v1 Announce Type: new -Abstract: We take the first step toward a structure theory that includes both operations of a ring $\mathcal{R}$. More precisely, we prove a series of inverse results for the structure of sets $A\subseteq \mathbf{F}_p$ such that, under certain conditions on integers $r_1, \dots, r_k$, one has $|A^{r_1} + \dots + A^{r_k}| \ll \sqrt[k]{p^{k-1} |A|}$. - oai:arXiv.org:2601.12457v1 + Rainbow spanning structures in strongly edge-colored graphs + https://arxiv.org/abs/2601.16084 + arXiv:2601.16084v1 Announce Type: new +Abstract: An edge-colored graph is a graph in which each edge is assigned a color. Such a graph is called strongly edge-colored if each color class forms an induced matching, and called rainbow if all edges receive pairwise distinct colors. In this paper, by establishing a connection with $\mu n$-bounded graphs, we prove that for all sufficiently large integers $n$, every strongly edge-colored graph $G$ on $n$ vertices with minimum degree at least $\frac{n+1}{2}$ contains a rainbow Hamilton cycle. We also characterize all strongly edge-colored graphs on $n$ vertices with minimum degree exactly $\frac{n}{2}$ that do not contain a rainbow Hamilton cycle. As an application, we determine the optimal minimum degree conditions for the existence of rainbow Hamilton paths and rainbow perfect matchings in strongly edge-colored graphs. Together, these results verify three conjectures concerning strongly edge-colored graphs for sufficiently large $n$. + oai:arXiv.org:2601.16084v1 math.CO - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Aliaksei Semchankau, Ilya Shkredov - - - Symmetric preparation of systems - https://arxiv.org/abs/2601.12458 - arXiv:2601.12458v1 Announce Type: new -Abstract: In this paper we generalize the Weierstrass and Malgrange preparation theorems to the symmetric matrix valued case, proving symmetric preparation of analytic and smooth symmetric systems that vanish of first order. - oai:arXiv.org:2601.12458v1 - math.CA - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Nils Dencker - - - Homological $(n-2)$-systole in $n$-manifolds with positive triRic curvature - https://arxiv.org/abs/2601.12461 - arXiv:2601.12461v1 Announce Type: new -Abstract: In this paper, we prove an optimal systolic inequality and characterize the case of equality on closed Riemannian manifolds with positive triRic curvature. This extends prior work of Bray-Brendle-Neves \cite{BrayBrenleNevesrigidity} and Chu-Lee-Zhu \cite{chuleezhu_n_systole} to higher codimensions. The proof relies on the notion of stable weighted $k$-slicing, a weighted volume comparison theorem and metric-deformation. - oai:arXiv.org:2601.12461v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Jingche Chen, Han Hong - - - Non-intersecting Squared Bessel Process: Spectral Moments and Dynamical Entanglement Entropy - https://arxiv.org/abs/2601.12484 - arXiv:2601.12484v1 Announce Type: new -Abstract: Statistical ensembles of reduced density matrices of bipartite quantum systems play a central role in entanglement estimation, but do not capture the non-stationary nature of entanglement relevant to realistic quantum information processing. To address this limitation, we propose a dynamical extension of the Hilbert-Schmidt ensemble, a baseline statistical model for entanglement estimation, arising from non-intersecting squared Bessel processes and perform entanglement estimation via average entanglement entropy and quantum purity. The investigation is enabled by finding spectral moments of the proposed dynamical ensemble, which serves as a new approach for systematic computation of entanglement metrics. Along the way, we also obtain new results for the underlying multiple orthogonal polynomials of modified Bessel weights, including structure and recurrence relations, and a Christoffel-Darboux formula for the correlation kernels. - oai:arXiv.org:2601.12484v1 - math-ph - math.MP - quant-ph - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Youyi Huang, Lu Wei + Laihao Ding, Xiaolan Hu, Suyun Jiang - Fast Computing Formulas for some Dirichlet L-Series - https://arxiv.org/abs/2601.12495 - arXiv:2601.12495v1 Announce Type: new -Abstract: For $\chi_k$ a self$-$dual primitive Dirichlet character mod $k$ several reduced identities of Dirichlet $L-$functions $L_k(s):=L(s,\chi_k)$, expressed as linear combinations of Hurwitz $\zeta$ functions, are found for $s=2,3$ and some selected values of $k$. By using a merged approach between the Wilf$-$Zeilberger method and a Dougall$'$s $_5H_5$ technique, new proven accelerated series of hypergeometric$-$type are derived for specific Hurwitz $\zeta$ function values. These fast series that are computed by means of the binary splitting algorithm, enter into the reduced identities found producing very efficient formulas to compute these selected $L-$functions. The new algorithms include $\zeta(3):=L_1(3)$, (Apery$'$s constant), $G:=L_\text{-4}(2)$ (Catalan$'$s constant) as well as $\text{}L_\text{k}(2)\text{}$ for $k=-7, -8, -15, -20, -24$ together with $L_k(3)$ for $k=5, 8, 12$. Formulas were tested and verified up to 100 million decimal places for each $L-$value. - oai:arXiv.org:2601.12495v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 + Birational automorphism groups in families of hyper-K\"ahler manifolds + https://arxiv.org/abs/2601.16090 + arXiv:2601.16090v1 Announce Type: new +Abstract: We study the behavior of birational automorphism groups in families of projective hyper-K\"ahler manifolds. + oai:arXiv.org:2601.16090v1 + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jorge Zuniga + Francesco Antonio Denisi, Claudio Onorati, Francesca Rizzo, Sasha Viktorova - The $\ell$-modular local theta correspondence in type II and partial permutations - https://arxiv.org/abs/2601.12497 - arXiv:2601.12497v1 Announce Type: new -Abstract: In this paper we compute the multiplicities appearing in the ${\overline{\mathbb{F}}_\ell}$-modular theta correspondence in type II over a non-archimedean field $\mathrm{F}$, where $\ell$ is a prime not dividing the residue cardinality of $\mathrm{F}$. Unlike for representations with complex coefficients, highly non-trivial multiplicities can emerge. We show that these multiplicities are precisely governed by the action of symmetric groups on the set of partial permutations, and the ${\overline{\mathbb{F}}_\ell}$-representation of symmetric groups these give rise to. The problem is thus reduced to certain branching problems in the modular representation theory of symmetric groups. In particular, if $d$ is the order of the residue cardinality of $\mathrm{F}$ in ${\overline{\mathbb{F}}_\ell}$, and the rank of the involved general linear groups is bounded above by $ d\ell$, the behavior of the theta correspondence can be predicted via explicit algorithms coming from Pieri's Formula. - oai:arXiv.org:2601.12497v1 + Monoidal adjunctions and abelian envelopes + https://arxiv.org/abs/2601.16092 + arXiv:2601.16092v1 Announce Type: new +Abstract: We show how monoidal adjunctions can be used to prove the existence of monoidal abelian envelopes of pseudo-tensor categories, in particular, those admitting a combinatorial description with certain properties. We derive concrete general criteria that we demonstrate by giving relatively simple combinatorial proofs of the existence of new abelian envelopes for interpolation categories of the hyperoctahedral and of the modified symmetric groups. + oai:arXiv.org:2601.16092v1 math.RT - Wed, 21 Jan 2026 00:00:00 -0500 + math.CT + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Johannes Droschl + Johannes Flake, Robert Laugwitz, Sebastian Posur - Heun-function analysis of the Dirac spinor spectrum in a sine-Gordon soliton background - https://arxiv.org/abs/2601.12504 - arXiv:2601.12504v1 Announce Type: new -Abstract: We study the Dirac spectrum in a sine-Gordon soliton background, where the induced position-dependent mass reduces the spectral problem to a Heun-type differential equation. Bound and scattering sectors are treated within a unified framework, with spectral data encoded in Wronskians matching local Heun solutions and exhibiting explicit dependence on the soliton parameters and the bare fermion mass. This formulation enables a systematic analysis of spinor bound and scattering states, supported by analytic and numerical verification of wave function matching across the soliton domain. The present work is related to arXiv:2512.07658 and emphasizes a pedagogical treatment of scattering states within the Heun-equation formalism. - oai:arXiv.org:2601.12504v1 - math-ph - math.MP - nlin.SI - Wed, 21 Jan 2026 00:00:00 -0500 + On Seshadri constants of adjoint divisors on surfaces and threefolds in arbitrary characteristic + https://arxiv.org/abs/2601.16094 + arXiv:2601.16094v1 Announce Type: new +Abstract: We develop a new approach towards obtaining lower bounds of the Seshadri constants of ample adjoint divisors on smooth projective varieties $X$ in arbitrary characteristic. Let $x\in X$ be a closed point and $A$ an ample divisor on $X$. If $X$ is a surface, we recover some known lower bounds by proving, e.g., that $\varepsilon(K_X+4A;x)\geq 3/4$. If $X$ is a threefold, we prove that for all $\delta>0$ and all but finitely many curves $C$ through $x$, we have $\frac{(K_X+6A).C}{\operatorname{mult}_x C}\geq\frac{1}{2\sqrt{2}}-\delta$. In particular, if $\varepsilon(K_X+6A;x)<1/(2\sqrt{2})$, then $\varepsilon(K_X+6A;x)$ is a rational number, attained by a Seshadri curve $C$. + oai:arXiv.org:2601.16094v1 + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - H. Blas, R. P. N. Laeber Fleitas, J. Silva Barroso + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Linus R\"osler - Approximability for Lagrangian submanifolds - https://arxiv.org/abs/2601.12506 - arXiv:2601.12506v1 Announce Type: new -Abstract: This paper introduces a notion of categorical approximability for metric spaces that can be viewed as a categorification of approximability for metric groups, as defined by Turing in 1938. Approximability as introduced here is a property of metric spaces that is more general than precompactness. It is shown that several classes of Lagrangian submanifolds - closed Lagrangian submanifolds in a cotangent disk bundle; equators on the sphere; weakly exact Lagrangians on the torus-endowed with the spectral metric are approximable in this sense. Among other geometric applications, we show that there are such examples of spaces of Lagrangians that are approximable but are not precompact. - oai:arXiv.org:2601.12506v1 - math.SG - math.AT - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 + Intersections of Convex Hulls of Polynomial Shifts and Critical Points + https://arxiv.org/abs/2601.16102 + arXiv:2601.16102v1 Announce Type: new +Abstract: Let $p(z)$ be a complex polynomial of degree $n\ge 2$. For each $c\in\mathbb{C}$, let $K_c$ denote the convex hull of the zeros of $p(z)+c$, and let $K'$ denote the convex hull of the zeros of $p'(z)$. We prove that $$\bigcap_{c\in\mathbb{C}} K_c = K',$$ by combining a strict separating hyperplane argument with a half-plane non-surjectivity theorem for polynomials without critical points (proved via analytic continuation, the monodromy theorem and Liouville's Theorem). We also characterize when $K_0=K'$ in terms of the multiplicities of the zeros of $p(z)$ that form the vertices of $K_0$. As an application, we obtain a partial result toward the Schmeisser's conjecture: if all zeros of $p$ lie in the closed unit disk, then for every $\zeta\in K'$ the disk $|z-\zeta|\le \sqrt{1-|\zeta|^2}$ contains a critical point of $p(z)$. Finally, we refine a recent barycentric bound in \cite{Zha26+} by showing that there is always a critical point within distance $\sqrt{\frac{n-2}{n-1}}\sqrt{1-|G|^2}$ of the centroid $G$ of the zeros. + oai:arXiv.org:2601.16102v1 + math.CV + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Giovanni Ambrosioni, Paul Biran, Octav Cornea + Teng Zhang - Extending graph total colorings to cell complexes - https://arxiv.org/abs/2601.12514 - arXiv:2601.12514v1 Announce Type: new -Abstract: Let $2\le k\in\mathbb{Z}$. A total coloring of a simple connected regular graph via color set $ \{0,1,\ldots, k\}$ is said to be {\it efficient} if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. In this work, focus is set upon 2-cell complexes whose 1-skeletons, namely their induced 1-cell complexes, are toroidal graphs. Each such 2-cell complex is said to cover its induced 1-skeleton. An efficient total coloring of one such skeleton induces an efficient total cell coloring of its covering 2-cell complex if it assigns a vertex-and-edge $k$-color set to the border skeleton of each of its 2-cells, with the consequently missing color in $\{0,1,\ldots,k\}$ assigned to the 2-cell itself, so that the two adjacent 2-cells along any 1-cell are assigned different colors. Applications are given for plane tilings, cycle products, toroidal triangulations, honeycombs and star-of-David tilings. - oai:arXiv.org:2601.12514v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + The Eisenbud-Goto conjecture for projectively normal varieties with mild singularities + https://arxiv.org/abs/2601.16103 + arXiv:2601.16103v1 Announce Type: new +Abstract: For a nondegenerate projective variety $X$, the Eisenbud-Goto conjecture asserts that $\operatorname{reg}X\leq\operatorname{deg}X-\operatorname{codim}X+1$. Despite the existence of counterexamples, identifying the classes of varieties for which the conjecture holds remains a major open problem. In this paper, we prove that the Eisenbud-Goto conjecture holds for $2$-very ample projectively normal varieties with mild singularities. + oai:arXiv.org:2601.16103v1 + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Italo J. Dejter + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Jong In Han - Rigidity results in multi-bubble dynamics for non-radial energy-critical heat equation - https://arxiv.org/abs/2601.12517 - arXiv:2601.12517v1 Announce Type: new -Abstract: This paper concerns the classification of asymptotic behaviors in multi-bubble dynamics for the energy-critical nonlinear heat equations in large dimensions $N\geq7$ without symmetry. This multi-bubble dynamics appears naturally at least for a sequence of times in view of soliton resolution. We assume each bubble is given by the scalings and translations of $\pm W$ with (localized) non-colliding conditions for a sequence of times, where $W$ is the ground state. The case of one soliton was previously established and in particular there is no blow-up. We consider the case of $J\geq2$ solitons, where we expect only infinite-time blow-up. - We are able to identify three different scenarios, where we have a continuous-in-time resolution with an unexpected universal blow-up speed. The first one is when one scaling is much larger than the others. In this case, one bubble does not concentrate (hence stabilize) and the other bubbles concentrate with the universal blow-up speed $t^{-2/(N-6)}$ together with strong sign constraints. Next, assuming we are not in the first scenario, we establish a non-degenerate condition on the positions of bubbles to obtain that all bubbles concentrate with the universal blow-up speed $t^{-1/(N-4)}$. The last case we consider is a degenerate, but not too much degenerate, scenario. Here again, we obtain that all bubbles concentrate with the universal blow-up speed $t^{-1/(N-3)}$. This last rate has not been discovered before. Our theorem covers the case of four or less bubbles and we provide the construction of examples. To our knowledge, this is the first classification result in the non-radial multi-bubble dynamics, where both the scales, positions, and signs enter the dynamics nontrivially. - oai:arXiv.org:2601.12517v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + A Linear Bound on the Rich Flow Number for Graphs with a Given Maximum Degree + https://arxiv.org/abs/2601.16104 + arXiv:2601.16104v1 Announce Type: new +Abstract: A rich $k$-flow is a nowhere-zero $k$-flow $\phi$ such that, for every pair of adjacent edges $e$ and $f$, $|\phi(e)| \neq |\phi(f)|$. A graph is rich flow admissible if it admits a rich $k$-flow for some integer $k$. In this paper, we prove that if $G$ is a rich flow admissible graph with maximum degree $\Delta$, then $G$ admits a rich $(264\Delta - 445)$-flow. + oai:arXiv.org:2601.16104v1 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kihyun Kim, Frank Merle + Robert Luko\v{t}ka - Quasigeodesic languages are not context-free in some non-hyperbolic groups - https://arxiv.org/abs/2601.12520 - arXiv:2601.12520v1 Announce Type: new -Abstract: We study the full language of quasigeodesics in Cayley graphs, with fixed error constants. We show that, given a non-virtually-cyclic nilpotent group or Baumslag--Solitar group, and any finite generating set, such languages fail to be context-free for sufficiently large error constants. In fact, this conclusion holds for any finitely generated group which contains one of these groups as an undistorted subgroup. This strengthens a recent theorem of Hughes, Nairne, and Spriano, who showed that such languages fail to be regular in any non-hyperbolic group, for sufficiently large error constants. - oai:arXiv.org:2601.12520v1 - math.GR - Wed, 21 Jan 2026 00:00:00 -0500 + Algorithms for Algebraic and Arithmetic Attributes of Hypergeometric Functions + https://arxiv.org/abs/2601.16105 + arXiv:2601.16105v1 Announce Type: new +Abstract: We discuss algorithms for arithmetic properties of hypergeometric functions. Most notably, we are able to compute the p-adic valuation of a hypergeometric function on any disk of radius smaller than the p-adic radius of convergence. This we use, building on work of Christol, to determine the set of prime numbers modulo which it can be reduced. Moreover, we describe an algorithm to find an annihilating polynomial of the reduction of a hypergeometric function modulo p. + oai:arXiv.org:2601.16105v1 + math.NT + cs.SC + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Arya Saranathan + Xavier Caruso, Florian F\"urnsinn - Derived equivalences via Tate resolutions - https://arxiv.org/abs/2601.12531 - arXiv:2601.12531v1 Announce Type: new -Abstract: For any finite sequence of elements $s_1, \ldots , s_d$ in a commutative noetherian ring $R$, we show that for $n \gg 0$, the natural map from the Koszul complex $K(s_1^n, \ldots , s_d^n)$ to the Koszul complex $K(s_1, \ldots , s_d)$ factors through the Tate resolution on $s_1^n, \ldots , s_d^n$. Using this, for any resolving subcategory $\mathcal A$ of mod($R$) and any ideal $I$ such that it has a filtration $\{ I_n \}$ which is equivalent to the $I$-adic filtration and $\textrm{dim}_{\mathcal A}(R/I_n) < \infty$, we show a derived equivalence between the bounded derived category of finitely generated modules supported on $V(I)$ having finite $\mathcal A$-dimension and the bounded derived category of $\mathcal A$ with homologies supported on $V(I)$. As a special case, when $R$ is of prime characteristic and $I$ is of finite projective dimension, we obtain a derived equivalence between the bounded derived category of finite projective dimension modules supported on $V(I)$ and the bounded derived category of projective modules with homologies supported on $V(I)$. - oai:arXiv.org:2601.12531v1 - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 + Stability and Decay for the 2D Anisotropic Navier-Stokes Equations with Fractional Horizontal Dissipation on $\mathbb{R}^2$ + https://arxiv.org/abs/2601.16110 + arXiv:2601.16110v1 Announce Type: new +Abstract: The stability problem for the 2D Navier-Stokes equations with dissipation in only one direction on $\mathbb R^2$ is not fully understood. This dissipation is in the intermediate regime between the fully dissipative Navier-Stokes and the inviscid Euler. Navier-Stokes solutions in $\mathbb R^2$ decay algebraically in time while Euler solutions can grow rather rapidly in time. This paper solves the fundamental stability and large-time behavior problem on the anisotropic Navier-Stokes with fractional dissipation $\Lambda_1^{2s}$ for all $0\leq s<1$. The case $s=1$ corresponds to the standard one directional dissipation $\partial_1^2$. Different techniques are developed to treat different ranges of fractional exponents: $0\leq s\leq \frac34$, $\frac34<s<\frac{11}{12}$, and $\frac{11}{12} \leq s <1$. The final range is the most difficult case, for which we introduce the spatial polynomial $A_2$ weights and exploit the boundedness of Riesz transforms on weighted $L^2$-spaces. + oai:arXiv.org:2601.16110v1 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.jalgebra.2025.02.015 - K. Ganapathy, Sarang Sane + Zhibin Wang, Jiahong Wu, Ning Zhu - Examples and counterexamples of injective types - https://arxiv.org/abs/2601.12536 - arXiv:2601.12536v1 Announce Type: new -Abstract: It is known that, in univalent mathematics, type universes, the type of $n$-types in a universe, reflective subuniverses, and the underlying type of any algebra of the lifting monad are all (algebraically) injective. Here, we further show that the type of ordinals, the type of iterative (multi)sets, the underlying type of any pointed directed complete poset, as well as the types of (small) $\infty$-magmas, monoids, and groups are all injective, among other examples. Not all types of mathematical structures are injective in general. For example, the type of inhabited types is injective if and only if all propositions are projective. In contrast, the type of pointed types and the type of non-empty types are always injective. The injectivity of the type of two-element types implies Fourman and \v{S}\v{c}edrov's world's simplest axiom of choice. We also show that there are no nontrivial small injective types unless a weak propositional resizing principle holds. Other counterexamples include the type of booleans, the simple types, the type of Dedekind reals, and the type of conatural numbers, whose injectivity implies weak excluded middle. More generally, any type with an apartness relation and two points apart cannot be injective unless weak excluded middle holds. Finally, we show that injective types have no non-trivial decidable properties, unless weak excluded middle holds, which amounts to a Rice-like theorem for injective types. - oai:arXiv.org:2601.12536v1 - math.LO - cs.LO - Wed, 21 Jan 2026 00:00:00 -0500 + Equivariant Morse-Bott cohomology through stabilization + https://arxiv.org/abs/2601.16119 + arXiv:2601.16119v1 Announce Type: new +Abstract: For closed manifolds with compact Lie group actions, we study Austin-Braam's Morse-theoretic construction of Borel equivariant cohomology using the technique of stabilization. We show that a $C^1$-small equivariant perturbation produces stable invariant Morse-Bott functions. This allows us to realize the equivariant transversality and orientability assumptions in Austin-Braam's framework by choosing generic invariant Riemannian metrics. + oai:arXiv.org:2601.16119v1 + math.DG + math.AT + math.DS + math.GT + math.SG + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Tom de Jong, Mart\'in H\"otzel Escard\'o + Erkao Bao, Robi Huq, Shengzhen Ning - Connections and na\"{i}ve lifting of DG modules - https://arxiv.org/abs/2601.12550 - arXiv:2601.12550v1 Announce Type: new -Abstract: In this paper, we generalize the notion of connections, which was introduced by Alain Connes in noncommutative differential geometry, to the differential graded (DG) homological algebra setting. Then, along a DG algebra homomorphism $A \to B$, where $B$ is assumed to be projective as an underlying graded $A$-module, we give necessary and sufficient conditions for a semifree DG $B$-module to be na\"{i}vely liftable to $A$ in terms of connections. - oai:arXiv.org:2601.12550v1 - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 + Canonical structure of the LLG equation for exponential updates in micromagnetism + https://arxiv.org/abs/2601.16122 + arXiv:2601.16122v1 Announce Type: new +Abstract: In this contribution we propose an exponential update algorithm for magnetic moments appearing in the framework of micromagnetics and the Landau-Lifshitz-Gilbert (LLG) equation. This algorithm can be interpreted as the geometric integration on spheres, that a priori satisfy the unit length constraint of the normalized magnetization vector. Even though the geometric structures for this are obvious and some works already use an exponential algorithm, to the best of the authors' knowledge, there is no canonical structure of the LLG equation for the exponential update algorithm in micromagnetism. Tensor algebraic reformulations of the LLG equation allow the canonical representation of the evolution equation for the magnetization, which serves as the basis for different integrators. Based on the specific structure of the exponential of skew symmetric matrices an efficient update scheme is derived. The excellent performance of the proposed exponential update algorithm is demonstrated in representative examples. + oai:arXiv.org:2601.16122v1 + math.NA + cs.NA + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Saeed Nasseh, Maiko Ono, Yuji Yoshino + http://creativecommons.org/licenses/by/4.0/ + J\"org Schr\"oder, Maximilian Vorwerk - Remarks on the second Chern class of a foliation - https://arxiv.org/abs/2601.12558 - arXiv:2601.12558v1 Announce Type: new -Abstract: We bound the second Chern class of the tangent sheaf of a codimension-one foliation. Equivalently, we bound the degree of the pure codimension-two part of the singular scheme. In particular, for a degree-$d$ foliation on the projective space, the codimension-two part of its singular scheme must have degree at least $d+1$. Moreover, equality holds only for rational foliations of type $(1,d+1)$. These bounds involve counting an invariant related to first-order unfoldings of 2-dimensional foliated singularities. - oai:arXiv.org:2601.12558v1 - math.AG - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 + A hybrid reconstruction of piece-wise smooth functions from non-uniform Fourier data + https://arxiv.org/abs/2601.16124 + arXiv:2601.16124v1 Announce Type: new +Abstract: In this paper, we consider the problem of reconstructing piece-wise smooth functions from their non-uniform Fourier data. We first extend the filter method for uniform Fourier data to the non-uniform setting by using the techniques of admissible frames. We show that the proposed non-uniform filter method converges exponentially away from the jump discontinuities. However, the convergence rate is significantly slower near the jump discontinuities due to the Gibbs phenomenon. To overcome this issue, we combine the non-uniform filter method with a stable extrapolation method to recover the function values near the jump discontinuities. We show that the proposed hybrid method could achieve exponential accuracy uniformly on the entire domain. Numerical experiments are provided to demonstrate the performance of the proposed method. + oai:arXiv.org:2601.16124v1 + math.NA + cs.NA + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Alan Muniz + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Guohui Song, Congzhi Xia - Sheared displays and $p$-divisible groups - https://arxiv.org/abs/2601.12565 - arXiv:2601.12565v1 Announce Type: new -Abstract: We develop a Dieudonn\'e theory for $p$-divisible groups using sheared Witt vectors. - oai:arXiv.org:2601.12565v1 - math.NT - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 + Proximity Operator of the $\ell_1$ over $\ell_2$ Function + https://arxiv.org/abs/2601.16128 + arXiv:2601.16128v1 Announce Type: new +Abstract: We study the proximity operator of the nonconvex, scale-invariant ratio $h(\vx)=\|\vx\|_{1}/\|\vx\|_{2}$ and show it can be computed exactly in any dimension. By expressing $\vx=r\vu$ and exploiting sign and permutation invariance, we reduce the proximal step to a smooth optimization of a rank-one quadratic over the nonnegative orthant of the unit sphere. We prove that every proximal point arises from a finite candidate set indexed by $k\in\{1,\dots,n\}$: the active subvector is a local, but nonglobal, minimizer on $\mathbb{S}^{k-1}$ characterized by the roots of an explicit quartic. This yields closed-form candidates, an exact selection rule, and a necessary and sufficient existence test. Building on these characterizations, we develop practical algorithms, including an $O(n)$ implementation via prefix sums and a pruning criterion that avoids unnecessary quartic solves. The method returns all proximal points when the prox is non-unique, and in experiments it attains strictly lower objective values than approaches that guess sparsity or rely on sphere projections with limited scalability. + oai:arXiv.org:2601.16128v1 + math.OC + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Manuel Hoff, Eike Lau + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Lixin Shen, Guohui Song - Self-avoiding walk, connective constant, cubic graph, Fisher transformation, quasi-transitive graph - https://arxiv.org/abs/2601.12571 - arXiv:2601.12571v1 Announce Type: new -Abstract: We study self-avoiding walks (SAWs) on infinite quasi-transitive cubic graphs under \emph{local transformations} that replace each degree-$3$ vertex by a finite, symmetric three-port gadget. To each gadget we associate a two-port SAW generating function $g(x)$, defined by counting SAWs that enter and exit the gadget through prescribed ports. Our first main result shows that, if $G$ is cubic and $G_1=\phi(G)$ is obtained by applying the local transformation at every vertex, then the connective constants $\mu(G)$ and $\mu(G_1)$ satisfy the functional relation \[ \mu(G)^{-1}=g\bigl(\mu(G_1)^{-1}\bigr). \] We next consider critical exponents defined via susceptibility-type series that do not rely on an ambient Euclidean dimension, and prove that the exponents $\gamma$ and $\eta$ are invariant under local transformations; moreover $\nu$ is invariant under a standard regularity hypothesis on SAW counts (a common slowly varying function). - Our second set of results concerns bipartite graphs, where the local transformation is applied to one colour class (or to both classes, possibly with different gadgets). In this setting we obtain an analogous relation \[ \mu(G)^{-2}=h\bigl(\mu(G_{\mathrm e})^{-1}\bigr), \] with $h(x)=xg(x)$ when only one class is transformed and $h(x)=g_{\phi_1}(x)\,g_{\phi_2}(x)$ when both are transformed. We further present explicit families of examples, including replacing each degree-3 vertex by a complete-graph gadget $K_N$. - oai:arXiv.org:2601.12571v1 - math.CO + A pseudo-bosonic Klein-Gordon field with finite two-points function + https://arxiv.org/abs/2601.16131 + arXiv:2601.16131v1 Announce Type: new +Abstract: We introduce a class of pseudo-bosonic Klein-Gordon fields in 1+1 dimensions and we discuss some of their properties. This work originates from non Hermitian quantum mechanics and deformed canonical commutation relations. We show that, within this class of fields, there exist a specific subclass with the interesting feature of having finite equal space-time two-points function, contrarily to what happens for {\em standard} Klein-Gordon fields. This, in our opinion, is a relevant aspect of our proposal which is a good motivation to undertake a deeper analysis of this (and related) quantum fields. + oai:arXiv.org:2601.16131v1 math-ph + hep-th math.MP - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 + quant-ph + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Benjamin Grant, Zhongyang Li + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Fabio Bagarello - A Functorial Approach to Multi-Space Interpolation with Function Parameters - https://arxiv.org/abs/2601.12572 - arXiv:2601.12572v1 Announce Type: new -Abstract: We introduce an extension of interpolation theory to more than two spaces by employing a functional parameter, while retaining a fully functorial and systematic framework. This approach allows for the construction of generalized intermediate spaces and ensures stability under natural operations such as powers and convex combinations. As a significant application, we demonstrate that the interpolation of multiple generalized Sobolev spaces yields a generalized Besov space. Our framework provides explicit tools for handling multi-parameter interpolation, highlighting both its theoretical robustness and practical relevance. - oai:arXiv.org:2601.12572v1 - math.FA - cs.NA - math.NA - Wed, 21 Jan 2026 00:00:00 -0500 + Modular Weil representation and compatibility of cuspidals with congruences + https://arxiv.org/abs/2601.16132 + arXiv:2601.16132v1 Announce Type: new +Abstract: Let $F$ be a non-archimedean local field of characteristic different from $2$ and of residual characteristic $p$. We generalise the theory of the Weil representation over $F$ with complex coefficients to $\ell$-modular representations \textit{i.e.} when the complex coefficients are replaced by a coefficient field $R$ of characteristic $\ell \neq p$. We obtain along the way a generalisation of the Stone-von Neumann theorem to the $\ell$-modular setting, together with the Weil representation with coefficients in $R$ on the $R$-metaplectic group. Surprisingly enough, the latter $R$-metaplectic group happens to be split over the symplectic group if $\ell = 2$. The theory also makes sense when $F$ is a finite field of odd characteristic. We also establish the irreducibility of the theta lift in the cuspidal case as long as $\ell$ does not divide the pro-orders of the groups at stake and we provide a compatibility to congruences in this setting via an integral version of the theta lift. + oai:arXiv.org:2601.16132v1 + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Thomas Lamby, Samuel Nicolay + Justin Trias - L(3,2,1)-labelings of three classes of 4-valent circulants - https://arxiv.org/abs/2601.12574 - arXiv:2601.12574v1 Announce Type: new -Abstract: An $L(3,2,1)$-labeling of a graph $G$ is an assignment $f$ of nonnegative integers to vertices such that $\vert f(x)-f(y)\vert > 3-\mbox{dist}_G(x,y)$ for every pair $x,y$ of vertices of $G$, where $\mbox{dist}_G(x,y)$ denotes the distance between $x$ and $y$ in $G$. The minimum span (i.e., the difference between the largest and the smallest value) among all $L(3,2,1)$-labelings of $G$ is denoted by $\lambda_{(3,2,1)}(G)$. In this paper, we study $L(3,2,1)$-labelings of three classes of circulant graphs. Namely, we investigate $\lambda_{(3,2,1)}$ of $C_n(\{1,s_2,n-s_2,n-1\})$, where $s_2\in\{3,4,5\}$. This paper is a continuation of a recent publication of T. Calamoneri who studied the square of cycles, i.e., circulants $C_n(\{1,2,n-2,n-1\})$. - oai:arXiv.org:2601.12574v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + Pointwise Ergodic Averages Along the Omega Function in Number Fields + https://arxiv.org/abs/2601.16136 + arXiv:2601.16136v1 Announce Type: new +Abstract: We show the failure of the pointwise convergence of averages along the Omega function in a number field. As a consequence, we show, for instance, that the averages \[ \frac{1}{N^2}\sum_{1\leq m,n \leq N} f(T^{\Omega(m^2+n^2)}x)\] do not converge pointwise in ergodic systems, addressing a question posed by Le, Moreira, Sun, and the second author. On the other hand, using number-theoretic methods, we establish the pointwise convergence of averages along the $\Omega$ function defined on the ideals of a number field in uniquely ergodic systems. Using this dynamical framework, we also derive several natural number-theoretic consequences of independent interest. + oai:arXiv.org:2601.16136v1 + math.DS + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - P\v{r}emysl Holub, Martin Kop\v{r}iva + http://creativecommons.org/licenses/by/4.0/ + Diego C\'espedes, Sebasti\'an Donoso - The Origin of the Inaccessible Game - https://arxiv.org/abs/2601.12576 - arXiv:2601.12576v1 Announce Type: new -Abstract: The inaccessible game is an information-geometric framework where dynamics of information loss emerge from maximum entropy production under marginal-entropy conservation. - We study the game's starting state, the origin. Classical Shannon entropy forbids a representation with zero joint entropy and positive marginal entropies: non-negativity of conditional entropy rules this out. Replacing Shannon with von Neumann entropy within the Baez Fritz Leinster Parzygnat categorical framework removes this obstruction and admits a well-defined origin: a globally pure state with maximally mixed marginals, selected up to local-unitary equivalence. At this LME origin, marginal-entropy conservation becomes a second-order geometric condition. Because the marginal-entropy sum is saturated termwise, the constraint gradient vanishes and first-order tangency is vacuous; admissible directions are selected by the kernel of the constraint Hessian, characterised by the marginal-preserving tangent space. - We derive the constrained gradient flow in the matrix exponential family and show that, as the origin is approached, the affine time parameter degenerates. This motivates an axiomatically distinguished reparametrisation, entropy time $t$, defined by $dH/dt = c$ for fixed constant $c>0$. In this parametrisation, the infinite affine-time approach to the boundary maps to a finite entropy-time interval. The constrained dynamics split into a symmetric dissipative component realising SEA and a reversible component represented as unitary evolution. - As in the classical game, marginal-entropy conservation is equivalent to conservation of a sum of local modular Hamiltonian expectations, a state-dependent "modular energy"; in Gibbs regimes where local modular generators become approximately parameter-invariant, this reduces to familiar fixed-energy constraints from nonequilibrium thermodynamics. - oai:arXiv.org:2601.12576v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + On the rationality of the Weil Representation and the local theta correspondence + https://arxiv.org/abs/2601.16141 + arXiv:2601.16141v1 Announce Type: new +Abstract: We prove that the Weil representation over a non-archimedean local field can be realised with coefficients in a number field. We give an explicit descent argument to describe precisely which number field the Weil representation descends to. Our methods also apply over more general coefficient fields, such as $\ell$-modular coefficient fields, as well as coefficient rings such as rings of integers i.e. in families. We also prove that the theta correspondence over a perfect field is valid if and only if it is valid over the algebraic closure of this perfect field. These two results together show that the classical local theta correspondence is rational. + oai:arXiv.org:2601.16141v1 + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Neil D. Lawrence + Justin Trias - A semigroup approach to iterated binomial transforms - https://arxiv.org/abs/2601.12579 - arXiv:2601.12579v1 Announce Type: new -Abstract: We study a one-parameter family of binomial-convolution operators acting on sequences. These operators form an additive semigroup with an explicit inverse, and they subsume iterated classical binomial transforms as a special case. We describe the action in terms of ordinary and exponential generating functions, interpret the transform in the Riordan-array framework, and prove a general root-shift principle for constant-coefficient linear recurrences: applying the transform shifts the characteristic roots by a fixed amount. Several classical families (Fibonacci, Lucas, Pell, Jacobsthal, Mersenne) are treated uniformly as illustrative examples. - oai:arXiv.org:2601.12579v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + High-Degree Polynomial Approximations for Solving Linear Integral, Integro-Differential, and Ordinary Differential Equations + https://arxiv.org/abs/2601.16143 + arXiv:2601.16143v1 Announce Type: new +Abstract: This paper presents a universal numerical scheme tailored for tackling linear integral, integro-differential, and both initial and boundary value problems of ordinary differential equations. The numerical scheme is readily adapted for resolving ill-posed problems. Central to our approach is high-degree piecewise-polynomial approximation to the exact solution. We illustrate the accuracy and stability of our numerical solutions in the presence of noise through illustrative examples. Additionally, we demonstrate that proposed regularization being applied to high-degree interpolation, effectively eliminates Runge's phenomenon. + oai:arXiv.org:2601.16143v1 + math.GM + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Johann Verwee + http://creativecommons.org/licenses/by/4.0/ + 10.3934/cac.2024009 + Communication on Analysis and Computation 2024, Volume 2, Issue 2: 180 - 198 + Vladimir Kryzhniy - Integrals of products of four modified Bessel functions - https://arxiv.org/abs/2601.12590 - arXiv:2601.12590v1 Announce Type: new -Abstract: We evaluate definite integrals involving the product of four modified Bessel functions of the first and second kind and a power function. We provide general formulas expressed in terms of the Meijer $G$-function and generalized hypergeometric and Lauricella $F_C$ functions, and study a number of special cases in which the integrals can be evaluated in terms of simpler special functions or indeed take an elementary form. As a consequence, we deduce some new formulas for definite integrals of products of four Airy functions. - oai:arXiv.org:2601.12590v1 - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 + On the Ginzburg-Landau approximation for quasilinear pattern forming reaction-diffusion-advection systems + https://arxiv.org/abs/2601.16145 + arXiv:2601.16145v1 Announce Type: new +Abstract: We prove that the Ginzburg-Landau equation correctly predicts the dynamics of quasilinear pattern-forming reaction-diffusion-advection systems, close to the first instability. We present a simple theorem which is easily applicable for such systems and relies on key maximal regularity results. The theorem is applied to the Gray-Scott-Klausmeier vegetation-water interaction model and its application to general reaction-diffusion-advection systems is discussed. + oai:arXiv.org:2601.16145v1 + math.AP + math.DS + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Robert E. Gaunt + http://creativecommons.org/licenses/by/4.0/ + Th\'eo Belin (Institute f\"ur Analysis und Modellierung, University of Stuttgart), Guido Schneider (Institute f\"ur Analysis und Modellierung, University of Stuttgart) - Ehrhart quasi-polynomials via Barnes polynomials and discrete moments of parallelepipeds - https://arxiv.org/abs/2601.12596 - arXiv:2601.12596v1 Announce Type: new -Abstract: We give novel and explicit formulas for the Ehrhart quasi-polynomials of rational simple polytopes, in terms of Barnes polynomials and discrete moments of half-open parallelepipeds. These formulas also hold for all positive dilations of a rational polytope. There is an interesting appearance of an extra complex z-parameter, which seems to allow for more compact formulations. We also give similar formulas for discrete moments of rational polytopes, and their positive dilates, objects known in the literature as sums of polynomials over a polytope. The appearance of the Barnes polynomials and the Barnes numbers allow for explicit computations. From this work, it is clear that the complexity of computing Ehrhart quasi-polynomials lies mainly in the computation of various discrete moments of parallelepipeds. These discrete moments are in general summed over a particular lattice flow on a closed torus, defined in this paper. Some of the consequences involve novel vanishing identities for rational polytopes, novel formulations of Ehrhart polynomials of unimodular polytopes, and a differential equation that extends the work of Eva Linke. - oai:arXiv.org:2601.12596v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + On the structural properties of Lie algebras via associated labeled directed graphs + https://arxiv.org/abs/2601.16161 + arXiv:2601.16161v1 Announce Type: new +Abstract: We present a method for associating labeled directed graphs to finite-dimensional Lie algebras, thereby enabling rapid identification of key structural algebraic features. To formalize this approach, we introduce the concept of graph-admissible Lie algebras and analyze properties of valid graphs given the antisymmetry property of the Lie bracket as well as the Jacobi identity. Based on these foundations, we develop graph-theoretic criteria for solvability, nilpotency, presence of ideals, simplicity, semisimplicity, and reductiveness of an algebra. Practical algorithms are provided for constructing such graphs and those associated with the lower central series and derived series via an iterative pruning procedure. This visual framework allows for an intuitive understanding of Lie algebraic structures that goes beyond purely visual advantages, since it enables a simpler and swifter grasping of the algebras of interest beyond computational-heavy approaches. Examples, which include the Schr\"odinger and Lorentz algebra, illustrate the applicability of these tools to physically relevant cases. We further explore applications in physics, where the method facilitates computation of similtude relations essential for determining quantum mechanical time evolution via the Lie algebraic factorization method. Extensions to graded Lie algebras and related conjectures are discussed. Our approach bridges algebraic and combinatorial perspectives, offering both theoretical insights and computational tools into this area of mathematical physics. + oai:arXiv.org:2601.16161v1 + math-ph + math.MP + quant-ph + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Sinai Robins + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Tim Heib, David Edward Bruschi - Conjugating full cycles by adjacent transpositions: diameter and sorting time - https://arxiv.org/abs/2601.12597 - arXiv:2601.12597v1 Announce Type: new -Abstract: We establish upper and lower bounds on the maximal number of steps needed to transform a cyclic permutation to the canonical cyclic permutation using conjugation by adjacent transpositions, and on the diameter of the underlying Schreier graph. - oai:arXiv.org:2601.12597v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + Maximal toroids and Cartan subgroups of algebraic groups + https://arxiv.org/abs/2601.16162 + arXiv:2601.16162v1 Announce Type: new +Abstract: We introduce a unified theory of Cartan subgroups and maximal toroids - defined as connected multiplicative type subgroups that are maximal amongst all such subgroups - which holds for all affine algebraic groups over a field, regardless of smoothness. For instance we show that maximal toroids always exist, that they are invariant under base change, and that they are in natural 1-1 correspondence with Cartan subgroups. Our results generalise known results for Cartan subgroups and maximal tori of smooth affine algebraic groups, as well as their analogues for restricted Lie algebras. We conclude with some applications to, and a brief discussion of, some generation problems for algebraic groups. + oai:arXiv.org:2601.16162v1 + math.GR + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Ron M. Adin, Eli Bagno, Yuval Roichman + Damian Sercombe - Elementary proofs of ring commutativity theorems - https://arxiv.org/abs/2601.12599 - arXiv:2601.12599v1 Announce Type: new -Abstract: Jacobson's commutativity theorem says that a ring is commutative if, for each $x$, $x^n = x$ for some $n > 1$. Herstein's generalization says that the condition can be weakened to $x^n-x$ being central. In both theorems, $n$ may depend on $x$. In this paper, in certain cases where $n$ is a fixed constant, we find equational proofs of each theorem. For the odd exponent cases $n = 2k+1$ of Jacobson's theorem, our main tool is a lemma stating that for each $x$, $x^k$ is central. For Herstein's theorem, we consider the cases $n=4$ and $n=8$, obtaining proofs with the assistance of the automated theorem prover Prover9. - oai:arXiv.org:2601.12599v1 - math.RA - Wed, 21 Jan 2026 00:00:00 -0500 + Tensor Reed-Muller Codes: Achieving Capacity with Quasilinear Decoding Time + https://arxiv.org/abs/2601.16164 + arXiv:2601.16164v1 Announce Type: new +Abstract: Define the codewords of the Tensor Reed-Muller code $\mathsf{TRM}(r_1,m_1;r_2,m_2;\dots;r_t,m_t)$ to be the evaluation vectors of all multivariate polynomials in the variables $\left\{x_{ij}\right\}_{i=1,\dots,t}^{j=1,\dots m_i}$ with degree at most $r_i$ in the variables $x_{i1},x_{i2},\dots,x_{im_i}$. The generator matrix of $\mathsf{TRM}(r_1,m_1;\dots;r_t,m_t)$ is thus the tensor product of the generator matrices of the Reed-Muller codes $\mathsf{RM}(r_1,m_1),\dots, \mathsf{RM}(r_t,m_t)$. + We show that for any constant rate $R$ below capacity, one can construct a Tensor Reed-Muller code $\mathsf{TRM}(r_1,m_1;\dotsc;r_t,m_t)$ of rate $R$ that is decodable in quasilinear time. For any blocklength $n$, we provide two constructions of such codes: + 1) Our first construction (with $t=3$) has error probability $n^{-\omega(\log n)}$ and decoding time $O(n\log\log n)$. + 2) Our second construction, for any $t\geq 4$, has error probability $2^{-n^{\frac{1}{2}-\frac{1}{2(t-2)}-o(1)}}$ and decoding time $O(n\log n)$. + One of our main tools is a polynomial-time algorithm for decoding an arbitrary tensor code $C=C_1\otimes\dotsc\otimes C_t$ from $\frac{d_{\min}(C)}{2\max\{d_{\min}(C_1),\dotsc,d_{\min}(C_t) \}}-1$ adversarial errors. Crucially, this algorithm does not require the codes $C_1,\dotsc,C_t$ to themselves be decodable in polynomial time. + oai:arXiv.org:2601.16164v1 + cs.IT + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Michael Kinyon, Desmond MacHale + Emmanuel Abbe, Colin Sandon, Oscar Sprumont - Improved Averaged Distribution of $d_3(n)$ in Prime Arithmetic Progressions - https://arxiv.org/abs/2601.12601 - arXiv:2601.12601v1 Announce Type: new -Abstract: We say that $d_3(n)$ has exponent of distribution $\theta$ if, for all $\varepsilon>0$, the expected asymptotic holds uniformly for all moduli $q \le x^{\theta-\varepsilon}$. Nguyen proved that, after averaging over reduced residue classes $a \bmod q$, the function $d_3(n)$ has exponent of distribution $2/3$, following earlier work of Banks et al. Using the Petrow--Young subconvexity bound for Dirichlet $L$-functions, we improve this to an exponent of distribution $8/11$ when averaging over residue classes modulo a prime $q$. - oai:arXiv.org:2601.12601v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 + On the dimension drop for harmonic measure on uniformly non-flat Ahlfors-David regular boundaries + https://arxiv.org/abs/2601.16167 + arXiv:2601.16167v1 Announce Type: new +Abstract: We extend earlier results of Azzam on the dimension drop of the harmonic measure for a domain $\Omega\subset \R^{n}$ with $n\geq 3$, with dimensional Ahlfors regular boundary $\partial\Omega$ of dimension $s$ with $n-1-\delta_0 \leq s\leq n-1$, that is uniformly non flat. Here $\delta_0$ is a small positive constant dependent on the parameters of the problem. Our novel construction relies on elementary geometric and potential theoretic considerations. We avoid the use of Riesz transforms and compactness arguments, and also give quantitative bounds on the $\delta_0$ parameter. + oai:arXiv.org:2601.16167v1 + math.AP + math.CA + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Metin Can Aydemir, Muhammet Boran + Aritro Pathak - An unbounded number of canard limit cycles in linear regularizations of piecewise linear systems - https://arxiv.org/abs/2601.12602 - arXiv:2601.12602v1 Announce Type: new -Abstract: The purpose of this paper is to study the number of limit cycles of canard type in linear regularizations of piecewise linear systems with non-monotonic transition functions. Using the notion of slow divergence integral and elementary breaking mechanisms, we construct systems with an arbitrary finite number of hyperbolic limit cycles. The Hopf breaking mechanism deals with transition functions with precisely one critical point in the interval $(-1,1)$. On the other hand, the jump breaking mechanism produces any number of limit cycles using transition functions with precisely three critical points in $(-1,1)$. - oai:arXiv.org:2601.12602v1 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 + Non-Linearly Separable Distributed Computing: A Sparse Tensor Factorization Approach + https://arxiv.org/abs/2601.16171 + arXiv:2601.16171v1 Announce Type: new +Abstract: The work considers the $N$-server distributed computing setting with $K$ users requesting functions that are arbitrary multi-variable polynomial evaluations of $L$ real (potentially non-linear) basis subfunctions. Our aim is to seek efficient task-allocation and data-communication techniques that reduce computation and communication costs. Towards this, we take a tensor-theoretic approach, in which we represent the requested non-linearly decomposable functions using a properly designed tensor $\bar{\mathcal{F}}$, whose sparse decomposition into a tensor $\bar{\mathcal{E}}$ and matrix $\mathbf{D}$ directly defines the task assignment, connectivity, and communication patterns. We here design an achievable scheme, employing novel fixed-support SVD-based tensor factorization methods and careful multi-dimensional tiling of subtensors, yielding computation and communication protocols whose costs are derived here, and which are shown to perform substantially better than the state of art. + oai:arXiv.org:2601.16171v1 + cs.IT + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Renato Huzak, Otavio Henrique Perez + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Ali Khalesi, Ahmad Tanha, Derya Malak, Petros Elia - On the second homology of the genus 3 hyperelliptic Torelli group - https://arxiv.org/abs/2601.12605 - arXiv:2601.12605v1 Announce Type: new -Abstract: Let $s$ be a fixed hyperelliptic involution of the closed, oriented genus $g$ surface $\Sigma_g$. The hyperelliptic Torelli group $\mathcal{SI}_g$ is the subgroup of the mapping class group $\mathrm{Mod}(\Sigma_g)$ consisting of elements that act trivially on $\mathrm{H}_1(\Sigma_g;\mathbb{Z})$ and commute with $s$. It is generated by Dehn twists about $s$-invariant separating curves, and its cohomological dimension is $g-1$. In this paper we study the top homology group $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$. For each pair of disjoint $s$-invariant separating curves there is a naturally associated abelian cycle in $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$; we call such cycles \emph{simple}. We show that simple abelian cycles are in bijection with orthogonal (with respect to the intersection form) splittings of $\mathrm{H}_1(\Sigma_3;\mathbb{Z})$ satisfying a simple algebraic condition, and prove that these abelian cycles are linearly independent in $\mathrm{H}_2(\mathcal{SI}_3;\mathbb{Z})$. - oai:arXiv.org:2601.12605v1 - math.GT + Fixed-point proportion of geometric iterated Galois groups + https://arxiv.org/abs/2601.16173 + arXiv:2601.16173v1 Announce Type: new +Abstract: In 1980, Odoni initiated the study of the fixed-point proportion of iterated Galois groups of polynomials motivated by prime density problems in arithmetic dynamics. + The main goal of the present paper is to completely settle the longstanding open problem of computing the fixed-point proportion of geometric iterated Galois groups of polynomials. Indeed, we confirm the well-known conjecture that Chebyshev polynomials are the only complex polynomials whose geometric iterated Galois groups have positive fixed-point proportion. Our proof relies on methods from group theory, ergodic theory, martingale theory and complex dynamics. This result has direct applications to the proportion of periodic points of polynomials over finite fields. + The general framework developed in this paper applies more generally to rational functions over arbitrary fields and generalizes, via a unified approach, previous partial results, which have all been proved with very different methods. + oai:arXiv.org:2601.16173v1 + math.NT + math.DS math.GR - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Igor Spiridonov + Jorge Fari\~na-Asategui, Santiago Radi - On the second Bohr radius for vector valued pluriharmonic functions - https://arxiv.org/abs/2601.12608 - arXiv:2601.12608v1 Announce Type: new -Abstract: In this paper, we introduce the notion of the second Bohr radius for vector valued pluriharmonic functions on complete Reinhardt domains in $\mathbb{C}^n$. This investigation is motivated by the work of Lev Aizenberg [Proc. Amer. Math. Soc. 128 (2000), 1147-1155], where the corresponding problem was studied for complex valued holomorphic functions. We show that the second Bohr radius constant for pluriharmonic functions is strictly positive under suitable condition. In addition, we obtain its asymptotic behavior in the finite-dimensional settings using invariants from local Banach space theory. Asymptotic estimates for this constant are obtained on both convex and non-convex complete Reinhardt domains. Our results also apply to a broad class of Banach sequence spaces, including symmetric and convex Banach spaces. The framework developed here also includes the second Bohr radius problem for vector valued holomorphic functions. As an application of our results, we derive several consequences that extend known results in the scalar valued setting as well as existing results in the literature. - oai:arXiv.org:2601.12608v1 - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 + Metric-uniform spectral inequality for the Laplacian on manifolds with bounded sectional curvature + https://arxiv.org/abs/2601.16176 + arXiv:2601.16176v1 Announce Type: new +Abstract: Given a Riemannian manifold $M$ endowed with a smooth metric $g$ satisfying upper and lower sectional curvature bounds, we show an equivalence property between the $\mathrm{L}^2$ norm on $M$ and the $\mathrm{L}^2$ norm on subsets $\omega$ satisfying a thickness condition, for functions in the range of a spectral projector. The thickness condition is known to be optimal in this setting. The constant appearing in the equivalence of norms property depends only on the dimension of the manifold, curvature bounds, and frequency threshold of the spectral cutoff, but, crucially, not on the injectivity radius. + oai:arXiv.org:2601.16176v1 + math.AP + math.DG + math.OC + math.SP + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Himadri Halder + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Alix Deleporte, Jean Lagac\'e, Marc Rouveyrol - A class of non-cylindrical domains for parabolic equations - https://arxiv.org/abs/2601.12609 - arXiv:2601.12609v1 Announce Type: new -Abstract: We present a class of non-cylindrical domains where Dirichlet-type problems for parabolic equations, such as the heat equation, can be posed and solved. The regularity for the boundary of this class of domains is a mixed Lipschitz condition, as described in the bulk of the paper. The main tool is an adequate version of the implicit function theorem for functions with this kind of regularity. It is proved that the class introduced herein is of the same type as domains previously considered by several authors. - oai:arXiv.org:2601.12609v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Mild Solutions for Path-Dependent Parabolic PDEs with Neumann Boundary Conditions via Generalized BSDEs + https://arxiv.org/abs/2601.16178 + arXiv:2601.16178v1 Announce Type: new +Abstract: We study a system of Forward-Backward Stochastic Differential Equations (FBSDEs) with time-delayed generators. The forward process includes a reflection component expressed via a Stieltjes integral, while the backward process takes the form of a Generalized BSDE. We establish the connection between this FBSDE system and non-linear path-dependent PDEs with Neumann boundary conditions by deriving a representation formula. + oai:arXiv.org:2601.16178v1 + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Lect. Mat. 38 (2017), no. 2, 49-63 - Alberto Dom\'inguez Corella, Jorge Rivera-Noriega + Luca Di Persio, Matteo Garbelli, Adrian Zalinescu - An Eventown Result for Permutations - https://arxiv.org/abs/2601.12613 - arXiv:2601.12613v1 Announce Type: new -Abstract: A family of permutations $\mathcal{F} \subseteq S_n$ is even-cycle-intersecting if $\sigma \pi^{-1}$ has an even cycle for all $\sigma,\pi \in \mathcal{F}$. We show that if $\mathcal{F} \subseteq S_n$ is an even-cycle-intersecting family of permutations, then $|\mathcal{F}| \leq 2^{n-1}$, and that equality holds when $n$ is a power of 2 and $\mathcal{F}$ is a double-translate of a Sylow 2-subgroup of $S_n$. This result can be seen as an analogue of the classical eventown problem for subsets and it confirms a conjecture of J\'anos K\"orner on maximum reversing families of the symmetric group. Along the way, we show that the canonically intersecting families of $S_n$ are also the extremal odd-cycle-intersecting families of $S_n$ for all even $n$. While the latter result has less combinatorial significance, its proof uses an interesting new character-theoretic identity that might be of independent interest in algebraic combinatorics. - oai:arXiv.org:2601.12613v1 - math.CO - cs.DM - math.GR - Wed, 21 Jan 2026 00:00:00 -0500 + Gaussian maps on trigonal curves + https://arxiv.org/abs/2601.16183 + arXiv:2601.16183v1 Announce Type: new +Abstract: In this paper we study higher even Gaussian maps of the canonical bundle for cyclic trigonal curves. More precisely, we study suitable restrictions of these maps determining a lower bound for the rank, and more generally, a lower bound for the rank for the general trigonal curve. We also manage to give the explicit description of the kernel of the second Gaussian map. Finally, we use these results to show the non existence of "extra" asymptotic directions for cyclic trigonal curves in some spaces generated by higher Schiffer variations. + oai:arXiv.org:2601.16183v1 + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Nathan Lindzey + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Antonio Lacopo - Density of growth rates of subgroups of a free group -- an alternative proof - https://arxiv.org/abs/2601.12620 - arXiv:2601.12620v1 Announce Type: new -Abstract: We give an alternative proof to the theorem recently proved by Louvaris, Wise and Yehuda, that the growth rates of finitely generated subgroups of $F_r$ are dense in $[1,2r-1]$. - oai:arXiv.org:2601.12620v1 - math.GR - Wed, 21 Jan 2026 00:00:00 -0500 + The Pohozaev identity for the Spectral Fractional Laplacian + https://arxiv.org/abs/2601.16185 + arXiv:2601.16185v1 Announce Type: new +Abstract: In this paper, we prove a Pohozaev identity for the Spectral Fractional Laplacian (SFL). This identity allows us to establish non-existence results for the semilinear Dirichlet problem $(-\Delta|_{\Omega})^su = f(u)$ in star-shaped domains. The first such identity for non-local operators was established by Ros-Oton and Serra in 2014 for the Restricted Fractional Laplacian (RFL). However, the SFL differs fundamentally from the RFL, and the integration by parts strategy of Ros-Oton and Serra cannot be applied. Instead, we develop a novel spectral approach that exploits the underlying quadratic structure. Our main result expresses the identity as a Schur product of the classical Pohozaev quadratic form and a transition matrix that depends on the eigenvalues of the Laplacian and the fractional exponent. + oai:arXiv.org:2601.16185v1 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - \'Ad\'am Tim\'ar + Itahisa Barrios-Cubas, Matteo Bonforte, Mar\'ia del Mar Gonz\'alez, Clara Torres-Latorre - New Trends in the Stability of Sinkhorn Semigroups - https://arxiv.org/abs/2601.12633 - arXiv:2601.12633v1 Announce Type: new -Abstract: Entropic optimal transport problems play an increasingly important role in machine learning and generative modelling. In contrast with optimal transport maps which often have limited applicability in high dimensions, Schrodinger bridges can be solved using the celebrated Sinkhorn's algorithm, a.k.a. the iterative proportional fitting procedure. The stability properties of Sinkhorn bridges when the number of iterations tends to infinity is a very active research area in applied probability and machine learning. Traditional proofs of convergence are mainly based on nonlinear versions of Perron-Frobenius theory and related Hilbert projective metric techniques, gradient descent, Bregman divergence techniques and Hamilton-Jacobi-Bellman equations, including propagation of convexity profiles based on coupling diffusions by reflection methods. The objective of this review article is to present, in a self-contained manner, recently developed Sinkhorn/Gibbs-type semigroup analysis based upon contraction coefficients and Lyapunov-type operator-theoretic techniques. These powerful, off-the-shelf semigroup methods are based upon transportation cost inequalities (e.g. log-Sobolev, Talagrand quadratic inequality, curvature estimates), $\phi$-divergences, Kantorovich-type criteria and Dobrushin contraction-type coefficients on weighted Banach spaces as well as Wasserstein distances. This novel semigroup analysis allows one to unify and simplify many arguments in the stability of Sinkhorn algorithm. It also yields new contraction estimates w.r.t. generalized $\phi$-entropies, as well as weighted total variation norms, Kantorovich criteria and Wasserstein distances. - oai:arXiv.org:2601.12633v1 - math.PR - cs.NA - math.NA - stat.ML - Wed, 21 Jan 2026 00:00:00 -0500 + Inversion problem in algebras of integrable functions with summable Fourier transforms + https://arxiv.org/abs/2601.16186 + arXiv:2601.16186v1 Announce Type: new +Abstract: In this paper, we study the norm-controlled inversion problem in two classes of algebras of integrable functions. In contrast of the classical case of $L^{1}(G)$, we prove that this problem has a positive solution in our setting without any additional restrictions. + oai:arXiv.org:2601.16186v1 + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pierre Del Moral, Ajay Jasra + Przemys{\l}aw Ohrysko - Beyond Identification: Computing Boolean Functions via Channels - https://arxiv.org/abs/2601.12640 - arXiv:2601.12640v1 Announce Type: new -Abstract: Consider a point-to-point communication system in which the transmitter holds a binary message of length $m$ and transmits a corresponding codeword of length $n$. The receiver's goal is to recover a Boolean function of that message, where the function is unknown to the transmitter, but chosen from a known class $F$. We are interested in the asymptotic relationship of $m$ and $n$: given $n$, how large can $m$ be (asymptotically), such that the value of the Boolean function can be recovered reliably? This problem generalizes the identification-via-channels framework introduced by Ahlswede and Dueck. We formulate the notion of computation capacity, and derive achievability and converse results for selected classes of functions $F$, characterized by the Hamming weight of functions. Our obtained results are tight in the sense of the scaling behavior for all cases of $F$ considered in the paper. - oai:arXiv.org:2601.12640v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + Ergodic averages for commutative transformations along return times + https://arxiv.org/abs/2601.16188 + arXiv:2601.16188v1 Announce Type: new +Abstract: In this paper, we extend recent results on the convergence of ergodic averages along sequences generated by return times to shrinking targets in rapidly mixing systems, partially answering questions posed by the first author, Maass and the third author in [6]. In particular, for a fixed parameter $a\in (0,1)$ and for generic $y\in [0,1]$, we establish both $L^2$ and pointwise convergence for single averages and multiple averages for commuting transformations along the sequences $(a_n(y))_{n\in \mathbb{N}}$, obtained by arranging the set $$\Big\{n\in\mathbb{N}: 0<2^ny \mod{1}<n^{-a} \Big\}$$ in an increasing order. We also obtain new results for semi-random ergodic averages along sequences of similar type. + oai:arXiv.org:2601.16188v1 + math.DS + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 new http://creativecommons.org/licenses/by/4.0/ - Jingge Zhu, Matthias Frey + Sebasti\'an Donoso, Sovanlal Mondal, Vicente Saavedra-Araya - Torsion points of small order on cyclic covers of $\mathbb{P}^1$. III - https://arxiv.org/abs/2601.12643 - arXiv:2601.12643v1 Announce Type: new -Abstract: Let $d>1$ be an integer and $K_0$ a perfect field such that $char(K_0)$ does not divide $d$. Let $n>d$ be an integer that is prime to $d$. Let $f(x)\in K_0[x]$ be a degree $n$ monic polynomial without repeated roots, and $\mathcal{C}_{f,d}$ a smooth projective model of the affine curve $y^d=f(x)$. Let $J(\mathcal{C}_{f,d})$ be the Jacobian of the $K_0$-curve $\mathcal{C}_{f,d} $. As usual, we identify $\mathcal{C}_{f,d}$ with its canonical image in $J(\mathcal{C}_{f,d})$ (such that the only ``infinite point'' of $\mathcal{C}_{f,d}$ goes to the zero of the group law on $J(\mathcal{C}_{f,d})$). - We say that an integer $m>1$ is $(n,d)$-reachable over $K_0$ if there exists a polynomial $f(x)$ as above such that $\mathcal{C}_{f,d}(K_0)$ contains a torsion point of order $m$. - Let us put $\ell_0:=[(n+d)/d], \ m_0:=\ell_0 d$. Earlier we proved that if $m$ is $(n,d)$-reachable, then either $m=d$ or $m = n$ or $m \ge m_0$ (in addition, both $d$ and $n$ are $(n,d)$-reachable over every $K_0$). We also proved that if $m_0$ is $(n,d)$-reachable over some $K_0$ then $n-m_0+\ell_0\ge 0$. - In the present paper we discuss the $(n,d)$-reachability of $m_0$ when $n-m_0+\ell_0=0$ or $1$. - oai:arXiv.org:2601.12643v1 + Density-based structural frameworks for prime numbers, prime gaps, and Euler products + https://arxiv.org/abs/2601.16193 + arXiv:2601.16193v1 Announce Type: new +Abstract: We develop a unified density-based framework for primality, coprimality, and prime pairs, and introduce an intrinsic normalized model for prime gaps constrained by the Prime Number Theorem. Within this setting, a structural tension between Hardy-Littlewood, Cramer, and PNT predictions emerges, leading to quantitative estimates on the rarity of extreme gaps. Additive representations of even integers are reformulated as local density problems, yielding non-conjectural upper and lower bounds compatible with Hardy-Littlewood heuristics. Finally, the Riemann zeta function is analyzed via truncated Euler products, whose stability and oscillatory structure provide a coherent interpretation of the critical line and prime-based numerical criteria for the localization of non-trivial zeros. + oai:arXiv.org:2601.16193v1 math.NT - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 + math.CV + Fri, 23 Jan 2026 00:00:00 -0500 new http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Boris M. Bekker, Yuri G. Zarhin + Gregorio Vettori - On Some Properties of Matrices with Entries Defined by Products of $k$-Fibonacci and $k$-Lucas Numbers - https://arxiv.org/abs/2601.12644 - arXiv:2601.12644v1 Announce Type: new -Abstract: In this paper, we employ combinatorial and algebraic tools to derive closed-form expressions for several classical matrix invariants, including the determinant, inverse, trace, and powers, for a family of matrices whose entries are given by products of $k$-Fibonacci and $k$-Lucas numbers. Moreover, we compute the spectral radius and the energy of the graphs associated with this family of matrices. Finally, we investigate connections between the obtained formulas and certain integer sequences listed in the On-Line Encyclopedia of Integer Sequences (OEIS). - oai:arXiv.org:2601.12644v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 + A Rolling-Space Branch-and-Price Algorithm for the Multi-Compartment Vehicle Routing Problem with Multiple Time Windows + https://arxiv.org/abs/2601.16194 + arXiv:2601.16194v1 Announce Type: new +Abstract: This paper investigates the multi-compartment vehicle routing problem with multiple time windows (MCVRPMTW), an extension of the classical vehicle routing problem with time windows that considers vehicles equipped with multiple compartments and customers requiring service across several delivery time windows. The problem incorporates three key compartment-related features: (i) compartment flexibility in the number of compartments, (ii) item-to-compartment compatibility, and (iii) item-to-item compatibility. The problem also accommodates practical operational requirements such as driver breaks. To solve the MCVRPMTW, we develop an exact branch-and-price (B&P) algorithm in which the pricing problem is solved using a labeling algorithm. Several acceleration strategies are introduced to limit symmetry during label extensions, improve the stability of dual solutions in column generation, and enhance the branching process. To handle large-scale instances, we propose a rolling-space B&P algorithm that integrates clustering techniques into the solution framework. Extensive computational experiments on instances inspired by a real-world industrial application demonstrate the effectiveness of the proposed approach and provide useful managerial insights for practical implementation. + oai:arXiv.org:2601.16194v1 + math.OC + cs.LG + Fri, 23 Jan 2026 00:00:00 -0500 new - http://creativecommons.org/licenses/by/4.0/ - Pedro Fernando Fern\'andez Espinosa, Maritza Liliana Arciniegas Torres, Camilo Andr\'es Acevedo Cadena + http://creativecommons.org/licenses/by-nc-sa/4.0/ + El Mehdi Er Raqabi, Kevin Dalmeijer, Pascal Van Hentenryck - A Landau-de Gennes Type Theory for Cholesteric-Helical Smectic-Smectic C* Liquid Crystal Phase Transitions - https://arxiv.org/abs/2601.12653 - arXiv:2601.12653v1 Announce Type: new -Abstract: We present a rigorous mathematical analysis of a modified Landau-de Gennes (LdG) theory modeling temperature-driven phase transitions between cholesteric, helical smectic, and smectic C* phases. This model couples a tensor-valued order parameter (nematic orientational order) with a real-valued order parameter (smectic layer modulation). We establish the existence of energy minimizers of the modified LdG energy in three dimensions, subject to Dirichlet conditions, and rigorously analyze the energy minimizers in two asymptotic limits. First, in the Oseen--Frank limit, we show that the global minimizer strongly converges to a minimizer of the Landau-de Gennes bulk energy. Second, in the limit of dominant elastic constants, we prove that the global minimizers converge to a classical helical director profile. Finally, through stability analysis and bifurcation theory, we derive the complete sequence of symmetry-breaking transitions with decreasing temperature-from the cholesteric phase (with in-plane twist and no layering) to an intermediate helical smectic phase (with in-plane twist and layering), and ultimately to the smectic C* phase (with out-of-plane twist and layering). These theoretical results are supported by numerical simulations. - oai:arXiv.org:2601.12653v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 + Generalized Bassian Modules over Non-primitive Dedekind Prime Rings + https://arxiv.org/abs/2601.16201 + arXiv:2601.16201v1 Announce Type: new +Abstract: A right $A$-module $M$ is said to be generalized bassian if the existence of an injective homomorphism $M\to M/N$ for some submodule $N$ of $M$ implies that $N$ is a direct summand of $M$. We describe singular generalized bassian modules over non-primitive Dedekind prime rings.\\ The study is supported by grant of Russian Science Foundation. + oai:arXiv.org:2601.16201v1 + math.RA + Fri, 23 Jan 2026 00:00:00 -0500 new + http://creativecommons.org/licenses/by/4.0/ + Askar Tuganbaev + + + Real-Time HAP-Assisted Vehicular Edge Computing for Rural Areas + https://arxiv.org/abs/2301.09957 + arXiv:2301.09957v1 Announce Type: cross +Abstract: Non-Terrestrial Networks (NTNs) are expected to be a key component of 6th generation (6G) networks to support broadband seamless Internet connectivity and expand the coverage even in rural and remote areas. In this context, High Altitude Platforms (HAPs) can act as edge servers to process computational tasks offloaded by energy-constrained terrestrial devices such as Internet of Things (IoT) sensors and ground vehicles (GVs). In this paper, we analyze the opportunity to support Vehicular Edge Computing (VEC) via HAP in a rural scenario where GVs can decide whether to process data onboard or offload them to a HAP. We characterize the system as a set of queues in which computational tasks arrive according to a Poisson arrival process. Then, we assess the optimal VEC offloading factor to maximize the probability of real-time service, given latency and computational capacity constraints. + oai:arXiv.org:2301.09957v1 + cs.NI + cs.IT + eess.SP + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 + cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Apala Majumdar, Baoming Shi, Dawei Wu, Jingmin Xia, Lei Zhang + 10.1109/LWC.2023.3238851 + Alessandro Traspadini, Marco Giordani, Giovanni Giambene, Michele Zorzi - Physics-informed machine learning for reconstruction of dynamical systems with invariant measure score matching - https://arxiv.org/abs/2601.12675 - arXiv:2601.12675v1 Announce Type: new -Abstract: In this paper, we develop a novel mesh-free framework, termed physics-informed neural networks with invariant measure score matching (PINN-IMSM), for reconstructing dynamical systems from unlabeled point-cloud data that capture the system's invariant measure. The invariant density satisfies the steady-state Fokker-Planck (FP) equation. We reformulate this equation in terms of its score function (the gradient of the log-density), which is estimated directly from data via denoising score matching, thereby bypassing explicit density estimation. This learned score is then embedded into a physics-informed neural network (PINN) to reconstruct the drift velocity field under the resulting score-based FP equation. The mesh-free nature of PINNs allows the framework to scale to higher dimensions, avoiding the curse of dimensionality inherent in mesh-based methods. To address the ill-posedness of high-dimensional inverse problems, we recast the problem as a PDE-constrained optimization that seeks the minimal-energy velocity field. Under suitable conditions, we prove that this problem admits a unique solution that depends continuously on the score function. The constrained formulation is solved using a stochastic augmented Lagrangian method. Numerical experiments on representative dynamical systems, including the Van der Pol oscillator, an active swimmer in an anharmonic trap, and the chaotic Lorenz-63 and Lorenz-96 systems, demonstrate that PINN-IMSM accurately recovers invariant measures and reconstructs faithful dynamical behavior for problems in up to five dimensions. - oai:arXiv.org:2601.12675v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new + Ecosystem Competition and Cross-Market Subsidization: A Dynamic Theory of Platform Pricing + https://arxiv.org/abs/2601.15303 + arXiv:2601.15303v1 Announce Type: cross +Abstract: Platform giants in China have operated with persistently compressed margins in highly concentrated markets for much of the past decade, despite market shares exceeding 60\% in core segments. Standard theory predicts otherwise: either the weaker firm exits, or survivors raise prices to monopoly levels. We argue the puzzle dissolves once firms are viewed as ecosystem optimizers rather than single-market profit maximizers. We develop a dynamic game in which a firm's willingness to subsidize depends on the spillover value its users generate in adjacent markets -- what we call \textit{ecosystem complementarity}. When this complementarity is strong enough, perpetual below-cost pricing emerges as the unique stable equilibrium. The result is not predation in the classical sense; there is no recoupment phase. It is a permanent state of subsidized competition, rational for each firm individually but potentially inefficient in aggregate. We characterize the equilibrium, establish its dynamic stability, and show that welfare losses compound over time as capital flows into subsidy wars rather than innovation. The model's predictions are consistent with observed patterns in Chinese platform markets and suggest that effective antitrust intervention should target cross-market capital flows rather than prices. + oai:arXiv.org:2601.15303v1 + econ.TH + cs.GT + cs.IT + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 + cross http://creativecommons.org/licenses/by/4.0/ - Yongsheng Chen, Suddhasattwa Das, Wei Guo, Xinghui Zhong + Liang Chen - Stable and Fr\'echet limit theorem for subgraph functionals in the hyperbolic random geometric graph - https://arxiv.org/abs/2601.12677 - arXiv:2601.12677v1 Announce Type: new -Abstract: We study the fluctuations of subgraph counts in hyperbolic random geometric graphs on the $d$-dimensional Poincar\'e ball in the heterogeneous, heavy-tailed degree regime. In a hyperbolic random geometric graph whose vertices are given by a Poisson point process on a growing hyperbolic ball, we consider two basic families of subgraphs: star shape counts and clique counts, and we analyze their global counts and maxima over the vertex set. Working in the parameter regime where a small number of vertices close to the center of the Poincar\'e ball carry very large degrees and act as hubs, we establish joint functional limit theorems for suitably normalized star shape and clique count processes together with the associated maxima processes. The limits are given by a two-dimensional dependent process whose components are a stable L\'evy process and an extremal Fr\'echet process, reflecting the fact that a small number of hubs dominates both the total number of local subgraphs and their extremes. As an application, we derive fluctuation results for the global clustering coefficient, showing that its asymptotic behavior is described by the ratio of the components of a bivariate L\'evy process with perfectly dependent stable jumps. - oai:arXiv.org:2601.12677v1 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Christian Hirsch, Takashi Owada, Ruiting Tong + Pairwise Beats All-at-Once: Behavioral Gains from Sequential Choice Presentation + https://arxiv.org/abs/2601.15332 + arXiv:2601.15332v1 Announce Type: cross +Abstract: This paper presents the Sequential Rationality Hypothesis, which argues that consumers are better able to make utility-maximizing decisions when products appear in sequential pairwise comparisons rather than in simultaneous multi-option displays. Although this involves higher cognitive costs than the all-at-once format, the current digital market, with its diverse products listed by review ratings, pricing, and paid products, often creates inconsistent choices. The present work shows that preparing the list sequentially supports more rational choice, as the consumer tries to minimize cognitive costs and may otherwise make an irrational decision. If the decision remains the same on both offers, then that is a consistent preference. The platform uses this approach by reducing cognitive costs while still providing the list in an all-at-once format rather than sequentially. To show how sequential exposure reduces cognitive overload and prevents context-dependent errors, we develop a bounded attention model and extend the monotonic attention rule of the random attention model to theorize the sequential rational hypothesis. Using a theoretical design with common consumer goods, we test these hypotheses. This theoretical model helps policymakers in digital market laws, behavioral economics, marketing, and digital platform design consider how choice architectures may improve consumer choices and encourage rational decision-making. + oai:arXiv.org:2601.15332v1 + econ.TH + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Dipankar Das - Non-parabolic Spatial Hybrid Framed Curves and Their Applications in the Spatial Hybrid Number Space - https://arxiv.org/abs/2601.12679 - arXiv:2601.12679v1 Announce Type: new -Abstract: In this paper, we define non-parabolic spatial hybrid framed curves in the spatial hybrid number space, which may have singularities, and prove the existence and uniqueness theorem for non-parabolic spatial hybrid framed curves. As appliciations, we define evolutes, involutes, pedal and contrapedal curves of non-parabolic spatial hybrid framed curves and discuss their relations. - oai:arXiv.org:2601.12679v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new + Exactly solvable topological phase transition in a quantum dimer model + https://arxiv.org/abs/2601.15377 + arXiv:2601.15377v1 Announce Type: cross +Abstract: We introduce a family of generalized Rokhsar-Kivelson (RK) Hamiltonians, which are reverse-engineered to have an arbitrary edge-weighted superposition of dimer coverings as their exact ground state at the RK point. We then focus on a quantum dimer model on the triangular lattice, with doubly-periodic edge weights. For simplicity we consider a $2\times1$ periodic model in which all weights are set to one except for a tunable horizontal edge weight labeled $\alpha$. We analytically show that the model exhibits a continuous quantum phase transition at $\alpha=3$, changing from a topological $\mathbb{Z}_2$ quantum spin liquid ($\alpha<3$) to a columnar ordered state ($\alpha>3$). The dimer-dimer correlator decays exponentially on both sides of the transition with the correlation length $\xi\propto1/|\alpha-3|$ and as a power-law at criticality. The vison correlator exhibits an exponential decay in the spin liquid phase, but becomes a constant in the ordered phase. We explain the constant vison correlator in terms of loops statistics of the double-dimer model. Using finite-size scaling of the vison correlator, we extract critical exponents consistent with the 2D Ising universality class. + oai:arXiv.org:2601.15377v1 + cond-mat.str-el + cond-mat.stat-mech + math-ph + math.MP + quant-ph + Fri, 23 Jan 2026 00:00:00 -0500 + cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kaixin Yao - - - Igusa-Todorov properties of recollements of abelian categories - https://arxiv.org/abs/2601.12702 - arXiv:2601.12702v1 Announce Type: new -Abstract: In this paper, we investigate the behavior of Igusa-Todorov properties under recollements of abelian categories. In particular, we study how the Igusa-Todorov distances of the categories involved in a recollement are related. Applications are given to Artin algebras, especially to Morita context rings. - oai:arXiv.org:2601.12702v1 - math.RT - math.CT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-sa/4.0/ - Peiru Yang, Yajun Ma, Yu-Zhe Liu + Laura Shou, Jeet Shah, Matthew Lerner-Brecher, Amol Aggarwal, Alexei Borodin, Victor Galitski - Explicit Entropic Constructions for Coverage, Facility Location, and Graph Cuts - https://arxiv.org/abs/2601.12724 - arXiv:2601.12724v1 Announce Type: new -Abstract: Shannon entropy is a polymatroidal set function and lies at the foundation of information theory, yet the class of entropic polymatroids is strictly smaller than the class of all submodular functions. In parallel, submodular and combinatorial information measures (SIMs) have recently been proposed as a principled framework for extending entropy, mutual information, and conditional mutual information to general submodular functions, and have been used extensively in data subset selection, active learning, domain adaptation, and representation learning. This raises a natural and fundamental question: are the monotone submodular functions most commonly used in practice entropic? - In this paper, we answer this question in the affirmative for a broad class of widely used polymatroid functions. We provide explicit entropic constructions for set cover and coverage functions, facility location, saturated coverage, concave-over-modular functions via truncations, and monotone graph-cut-type objectives. Our results show that these functions can be realized exactly as Shannon entropies of appropriately constructed random variables. As a consequence, for these functions, submodular mutual information coincides with classical mutual information, conditional gain specializes to conditional entropy, and submodular conditional mutual information reduces to standard conditional mutual information in the entropic sense. These results establish a direct bridge between combinatorial information measures and classical information theory for many of the most common submodular objectives used in applications. - oai:arXiv.org:2601.12724v1 + The computational two-way quantum capacity + https://arxiv.org/abs/2601.15393 + arXiv:2601.15393v1 Announce Type: cross +Abstract: Quantum channel capacities are fundamental to quantum information theory. Their definition, however, does not limit the computational resources of sender and receiver. In this work, we initiate the study of computational quantum capacities. These quantify how much information can be reliably transmitted when imposing the natural requirement that en- and decoding have to be computationally efficient. We focus on the computational two-way quantum capacity and showcase that it is closely related to the computational distillable entanglement of the Choi state of the channel. This connection allows us to show a stark computational capacity separation. Under standard cryptographic assumptions, there exists a quantum channel of polynomial complexity whose computational two-way quantum capacity vanishes while its unbounded counterpart is nearly maximal. More so, we show that there exists a sharp transition in computational quantum capacity from nearly maximal to zero when the channel complexity leaves the polynomial realm. Our results demonstrate that the natural requirement of computational efficiency can radically alter the limits of quantum communication. + oai:arXiv.org:2601.15393v1 + quant-ph + cs.CC + cs.CR cs.IT - math.CO math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Johannes Jakob Meyer, Jacopo Rizzo, Asad Raza, Lorenzo Leone, Sofiene Jerbi, Jens Eisert + + + Quadratic tensors as a unification of Clifford, Gaussian, and free-fermion physics + https://arxiv.org/abs/2601.15396 + arXiv:2601.15396v1 Announce Type: cross +Abstract: Certain families of quantum mechanical models can be described and solved efficiently on a classical computer, including qubit or qudit Clifford circuits and stabilizer codes, free-boson or free-fermion models, and certain rotor and GKP codes. We show that all of these families can be described as instances of the same algebraic structure, namely quadratic functions over abelian groups, or more generally over (super) Hopf algebras. Different kinds of degrees of freedom correspond to different "elementary" abelian groups or Hopf algebras: $\mathbb{Z}_2$ for qubits, $\mathbb{Z}_d$ for qudits, $\mathbb{R}$ for continuous variables, both $\mathbb{Z}$ and $\mathbb{R}/\mathbb{Z}$ for rotors, and a super Hopf algebra $\mathcal F$ for fermionic modes. Objects such as states, operators, superoperators, or projection-operator valued measures, etc, are tensors. For the solvable models above, these tensors are quadratic tensors based on quadratic functions. Quadratic tensors with $n$ degrees of freedom are fully specified by only $O(n^2)$ coefficients. Tensor networks of quadratic tensors can be contracted efficiently on the level of these coefficients, using an operation reminiscent of the Schur complement. Our formalism naturally includes models with mixed degrees of freedom, such as qudits of different dimensions. We also use quadratic functions to define generalized stabilizer codes and Clifford gates for arbitrary abelian groups. Finally, we give a generalization from quadratic (or 2nd order) to $i$th order tensors, which are specified by $O(n^i)$ coefficients but cannot be contracted efficiently in general. + oai:arXiv.org:2601.15396v1 + quant-ph + math-ph + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 + cross http://creativecommons.org/licenses/by/4.0/ - Rishabh Iyer + Andreas Bauer, Seth Lloyd - On a class of logarithmic Schr\"odinger equations via perturbation method - https://arxiv.org/abs/2601.12732 - arXiv:2601.12732v1 Announce Type: new -Abstract: In this paper, we consider the following logarithmic Schr\"odinger equation \[ -\Delta u + V(x)u = u \log u^{2} \quad \text{in }\ \mathbb{R}^{N}. \] Assuming that \(V(x)\in C(\mathbb{R}^{N})\) and \(V(x)\to+\infty\) as \(|x|\to\infty\), we develop a new perturbative variational approach to overcome the lack of \(C^{1}\)-smoothness of the associated functional and prove the existence and multiplicity of nontrivial weak solutions. - oai:arXiv.org:2601.12732v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chen Huang, Zhipeng Yang - - - A Sharp Global Boundedness Result for Keller--Segel--(Navier--)Stokes Systems with Rapid Diffusion and Saturated Sensitivities - https://arxiv.org/abs/2601.12733 - arXiv:2601.12733v1 Announce Type: new -Abstract: We investigate the Keller--Segel--(Navier--)Stokes system posed in a smooth bounded domain \(\Omega \subset \mathbb{R}^N\) with \(N = 2,3\): \begin{equation*} \begin{cases} n_t + u \cdot \nabla n = \Delta n - \nabla \cdot \big( n S(n)\nabla c \big), \\[2mm] u \cdot \nabla c = \Delta c - c + n, \\[2mm] u_t + \kappa (u \cdot \nabla) u = \Delta u - \nabla P + n \nabla \phi, \\[2mm] \nabla \cdot u = 0, \end{cases} \end{equation*} where \(\kappa \in \left \{0,1 \right \} \), the given gravitational potential \(\phi \in W^{2, \infty}(\Omega)\), and the chemotactic sensitivity function \(S \in C^2([0,\infty))\). - Under no-flux boundary conditions for \(n\) and \(c\), together with the Dirichlet boundary condition for \(u\), we show that, provided the initial data satisfy suitable regularity assumptions, the following results hold: \begin{itemize} - \item If \(N = 2\), \(\kappa = 1\), and the sensitivity function satisfies - \(\lim_{\xi \to \infty} S(\xi) = 0\), then the Keller--Segel--Navier--Stokes system admits - a global classical solution that remains uniformly bounded in time. - \item If \(N = 3\), \(\kappa = 0\), and \(S\) satisfies - \[ - |S(\xi)| \le K_S (\xi + 1)^{-\alpha} \quad \text{for all } \xi \ge 0, - \] - with some constants \(K_S > 0\) and \(\alpha > \frac{1}{3}\), then the - Keller--Segel--Stokes system possesses a global bounded classical solution. \end{itemize} Our results are optimal, since it is well established that, in the absence of fluid effects, blow-up can occur when $S \equiv \mathrm{const}$ in two dimensions, or when $\alpha < \tfrac{1}{3}$ in three dimensions. - oai:arXiv.org:2601.12733v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Minh Le + Problems with fixpoints of polynomials of polynomials + https://arxiv.org/abs/2601.15420 + arXiv:2601.15420v1 Announce Type: cross +Abstract: Motivated by applications in computable analysis, we study fixpoints of certain endofunctors over categories of containers. More specifically, we focus on fibred endofunctors over the fibrewise opposite of the codomain fibration that can be themselves be represented by families of polynomial endofunctors. In this setting, we show how to compute initial algebras, terminal coalgebras and another kind of fixpoint $\zeta$. We then explore a number of examples of derived operators inspired by Weihrauch complexity and the usual construction of the free polynomial monad. + We introduce $\zeta$-expressions as the syntax of $\mu$-bicomplete categories, extended with $\zeta$-binders and parallel products, which thus have a natural denotation in containers. By interpreting certain $\zeta$-expressions in a category of type 2 computable maps, we are able to capture a number of meaningful Weihrauch degrees, ranging from closed choice on $\{0, 1\}$ to determinacy of infinite parity games, via an "answerable part" operator. + oai:arXiv.org:2601.15420v1 + cs.LO + math.LO + Fri, 23 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/publicdomain/zero/1.0/ + C\'ecilia Pradic, Ian Price - Optimal Error Estimates of a Linearized Backward Euler Localized Orthogonal Decomposition for the Landau-Lifshitz Equation - https://arxiv.org/abs/2601.12734 - arXiv:2601.12734v1 Announce Type: new -Abstract: We introduce a novel spatial discretization technique for the reliable and efficient simulation of magnetization dynamics governed by the Landau-Lifshitz (LL) equation. The overall discretization error is systematically decomposed into temporal and spatial components. The spatial error analysis is conducted by formulating the LL equation within the framework of the Localized Orthogonal Decomposition (LOD) method. Numerical examples are presented to validate the accuracy and approximation properties of the proposed scheme. - oai:arXiv.org:2601.12734v1 - math.NA + A tensor network formalism for neuro-symbolic AI + https://arxiv.org/abs/2601.15442 + arXiv:2601.15442v1 Announce Type: cross +Abstract: The unification of neural and symbolic approaches to artificial intelligence remains a central open challenge. In this work, we introduce a tensor network formalism, which captures sparsity principles originating in the different approaches in tensor decompositions. In particular, we describe a basis encoding scheme for functions and model neural decompositions as tensor decompositions. The proposed formalism can be applied to represent logical formulas and probability distributions as structured tensor decompositions. This unified treatment identifies tensor network contractions as a fundamental inference class and formulates efficiently scaling reasoning algorithms, originating from probability theory and propositional logic, as contraction message passing schemes. The framework enables the definition and training of hybrid logical and probabilistic models, which we call Hybrid Logic Network. The theoretical concepts are accompanied by the python library tnreason, which enables the implementation and practical use of the proposed architectures. + oai:arXiv.org:2601.15442v1 + cs.AI + cs.LG + cs.LO cs.NA - physics.comp-ph - Wed, 21 Jan 2026 00:00:00 -0500 - new + math.NA + stat.ML + Fri, 23 Jan 2026 00:00:00 -0500 + cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zetao Ma, Rui Du, Lei Zhang + Alex Goessmann, Janina Sch\"utte, Maximilian Fr\"ohlich, Martin Eigel - Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions - https://arxiv.org/abs/2601.12737 - arXiv:2601.12737v1 Announce Type: new -Abstract: Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question is whether it contains arbitrarily long arithmetic progressions. In this paper we study the fractal structure of irrational numbers whose sequences of partial quotients are strictly increasing and contain arbitrarily long, quantified arithmetic progressions. - oai:arXiv.org:2601.12737v1 - math.NT + Dynamic Mean Field Theories for Nonlinear Noise in Recurrent Neuronal Networks + https://arxiv.org/abs/2601.15462 + arXiv:2601.15462v1 Announce Type: cross +Abstract: Strong, correlated noise in recurrent neural circuits often passes through nonlinear transfer functions, complicating dynamical mean-field analyses of complex phenomena such as transients and bifurcations. We introduce a method that replaces nonlinear functions of Ornstein-Uhlenbeck (OU) noise with a Gaussian-equivalent process matched in mean and covariance, and combine this with a lognormal moment closure for expansive nonlinearities to derive a closed dynamical mean-field theory for recurrent neuronal networks. The resulting theory captures order-one transients, fixed points, and noise-induced shifts of bifurcation structure, and outperforms standard linearization-based approximations in the strong-fluctuation regime. More broadly, the approach applies whenever dynamics depend smoothly on OU processes via nonlinear transformations, offering a tractable route to noise-dependent phase diagrams in computational neuroscience models. + oai:arXiv.org:2601.15462v1 + q-bio.NC math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yuto Nakajima, Hiroki Takahasi, Baowei Wang + Fri, 23 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Shoshana Chipman, Brent Doiron - Monotonicity of Pairs of Operators and Generalized Inertial Proximal Method - https://arxiv.org/abs/2601.12738 - arXiv:2601.12738v1 Announce Type: new -Abstract: Monotonicity of pairs of operators is an extension of monotonicity of operators, which plays an important role in solving non-monotone inclusions. One of challenging problems in this new tool is how to design the associated mappings to obtain the monotone pairs. In this paper, we solve this problem and propose a Generalized Inertial Proximal Point Algorithm (GIPPA) using warped resolvents under the monotonicity of pairs. The weak, strong and linear convergence of the algorithm under some mild assumptions are established. We also provide numerical examples illustrating the implementability and effectiveness of the proposed method. - oai:arXiv.org:2601.12738v1 + Early predicting of hospital admission using machine learning algorithms: Priority queues approach + https://arxiv.org/abs/2601.15481 + arXiv:2601.15481v1 Announce Type: cross +Abstract: Emergency Department overcrowding is a critical issue that compromises patient safety and operational efficiency, necessitating accurate demand forecasting for effective resource allocation. This study evaluates and compares three distinct predictive models: Seasonal AutoRegressive Integrated Moving Average with eXogenous regressors (SARIMAX), EXtreme Gradient Boosting (XGBoost) and Long Short-Term Memory (LSTM) networks for forecasting daily ED arrivals over a seven-day horizon. Utilizing data from an Australian tertiary referral hospital spanning January 2017 to December 2021, this research distinguishes itself by decomposing demand into eight specific ward categories and stratifying patients by clinical complexity. To address data distortions caused by the COVID-19 pandemic, the study employs the Prophet model to generate synthetic counterfactual values for the anomalous period. Experimental results demonstrate that all three proposed models consistently outperform a seasonal naive baseline. XGBoost demonstrated the highest accuracy for predicting total daily admissions with a Mean Absolute Error of 6.63, while the statistical SARIMAX model proved marginally superior for forecasting major complexity cases with an MAE of 3.77. The study concludes that while these techniques successfully reproduce regular day-to-day patterns, they share a common limitation in underestimating sudden, infrequent surges in patient volume. + oai:arXiv.org:2601.15481v1 + cs.LG math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Ba Khiet Le, Zakaria Mazgouri, Michel Th\'era - - - When all directed cycles have the same weight - https://arxiv.org/abs/2601.12746 - arXiv:2601.12746v1 Announce Type: new -Abstract: A digraph $G$ is weightable if its edges can be weighted with real numbers such that the total weight in each directed cycle equals 1. There are several equivalent conditions: that $G$ admits a 0/1-weighting with the same property, or that $G$ contains no subdivided "double-cycle" as a subdigraph, or that for every triple of vertices, all directed cycles containing all three pass through them in the same cyclic order. And there is quite a rich supply of such digraphs: for instance, any digraph drawn in the plane such that each of its directed cycles rotates clockwise around the origin is weightable (let us call such digraphs "circular"), and there are weightable planar digraphs with much more complicated structure than this. - Until now the general structure of weightable digraphs was not known, and that is our objective in this paper. We will show that: - - there is a construction that builds every planar weightable digraph from circular digraphs; and - - there is a (different) construction that builds every weightable digraph from planar ones. - We derive a poly-time algorithm to test if a digraph is weightable. - oai:arXiv.org:2601.12746v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + cross http://creativecommons.org/licenses/by/4.0/ - Eli Berger, Daniel Carter, Paul Seymour + Jakub Antczak, James Montgomery, Ma{\l}gorzata O'Reilly, Zbigniew Palmowski, Richard Turner - Non-Wieferich property of prime ideals and a conjecture of Erd\"os - https://arxiv.org/abs/2601.12753 - arXiv:2601.12753v1 Announce Type: new -Abstract: Let $K$ be a number field with ring of integers $\mathcal{O}$ and $\alpha\in\mathcal{O}$. For any prime ideal $\mathfrak{p}$ of $\mathcal{O}$, we obtain its higher $\alpha$-Wieferich property, which implies a nonexistence theorem for higher Wieferich unramified prime ideals. If $\beta\in\mathcal{O}$ is relatively prime to $\alpha$ and all prime ideal factors of $(\beta)$ are unramified and have residue degree $1$, we apply our higher $\alpha$-Wieferich property to establish the asymptotic equidistribution of digits in $\beta$-adic expansions of $\alpha^n$, which is a generalization of the Dupuy-Weirich theorem. When $(\beta)$ have ramified prime ideal factors, we also obtain a result on the block complexity of $\beta$-adic expansions of $\alpha^n$. - oai:arXiv.org:2601.12753v1 - math.NT - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - new + Low-Dimensional Adaptation of Rectified Flow: A New Perspective through the Lens of Diffusion and Stochastic Localization + https://arxiv.org/abs/2601.15500 + arXiv:2601.15500v1 Announce Type: cross +Abstract: In recent years, Rectified flow (RF) has gained considerable popularity largely due to its generation efficiency and state-of-the-art performance. In this paper, we investigate the degree to which RF automatically adapts to the intrinsic low dimensionality of the support of the target distribution to accelerate sampling. We show that, using a carefully designed choice of the time-discretization scheme and with sufficiently accurate drift estimates, the RF sampler enjoys an iteration complexity of order $O(k/\varepsilon)$ (up to log factors), where $\varepsilon$ is the precision in total variation distance and $k$ is the intrinsic dimension of + the target distribution. In addition, we show that the denoising diffusion probabilistic model (DDPM) procedure is equivalent to a stochastic version of RF by establishing a novel connection between these processes and stochastic localization. Building on this connection, we further design a stochastic RF sampler that also adapts to the low-dimensionality of the target distribution under milder requirements on the accuracy of the drift estimates, and also with a specific time schedule. We illustrate with simulations on the synthetic data and text-to-image data experiments the improved performance of the proposed samplers implementing the newly designed time-discretization schedules. + oai:arXiv.org:2601.15500v1 + stat.ML + cs.AI + cs.LG + math.ST + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 + cross http://creativecommons.org/licenses/by/4.0/ - Ruofan Li, Jiuzhou Zhao + Saptarshi Roy, Alessandro Rinaldo, Purnamrita Sarkar - Relativistic Hamiltonian as an emergent structure from information geometry - https://arxiv.org/abs/2601.12764 - arXiv:2601.12764v1 Announce Type: new -Abstract: We show that the relativistic energy-momentum relation can emerge as an effective ensemble-averaged structure from a multiplicative Hamiltonian when fluctuations of an auxiliary parameter are treated using maximum entropy inference. The resulting probability distribution is uniquely fixed by scale-invariant constraints, which are shown to arise naturally from the Fisher-Rao geometry of the associated statistical manifold. Within this information-geometric framework, the relativistic dispersion relation appears without initially imposing Lorentz symmetry, but as a consequence of statistical averaging and geometric invariance. - oai:arXiv.org:2601.12764v1 + A Modified Center-of-Mass Conservation Law in Finite-Domain Simulations of the Zakharov--Kuznetsov Equation + https://arxiv.org/abs/2601.15573 + arXiv:2601.15573v1 Announce Type: cross +Abstract: We investigate conservation laws of the two-dimensional Zakharov-Kuznetsov (ZK) equation, a natural higher-dimensional and non-integrable extension of the Korteweg--de Vries equation. The ZK equation admits three scalar conserved quantities -- mass, momentum, and energy -- represented as $I_1$, $I_2$, and $I_3$, as well as a vector-valued quantity $\bm{I}_4$. In high-accuracy numerical simulations on a finite double-periodic domain, most of these quantities are well preserved, while a systematic temporal drift is observed only in the $x$-component $I_{4x}$. We show that the nontrivial evolution of $I_{4x}$ originates from an explicit boundary-flux contribution, which is induced by fluctuations of the solution and its spatial derivatives at the domain boundaries. We successfully identify the source of the inaccuracy in the numerical solutions. Motivated by this analysis, we define a modified center-of-mass quantity $I_{4x}^{\mathrm{mod}}$ and demonstrate its conservation numerically for single-pulse configurations. The modified quantity thus provides a consistent conservation law for the ZK equation and yields an appropriate description of center-of-mass motion in finite-domain numerical simulations. + oai:arXiv.org:2601.15573v1 + nlin.SI + hep-th math-ph - cs.IT - math.IT math.MP - physics.class-ph - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Sikarin Yoo-Kong - - - 2-Adic Obstructions to Presburger-Definable Characterizations of Collatz Cycles - https://arxiv.org/abs/2601.12772 - arXiv:2601.12772v1 Announce Type: new -Abstract: I investigate structural limitations of Presburger-arithmetic-based approaches to the Collatz problem. I show that the Collatz cycle equation admits a unique solution in the $2$-adic integers, which I term a \emph{ghost cycle}. These ghost cycles are shown to be genuine periodic orbits of the $2$-adic Collatz map, satisfying all local parity constraints. - I prove unconditionally that the divisibility predicate $\mathcal{D}_y = \{(x, C) \in \mathbb{N}^2: (2^x - 3^y) \mid C\}$, which acts as the algebraic necessary condition for integrality, is not semilinear for any fixed number of odd steps $y \ge 1$. This result is established by demonstrating that the fibers of $\mathcal{D}_y$ exhibit unbounded periods, an obstruction to Presburger definability. Consequently, strategies relying solely on Presburger arithmetic or finite automata to define the integrality constraint cannot capture the distinction between ghost cycles and genuine integer cycles. I conclude with a heuristic argument suggesting that because ghost cycles satisfy the algebraic cycle equation, the non-existence of integer cycles cannot be proven solely through algebraic manipulation of the cycle equation itself. - oai:arXiv.org:2601.12772v1 - math.NT - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Madhav Dhiman, Rohan Pandey - - - High-order Lagrange multiplier schemes for general Hamiltonian PDEs - https://arxiv.org/abs/2601.12776 - arXiv:2601.12776v1 Announce Type: new -Abstract: In this paper, we introduce a Lagrange multiplier approach to construct linearly implicit energy-preserving schemes of arbitrary order for general Hamiltonian PDEs. Unlike the widely used auxiliary variable methods, this novel approach does not require the nonlinear part of the energy to be bounded from below, thereby offering broader applicability. Moreover, this approach preserves the original energy exactly at both the continuous and discrete levels, as opposed to a modified energy preserved by the auxiliary variable methods. Rigorous proofs are provided for the energy conservation and numerical accuracy of all derived schemes. The trade-off for these advantages is the need to solve a nonlinear algebraic equation to determine the Lagrange multiplier. Nevertheless, numerical experiments show that the associated computational cost is generally not dominant, indicating that the new schemes retain computational efficiency comparable to the auxiliary variable-based schemes. Numerical results demonstrate the efficiency, accuracy, and structure-preserving properties of the proposed schemes. - oai:arXiv.org:2601.12776v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Yonghui Bo, Yushun Wang + Fri, 23 Jan 2026 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Nobuyuki Sawado, Yuichiro Shimazaki - Extended Gabidulin-Kronecker Product Codes and Their Application to Cryptosystems - https://arxiv.org/abs/2601.12780 - arXiv:2601.12780v1 Announce Type: new -Abstract: In this paper, we initiate the study of Extended Gabidulin codes with a Kronecker product structure and propose three enhanced variants of the Rank Quasi-Cyclic (RQC) (Melchor et.al., IEEE IT, 2018) cryptosystem. First, we establish precise bounds on the minimum rank distance of Gabidulin-Kronecker product codes under two distinct parameter regimes. Specifically, when $n_{1}=k_{1}$ and $n_{2}=m<n_{1}n_{2}$, the minimum rank distance is exactly $n_{2}-k_{2}+1$. This yields a new family of Maximum Rank Distance (MRD) codes, which are distinct from classical Gabidulin codes. For the case of $k_{1}\leq n_{1},k_{2}\leq n_{2},n_{1}n_{2}\leq m$, the minimum rank distance $d$ of Gabidulin-Kronecker product codes satisfies a tight upper and lower bound, i.e., $n_{2}-k_{2}+1 \leq d \leq (n_{1}-k_{1}+1)(n_{2}-k_{2}+1)$. Second, we introduce a new class of decodable rank-metric codes, namely Extended Gabidulin-Kronecker product (EGK) codes, which generalize the structure of Gabidulin-Kronecker product (GK) codes. We also propose a decoding algorithm that directly retrieves the codeword without recovering the error vector, thus improving efficiency. This algorithm achieves zero decoding failure probability when the error weight is within its correction capability. Third, we propose three enhanced variants of the RQC cryptosystem based on EGK codes, each offering a distinct trade-off between security and efficiency. For 128-bit security, all variants achieve significant reductions in public key size compared to the Multi-UR-AG (Bidoux et.al., IEEE IT, 2024) while ensuring zero decryption failure probability--a key security advantage over many existing rank-based schemes. - oai:arXiv.org:2601.12780v1 + Does 6G Need a New Waveform: Comparing Zak-OTFS with CP-OFDM + https://arxiv.org/abs/2601.15602 + arXiv:2601.15602v1 Announce Type: cross +Abstract: Across the world, there is growing interest in new waveforms, Zak-OTFS in particular, and over-the-air implementations are starting to appear. The choice between OFDM and Zak-OTFS is not so much a choice between waveforms as it is an architectural choice between preventing inter-carrier interference (ICI) and embracing ICI. In OFDM, once the Input-Output (I/O) relation is known, equalization is relatively simple, at least when there is no ICI. However, in the presence of ICI the I/O relation is non-predictable and its acquisition is non-trivial. In contrast, equalization is more involved in Zak-OTFS due to inter-symbol-interference (ISI), however the I/O relation is predictable and its acquisition is simple. {Zak-OTFS exhibits superior performance in doubly-spread 6G use cases with high delay/Doppler channel spreads (i.e., high mobility and/or large cells), but architectural choice is governed by the typical use case, today and in the future. What is typical depends to some degree on geography, since large delay spread is a characteristic of large cells which are the rule rather than the exception in many important wireless markets.} This paper provides a comprehensive performance comparison of cyclic prefix OFDM (CP-OFDM) and Zak-OTFS across the full range of 6G propagation environments. The performance results provide insights into the fundamental architectural choice. + oai:arXiv.org:2601.15602v1 + eess.SP cs.IT math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Zhe Sun, Terry Shue Chien Lau, Mengying Zhao, Zimeng Zhou, Fang-Wei Fu - - - Graph Laplacian assisted regularization method under noise level free heuristic and statistical stopping rule - https://arxiv.org/abs/2601.12792 - arXiv:2601.12792v1 Announce Type: new -Abstract: In this work, we address the solution of both linear and nonlinear ill-posed inverse problems by developing a novel graph-based regularization framework, where the regularization term is formulated through an iteratively updated graph Laplacian. The proposed approach operates without prior knowledge of the noise level and employs two distinct stopping criteria namely, the heuristic rule and the statistical discrepancy principle. To facilitate the latter, we utilize averaged measurements derived from multiple repeated observations. We provide a detailed convergence analysis of the method in statistical prospective, establishing its stability and regularization properties under both stopping strategies. The algorithm begins with the computation of an initial reconstruction using any suitable techniques like Tikhonov regularization (Tik), filtered back projection (FBP) or total variation (TV), which is used as the foundation for generating the initial graph Laplacian. The reconstruction is made better step by step using an iterative process, during which the graph Laplacian is dynamically re-calibrated to reflect how the solution's structure is changing. Finally, we present numerical experiments on X-ray Computed Tomography (CT) and phase retrieval CT, demonstrating the effectiveness and robustness of the proposed method and comparing its reconstruction performance under both stopping rules. - oai:arXiv.org:2601.12792v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Harshit Bajpai, Ankik Kumar Giri + Imran Ali Khan, Saif Khan Mohammed, Ronny Hadani, Ananthanarayanan Chockalingam, Robert Calderbank, Anton Monk, Shachar Kons, Shlomo Rakib, Yoav Hebron - Two Frameworks and their Fourth Order Implicit Schemes for Time Discretization of Maxwell's Equations - https://arxiv.org/abs/2601.12793 - arXiv:2601.12793v1 Announce Type: new -Abstract: Our work is about energy conserving fourth-order time discretizations of a three-field formulation of Maxwell's equations in conjunction with a spatial discretization using higher-order and compatible de Rham finite element spaces. Toward this end, we delineate two broad classes of strategies for general higher-order time discretizations which we term spatial and temporal strategies. We provide a description of these two strategies and develop fourth-order time accurate schemes in the context of our Maxwell's system. However, our description can be used to prescribe similar fourth- or even higher-order time-integration methods for any linear (or quasi-linear) system of time-dependent partial differential equations. Our organizing principle in our proposed two strategies is to Taylor expand the unknown solution in time by assuming sufficient regularity. Then, in the spatial strategy, we use Maxwell's equations themselves to replace the fourth-order time derivatives in an appropriately truncated Taylor expansion with corresponding higher-order spatial derivatives. On the other hand, in the temporal strategy, we simply use higher-order finite difference schemes for the various higher-order time derivative terms in the truncated Taylor approximation. In both cases, we then defer to a standard finite element exterior calculus manner of compatible discretization for the spatial component of the Maxwell's solution. For our proposed schemes corresponding to the two strategies, we show that they are both stable and convergent and provide some validating numerical examples in $\mathbb{R}^2$. Our main contributions are in the development of the fourth-order time discretization methods that are energy conserving using our two outlined strategies and proofs of their convergence for semi- and full-discretizations of our three-field system of Maxwell's equations. - oai:arXiv.org:2601.12793v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new + Algebraic Statistics in OSCAR + https://arxiv.org/abs/2601.15807 + arXiv:2601.15807v1 Announce Type: cross +Abstract: We introduce the AlgebraicStatistics section of the OSCAR computer algebra system. We give an overview of its extensible design and highlight its features including serialization of data types for sharing results and creating databases, and state-of-the-art implicitization algorithms. + oai:arXiv.org:2601.15807v1 + stat.CO + cs.NE + math.AC + math.ST + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 + cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Archana Arya, Kaushik Kalyanaraman + Tobias Boege, Antony Della Vecchia, Marina Garrote-L\'opez, Benjamin Hollering - Probabilistic degenerate logarithm and heterogeneous stirling numbers - https://arxiv.org/abs/2601.12794 - arXiv:2601.12794v1 Announce Type: new -Abstract: Let Y be a random variable whose moment-generating function exists in some neighborhood of the origin. While probabilistic Stirling numbers of the first and second kind have been introduced, early definitions often failed to satisfy fundamental orthogonality and inverse relations or lacked consistency with classical forms in the case when Y = 1. This paper addresses these limitations by utilizing redefined probabilistic Stirling numbers of the first kind and the second kind alongside their degenerate counterparts. Our primary objective is twofold: first,to introduce the probabilistic (degenerate) logarithm associated with Y, providing explicit - expressions for various random variables and defining new probabilistic degenerate Daehee and Cauchy numbers; and second, to investigate probabilistic heterogeneous Stirling numbers and establish a probabilistic degenerate version of the Schlomilch formula, demonstrating that these new frameworks maintain the essential algebraic properties of their classical counterparts. - oai:arXiv.org:2601.12794v1 - math.NT - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new + A two-sample pseudo-observation-based regression approach for the relative treatment effect + https://arxiv.org/abs/2601.15880 + arXiv:2601.15880v1 Announce Type: cross +Abstract: The relative treatment effect is an effect measure for the order of two sample-specific outcome variables. It has the interpretation of a probability and also a connection to the area under the ROC curve. In the literature it has been considered for both ordinal or right-censored time-to-event outcomes. For both cases, the present paper introduces a distribution-free regression model that relates the relative treatment effect to a linear combination of covariates. To fit the model, we develop a pseudo-observation-based procedure yielding consistent and asymptotically normal coefficient estimates. In addition, we propose bootstrap-based hypothesis tests to infer the effects of the covariates on the relative treatment effect. A simulation study compares the novel method to Cox regression, demonstrating that the proposed hypothesis tests have high power and keep up with the z-test of the Cox model even in scenarios where the latter is specified correctly. The new methods are used to re-analyze data from the SUCCESS-A trial for progression-free survival of breast cancer patients. + oai:arXiv.org:2601.15880v1 + stat.ME + math.ST + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 + cross http://creativecommons.org/licenses/by/4.0/ - Dae San Kim, Taekyun Kim + Dennis Dobler, Alina Schenk, Matthias Schmid - Llarull's theorem on noncompact manifolds with boundary - https://arxiv.org/abs/2601.12803 - arXiv:2601.12803v1 Announce Type: new -Abstract: Recently, Zhang \cite{Zh20} and Li-Su-Wang-Zhang \cite{LSWZ24+} generalized Llarull's theorem to the noncompact complete spin manifold. In this paper, we further extend their results to the noncompact manifold with compact boundary. - oai:arXiv.org:2601.12803v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bo Liu, Daoqiang Liu + Existential Positive Transductions of Sparse Graphs + https://arxiv.org/abs/2601.15890 + arXiv:2601.15890v1 Announce Type: cross +Abstract: Monadic stability generalizes many tameness notions from structural graph theory such as planarity, bounded degree, bounded tree-width, and nowhere density. The sparsification conjecture predicts that the (possibly dense) monadically stable graph classes are exactly those that can be logically encoded by first-order (FO) transductions in the (always sparse) nowhere dense classes. So far this conjecture has been verified for several special cases, such as for classes of bounded shrub-depth, and for the monadically stable fragments of bounded (linear) clique-width, twin-width, and merge-width. + In this work we propose the existential positive sparsification conjecture, predicting that the more restricted co-matching-free, monadically stable classes are exactly those that can be transduced from nowhere dense classes using only existential positive FO formulas. While the general conjecture remains open, we verify its truth for all known special cases of the original conjecture. Even stronger, we find the sparse preimages as subgraphs of the dense input graphs. + As a key ingredient, we introduce a new combinatorial operation, called subflip, that arises as the natural co-matching-free analog of the flip operation, which is a central tool in the characterization of monadic stability. Using subflips, we characterize the co-matching-free fragment of monadic stability by appropriate strengthenings of the known flip-flatness and flipper game characterizations for monadic stability. In an attempt to generalize our results to the more expressive MSO logic, we discover (rediscover?) that on relational structures (existential) positive MSO has the same expressive power as (existential) positive FO. + oai:arXiv.org:2601.15890v1 + cs.DM + cs.LO + math.CO + math.LO + Fri, 23 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Nikolas M\"ahlmann, Sebastian Siebertz - Joint Source-Channel-Generation Coding: From Distortion-oriented Reconstruction to Semantic-consistent Generation - https://arxiv.org/abs/2601.12808 - arXiv:2601.12808v1 Announce Type: new -Abstract: Conventional communication systems, including both separation-based coding and AI-driven joint source-channel coding (JSCC), are largely guided by Shannon's rate-distortion theory. However, relying on generic distortion metrics fails to capture complex human visual perception, often resulting in blurred or unrealistic reconstructions. In this paper, we propose Joint Source-Channel-Generation Coding (JSCGC), a novel paradigm that shifts the focus from deterministic reconstruction to probabilistic generation. JSCGC leverages a generative model at the receiver as a generator rather than a conventional decoder to parameterize the data distribution, enabling direct maximization of mutual information under channel constraints while controlling stochastic sampling to produce outputs residing on the authentic data manifold with high fidelity. We further derive a theoretical lower bound on the maximum semantic inconsistency with given transmitted mutual information, elucidating the fundamental limits of communication in controlling the generative process. Extensive experiments on image transmission demonstrate that JSCGC substantially improves perceptual quality and semantic fidelity, significantly outperforming conventional distortion-oriented JSCC methods. - oai:arXiv.org:2601.12808v1 + Performance Scaling Laws for PD Array-based Receivers in IM/DD Optical Wireless Communication Systems + https://arxiv.org/abs/2601.15973 + arXiv:2601.15973v1 Announce Type: cross +Abstract: We study the performance scaling laws for electrical-domain combining in photodetector (PD) array-based receivers employing intensity modulation and direct detection, taking into account the inherent square-law relationship between the optical and electrical received powers. The performance of PD array-based systems is compared, in terms of signal-to-noise ratio (SNR) and achievable rate, to that of a reference receiver employing a single PD. Analytical and numerical results show that PD arrays provide performance gains for sufficiently narrow beams and above an SNR threshold. Furthermore, increasing the number of PDs alone does not enhance performance, and joint optimization of beam pattern, transverse electromagnetic mode, received power, and PD positions is necessary. Our model and derived insights provide practical guidelines and highlight the trade-offs for the design of next-generation high-bandwidth PD array receivers. + oai:arXiv.org:2601.15973v1 + eess.SP cs.IT - cs.CV - cs.LG math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Tong Wu, Zhiyong Chen, Guo Lu, Li Song, Feng Yang, Meixia Tao, Wenjun Zhang + Fri, 23 Jan 2026 00:00:00 -0500 + cross + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Aravindh Krishnamoorthy, Robert Schober, Harald Haas - Optimal bounds for the boundary control cost of one-dimensional fractional Schr\"odinger and heat equations - https://arxiv.org/abs/2601.12810 - arXiv:2601.12810v1 Announce Type: new -Abstract: We derive sharp bounds for the boundary control cost of the one-dimensional fractional Schr\"odinger and heat equations. The analysis of the lower bound is based on the study of the control cost of a related singular boundary control problem in finite time, using tools from complex analysis. The analysis of the upper bound relies on the moment method, involving estimates of the Fourier transform of a class of compactly supported functions. - oai:arXiv.org:2601.12810v1 - math.OC - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Hoai-Minh Nguyen + Characterizations of monadically dependent tree-ordered weakly sparse structures + https://arxiv.org/abs/2601.16039 + arXiv:2601.16039v1 Announce Type: cross +Abstract: A class of structures is monadically dependent if one cannot interpret all graphs in colored expansions from the class using a fixed first-order formula. A tree-ordered $\sigma$-structure is the expansion of a $\sigma$-structure with a tree-order. A tree-ordered $\sigma$-structure is weakly sparse if the Gaifman graph of its $\sigma$-reduct excludes some biclique (of a given fixed size) as a subgraph. Tree-ordered weakly sparse graphs are commonly used as tree-models (for example for classes with bounded shrubdepth, structurally bounded expansion, bounded cliquewidth, or bounded twin-width), motivating their study on their own. In this paper, we consider several constructions on tree-ordered structures, such as tree-ordered variants of the Gaifman graph and of the incidence graph, induced and non-induced tree-ordered minors, and generalized fundamental graphs. + We provide characterizations of monadically dependent classes of tree-ordered weakly sparse $\sigma$-structures based on each of these constructions, some of them establishing unexpected bridges with sparsity theory. As an application, we prove that a class of tree-ordered weakly sparse structures is monadically dependent if and only if its sparsification is nowhere-dense. Moreover, the sparsification transduction translates boundedness of clique-width and linear clique-width into boundedness of tree-width and path-width. We also prove that first-order model checking is not fixed parameter tractable on independent hereditary classes of tree-ordered weakly sparse graphs (assuming $AW[*]\neq FPT$) and give what we believe is the first model-theoretical characterization of classes of graphs excluding a minor, thus opening a new perspective of structural graph theory. + oai:arXiv.org:2601.16039v1 + cs.DM + cs.LO + math.CO + math.LO + Fri, 23 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by/4.0/ + Hector Buffi\`ere, Yuquan Lin, Jaroslav Ne\v{s}et\v{r}il, Patrice Ossona de Mendez, Sebastian Siebertz - Perfect codes in weakly metric association schemes - https://arxiv.org/abs/2601.12818 - arXiv:2601.12818v1 Announce Type: new -Abstract: The Lloyd Theorem of (Sol\'e, 1989) is combined with the Schwartz-Zippel Lemma of theoretical computer science to derive non-existence results for perfect codes in the Lee metric, NRT metric, mixed Hamming metric, and for the sum-rank distance. The proofs are based on asymptotic enumeration of integer partitions. The framework is the new concept of {\em polynomial} weakly metric association schemes. - A connection between this notion and the recent theory of multivariate P-polynomial schemes of ( Bannai et al. 2025) and of $m$-distance regular graphs ( Bernard et al 2025) is pointed out. - oai:arXiv.org:2601.12818v1 - math.CO - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new + On damage of interpolation to adversarial robustness in regression + https://arxiv.org/abs/2601.16070 + arXiv:2601.16070v1 Announce Type: cross +Abstract: Deep neural networks (DNNs) typically involve a large number of parameters and are trained to achieve zero or near-zero training error. Despite such interpolation, they often exhibit strong generalization performance on unseen data, a phenomenon that has motivated extensive theoretical investigations. Comforting results show that interpolation indeed may not affect the minimax rate of convergence under the squared error loss. In the mean time, DNNs are well known to be highly vulnerable to adversarial perturbations in future inputs. A natural question then arises: Can interpolation also escape from suboptimal performance under a future $X$-attack? In this paper, we investigate the adversarial robustness of interpolating estimators in a framework of nonparametric regression. A finding is that interpolating estimators must be suboptimal even under a subtle future $X$-attack, and achieving perfect fitting can substantially damage their robustness. An interesting phenomenon in the high interpolation regime, which we term the curse of simple size, is also revealed and discussed. Numerical experiments support our theoretical findings. + oai:arXiv.org:2601.16070v1 + stat.ML + cs.LG + math.ST + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 + cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Minjia Shi, Jing Wang, Patrick Sol\'e + Jingfu Peng, Yuhong Yang - Sub-wavelength resonances in two-dimensional multi-layer elastic media - https://arxiv.org/abs/2601.12821 - arXiv:2601.12821v1 Announce Type: new -Abstract: In this paper, we focus on the sub-wavelength resonances in two-dimensional elastic media characterized by high contrasts in both Lam\'e parameters and density. Our contributions are fourfold. First, it is proved that the operator $\hat{\mathbf{S}}_{\partial D}^{\omega}$, which serves as a leading order approximation to $\mathbf{S}_{\partial D}^{\omega}$ as $\omega\rightarrow0$, is invertible in the space $\mathcal{L}(L^{2}\left(\partial D)^{2},H^{1}(\partial D)^{2}\right)$. Second, based on layer potential techniques in combination with asymptotic analysis, we derive an original formula for the leading-order terms of sub-wavelength resonance frequencies, which are controlled by the determinant of the $3N \times 3N$ matrices. Specifically, there are $3N$ resonance frequencies within an $N$-nested layer structure. In addition, the scattering field exhibits an enhancement coefficient on the order of $\mathcal{O}(\omega^{-2})$ as the incident frequency $\omega$ approaches the resonance frequency. Third, by applying spectral properties to solve the corresponding eigenvalue problem, we compute the quantitative expressions for sub-wavelength resonance frequencies within a disk. Finally, some numerical experiments are provided to illustrate theoretical results and demonstrate the existence of the sub-wavelength resonance modes. - oai:arXiv.org:2601.12821v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Yan Jiang, Hongyu Liu, Fanbo Sun, Yajuan Wang + CLASP: An online learning algorithm for Convex Losses And Squared Penalties + https://arxiv.org/abs/2601.16072 + arXiv:2601.16072v1 Announce Type: cross +Abstract: We study Constrained Online Convex Optimization (COCO), where a learner chooses actions iteratively, observes both unanticipated convex loss and convex constraint, and accumulates loss while incurring penalties for constraint violations. We introduce CLASP (Convex Losses And Squared Penalties), an algorithm that minimizes cumulative loss together with squared constraint violations. Our analysis departs from prior work by fully leveraging the firm non-expansiveness of convex projectors, a proof strategy not previously applied in this setting. For convex losses, CLASP achieves regret $O\left(T^{\max\{\beta,1-\beta\}}\right)$ and cumulative squared penalty $O\left(T^{1-\beta}\right)$ for any $\beta \in (0,1)$. Most importantly, for strongly convex problems, CLASP provides the first logarithmic guarantees on both regret and cumulative squared penalty. In the strongly convex case, the regret is upper bounded by $O( \log T )$ and the cumulative squared penalty is also upper bounded by $O( \log T )$. + oai:arXiv.org:2601.16072v1 + cs.LG + math.OC + Fri, 23 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Ricardo N. Ferreira, Cl\'audia Soares, Jo\~ao Xavier - A converse of Berndtsson's theorem on the positivity of direct images - https://arxiv.org/abs/2601.12825 - arXiv:2601.12825v1 Announce Type: new -Abstract: Berndtsson's famous theorem asserts that, for a compact K\"ahler fibration $p:X\to Y$, the direct image bundle $p_*(K_{X/Y}\otimes L)$ of a semi-positive Hermitian holomorphic line bundle $L\to X$ is Nakano semi-positive. As a continuation of our previous work, we prove a converse of Berndtsson's theorem in the case of a projective fibration: if $p_*(K_{X/Y}\otimes L\otimes E)$ is Griffiths semi-positive for every semi-positive Hermitian holomorphic line bundle $E\to X$, then the curvature of $L$ must be semi-positive. - oai:arXiv.org:2601.12825v1 - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Wang Xu, Hui Yang + On the spherical cardioid distribution and its goodness-of-fit + https://arxiv.org/abs/2601.16095 + arXiv:2601.16095v1 Announce Type: cross +Abstract: In this paper, we study the spherical cardioid distribution, a higher-dimensional and higher-order generalization of the circular cardioid distribution. This distribution is rotationally symmetric and generates unimodal, multimodal, axial, and girdle-like densities. We show several characteristics of the spherical cardioid that make it highly tractable: simple density evaluation, closedness under convolution, explicit expressions for vectorized moments, and efficient simulation. The moments of the spherical cardioid up to a given order coincide with those of the uniform distribution on the sphere, highlighting its closeness to the latter. We derive estimators by the method of moments and maximum likelihood, their asymptotic distributions, and their asymptotic relative efficiencies. We give the machinery for a bootstrap goodness-of-fit test based on the projected-ecdf approach, including the projected distribution and closed-form expressions for test statistics. An application to modeling the orbits of long-period comets shows the usefulness of the spherical cardioid distribution in real data analyses. + oai:arXiv.org:2601.16095v1 + stat.ME + math.ST + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Eduardo Garc\'ia-Portugu\'es - Data-Consistent Learning of Inverse Problems - https://arxiv.org/abs/2601.12831 - arXiv:2601.12831v1 Announce Type: new -Abstract: Inverse problems are inherently ill-posed, suffering from non-uniqueness and instability. Classical regularization methods provide mathematically well-founded solutions, ensuring stability and convergence, but often at the cost of reduced flexibility or visual quality. Learned reconstruction methods, such as convolutional neural networks, can produce visually compelling results, yet they typically lack rigorous theoretical guarantees. DC (DC) networks address this gap by enforcing the measurement model within the network architecture. In particular, null-space networks combined with a classical regularization method as an initial reconstruction define a convergent regularization method. This approach preserves the theoretical reliability of classical schemes while leveraging the expressive power of data-driven learning, yielding reconstructions that are both accurate and visually appealing. - oai:arXiv.org:2601.12831v1 - math.NA - cs.CV - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Markus Haltmeier, Gyeongha Hwang + Exceptional points in Gaussian channels: diffusion gauging and drift-governed spectrum + https://arxiv.org/abs/2601.16121 + arXiv:2601.16121v1 Announce Type: cross +Abstract: McDonald and Clerk [Phys.\ Rev.\ Research 5, 033107 (2023)] showed that for linear open quantum systems the Liouvillian spectrum is independent of the noise strength. We first make this noise-independence principle precise in continuous time for multimode bosonic Gaussian Markov semigroups: for Hurwitz drift, a time-independent Gaussian similarity fixed by the Lyapunov equation gauges away diffusion for all times, so eigenvalues and non-diagonalizability are controlled entirely by the drift, while diffusion determines steady states and the structure of eigenoperators. We then extend the same separation to discrete time for general stable multimode bosonic Gaussian channels: for any stable Gaussian channel, we construct an explicit Gaussian similarity transformation that gauges away diffusion at the level of the channel parametrization. We illustrate the method with a single-mode squeezed-reservoir Lindbladian and with a non-Markovian family of single-mode Gaussian channels, where the exceptional-point manifolds and the associated gauging covariances can be obtained analytically. + oai:arXiv.org:2601.16121v1 + quant-ph + math-ph + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 + cross + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Frank Ernesto Quintela Rodr\'iguez - Cofiniteness and $P(z)$-tensor product bifunctors in orbifold theories associated to abelian but not-necessarily-finite groups - https://arxiv.org/abs/2601.12834 - arXiv:2601.12834v1 Announce Type: new -Abstract: Let $V$ be a M\"{o}bius vertex algebra and $G$ an abelian group of automorphisms of $V$. We construct $P(z)$-tensor product bifunctors for the category of $C_{n}$-cofinite grading-restricted generalized $g$-twisted $V$-modules (without $g$-actions) for $g\in G$ and the category of $C_{n}$-cofinite grading-restricted generalized $g$-twisted $V$-modules with $G$-actions for $g\in G$. In this paper, an automorphism $g$ of $V$ can be of infinite order and does not have to act semisimply on $V$, and the group $G$ can be an infinite abelian group containing nonsemisimple automorphisms of $V$. - oai:arXiv.org:2601.12834v1 - math.QA - hep-th - math.CT - Wed, 21 Jan 2026 00:00:00 -0500 - new + Interconnection-based Model Reduction for Linear Hybrid Systems + https://arxiv.org/abs/2601.16149 + arXiv:2601.16149v1 Announce Type: cross +Abstract: In this paper, we address the model reduction problem for linear hybrid systems via the interconnection-based technique called moment matching. We consider two classical interconnections, namely the direct and swapped interconnections, in the hybrid setting, and we present families of reduced-order models for each interconnection via a hybrid characterisation of the steady-state responses. By combining the results for each interconnection, the design of a reduced-order model that achieves moment matching simultaneously for both interconnections is studied. In addition, we show that the presented results have simplified counterparts when the jumps of the hybrid system are periodic. A numerical simulation is finally given to illustrate the results. + oai:arXiv.org:2601.16149v1 + eess.SY + cs.SY + math.OC + Fri, 23 Jan 2026 00:00:00 -0500 + cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yi-Zhi Huang + Zirui Niu, Giordano Scarciotti, Alessandro Astolfi - A diagrammatic approach to the three-page index - https://arxiv.org/abs/2601.12846 - arXiv:2601.12846v1 Announce Type: new -Abstract: The three-page index $\alpha_3(L)$ is an invariant that measures the complexity of representing a link $L$ in a three-page book. It is known that $\alpha_3(L)$ admits a linear upper bound in terms of the crossing number, with equality realized by the Hopf link. - In this paper, we investigate the equality case of this bound from a diagrammatic viewpoint. Starting from a reduced link diagram, we construct three-page presentations via binding circles arising as boundaries of suitable contractible subcomplexes of the induced cell decomposition of the $2$-sphere. This approach allows a refined control of the number of arcs in the resulting three-page presentation. - As a consequence, we prove that for any non-split, nontrivial link $L$ other than the Hopf link, \[ \alpha_3(L)\le 3c(L)-1, \] and hence characterize completely the links for which $\alpha_3(L)=3c(L)$. - oai:arXiv.org:2601.12846v1 + Gauge Theory and Skein Modules + https://arxiv.org/abs/2601.16213 + arXiv:2601.16213v1 Announce Type: cross +Abstract: We study skein modules of 3-manifolds by embedding them into the Hilbert spaces of 4d ${\cal N}=4$ super-Yang-Mills theories. When the 3-manifold has reduced holonomy, we present an algorithm to determine the dimension and the list of generators of the skein module with a general gauge group. The analysis uses a deformation preserving ${\cal N}=1$ supersymmetry to express the dimension as a sum over nilpotent orbits. We find that the dimensions often differ between Langlands-dual pairs beyond the A-series, for which we provide a physical explanation involving chiral symmetry breaking and 't Hooft operators. We also relate our results to the structure of $\mathbb{C}^*$-fixed loci in the moduli space of Higgs bundles. This approach helps to clarify the relation between the gauge-theoretic framework of Kapustin and Witten with other versions of the geometric Langlands program, explains why the dimensions of skein modules do not exhibit a TQFT-like behavior, and provides a physical interpretation of the skein-valued curve counting of Ekholm and Shende. + oai:arXiv.org:2601.16213v1 + hep-th + math.AG math.GT - Wed, 21 Jan 2026 00:00:00 -0500 - new + math.QA + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 + cross http://creativecommons.org/licenses/by/4.0/ - Hyungkee Yoo + Du Pei - Rankin-Cohen Bracket for Vector-Valued Modular Forms - https://arxiv.org/abs/2601.12860 - arXiv:2601.12860v1 Announce Type: new -Abstract: In this paper, we explore the relationship between Rankin-Cohen brackets for vector-valued modular forms and Petersson's inner products, deriving an explicit description of the adjoint map for the bracket operator. The study extends to the cases of Jacobi forms and skew-holomorphic Jacobi forms, establishing connections between their respective Rankin-Cohen brackets and those defined for vector-valued modular forms through an isomorphism. Adjoint maps for these extended bracket operators are also examined. - oai:arXiv.org:2601.12860v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - new + Deterministic Structures in the Stopping Time Dynamics of the 3x+1 Problem + https://arxiv.org/abs/1709.03385 + arXiv:1709.03385v5 Announce Type: replace +Abstract: The $3x+1$ problem concerns the iteration of the map $T:\mathbb{Z}\to\mathbb{Z}$ defined by $T(x)=x/2$ for even $x$ and $T(x)=(3x+1)/2$ for odd $x$. This paper investigates the stopping time dynamics associated with $T$ within a deterministic and algebraic framework. By relating the parity vectors of Collatz trajectories to exponential Diophantine equations, we construct a recursively generated tree of congruence classes $\bmod\, 2^{\sigma_n}$ that characterizes the stopping time classes $\sigma(x)=\sigma_n$. We demonstrate that the generation of these classes follows an explicit deterministic recursion and derive arithmetic transition rules between neighboring congruence classes, based on the differences of the associated Diophantine sums. Finally, we prove that the union of stopping time congruence classes generated up to a fixed order $N$ is periodic, establishing a computable finite-range coverage bound. + oai:arXiv.org:1709.03385v5 + math.GM + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Youngmin Lee, Subong Lim, Wissam Raji + Mike Winkler - Traveling waves for monostable reaction-diffusion-convection equations with discontinuous density-dependent coefficients - https://arxiv.org/abs/2601.12869 - arXiv:2601.12869v1 Announce Type: new -Abstract: This paper concerns wave propagation in a class of scalar reaction-diffusion-convection equations with $p$-Laplacian-type diffusion and monostable reaction. We introduce a new concept of a non-smooth traveling wave profile, which allows us to treat discontinuous diffusion with possible degenerations and singularities at 0 and 1, as well as only piecewise continuous convective velocity. Our approach is based on comparison arguments for an equivalent non-Lipschitz first-order ODE. We formulate sufficient conditions for the existence and non-existence of these generalized solutions and discuss how the convective velocity affects the minimal wave speed compared to the problem without convection. We also provide brief asymptotic analysis of the profiles, for which we need to assume power-type behavior of the diffusion and reaction terms. - oai:arXiv.org:2601.12869v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new + Unbounded field operators in categorical extensions of conformal nets + https://arxiv.org/abs/2001.03095 + arXiv:2001.03095v5 Announce Type: replace +Abstract: We prove the equivalence of VOA tensor categories and conformal net tensor categories for the following examples: all WZW models; all lattice VOAs; all unitary parafermion VOAs; type $ADE$ discrete series $W$-algebras; their tensor products; their regular cosets. A new proof of the complete rationality of conformal nets is also given. + oai:arXiv.org:2001.03095v5 + math.QA + math-ph + math.MP + math.OA + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - J. Math. Anal. Appl. 539(2024), no. 1, part 1, Paper No. 128481, 26 pp - Pavel Dr\'abek, Soyeun Jung, Eunkyung Ko, Michaela Zahradn\'ikov\'a + 10.1007/s00222-026-01407-7 + Bin Gui - Residues and Infinitesimal Torelli for Equisingular Curves - https://arxiv.org/abs/2601.12873 - arXiv:2601.12873v1 Announce Type: new -Abstract: We study infinitesimal Torelli problems and infinitesimal variations of Hodge structure for families of curves arising in singular and extrinsically constrained geometric settings. Motivated by the Green--Voisin philosophy, we develop an explicit approach based on Poincar\'e residue calculus, allowing a uniform treatment of smooth, singular, and equisingular situations. In particular, we prove infinitesimal Torelli theorems for general equisingular plane curves of sufficiently high degree and construct relative IVHS exact sequences for curves lying on smooth projective threefolds. - Our results show that maximal infinitesimal variation of Hodge structure persists even after imposing strong extrinsic conditions, such as fixed degree and prescribed singularities, and in the presence of isolated planar singularities. The methods presented here provide a concrete and geometric realization of Jacobian-type constructions and extend the Green--Voisin philosophy to singular and equisingular settings and provide a unified residue--theoretic framework for Torelli--type problems across dimensions and codimensions. - oai:arXiv.org:2601.12873v1 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - new + Control Occupation Kernel Regression for Nonlinear Control-Affine Systems + https://arxiv.org/abs/2106.00103 + arXiv:2106.00103v2 Announce Type: replace +Abstract: This manuscript presents an algorithm for obtaining an approximation of a nonlinear high order control affine dynamical system. Controlled trajectories of the system are leveraged as the central unit of information via embedding them in vector-valued reproducing kernel Hilbert space (vvRKHS). The trajectories are embedded as the so-called higher order control occupation kernels which represent an operator on the vvRKHS corresponding to iterated integration after multiplication by a given controller. The solution to the system identification problem is then the unique solution of an infinite dimensional regularized regression problem. The representer theorem is then used to express the solution as finite linear combination of these occupation kernels, which converts an infinite dimensional optimization problem to a finite dimensional optimization problem. The vector valued structure of the Hilbert space allows for simultaneous approximation of the drift and control effectiveness components of the control affine system. Several experiments are performed to demonstrate the effectiveness of the developed approach. + oai:arXiv.org:2106.00103v2 + math.OC + cs.LG + cs.SY + eess.SY + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mounir Nisse + Moad Abudia, Tejasvi Channagiri, Joel A. Rosenfeld, Rushikesh Kamalapurkar - A hierarchical splitting approach for N-split differential equations - https://arxiv.org/abs/2601.12878 - arXiv:2601.12878v1 Announce Type: new -Abstract: We propose a hierarchical splitting approach to differential equations that provides a design principle for constructing splitting methods for $N$-split systems by iteratively applying splitting methods for two-split systems. We analyze the convergence order, derive explicit formulas for the leading-order error terms, and investigate self-adjointness. Moreover, we discuss compositions of hierarchical splitting methods in detail. We further augment the hierarchical splitting approach with multiple time-stepping techniques, turning the class into a promising framework at the intersection of geometric numerical integration and multirate integration. In this context, we characterize the computational order of a multirate integrator and establish conditions on the multirate factors that guarantee an increased convergence rate in practical computations up to a certain step size. Finally, we design several hierarchical splitting methods and perform numerical simulations for rigid body equations and a separable Hamiltonian system with multirate potential, confirming the theoretical findings and showcasing the computational efficiency of hierarchical splitting methods. - oai:arXiv.org:2601.12878v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Kevin Sch\"afers, Michael G\"unther + On rationality for $C_2$-cofinite vertex operator algebras + https://arxiv.org/abs/2108.01898 + arXiv:2108.01898v3 Announce Type: replace +Abstract: Let $V$ be an $\mathbb{N}$-graded, simple, self-contragredient, $C_2$-cofinite vertex operator algebra. We show that if the $S$-transformation of the character of $V$ is a linear combination of characters of $V$-modules, then the category $\mathcal{C}$ of grading-restricted generalized $V$-modules is a rigid tensor category. We further show, without any assumption on the character of $V$ but assuming that $\mathcal{C}$ is rigid, that $\mathcal{C}$ is a factorizable finite ribbon category, that is, a not-necessarily-semisimple modular tensor category. As a consequence, we show that if the Zhu algebra of $V$ is semisimple, then $\mathcal{C}$ is semisimple and thus $V$ is rational. The proofs of these theorems use techniques and results from tensor categories together with the method of Moore-Seiberg and Huang for deriving identities of two-point genus-one correlation functions associated to $V$. We give two main applications. First, we prove the conjecture of Kac-Wakimoto and Arakawa that $C_2$-cofinite affine $W$-algebras obtained via quantum Drinfeld-Sokolov reduction of admissible-level affine vertex algebras are strongly rational. The proof uses the recent result of Arakawa and van Ekeren that such $W$-algebras have semisimple (Ramond twisted) Zhu algebras. Second, we use our rigidity results to reduce the "coset rationality problem" to the problem of $C_2$-cofiniteness for the coset. That is, given a vertex operator algebra inclusion $U\otimes V\hookrightarrow A$ with $A$, $U$ strongly rational and $U$, $V$ a pair of mutual commutant subalgebras in $A$, we show that $V$ is also strongly rational provided it is $C_2$-cofinite. + oai:arXiv.org:2108.01898v3 + math.QA + math-ph + math.CT + math.MP + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Robert McRae + + + Parameterising the effect of a continuous treatment using average derivative effects + https://arxiv.org/abs/2109.13124 + arXiv:2109.13124v2 Announce Type: replace +Abstract: The average treatment effect (ATE) is commonly used to quantify the main effect of a binary treatment on an outcome. Extensions to continuous treatments are usually based on the dose-response curve or shift interventions, but both require strong overlap conditions and the resulting curves may be difficult to summarise. We focus instead on average derivative effects (ADEs) that are scalar estimands related to infinitesimal shift interventions requiring only local overlap assumptions. ADEs, however, are rarely used in practice because their estimation usually requires estimating conditional density functions. By characterising the Riesz representers of weighted ADEs, we propose a new class of estimands that provides a unified view of weighted ADEs/ATEs when the treatment is continuous/binary. We derive the estimand in our class that minimises the nonparametric efficiency bound, thereby extending optimal weighting results from the binary treatment literature to the continuous setting. We develop efficient estimators for two weighted ADEs that avoid density estimation and are amenable to modern machine learning methods, which we evaluate in simulations and an applied analysis of Warfarin dosage effects. + oai:arXiv.org:2109.13124v2 + math.ST + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Oliver J. Hines, Karla Diaz-Ordaz, Stijn Vansteelandt - Hunting The Poles in the Staircases - https://arxiv.org/abs/2601.12881 - arXiv:2601.12881v1 Announce Type: new -Abstract: Motivated by applications to the fractional quantum Hall effect and, in particular, to the Bernevig-Haldane conjectures, we investigates the behavior of Macdonald polynomials under specializations of the form q a t b = 1. Our main focus is to explain, in a simple and purely combinatorial way, why certain nonsymmetric Macdonald polynomials indexed by staircase vectors with steps of height a and width b remain regular at the specialization q a t b+1 = 1, despite the presence of potential poles in their rational coefficients. To this end, we introduce a set of combinatorial tools that track how poles are created or cancelled along paths in the Yang-Baxter graph. By carefully constructing paths from the zero vector to the staircase and analyzing the resulting denominators, we show that the absence of certain poles follows from intrinsic symmetries and cancellations encoded in the Yang-Baxter graph. - oai:arXiv.org:2601.12881v1 + Uni-width subgroups, universal elements, and lambda number of finite groups + https://arxiv.org/abs/2202.09818 + arXiv:2202.09818v2 Announce Type: replace +Abstract: A cyclic subgroup $N$ of a finite group $G$ is called a uni-width subgroup of $G$ if $N$ is the unique cyclic subgroup of $G$ of order $|N|$. In this article, we prove that a finite group $G$ admits a unique largest uni-width subgroup denoted by $U(1;G)$. We then show that the prime factors of the order of $U(1;G)$ influence the structure decomposition of its Fitting subgroup ${\mathrm{Fit}}(G)$. A power graph $\Gamma_G$ of a finite group is defined by $G$ being its set of vertices, and a pair of distinct elements $x,y \in G$ are connected by an edge if either $x \in \langle y \rangle$ or $y \in \langle x \rangle$. A universal element of a graph is a vertex that is adjacent to each of the remaining vertices. Our following result shows that a power graph $\Gamma_G$ of a finite non-trivial group admits a non-identity universal element if and only if it is either cyclic or a generalized quaternion $2$-group. The lambda number $\lambda(G)$ of a finite group $G$ is a measure of the least number of colors required for an $L(2,1)$-type of vertex coloring on $\Gamma_G$, which is known to be $\geq |G|$. Generalizing an earlier result, we then derive a necessary condition on a finite group $G$ such that $\lambda(G) = |G|$. Finally, we show that this result is best possible by exhibiting a family of groups without the necessary condition for which $\lambda(G) > |G|$. + oai:arXiv.org:2202.09818v2 + math.GR math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Christophe Carr\'e (GR2IF), Ulysse Goncalves (GR2IF), Jean-Gabriel Luque (GR2IF) + Siddhartha Sarkar - A formula for the local Heun solution - https://arxiv.org/abs/2601.12888 - arXiv:2601.12888v1 Announce Type: new -Abstract: The local Heun solution is the unique solution to Heun's equation which is analytic in the unit disk centered at $0\in\mathbb{C}$ and taking the value $1$ at the center of the disk. In this paper, as an application of the theory of orthogonal polynomials, we are able to express the coefficients in the corresponding power series as finite multiple sums. In addition, the obtained formula can be used to derive an explicit estimate on the coefficients giving a hint on their asymptotic behavior for large indices. - oai:arXiv.org:2601.12888v1 + Proper harmonic embeddings of open Riemann surfaces into $\mathbb{R}^4$ + https://arxiv.org/abs/2206.03566 + arXiv:2206.03566v2 Announce Type: replace +Abstract: We prove that every open Riemann surface admits a proper embedding into $\mathbb{R}^4$ by harmonic functions. This reduces by one the previously known embedding dimension in this framework, dating back to a theorem by Greene and Wu from 1975. + oai:arXiv.org:2206.03566v2 + math.DG math.CV - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pavel \v{S}\v{t}ov\'i\v{c}ek + Antonio Alarcon, Francisco J. Lopez - Bi-Lipschitz invariance of Newton polygons along gradient canyons - https://arxiv.org/abs/2601.12897 - arXiv:2601.12897v1 Announce Type: new -Abstract: We study bi-Lipschitz right-equivalence of holomorphic function germs $f:(\mathbb{C}^2,0)\to(\mathbb{C},0)$ via polar arcs and gradient canyons. For a polar arc $\gamma$ we consider the Newton polygon of $f_x(X+\gamma(Y),Y)$ and define its augmentation by adjoining the point $(0,\text{ord } f(\gamma(y),y)-1)$. We prove that the resulting augmented Newton polygon is constant along each gradient canyon of degree $>1$ and is invariant under bi-Lipschitz right-equivalence. Moreover, its compact edges decompose into a topological part and a Lipschitz part: the latter encodes, through simple intercept relations, the second-level Henry-Parusi\'nski type invariants. As an application we introduce the polar multiplicity of a canyon and identify it with the horizontal length of the top edge of the augmented polygon, yielding a new discrete bi-Lipschitz invariant. - oai:arXiv.org:2601.12897v1 - math.CV + On the additivity of Newton-Okounkov bodies + https://arxiv.org/abs/2207.09229 + arXiv:2207.09229v3 Announce Type: replace +Abstract: We study the additivity of Newton-Okounkov bodies. Our main result states that on two-dimensional subcones of the ample cone the Newto-Okounkov body associated to an appropriate flag acts additively. We prove this by induction relying on the slice formula for Newton-Okounkov bodies. Moreover, we discuss a necessary condition for the additivity showing that our result is optimal in general situations. As an application, we deduce an inequality between intersection numbers of nef line bundles. + oai:arXiv.org:2207.09229v3 math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Piotr Migus, Lauren\c{t}iu P\u{a}unescu, Mihai Tib\u{a}r + 10.1007/s13366-025-00807-9 + Robert Wilms - On the number of spanning trees of bicirculant graphs - https://arxiv.org/abs/2601.12899 - arXiv:2601.12899v1 Announce Type: new -Abstract: A bi-Cayley graph over a cyclic group $\mathbb{Z}_n$ is called a bicirculant graph. Let - $\Gamma=BC(\mathbb{Z}_n; R,T,S)$ be a bicirculant graph with $R=R^{-1}\subseteq \mathbb{Z}_n\setminus \{0\}$ and $T=T^{-1}\subseteq \mathbb{Z}_n\setminus \{0\}$ and $S\subseteq \mathbb{Z}_n$. In this paper, using Chebyshev polynomials, we obtain a closed formula - for the number of spanning trees of bicirculant graph $\Gamma$, investigate some arithmetic properties of the number of spanning trees of $\Gamma$, and find its asymptotic behaviour as $n$ tends infinity. In addition, - we show that $F(x)=\sum_{n=1}^{\infty}\tau(\Gamma)x^n$ is a rational function with integer coefficients. - oai:arXiv.org:2601.12899v1 + Mixing times of a Burnside process Markov chain on set partitions + https://arxiv.org/abs/2207.14269 + arXiv:2207.14269v3 Announce Type: replace +Abstract: Let $X$ be a finite set and let $G$ be a finite group acting on $X$. The group action splits $X$ into disjoint orbits. The Burnside process is a Markov chain on $X$ which has a uniform stationary distribution when the chain is lumped to orbits. We consider the case where $X = [k]^n$ with $k \geq n$ and $G = S_k$ is the symmetric group on $[k]$, such that $G$ acts on $X$ by permuting the value of each coordinate. The resulting Burnside process gives a novel algorithm for sampling a set partition of $[n]$ uniformly at random. We obtain bounds on the mixing time and show that the chain is rapidly mixing. For the case $k < n$, the algorithm corresponds to sampling a set partition of $[n]$ with at most $k$ blocks, and we obtain a mixing time bound which is independent of $n$. Along the way, we obtain explicit formulas for the transition probabilities and bounds on the second largest eigenvalue for both the original process and the lumped chain. + oai:arXiv.org:2207.14269v3 + math.PR math.CO - math.SP - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jing Yang, Fangming Xian + J. E. Paguyo - Machine Learning for highly oscillatory differential equations - https://arxiv.org/abs/2601.12907 - arXiv:2601.12907v1 Announce Type: new -Abstract: Highly oscillatory differential equations, commonly encountered in multi-scale problems, are often too complex to solve analytically. However, several numerical methods have been developed to approximate their solutions. Although these methods have shown their efficiency, the first part of the strategy often involves heavy pre-computations from averaging theory. In this paper, we leverage neural networks (machine learning) to approximate the vector fields required by the pre-computations in the first part, and combine this with micro-macro techniques to efficiently solve the oscillatory problem. We illustrate our work by numerical simulations. - oai:arXiv.org:2601.12907v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new + Conformal and extrinsic upper bounds for the harmonic mean of Neumann and Steklov eigenvalues + https://arxiv.org/abs/2208.13959 + arXiv:2208.13959v5 Announce Type: replace +Abstract: Let $M$ be an $m$-dimensional compact Riemannian manifold with boundary. We obtain the upper bound of the harmonic mean of the first $m$ nonzero Neumann eigenvalues and Steklov eigenvalues involving the conformal volume and relative conformal volume, respectively. We also give an optimal sharp extrinsic upper bound for closed submanifolds in space forms. These extend the previous related results for the first nonzero eigenvalues. + oai:arXiv.org:2208.13959v5 + math.DG + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Maxime Bouchereau (IRMAR) + 10.1016/j.jfa.2026.111361 + Hang Chen - Gender and assessment in mathematics: a comparative study of managing assessment episodes - https://arxiv.org/abs/2601.12908 - arXiv:2601.12908v1 Announce Type: new -Abstract: The article focuses on the differences in mathematics performance between girls and boys visible from the first four months of compulsory schooling in the French education system. The influence of gender stereotypes in the evaluation practices of teachers and the threat of the gender stereotype on student performance are questioned. To obtain answers, a comparative study of management of evaluative episodes is proposed based on different theoretical tools. - oai:arXiv.org:2601.12908v1 - math.HO - Wed, 21 Jan 2026 00:00:00 -0500 - new + An analogue of Bonami's Lemma for functions on spaces of linear maps, and 2-2 Games + https://arxiv.org/abs/2209.04243 + arXiv:2209.04243v2 Announce Type: replace +Abstract: We prove an analogue of Bonami's (hypercontractive) lemma for complex-valued functions on $\mathcal{L}(V,W)$, where $V$ and $W$ are vector spaces over a finite field. This inequality is useful for functions on $\mathcal{L}(V,W)$ whose `generalised influences' are small, in an appropriate sense. It leads to a significant shortening of the proof of a recent seminal result by Khot, Minzer and Safra that pseudorandom sets in Grassmann graphs have near-perfect expansion, which (in combination with the work of Dinur, Khot, Kindler, Minzer and Safra) implies the 2-2 Games conjecture (the variant, that is, with imperfect completeness). + oai:arXiv.org:2209.04243v2 + math.CO + math.FA + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Espace Math{\'e}matique Francophone 2025, May 2025, Montr{\'e}al, Canada. pp.1273-1282 - Chlo\'e Brismontier (LDAR, UFR Math\'ematiques UPCit\'e, UPCit\'e) + David Ellis, Guy Kindler, Noam Lifshitz - A functional inequalities approach for the field-road diffusion model with (symmetric) nonlinear exchanges - https://arxiv.org/abs/2601.12909 - arXiv:2601.12909v1 Announce Type: new -Abstract: In this note, we consider the so-called field-road diffusion model in a bounded domain, consisting of two parabolic PDEs posed on sets of different dimensions and coupled through (symmetric) nonlinear exchange terms. We propose a new and rather direct functional inequalities approach to prove the exponential decay of a relative entropy, and thus the convergence of the solution towards the stationary state selected by the total mass of the initial datum. - oai:arXiv.org:2601.12909v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fibrantly-transferred model structures + https://arxiv.org/abs/2301.07801 + arXiv:2301.07801v2 Announce Type: replace +Abstract: We develop new techniques for constructing model structures from a given class of cofibrations, together with a class of fibrant objects and a choice of weak equivalences between them. As a special case, we obtain a more flexible version of the classical right-transfer theorem in the presence of an adjunction. Namely, instead of lifting the classes of fibrations and weak equivalences through the right adjoint, we now only do so between fibrant objects, which allows for a wider class of applications. + oai:arXiv.org:2301.07801v2 + math.AT + math.CT + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Matthieu Alfaro (LMRS), Claire Chainais-Hillairet (LPP), Flore Nabet (CMAP) + L\'eonard Guetta, Lyne Moser, Maru Sarazola, Paula Verdugo - Countable basis for free electromagnetic fields - https://arxiv.org/abs/2601.12911 - arXiv:2601.12911v1 Announce Type: new -Abstract: Polychromatic electromagnetic fields are typically expanded as integrals over monochromatic fields, such as plane waves, multipolar fields, or Bessel beams. However, monochromatic fields do not belong to the Hilbert space of free Maxwell fields, since their norms diverge. Moreover, the continuous frequency integrals involved in such expansions complicate the treatment of light--matter interactions via the scattering operator. Here, we identify and study a polychromatic basis for free Maxwell fields whose basis vectors belong to the Hilbert space. These vectors are defined as simultaneous eigenstates of four commuting operators with integer eigenvalues. As a consequence, the basis set is countable, and the Hilbert space is separable and isomorphic to $\ell^2$, the Hilbert space of square-summable sequences. Each basis vector represents a polychromatic single-photon wave with quantized energy and a wavelet--like temporal dependence. Three versions of this basis are defined: Regular, incoming, and outgoing. The fields of the regular basis are smooth in both space and time. The incoming and outgoing fields are likewise smooth, except at the spatial origin. These results support and motivate the use of countable bases for both the theoretical description and the practical computation of light--matter interactions. - oai:arXiv.org:2601.12911v1 + Determinantally equivalent nonzero functions + https://arxiv.org/abs/2302.02471 + arXiv:2302.02471v4 Announce Type: replace +Abstract: We study the problem raised in [Marco Stevens, Equivalent symmetric kernels of determinantal point processes, RMTA, 10(03):2150027, 2021] concerning the extension of its main result to the more general (potentially non-symmetric) setting. We construct a counterexample disproving the conjecture proposed in the paper, and subsequently solve it under some additional minor assumptions that preclude such counterexamples. + The problem is plainly stated as follows: Let $\Lambda$ be a set and $\mathbb{F}$ a field, and suppose that $K,Q:\Lambda^2\to\mathbb{F}$ are two functions such that for any $n\in\mathbb{N}$ and $x_1,x_2,\ldots,x_n\in\Lambda$, the determinants of matrices $(K(x_i,x_j))_{1\leq i,j\leq n}$ and $(Q(x_i,x_j))_{1\leq i,j\leq n}$ agree. What are all the possible transformations that transform $Q$ into $K$? In [Marco Stevens, Equivalent symmetric kernels of determinantal point processes, RMTA, 10(03):2150027, 2021] the following two were conjectured: $(Tf)(x,y)=f(y,x)$; and $(Tf)(x,y)=g(x)g(y)^{-1}f(x,y)$ for some nowhere-zero function $g$. In the same paper, this conjectured classification is verified in the case of symmetric functions $K$ and $Q$. By extending the graph-theoretic techniques of the paper, we show that under some surprisingly simple and natural conditions the conjecture remains valid even with the symmetry constraints relaxed. + By taking $\Lambda$ finite, the above problem, furthermore, reduces to that between two square matrices investigated in [Raphael Loewy, Principal minors and diagonal similarity of matrices, Linear Algebra and its Applications 78 (1986), 23--64]. Hence, our paper presents a simple non-linear-algebraic proof that uses only some elementary combinatorics and three simple algebraic identities involving $3$-cycles and $4$-cycles. + oai:arXiv.org:2302.02471v4 + math.CA math-ph math.MP - physics.optics - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ivan Fernandez-Corbaton - - - Sharp lower bound for the Monge-Amp\`ere torsion on convex sets - https://arxiv.org/abs/2601.12915 - arXiv:2601.12915v1 Announce Type: new -Abstract: The \emph{Monge-Amp\`ere} torsion deficit of an open, bounded convex set $\Omega\subset\R^n$ of class $C^2$ is the normalized gap between the value of the torsion functional evaluated on $\Omega$ and its value on the ball with the same $(n-1)$-quermassintegral as $\Omega$. Using the technique of the \emph{shape derivative}, we prove that the ratio between this deficit and to a geometric deficit arising from the \emph{Alexandrov-Fenchel inequality}, for any given family of open, bounded convex sets of $\R^n$ ($n\geq2$) of class $C^2$, smoothly converging to a ball, is bounded from below by a dimensional constant. We also show that this ratio is always bounded from above by a constant. - oai:arXiv.org:2601.12915v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Francesco Salerno + Harry Sapranidis Mantelos - On Kippenhahn curves of low rank partial isometries - https://arxiv.org/abs/2601.12923 - arXiv:2601.12923v1 Announce Type: new -Abstract: Conditions are established for rank three partial isometries to have circular components contained in their Kippenhahn curves. In particular, such matrices with circular numerical ranges are described. It is also established that the Gau-Wang-Wu conjecture holds for matrices under consideration. - oai:arXiv.org:2601.12923v1 - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - new + Hopf 2-algebras and Braided Monoidal 2-Categories + https://arxiv.org/abs/2304.07398 + arXiv:2304.07398v4 Announce Type: replace +Abstract: Following the theory of principal $\infty$-bundles of Niklaus-Schreiber-Steveson, we develop a homotopy categorification of Hopf algebras, which model quantum groups. We study their higher-representation theory in the setting of $\mathsf{2Vect}^{hBC}$, which is a homotopy refinement of the notion of 2-vector spaces due to Baez-Crans that allows for higher coherence data. We construct in particular the 2-quantum double as a homotopy double crossed product, and prove its duality and factorization properties. We also define and characterize "2-$R$-matrices", which can be seen as an extension of the usual notion of $R$-matrix in an ordinary Hopf algebra. We found that the 2-Yang-Baxter equations describe the braiding of extended defects in 4d, distinct from but not unlike the Zamolodchikov tetrahedron equations. The main results we prove in this paper is that the 2-representation 2-category of a weak 2-bialgebra is braided monoidal if it is equipped with a universal 2-$R$-matrix, and that our homotopy quantization admits the theory of Lie 2-bialgebras as a semiclassical limit. + oai:arXiv.org:2304.07398v4 + math.QA + math-ph + math.CT + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Nikita Popov, Eric Shen, Ilya M. Spitkovsky + Hank Chen, Florian Girelli - Stone Duality for Preordered Topological Spaces - https://arxiv.org/abs/2601.12932 - arXiv:2601.12932v1 Announce Type: new -Abstract: A preordered topological space is a topological space with a preordering. We exhibit a Stone-like duality for preordered topological spaces, Inspired by a similar duality for bitopological spaces, due to Jung-Moshier and Jakl, and by a duality for preordered sets due to Bonsangue, Jacobs and Kok. - oai:arXiv.org:2601.12932v1 - math.GN - math.CT - Wed, 21 Jan 2026 00:00:00 -0500 - new + Nijenhuis operators on 2D pre-Lie algebras and 3D associative algebras + https://arxiv.org/abs/2308.12121 + arXiv:2308.12121v3 Announce Type: replace +Abstract: In this paper, we describe all Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras. As an application, using these operators, we obtain solutions of the classical Yang-Baxter equation on the corresponding sub-adjacent Lie algebras. + oai:arXiv.org:2308.12121v3 + math-ph + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Jean Goubault-Larrecq + Xiaoguang Zou, Xiang Gao, Chuangchuang Kang, Jiafeng L\"u - On the Concavity of Tsallis Entropy along the Heat Flow - https://arxiv.org/abs/2601.12944 - arXiv:2601.12944v1 Announce Type: new -Abstract: We demonstrate the concavity of the Tsallis entropy along the heat flow for general dimensions, expanding upon the findings of Wu et al 2025 and Hung 2022, which were previously limited to the one-dimensional case. The core of the proof is a novel estimate of the terms in the second-order time derivative, and a rigorous validation of integration by parts. The resulting bound establishes a new functional inequality, which may be of interest for other areas of mathematical analysis. - oai:arXiv.org:2601.12944v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new + Beyond the Hodge Theorem: curl and asymmetric pseudodifferential projections + https://arxiv.org/abs/2309.02015 + arXiv:2309.02015v5 Announce Type: replace +Abstract: We develop a new approach to the study of spectral asymmetry. Working with the operator $\operatorname{curl}:=*\mathrm{d}$ on a connected oriented closed Riemannian 3-manifold, we construct, by means of microlocal analysis, the asymmetry operator -- a scalar pseudodifferential operator of order $-3$. The latter is completely determined by the Riemannian manifold and its orientation, and encodes information about spectral asymmetry. The asymmetry operator generalises and contains the classical eta invariant traditionally associated with the asymmetry of the spectrum, which can be recovered by computing its regularised operator trace. Remarkably, the whole construction is direct and explicit. + oai:arXiv.org:2309.02015v5 + math.DG + math-ph + math.AP + math.MP + math.SP + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Lukang Sun + 10.1112/jlms.70431 + Journal of the London Mathematical Society 113:1 (2026) e70431 + Matteo Capoferri, Dmitri Vassiliev - Random tree Besov priors: Data-driven regularisation parameter selection - https://arxiv.org/abs/2601.12957 - arXiv:2601.12957v1 Announce Type: new -Abstract: We develop a data-driven algorithm for automatically selecting the regularisation parameter in Bayesian inversion under random tree Besov priors. One of the key challenges in Bayesian inversion is the construction of priors that are both expressive and computationally feasible. Random tree Besov priors, introduced in Kekkonen et al. (2023), provide a flexible framework for capturing local regularity properties and sparsity patterns in a wavelet basis. In this paper, we extend this approach by introducing a hierarchical model that enables data-driven selection of the wavelet density parameter, allowing the regularisation strength to adapt across scales while retaining computational efficiency. We focus on nonparametric regression and also present preliminary plug-and-play results for a deconvolution problem. - oai:arXiv.org:2601.12957v1 - math.ST - math.PR - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - new + K\"{a}hler Solitons, Contact Structures, and Isoparametric Functions + https://arxiv.org/abs/2310.11328 + arXiv:2310.11328v3 Announce Type: replace +Abstract: All known examples of simply-connected gradient K\"{a}hler-Ricci soliton in real dimension four are toric, and the symmetry is intrinsically related to the potential function $f$ and the scalar curvature $\SS$. In this article, we consider the case that $f$ and $\SS$ are functionally dependent and deduce a complete classification, while the independence case is addressed elsewhere. The main theorem recovers all known examples of cohomogeneity one symmetry. We also discover a connection to the theory of isoparametric functions and contact geometry. Indeed, a key ingredient is a new characterization for a deformed Sasakian structure generalizing a classical result. + oai:arXiv.org:2310.11328v3 + math.DG + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Hung Tran + + + Sharp quantitative stability for the fractional Sobolev trace inequality + https://arxiv.org/abs/2312.01766 + arXiv:2312.01766v3 Announce Type: replace +Abstract: In this paper, we study the stability of fractional Sobolev trace inequality within both the functional and critical point settings. + In the functional setting, we establish the following sharp estimate: + $$C_{\mathrm{BE}}(n,m,\alpha)\inf_{v\in\mathcal{M}_{n,m,\alpha}}\left\Vert f-v\right\Vert_{D_\alpha(\mathbb{R}^n)}^2 \leq \left\Vert f\right\Vert_{D_\alpha(\mathbb{R}^n)}^2 - S(n,m,\alpha) \left\Vert\tau_mf\right\Vert_{L^{q}(\mathbb{R}^{n-m})}^2,$$ + where $0\leq m< n$, $\frac{m}{2}<\alpha<\frac{n}{2}, q=\frac{2(n-m)}{n-2\alpha}$ and $\mathcal{M}_{n,m,\alpha}$ denotes the manifold of extremal functions. Additionally, We find an explicit bound for the stability constant $C_{\mathrm{BE}}$ and establish a compactness result ensuring the existence of minimizers. + In the critical point setting, we investigate the validity of a sharp quantitative profile decomposition related to the Escobar trace inequality and establish a qualitative profile decomposition for the critical elliptic equation + \begin{equation*} + \Delta u= 0 \quad\text{in }\mathbb{R}_+^n,\quad\frac{\partial u}{\partial t}=-|u|^{\frac{2}{n-2}}u \quad\text{on }\partial\mathbb{R}_+^n. + \end{equation*} + We then derive the sharp stability estimate: + $$ + C_{\mathrm{CP}}(n,\nu)d(u,\mathcal{M}_{\mathrm{E}}^{\nu})\leq \left\Vert \Delta u +|u|^{\frac{2}{n-2}}u\right\Vert_{H^{-1}(\mathbb{R}_+^n)}, + $$ + where $\nu=1,n\geq 3$ or $\nu\geq2,n=3$ and $\mathcal{M}_{\mathrm{E}}^\nu$ represents the manifold consisting of $\nu$ weak-interacting Escobar bubbles. Through some refined estimates, we also give a strict upper bound for $C_{\mathrm{CP}}(n,1)$, which is $\frac{2}{n+2}$. + oai:arXiv.org:2312.01766v3 + math.AP + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Mathematische Zeitschrift (2025) + Yingfang Zhang, Yuxuan Zhou, Wenming Zou + + + Classification of positive solutions to the H\'enon-Sobolev critical systems + https://arxiv.org/abs/2312.01784 + arXiv:2312.01784v2 Announce Type: replace +Abstract: In this paper, we investigate positive solutions to the following H\'enon-Sobolev critical system: $$ + -\mathrm{div}(|x|^{-2a}\nabla u)=|x|^{-bp}|u|^{p-2}u+\nu\alpha|x|^{-bp}|u|^{\alpha-2}|v|^{\beta}u\quad\text{in }\mathbb{R}^n,$$ + $$ -\mathrm{div}(|x|^{-2a}\nabla v)=|x|^{-bp}|v|^{p-2}v+\nu\beta|x|^{-bp}|u|^{\alpha}|v|^{\beta-2}v\quad\text{in }\mathbb{R}^n,$$ + $$u,v\in D_a^{1,2}(\mathbb{R}^n),$$ + where $n\geq 3,-\infty< a<\frac{n-2}{2},a\leq b<a+1,p=\frac{2n}{n-2+2(b-a)},\nu>0$ and $\alpha>1,\beta>1$ satisfying $\alpha+\beta=p$. Our findings are divided into two parts, according to the sign of the parameter $a$. + For $a\geq 0$, we demonstrate that any positive solution $(u,v)$ is synchronized, indicating that $u$ and $v$ are constant multiples of positive solutions to the decoupled H\'enon equation: + \begin{equation*} + -\mathrm{div}(|x|^{-2a}\nabla w)=|x|^{-bp}|w|^{p-2}w. + \end{equation*} + For $a<0$ and $b>a$, we characterize all nonnegative ground states. Additionally, we study the nondegeneracy of nonnegative synchronized solutions. + This work also delves into some general $k$-coupled H\'enon-Sobolev critical systems. + oai:arXiv.org:2312.01784v2 + math.AP + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Hanne Kekkonen, Andreas Tataris + Journal of Differential Equations 422 (2025) + Yuxuan Zhou, Wenming Zou - On cohomological dimensions of totally disconnected locally compact groups - https://arxiv.org/abs/2601.12958 - arXiv:2601.12958v1 Announce Type: new -Abstract: In this paper, we introduce Mackey functors for a t.d.l.c. group and define the cohomological dimension of this group over the Mackey category. We then compare this dimension to the rational discrete cohomological dimension defined by Castellano and Weigel, as well as to the Bredon cohomological dimension of that t.d.l.c. group with respect to the family of compact open subgroups. We also extend results about the geometric dimension of a t.d.l.c. group. - oai:arXiv.org:2601.12958v1 - math.GR - Wed, 21 Jan 2026 00:00:00 -0500 - new + Differential operators on the base affine space of $SL_n$ and quantized Coulomb branches + https://arxiv.org/abs/2312.10278 + arXiv:2312.10278v2 Announce Type: replace +Abstract: We show that the algebra $D_\hbar(SL_n/U)$ of differential operators on the base affine space of $SL_n$ is the quantized Coulomb branch of a certain 3d $\mathcal{N} = 4$ quiver gauge theory. In the semiclassical limit this proves a conjecture of Dancer-Hanany-Kirwan about the universal hyperk\"ahler implosion of $SL_n$. We also formulate and prove a generalization identifying the Hamiltonian reduction of $T^* SL_n$ with respect to an arbitrary unipotent character as a Coulomb branch. As an application of our results, we provide a new interpretation of the Gelfand-Graev symmetric group action on $D_\hbar(SL_n/U)$. + oai:arXiv.org:2312.10278v2 + math.RT + math-ph + math.AG + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ilaria Castellano, Nadia Mazza, Brita Nucinkis + Tom Gannon, Harold Williams - Codes Correcting Few Restricted Errors - https://arxiv.org/abs/2601.12959 - arXiv:2601.12959v1 Announce Type: new -Abstract: We consider linear codes over a field in which the error values are restricted to a subgroup of its unit group. This scenario captures Lee distance codes as well as codes over the Gaussian or Eisenstein integers. Codes correcting restricted errors gained increased attention recently in the context of code-based cryptography. - In this work we provide new constructions of codes over the Gaussian or Eisenstein integers correcting two or three errors. We adapt some techniques from Roth and Siegel's work on codes for the Lee metric. We propose two construction methods, which may be seen of geometric and algebraic flavor, respectively. - oai:arXiv.org:2601.12959v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jens Zumbr\"agel + The stability on the Caffarelli-Kohn-Nirenberg and Hardy-type inequalities and beyond + https://arxiv.org/abs/2312.15735 + arXiv:2312.15735v3 Announce Type: replace +Abstract: In this paper, we establish several improved Caffarelli-Kohn-Nirenberg and Hardy-type inequalities. Our main results are divided into two parts. + In the first part, we consider the following Caffarelli-Kohn-Nirenberg inequality: \begin{equation*} + \left(\int_{\mathbb{R}^n}|x|^{-pa}|\nabla u|^pdx\right)^{\frac{1}{p}}\geq S(p,a,b)\left(\int_{\mathbb{R}^n}|x|^{-qb}|u|^qdx\right)^{\frac{1}{q}},\quad\forall\; u\in D_a^p(\mathbb{R}^n), \end{equation*} We establish gradient stability of this inequality in both functional and critical settings, and we derive some functional properties of the stability constant. Building on the gradient stability, we also obtain several refined Sobolev-type embeddings involving weak Lebesgue norms for functions supported in general domains. + In the second part, we focus on various classical Hardy-type inequalities, including the standard Hardy inequality, the $L^p$-logarithmic Sobolev inequality with weights, the logarithmic Hardy inequality, the Hardy-Morrey inequality, the Hardy-Sobolev interpolation inequality, and the interpolated Caffarelli-Kohn-Nirenberg inequality. We investigate their weighted versions and derive corresponding extremal functions, refinements, new remaining terms and stability constants. + oai:arXiv.org:2312.15735v3 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Journal of Differential Equations 450 (2026) + Yuxuan Zhou, Wenming Zou - Counting Irreducible polynomials with coefficients from thin subgroups - https://arxiv.org/abs/2601.12968 - arXiv:2601.12968v1 Announce Type: new -Abstract: L. Bary-Soroker and R. Shmueli (2026) have given an asymptotic formula for the number of irreducible polynomials over the finite fields $\mathbb F_q$ of $q$ elements, such that their coefficients are perfect squares in $\mathbb F_q$ and also extended this to classes of polynomials with coefficients described by finitely many unions of intersections of polynomial images. Here we use a different approach, which allows us to obtain another generalisation of this result to polynomials with coefficients from small subgroups of $\mathbb F_q^*$. As a demonstration of the power of our approach, we also use it to count such irreducible polynomials with an additional condition, namely, with a prescribed value of their discriminant. This generalisation seems to be unachievable via the approach of L. Bary-Soroker and R. Shmueli (2026). - oai:arXiv.org:2601.12968v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - new + Recursion relations and BPS-expansions in the HOMFLY-PT skein of the solid torus + https://arxiv.org/abs/2401.10730 + arXiv:2401.10730v2 Announce Type: replace +Abstract: Inspired by the skein valued open Gromov-Witten theory of Ekholm and Shende and the Gopakumar-Vafa formula, we associate to each pair of non-negative integers $(g,l)$ a formal power series with values in the HOMFLY-PT skein of a disjoint union of $l$ solid tori. The formal power series can be thought of as open BPS-states of genus $g$ with $l$ boundary components and reduces to the contribution of a single BPS state of genus $g$ for $l=0$. Using skein theoretic methods we show that the formal power series satisfy gluing identities and multi-cover skein relations corresponding to an elliptic boundary node of the underlying curves. For $(g,l)=(0,1)$ we prove a crossing formula which is the multi-cover skein relation corresponding to a hyperbolic boundary node, also known as the pentagon identity. + oai:arXiv.org:2401.10730v2 + math.QA + math.SG + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alina Ostafe, Igor E. Shparlinski - - - Bernstein type gradient estimate for system of weighted local heat equations with potential term - https://arxiv.org/abs/2601.12992 - arXiv:2601.12992v1 Announce Type: new -Abstract: In this article we provide Bernstein type gradient estimates for two system of local weighted heat type equations with potentials on a weighted Riemannian manifold. We derive all possible cases considering linear potential, exponential potential, combining with static manifold and evolving manifold. This work partially resolved the problem raised by Bhattacharyya et al. in \cite{SB-1}. - oai:arXiv.org:2601.12992v1 - math.AP - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-sa/4.0/ - Sujit Bhattacharyya + Lukas Nakamura - The distribution of the ratio of products of independent zero mean normal random variables - https://arxiv.org/abs/2601.12997 - arXiv:2601.12997v1 Announce Type: new -Abstract: Let $X_1,\ldots,X_M$ and $Y_1,\ldots,Y_N$ be independent zero mean normal random variables with variances $\sigma_{X_i}^2$, $i=1,\ldots,M$, and $\sigma_{Y_j}^2$, $j=1,\ldots,N$, respectively, and let $X=X_1\cdots X_M$ and $Y=Y_1\cdots Y_N$. In this paper, we derive the exact probability density function of the ratio $X/Y$. We apply this formula to derive exact formulas for the cumulative distribution function and the characteristic function. We also obtain further distributional properties, including asymptotic approximations for the probability density function, tail probabilities and the quantile function. - oai:arXiv.org:2601.12997v1 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Robert E. Gaunt, Heather L. Sutcliffe + Operational Methods Applied to the Spherical Mean and X-Ray Transform + https://arxiv.org/abs/2402.10272 + arXiv:2402.10272v4 Announce Type: replace +Abstract: We employ the framework of operational calculus to derive the operators associated with the spherical mean and a class of related averaging means of a function in $n$-dimensional space. Beginning with the classical definition of the spherical mean, we obtain a compact operator representation in terms of confluent hypergeometric functions of the Laplacian. This operator-based formulation provides a straightforward approach to the analysis of spherical means, allowing us to determine their power series expansions, construct series solutions to the corresponding inversion problems, derive the partial differential equations they satisfy, and give meaning to iterated and fractional spherical means. Finally, we apply the spherical mean operator to derive the inversion formula for the X-ray transform in an operational manner. + oai:arXiv.org:2402.10272v4 + math.CA + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Julius Lehmann - Weighted-Hamming Metric: Bounds and Codes - https://arxiv.org/abs/2601.12998 - arXiv:2601.12998v1 Announce Type: new -Abstract: The weighted-Hamming metric generalizes the Hamming metric by assigning different weights to blocks of coordinates. It is well-suited for applications such as coding over independent parallel channels, each of which has a different level of importance or noise. From a coding-theoretic perspective, the actual error-correction capability of a code under this metric can exceed half its minimum distance. In this work, we establish direct bounds on this capability, tightening those obtained via minimum-distance arguments. We also propose a flexible code construction based on generalized concatenation and show that these codes can be efficiently decoded up to a lower bound on the error-correction capability. - oai:arXiv.org:2601.12998v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new + Toughness and A{\alpha}-spectral radius in graphs + https://arxiv.org/abs/2402.17421 + arXiv:2402.17421v2 Announce Type: replace +Abstract: Let $\alpha\in[0,1)$, and let $G$ be a connected graph of order $n$ with $n\geq f(\alpha)$, where $f(\alpha)=6$ for $\alpha\in[0,\frac{2}{3}]$ and $f(\alpha)=\frac{4}{1-\alpha}$ for $\alpha\in(\frac{2}{3},1)$. A graph $G$ is said to be $t$-tough if $|S|\geq tc(G-S)$ for each subset $S$ of $V(G)$ with $c(G-S)\geq2$, where $c(G-S)$ is the number of connected components in $G-S$. The $A_{\alpha}$-spectral radius of $G$ is denoted by $\rho_{\alpha}(G)$. In this paper, it is verified that $G$ is a 1-tough graph unless $G=K_1\vee(K_{n-2}\cup K_1)$ if $\rho_{\alpha}(G)\geq\rho_{\alpha}(K_1\vee(K_{n-2}\cup K_1))$, where $\rho_{\alpha}(K_1\vee(K_{n-2}\cup K_1))$ equals the largest root of $x^{3}-((\alpha+1)n+\alpha-3)x^{2}+(\alpha n^{2}+(\alpha^{2}-\alpha-1)n-2\alpha+1)x-\alpha^{2}n^{2}+(3\alpha^{2}-\alpha+1)n-4\alpha^{2}+5\alpha-3=0$. Further, we present an $A_{\alpha}$-spectral radius condition for a graph to be a $t$-tough graph. + oai:arXiv.org:2402.17421v2 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sebastian Bitzer, Alberto Ravagnani, Violetta Weger + Sizhong Zhou, Yuli Zhang, Tao Zhang, Hongxia Liu - An iterative approach to a fluid-rigid body interaction problem - https://arxiv.org/abs/2601.13004 - arXiv:2601.13004v1 Announce Type: new -Abstract: We study a novel approach for the existence of solutions to an incompressible fluid-rigid body interaction problem in three dimensions. Our approach introduces an iteration based on a sequence of related problems posed on domains with prescribed evolution. In particular we prove the short-time existence of strong solutions to a system coupling the incompressible Navier--Stokes equations to the ordinary differential equations governing the motion of a rigid body, with no slip boundary conditions on the boundary of the rigid body, provided that the relative density $\frac{\rho}{\rho_B}$, is sufficiently small. We also discuss the use of our iterative approach in numerical methods for the moving boundary problem, and complement this with some numerical experiments in two dimensions which demonstrate the necessity of the smallness assumption on $\frac{\rho}{\rho_B}$. - oai:arXiv.org:2601.13004v1 - math.NA - cs.NA - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new + Unipotent normal subgroups of algebraic groups + https://arxiv.org/abs/2404.12221 + arXiv:2404.12221v2 Announce Type: replace +Abstract: Let $G$ be an affine algebraic group scheme over a field $k$. We show there exists a unipotent normal subgroup of $G$ which contains all other such subgroups; we call it the restricted unipotent radical $\mathrm{Rad}_u(G)$ of $G$. We investigate some properties of $\mathrm{Rad}_u(G)$, and study those $G$ for which $\mathrm{Rad}_u(G)$ is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine $k$-groups. + oai:arXiv.org:2404.12221v2 + math.GR + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Charles M. Elliott, Thomas Sales + Damian Sercombe - Isomorphism relations on classes of c.e. algebras - https://arxiv.org/abs/2601.13005 - arXiv:2601.13005v1 Announce Type: new -Abstract: We investigate the complexity of isomorphism relations for classes of finitely generated and n-generated computably enumerable (c.e.) algebras, presented via c.e. presentations -- that is, as quotients of term algebras over decidable sets of generators by c.e. congruences. Our goal is to develop a systematic framework for analyzing such isomorphism problems from a computability-theoretic perspective. To compare their complexity, we employ the notion of computable reducibility, measuring these relations against canonical benchmarks on c.e. sets, such as =^{ce}, E_0^{ce}, and the ordinal-indexed family E_min(\alpha). A central insight of our work is the interplay between the algebraic structure and the algorithmic complexity: we show that if every algebra in a class satisfies the ascending chain condition on its congruence lattice, then the corresponding isomorphism relation is computably reducible to =^{ce}. We also apply this framework to a range of concrete cases. In particular, we analyze the isomorphism relations for finitely generated commutative semigroups, monoids, and groups, positioning them within the broader landscape of classification problems. - oai:arXiv.org:2601.13005v1 - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fujita-Kato Solutions and Optimal Time Decay for the Vlasov-Navier-Stokes System in the Whole Space + https://arxiv.org/abs/2405.09937 + arXiv:2405.09937v2 Announce Type: replace +Abstract: We are concerned with the construction of global-in-time strong solutions for the incompressible Vlasov-Navier-Stokes system in the whole three-dimensional space. One of our goals is to establish that small initial velocities with critical Sobolev regularity and sufficiently well localized initial kinetic distribution functions give rise to global and unique solutions. This constitutes an extension of the celebrated result for the incompressible Navier-Stokes equations (NS) that has been established in 1964 by Fujita and Kato. If in addition the initial velocity is integrable, we establish that the total energy of the system decays to 0 with the optimal rate t^{-3/2}, like for the weak solutions of (NS). Our results partly rely on the use of a higher order energy functional that controls the regularity $H^1$ of the velocity and seems to have been first introduced by Li, Shou and Zhang in the context of nonhomogeneous Vlasov-Navier-Stokes system. In the small data case, we show that this energy functional decays with the rate t^{-5/2}. + oai:arXiv.org:2405.09937v2 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Meng-Che "Turbo" Ho, Martin Ritter, Luca San Mauro + Rapha\"el Danchin (LAMA) - Semi-infinite Lakshmibai--Seshadri paths and level-zero extremal weight modules over twisted quantum affine algebras - https://arxiv.org/abs/2601.13016 - arXiv:2601.13016v1 Announce Type: new -Abstract: In this paper, we study level-zero extremal weight modules over twisted quantum affine algebras. To this end, we introduce semi-infinite Lakshmibai--Seshadri paths associated with a level-zero dominant integral weight $\lambda$. We then show that the set $\tfrac{\infty}{2}\mathrm{LS}(\lambda)$ of semi-infinite LS paths of shape $\lambda$ is isomorphic, as a crystal, to the crystal basis $\mathcal{B}(\lambda)$ of the corresponding level-zero extremal weight module $V(\lambda)$. - oai:arXiv.org:2601.13016v1 - math.QA + A combinatorial interpretation of the Bernstein degree of unitary highest weight modules + https://arxiv.org/abs/2405.18766 + arXiv:2405.18766v2 Announce Type: replace +Abstract: Consider the $(\mathfrak{g}, K)$-modules $L_{\lambda}$ for unitary highest weight representations of the real reductive group $G_{\mathbb{R}} = \operatorname{U}(p,q)$, $\operatorname{Mp}(2n, \mathbb{R})$, or $\operatorname{O}^*(2n)$, where $\operatorname{Mp}(2n,\mathbb{R})$ denotes the metaplectic double cover of $\operatorname{Sp}(2n,\mathbb{R})$. Let $k$ be a positive integer. Corresponding to $G_{\mathbb{R}}$ via Howe duality is the compact group $\operatorname{U}(k)$, $\operatorname{O}(k)$, or $\operatorname{Sp}(k)$, respectively, for which every irreducible representation $\sigma$ corresponds to a unique $L_{\lambda} = L_{\lambda(\sigma)}$. Nishiyama-Ochiai-Taniguchi (2001) expressed the Bernstein degree $\operatorname{Deg} L_{\lambda(\sigma)}$ as the product of $\dim \sigma$ and the degree of the associated variety of $L_{\lambda(\sigma)}$; this result is valid when $k \leq r :=$ the real rank of $G_{\mathbb{R}}$. In this paper, for arbitrary $k$, we give a new combinatorial interpretation $\operatorname{Deg} L_{\lambda(\sigma)} = \#(\mathcal{Q}_k(\sigma) \times \mathcal{P}_k)$, where $\mathcal{Q}_k(\sigma)$ is a certain set of semistandard tableaux, whose cardinality (for $k \geq r$) interpolates between $\dim \sigma$ and the dimension of the simple $K$-module with highest weight $\lambda(\sigma)$. The set $\mathcal{P}_k$ consists of certain plane partitions that encode the Hilbert series of the associated variety. We exhibit analogous sets $\mathcal{P}_k$ of plane partitions for all real reductive groups of Hermitian type, including the exceptional groups. + oai:arXiv.org:2405.18766v2 + math.CO math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Shohei Adachi, Hayato Koike + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + William Q. Erickson, Markus Hunziker - Characterization of eigenfunctions of the Laplacian having exponential growth - https://arxiv.org/abs/2601.13017 - arXiv:2601.13017v1 Announce Type: new -Abstract: In 1993, Robert Strichartz proved a characterization for the bounded eigenfunctions of Laplacian $\Delta=-\sum_{j=1}^d \frac{\partial^2}{\partial x_j^2} $ on $\mathbb{R}^d$: If $\left\{f_k \right\}_{k\in \mathbb{Z}}$ be a doubly infinite sequence of functions on $\mathbb{R}^d$ such that $\Delta f_k=f_{k+1}$ and $ \|f_k\|_{L^{\infty}(\mathbb{R}^d)} \leq C$ for all $ k \in \mathbb{Z}$, for some $C>0$, then $f_0$ is an eigenfunction of $\Delta$. Observing the existence of unbounded eigenfunctions of the Laplacian, Howard and Reese generalized Strichartz's theorem to characterize eigenfunctions of the Laplacian having at most polynomial growth. In this article, we shall prove an extended version of Strichartz's theorem to characterize eigenfunctions of the Laplacian having exponential growth. - oai:arXiv.org:2601.13017v1 - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 - new + Crossed product splitting of intermediate operator algebras via 2-cocycles + https://arxiv.org/abs/2406.00304 + arXiv:2406.00304v2 Announce Type: replace +Abstract: We investigate the C*-algebra inclusions $B \subset A \rtimes_{\rm r} \Gamma$ arising from inclusions $B \subset A$ of $\Gamma$-C*-algebras. The main result shows that, when $B \subset A$ is C*-irreducible in the sense of R{\o}rdam, and is centrally $\Gamma$-free in the sense of the author, then after tensoring with the Cuntz algebra $\mathcal{O}_2$, all intermediate C*-algebras $B \subset C\subset A \rtimes_{\rm r} \Gamma$ enjoy a natural crossed product splitting \[\mathcal{O}_2\otimes C=(\mathcal{O}_2 \otimes D) \rtimes_{{\rm r}, \gamma, \mathfrak{w}} \Lambda\] for $D:= C \cap A$, some $\Lambda<\Gamma$, and a subsystem $(\gamma, \mathfrak{w})$ of a unitary perturbed cocycle action $\Lambda \curvearrowright \mathcal{O}_2\otimes A$. As an application, we give a new Galois's type theorem for the Bisch--Haagerup type inclusions \[A^K \subset A\rtimes_{\rm r} \Gamma\] for actions of compact-by-discrete groups $K \rtimes \Gamma$ on simple C*-algebras. + Due to a K-theoretical obstruction, the operation $\mathcal{O}_2\otimes -$ is necessary to obtain the clean splitting. Also, in general 2-cocycles $\mathfrak{w}$ appearing in the splitting cannot be removed even further tensoring with any unital (cocycle) action. We show them by examples, which further show that $\mathcal{O}_2$ is a minimal possible choice. + We also establish a von Neumann algebra analogue, where $\mathcal{O}_2$ is replaced by the type I factor $\mathbb{B}(\ell^2(\mathbb{N}))$. + oai:arXiv.org:2406.00304v2 + math.OA + math.DS + math.GR + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Basil Paul, Pradeep Boggarapu - - - The Reduced Phase Space of $N=1, D=4$ Supergravity in the BV-BFV formalism - https://arxiv.org/abs/2601.13025 - arXiv:2601.13025v1 Announce Type: new -Abstract: This paper describes the reduced phase space of $N=1$, $D=4$ supergravity in the fully off-shell Palatini--Cartan formalism. This is achieved through the KT construction, allowing an explicit description of first-class constraints on the boundary. The corresponding BFV description is obtained, and its relation with the BV one in the bulk is described by employing the BV pushforward in the particular example of a cylindrical spacetime. - oai:arXiv.org:2601.13025v1 - math-ph - gr-qc - hep-th - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Alberto S. Cattaneo, Filippo Fila-Robattino + Yuhei Suzuki - Multi-gear bandits, partial conservation laws, and indexability - https://arxiv.org/abs/2601.13026 - arXiv:2601.13026v1 Announce Type: new -Abstract: This paper considers what we propose to call multi-gear bandits, which are Markov decision processes modeling a generic dynamic and stochastic project fueled by a single resource and which admit multiple actions representing gears of operation naturally ordered by their increasing resource consumption. The optimal operation of a multi-gear bandit aims to strike a balance between project performance costs or rewards and resource usage costs, which depend on the resource price. A computationally convenient and intuitive optimal solution is available when such a model is indexable, meaning that its optimal policies are characterized by a dynamic allocation index (DAI), a function of state--action pairs representing critical resource prices. Motivated by the lack of general indexability conditions and efficient index-computing schemes, and focusing on the infinite-horizon finite-state and -action discounted case, we present a verification theorem ensuring that, if a model satisfies two proposed PCL-indexability conditions with respect to a postulated family of structured policies, then it is indexable and such policies are optimal, with its DAI being given by a marginal productivity index computed by a downshift adaptive-greedy algorithm in $A N$ steps, with $A+1$ actions and $N$ states. The DAI is further used as the basis of a new index policy for the multi-armed multi-gear bandit problem. - oai:arXiv.org:2601.13026v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.3390/math10142497 - \emph{Mathematics} \textbf{10}, 2497 (2022) - Jos\'e Ni\~no-Mora + Surface groups among cubulated hyperbolic and one-relator groups + https://arxiv.org/abs/2406.02121 + arXiv:2406.02121v3 Announce Type: replace +Abstract: Let $X$ be a non-positively curved cube complex with hyperbolic fundamental group. We prove that $\pi_1(X)$ has a non-free subgroup of infinite index unless $\pi_1(X)$ is either free or a surface group, answering questions of Gromov and Whyte (in a special case) and Wise. A similar result for one-relator groups follows, answering a question posed by several authors. The proof relies on a careful analysis of free and cyclic splittings of cubulated groups. + oai:arXiv.org:2406.02121v3 + math.GR + math.GT + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Henry Wilton - Optimality Conditions for Sparse Bilinear Least Squares Problems - https://arxiv.org/abs/2601.13027 - arXiv:2601.13027v1 Announce Type: new -Abstract: The first-order optimality conditions of sparse bilinear least squares problems are studied. The so-called T-type and N-type stationary points for this problem are characterized in terms of tangent cone and normal cone in Bouligand and Clarke senses, and another stationarity concept called the coordinate-wise minima is introduced and discussed. Moreover, the L-like stationary point for this problem is introduced and analyzed through the newly introduced concept of like-projection, and the M-stationary point is also investigated via a complementarity-type reformulation of the problem. The relationship between these stationary points is discussed as well. It turns out that all stationary points discussed in this work satisfy the necessary optimality conditions for the sparse bilinear least squares problem. - oai:arXiv.org:2601.13027v1 + Exact worst-case convergence rates of gradient descent: a complete analysis for all constant stepsizes over nonconvex and convex functions + https://arxiv.org/abs/2406.17506 + arXiv:2406.17506v2 Announce Type: replace +Abstract: We consider gradient descent with constant stepsizes and derive exact worst-case convergence rates on the minimum gradient norm of the iterates. Our analysis covers all possible stepsizes and arbitrary upper/lower bounds on the curvature of the objective function, thus including convex, strongly convex and weakly convex (hypoconvex) objective functions. + Among the challenging parts of the analysis, we note the necessity to exploit dependencies between non-consecutive iterates. While this complicates the proofs to some extent, it enables us to achieve an exact full-range analysis of gradient descent for any constant stepsize (covering, in particular, normalized stepsizes greater than one), whereas the literature contained only conjectured rates of this type. + In the nonconvex case, allowing arbitrary bounds on upper and lower curvatures extends existing partial results that are valid only for gradient Lipschitz functions (i.e., where lower and upper bounds on curvature are equal), leading to improved rates for weakly convex functions. + From our exact worst-case performance bounds, we deduce the optimal constant stepsize for gradient descent. Leveraging our analysis, we also introduce a new variant of gradient descent based on a unique, fixed sequence of variable stepsizes, demonstrating its superiority in the worst-case over any constant stepsize schedule. + oai:arXiv.org:2406.17506v2 math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zixin Deng, Zheng-Hai Huang, Yun-Bin Zhao - - - Generalized MICZ-Kepler systems on three-dimensional sphere and hyperboloid - https://arxiv.org/abs/2601.13028 - arXiv:2601.13028v1 Announce Type: new -Abstract: We propose analogs of the generalized MICZ-Kepler system on the three-dimensional sphere and (two-sheet) hyperboloid. We then construct their energy spectra and normalized wave functions, concluding that the suggested systems are minimally superintegrable. - oai:arXiv.org:2601.13028v1 - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Levon Mardoyan, Armen Nersessian + Teodor Rotaru, Fran\c{c}ois Glineur, Panagiotis Patrinos - Complete orbit equivalence relation and non-universal Polish groups - https://arxiv.org/abs/2601.13030 - arXiv:2601.13030v1 Announce Type: new -Abstract: We show that a non-universal Polish group can induce a complete orbit equivalence relation, which answers a question of Sabok from \cite{OPENPROBLEMS}. - oai:arXiv.org:2601.13030v1 - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 - new + Totally symmetric Grassmannian codes + https://arxiv.org/abs/2406.19542 + arXiv:2406.19542v2 Announce Type: replace +Abstract: We introduce a general technique to construct tight fusion frames with prescribed symmetries. Applying this technique with a prescription for "all the symmetries", we construct a new family of equi-isoclinic tight fusion frames (EITFFs), which consequently form optimal Grassmannian codes. By virtue of their construction, our EITFFs have the remarkable property of total symmetry: any permutation of subspaces can be achieved by an appropriate unitary. + oai:arXiv.org:2406.19542v2 + math.CO + cs.IT + math.FA + math.GR + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Longyun Ding, Ruiwen Li, Bo Peng + Matthew Fickus, Joseph W. Iverson, John Jasper, Dustin G. Mixon - On the additive index of the Diffie-Hellman mapping and the discrete logarithm - https://arxiv.org/abs/2601.13034 - arXiv:2601.13034v1 Announce Type: new -Abstract: Several complexity measures such as degree, sparsity and multiplicative index for cryptographic functions including the Diffie-Hellman mapping and the discrete logarithm in a finite field have been studied in the literature. In 2022, Reis and Wang introduced another complexity measure, the additive index, of a self-mapping of a finite field. In this paper, under certain conditions, we determine lower bounds on the additive index of the univariate Diffie-Hellman mapping and a self-mapping of $\mathbb{F}_q$ which can be identified with the discrete logarithm in a finite field. - oai:arXiv.org:2601.13034v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - new + The convergence and uniqueness of a discrete-time nonlinear Markov chain + https://arxiv.org/abs/2407.00314 + arXiv:2407.00314v2 Announce Type: replace +Abstract: In this paper, we prove the convergence and uniqueness of a general discrete-time nonlinear Markov chain with specific conditions. The results have important applications in discrete differential geometry. First, we prove the discrete-time Ollivier Ricci curvature flow $d_{n+1}:=(1-\alpha\kappa_{d_{n}})d_{n}$ converges to a constant curvature metric on a finite weighted graph. As shown in \cite[Theorem 5.1]{M23}, a Laplacian separation principle holds on a locally finite graph with nonnegative Ollivier curvature. We further prove that the Laplacian separation flow converges to the constant Laplacian solution and generalize the result to nonlinear $p$-Laplace operators. Moreover, our results can also be applied to study the long-time behavior in the nonlinear Dirichlet forms theory and nonlinear Perron-Frobenius theory. Finally, we define the Ollivier Ricci curvature of the nonlinear Markov chain which is consistent with the classical Ollivier Ricci curvature, sectional curvature \cite{CMS24}, coarse Ricci curvature on hypergraphs \cite{IKTU21} and the modified Ollivier Ricci curvature for $p$-Laplace. We also establish the convergence results for the nonlinear Markov chain with nonnegative Ollivier Ricci curvature. + oai:arXiv.org:2407.00314v2 + math.DS + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Pierre-Yves Bienvenu, Arne Winterhof - - - Classification of quaternionic skew-Hermitian symmetric spaces - https://arxiv.org/abs/2601.13036 - arXiv:2601.13036v1 Announce Type: new -Abstract: We provide a complete classification of quaternionic skew-Hermitian symmetric spaces, namely symmetric spaces that admit a torsion-free ${\rm SO}^{*}(2n){\rm Sp}(1)$-structure for arbitrary $n>1$. Moreover, we prove that any homogeneous quaternionic skew-Hermitian manifold is necessarily a symmetric space. - oai:arXiv.org:2601.13036v1 - math.DG - math.SG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ioannis Chrysikos, Jan Gregorovi\v{c} - - - Solving Generalized Lyapunov Equations with guarantees: application to the Model Reduction of Switched Linear Systems - https://arxiv.org/abs/2601.13039 - arXiv:2601.13039v1 Announce Type: new -Abstract: We present an efficient strategy to approximate the solutions of large-scale generalized Lyapunov equations (GLEs) with rigorous, computable error guarantees. This work is motivated by applications in model order reduction (MOR) of switched linear systems (SLS) in control form, where GLEs play a central role. We analyze how inaccuracies in the numerical solution of GLEs propagate through the MOR procedure and affect the accuracy and reliability of the reduced order model. Furthermore, the classical balanced-truncation error estimate for SLS is neither theoretically nor practically viable, as they rely on restrictive assumptions requiring several requiring several linear matrix inequalities (LMI) to be satisfied exactly by numerically computed solutions of the GLEs. To overcome these limitation, we propose a new MOR framework for SLS, called piecewise balanced reduction (PBR). The method is based on solving multiple GLEs and the construction of projection matrices that are piecewise constant in time to appropriately balance and subsequently reduce the SLS. We extend the standard balanced-truncation error bounds and demonstrate that the PBR formulation allows us to control the error arising from the inexact LMI. In addition, our new error bound accounts for the influence of the piecewise constant time-varying projection matrices. Altogether, this renders the PBR approach for SLS applicable to a broad and flexible class of SLS. Numerical experiments are provided to corroborate our theoretical results. - oai:arXiv.org:2601.13039v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Mattia Manucci, Benjamin Unger + Ruowei Li, Florentin M\"unch - Markovian restless bandits and index policies: A review - https://arxiv.org/abs/2601.13045 - arXiv:2601.13045v1 Announce Type: new -Abstract: The restless multi-armed bandit problem is a paradigmatic modeling framework for optimal dynamic priority allocation in stochastic models of wide-ranging applications that has been widely investigated and applied since its inception in a seminal paper by Whittle in the late 1980s. The problem has generated a vast and fast-growing literature from which a significant sample is thematically organized and reviewed in this paper. While the main focus is on priority-index policies due to their intuitive appeal, tractability, asymptotic optimality properties, and often strong empirical performance, other lines of work are also reviewed. Theoretical and algorithmic developments are discussed, along with diverse applications. The main goals are to highlight the remarkable breadth of work that has been carried out on the topic and to stimulate further research in the field. - oai:arXiv.org:2601.13045v1 + Integer programs with nearly totally unimodular matrices: the cographic case + https://arxiv.org/abs/2407.09477 + arXiv:2407.09477v2 Announce Type: replace +Abstract: It is a notorious open question whether integer programs (IPs), with an integer coefficient matrix $M$ whose subdeterminants are all bounded by a constant $\Delta$ in absolute value, can be solved in polynomial time. We answer this question in the affirmative if we further require that, by removing a constant number of rows and columns from $M$, one obtains a submatrix $A$ that is the transpose of a network matrix. + Our approach focuses on the case where $A$ arises from $M$ after removing $k$ rows only, where $k$ is a constant. We achieve our result in two main steps, the first related to the theory of IPs and the second related to graph minor theory. + First, we derive a strong proximity result for the case where $A$ is a general totally unimodular matrix: Given an optimal solution of the linear programming relaxation, an optimal solution to the IP can be obtained by finding a constant number of augmentations by circuits of $[A\; I]$. + Second, for the case where $A$ is transpose of a network matrix, we reformulate the problem as a maximum constrained integer potential problem on a graph $G$. We observe that if $G$ is $2$-connected, then it has no rooted $K_{2,t}$-minor for $t = \Omega(k \Delta)$. We leverage this to obtain a tree-decomposition of $G$ into highly structured graphs for which we can solve the problem locally. This allows us to solve the global problem via dynamic programming. + oai:arXiv.org:2407.09477v2 + math.CO + cs.DM + cs.DS math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.3390/math11071639 - Mathematics, vol. 11, 1639 (2023) - Jos\'e Ni\~no-Mora + Manuel Aprile, Samuel Fiorini, Gwena\"el Joret, Stefan Kober, Micha{\l} T. Seweryn, Stefan Weltge, Yelena Yuditsky - Optimal existence of weak solutions for the generalised Navier-Stokes-Voigt equations - https://arxiv.org/abs/2601.13051 - arXiv:2601.13051v1 Announce Type: new -Abstract: In this study, we investigate the incompressible generalised Navier-Stokes-Voigt equations within a bounded domain $\Omega \subset \mathbb{R}^d$, where $d \geq 2$. The governing momentum equation is expressed as: \begin{align*} \partial_t(\boldsymbol{v} - \kappa \Delta \boldsymbol{v}) + \nabla \cdot (\boldsymbol{v} \otimes \boldsymbol{v}) + \nabla \pi - \nu \nabla \cdot \big( |\mathbf{D}(\boldsymbol{v})|^{p-2} \mathbf{D}(\boldsymbol{v}) \big) = \boldsymbol{f}. \end{align*} Here, for $d \in \{2,3\}$, $\boldsymbol{v}$ represents the velocity field, $\pi$ denotes the pressure, and $\boldsymbol{f}$ is the external forcing term. The constants $\kappa$ and $\nu$ correspond to the relaxation time and kinematic viscosity, respectively. The parameter $p \in (1, \infty)$ characterizes the fluid's flow behavior, and $\mathbf{D}(\boldsymbol{v})$ denotes the symmetric part of the velocity gradient $\nabla \boldsymbol{v}$. For the power-law exponent $p \in \big( \frac{2d}{d+2}, \infty \big)$, we establish the existence of a weak solution to the generalised Navier-Stokes-Voigt equations. Furthermore, we demonstrate that the weak solution is unique for the same range of the exponent $p$. The optimality of our results lies in the framework's use of a Gelfand triple, which allows the Aubin-Dubinskii lemma to yield strong convergence of approximate solutions, essential for existence and valid precisely for $p > \frac{2d}{d+2}$. - oai:arXiv.org:2601.13051v1 + Degenerate stability of critical points of the Caffarelli-Kohn-Nirenberg inequality along the Felli-Schneider curve + https://arxiv.org/abs/2407.10849 + arXiv:2407.10849v2 Announce Type: replace +Abstract: In this paper, we investigate the validity of a quantitative version of stability for the critical Hardy-H\'enon equation \begin{equation*} + H(u):=\div(|x|^{-2a}\nabla u)+|x|^{-pb}|u|^{p-2}u=0,\quad u\in D_a^{1,2}(\R^n), \end{equation*} \begin{equation*} + n\geq 2,\quad a<b<a+1,\quad a<\frac{n-2}{2},\quad p=\frac{2n}{n-2+2(b-a)}, \end{equation*} which is well known as the Euler-Lagrange equation of the classical Caffarelli-Kohn-Nirenberg inequality. Establishing quantitative stability for this equation amounts to finding a nonnegative function $F$ such that the estimate \begin{equation*} + \inf_{\substack{U_i\in\mathcal{M} + 1\leq i\leq\nu}}\norm*{u-\sum_{i=1}^\nu U_i}_{D_a^{1,2}(\R^n)}\leq C(a,b,n)F(\norm*{H(u)}_{D_a^{-1,2}(\R^n)}) \end{equation*} holds for any nonnegative function $u$ satisfying \begin{equation*} + \left(\nu-\frac{1}{2}\right)S(a,b,n)^{\frac{p}{p-2}}\leq\int_{\R^n}|x|^{-2a}|\nabla u|^2\mathrm{d}x\leq \left(\nu+\frac{1}{2}\right)S(a,b,n)^{\frac{p}{p-2}}. \end{equation*} Here $\nu\in\N_+$ and $\mathcal{M}$ denotes the set of positive solutions to this equation. When $(a,b)$ falls above the Felli-Schneider curve, Wei and Wu \cite{Wei} found an optimal $F$. Their proof relies heavily on the fact that $\mathcal{M}$ is non-degenerate. When $(a,b)$ falls on the Felli-Schneider curve, due to the absence of the non-degeneracy condition, it becomes complicated and technical to find a suitable $F$. In this paper, we focus on this case. When $\nu=1$, we obtain an optimal $F$. When $\nu\geq2$ and $u$ is not too degenerate, we also derive an optimal $F$. To our knowledge, the results in this paper provide the first instance of degenerate stability in the critical point setting. We believe that our methods will be useful in other works on degenerate stability. + oai:arXiv.org:2407.10849v2 math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Ankit Kumar, Hermenegildo Borges de Oliveira, Manil T. Mohan + Yuxuan Zhou, Wenming Zou - Period growth and co-context-free groups - https://arxiv.org/abs/2601.13058 - arXiv:2601.13058v1 Announce Type: new -Abstract: We study period growth in co-context-free groups, giving general results and looking at specific examples such as Thompson groups $T$ and $V$ and the Houghton groups $H_m$. Along the way, we give a refined upper bound on the word metric in Thompson $V$, as well as efficient algorithms to determine if elements of $V$ are torsion, and compute their order. We also adapt our algorithm to compute the rotation number of elements of $T$ and answer a question of D. Calegari. - oai:arXiv.org:2601.13058v1 - math.GR - Wed, 21 Jan 2026 00:00:00 -0500 - new + Counting points on generic character varieties + https://arxiv.org/abs/2409.04735 + arXiv:2409.04735v4 Announce Type: replace +Abstract: We count points on character varieties associated with punctured surfaces and regular semisimple generic conjugacy classes in reductive groups. We find that the number of points are palindromic polynomials. This suggests a $P=W$ conjecture for these varieties. We also count points on the corresponding additive character varieties and find that the number of points are also polynomials, which we conjecture have non-negative coefficients. These polynomials can be considered as the reductive analogues of the Kac polynomials of comet-shaped quivers. + oai:arXiv.org:2409.04735v4 + math.AG + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Alex Bishop, Corentin Bodart, Letizia Issini, Davide Perego + Masoud Kamgarpour, GyeongHyeon Nam, Bailey Whitbread, Stefano Giannini - Generalized Reproducing Kernel Banach Spaces: A Functional Analytic Framework for Abstract Neural Networks - https://arxiv.org/abs/2601.13062 - arXiv:2601.13062v1 Announce Type: new -Abstract: In this paper, we introduce a generalization of Reproducing Kernel Banach Spaces (RKBS), which we term \emph{Generalized Reproducing Kernel Banach Spaces} (GRKBS). The motivation stems from recent results showing that classical fully connected neural networks can be understood as finite-dimensional subspaces of RKBS. Our generalization extends this perspective to settings with Banach-valued codomains, allowing the construction of \emph{abstract neural networks} (AbsNN) as compositions of GRKBS. This framework provides a natural pathway to model neural architectures that go beyond classical machine learning paradigms, including physically-informed structures governed by differential equations. We establish a unified definition of GRKBS, prove structural uniqueness results, and analyze the existence of sparse minimizers for the corresponding abstract training problem. This contributes to bridging functional analytic theory and the design of new neural architectures with applications in both approximation theory and mathematical modeling. - oai:arXiv.org:2601.13062v1 - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - new + Polynomials Counting Group Colorings in Graphs + https://arxiv.org/abs/2409.12404 + arXiv:2409.12404v5 Announce Type: replace +Abstract: Jaeger et al. in 1992 introduced group coloring as the dual concept to group connectivity in graphs. Let $A$ be an additive Abelian group, $ f: E(G)\to A$ and $D$ an orientation of a graph $G$. A vertex coloring $c:V(G)\to A$ is an $(A, f)$-coloring if $c(v)-c(u)\ne f(e)$ for each oriented edge $e=uv$ from $u$ to $v$ under $D$. Kochol recently introduced the assigning polynomial to count nowhere-zero chains in graphs--nonhomogeneous analogues of nowhere-zero flows in \cite{Kochol2022}, and later extended the approach to regular matroids in \cite{Kochol2024}. Motivated by Kochol's work, we define the $\alpha$-compatible graph and the cycle-assigning polynomial $P(G, \alpha; k)$ at $k$ in terms of $\alpha$-compatible spanning subgraphs, where $\alpha$ is an assigning of $G$ from its cycles to $\{0,1\}$. We prove that $P(G,\alpha;k)$ evaluates the number of $(A,f)$-colorings of $G$ for any Abelian group $A$ of order $k$ and $f:E(G)\to A$ such that the assigning $\alpha_{D,f}$ given by $f$ equals $\alpha$. Such an assigning is admissible. Based on Kochol's work, we derive that $k^{-c(G)}P(G,\alpha;k)$ is a polynomial enumerating $(A,f)$-tensions and counting specific nowhere-zero chains. + Furthermore, by extending Whitney's broken cycle concept to broken compatible cycles, we show that the absolute value of the coefficient of $k^{|V(G)|-i}$ in $P(G,\alpha;k)$ associated with admissible assignings $\alpha$ equals the number of $\alpha$-compatible spanning subgraphs that have $i$ edges and contain no broken $\alpha$-compatible cycles. According to the combinatorial explanation, we establish a unified order-preserving relation from admissible assignings to cycle-assigning polynomials, and further show that for any admissible assigning $\alpha$ of $G$ with $\alpha(e)=1$ for every loop $e$, the coefficients of $P(G,\alpha;k)$ are nonzero and alternate in sign. + oai:arXiv.org:2409.12404v5 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Raul Felipe-Sosa + Houshan Fu - Two-timescale Optimization for Hybrid Mechanically and Electronically Tunable 6DMA Aided Communication - https://arxiv.org/abs/2601.13064 - arXiv:2601.13064v1 Announce Type: new -Abstract: This letter proposes a hybrid mechanically and electronically tunable six-dimensional movable antenna (6DMA) base station (BS) architecture for future wireless communication networks. Such BS consists of multiple antenna arrays that are mechanically movable along a circular rail to adapt to the horizontal user hotspots, and each array is equipped with pattern reconfigurable antennas (PRAs) that are capable of electronically switching among a set of specified beam patterns to cater to the instantaneous user channels. The mechanical adjustment provides wide-angle coverage but suffers from slow response, while the electronic tuning enables rapid beam reconfiguration but with limited angular range. To effectively combine their complementary advantages, we propose to jointly design both mechanical and electronic configurations to maximize the average sum-rate of users via a two-timescale optimization approach, in which the array positions are optimized on the long timescale according to large-scale user distribution statistics, and the pattern selection vectors are optimized on the short timescale to enable fast beam alignment based on the instantaneous user locations. An alternating optimization algorithm based on the Monte Carlo sampling method is developed to solve the problem efficiently. Finally, simulation results show that our proposed design achieves significant performance gains over various benchmark schemes. - oai:arXiv.org:2601.13064v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new + The Hilbert scheme of points on a threefold: broken Gorenstein structures and linkage + https://arxiv.org/abs/2409.17009 + arXiv:2409.17009v2 Announce Type: replace +Abstract: We investigate the Hilbert scheme of points on a smooth threefold. We introduce a notion of broken Gorenstein structure for finite schemes, and show that its existence guarantees smoothness on the Hilbert scheme. Moreover, we conjecture that it is exhaustive: every smooth point admits a broken Gorenstein structure. We give an explicit characterization of the smooth points on the Hilbert scheme of A^3 corresponding to monomial ideals. We investigate the nature of the singular points, and prove several conjectures by Hu. Along the way, we obtain a number of additional results, related to linkage classes, nested Hilbert schemes, and a bundle on the Hilbert scheme of a surface. + oai:arXiv.org:2409.17009v2 + math.AG + math.AC + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yuyan Zhou, Haocheng Hua, Jie Xu, Rui Zhang + Joachim Jelisiejew, Ritvik Ramkumar, Alessio Sammartano - Non-abelian Hodge correspondence over singular K\"ahler spaces - https://arxiv.org/abs/2601.13071 - arXiv:2601.13071v1 Announce Type: new -Abstract: In this paper, we establish the non-abelian Hodge correspondence over compact K\"ahler spaces with Kawamata log terminal (klt) singularities as well as over their regular loci, thereby extending the result of Greb-Kebekus-Peternell-Taji for projective klt varieties to the context of compact K\"ahler klt spaces. The proof relies on two key ingredients: first, we establish an equivalence over the regular loci-via harmonic bundles-between polystable Higgs bundles with vanishing orbifold Chern numbers and semi-simple flat bundles; second, we prove a descent theorem for semistable Higgs bundles with vanishing Chern classes along resolutions of singularities. As an application of our framework, we obtain a quasi-uniformization theorem for projective klt varieties with big canonical divisor that satisfy the orbifold Miyaoka-Yau equality. - oai:arXiv.org:2601.13071v1 - math.DG - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 - new + On Ulam widths of finitely presented infinite simple groups + https://arxiv.org/abs/2410.07512 + arXiv:2410.07512v2 Announce Type: replace +Abstract: A fundamental notion in group theory, which originates in an article of Ulam and von Neumann from $1947$ is uniform simplicity. A group $G$ is said to be $n$-uniformly simple for $n \in \mathbf{N}$ if for every $f,g\in G\setminus \{id\}$, there is a product of no more than $n$ conjugates of $g$ and $g^{-1}$ that equals $f$. Then $G$ is uniformly simple if it is $n$-uniformly simple for some $n \in \mathbf{N}$, and we refer to the smallest such $n$ as the Ulam width, denoted as $\mathcal{R}(G)$. If $G$ is simple but not uniformly simple, one declares $\mathcal{R}(G)=\infty$. In this article, we construct for each $n\in \mathbf{N}$, a finitely presented infinite simple group $G$ such that $n<\mathcal{R}(G)<\infty$. These are the first such examples among the class of finitely presented infinite simple groups. For the class of finitely generated (but not finitely presentable) infinite simple groups, the existence of such examples was settled in the work of Muranov. However, this had remained open for the class of finitely presented infinite simple groups. Our examples are also of type $F_{\infty}$, which means that they are fundamental groups of aspherical CW complexes with finitely many cells in each dimension. Uniformly simple groups are in particular uniformly perfect: there is an $n\in \mathbf{N}$ such that every element of the group can be expressed as a product of at most $n$ commutators of elements in the group. We also show that the analogous notion of width for uniform perfection is unbounded for our family of finitely presented infinite simple groups. To our knowledge, this is also the first such family. + oai:arXiv.org:2410.07512v2 + math.GR + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chuanjing Zhang, Shiyu Zhang, Xi Zhang + James Hyde, Yash Lodha - Faster 3-colouring algorithm for graphs of diameter 3 - https://arxiv.org/abs/2601.13072 - arXiv:2601.13072v1 Announce Type: new -Abstract: We show that given an $n$-vertex graph $G$ of diameter 3 we can decide if $G$ is $3$-colourable in time $2^{O(n^{2/3-\varepsilon})}$ for any $\varepsilon < 1/33$. This improves on the previous best algorithm of $2^{O((n\log n)^{2/3})}$ from D\k{e}bski, Piecyk and Rz\k{a}\.zewski [Faster 3-coloring of small-diameter graphs, ESA 2021]. - oai:arXiv.org:2601.13072v1 - math.CO - cs.DM - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Carla Groenland, Hidde Koerts, Sophie Spirkl + Sobolev estimates for the Keller-Segel system and applications to the JKO scheme + https://arxiv.org/abs/2410.15095 + arXiv:2410.15095v3 Announce Type: replace +Abstract: We prove $L^{\infty}_{t}W^{1,p}_{x}$ Sobolev estimates in the Keller-Segel system with linear diffusion in any dimensionby proving a functional inequality, inspired by the Brezis-Gallou\"et-Wainger inequality. These estimates are also valid at the discrete level in the Jordan-Kinderlehrer-Otto (JKO) scheme. By coupling this result with the diffusion properties of a functional according to Bakry-Emery theory, we deduce the $L^2_t H^{2}_{x}$ convergence of the scheme, thereby extending the recent result of Santambrogio and Toshpulatov in the context of the Fokker-Planck equation to the Keller-Segel system. + oai:arXiv.org:2410.15095v3 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Charles Elbar - Wasserstein geometry of nonnegative measures on finite Markov chains I: Gradient flow - https://arxiv.org/abs/2601.13073 - arXiv:2601.13073v1 Announce Type: new -Abstract: We investigate a Benamou--Brenier type transportation metric for nonnegative measures on a finite reversible Markov chain, which endows the space of measures with a Riemannian structure. Using this geometric framework, we identify a generalized heat equation with source as the gradient flow of the discrete entropy. Moreover, by means of a local \L{}ojasiewicz inequality, we prove exponential convergence of the flow to a unique equilibrium. Our results clarify the role of the Benamou--Brenier formulation in discrete optimal transport for nonnegative measures and provide a coherent geometric interpretation of generalized diffusion equations with source terms. - oai:arXiv.org:2601.13073v1 + The Calder\'on problem for third order nonlocal wave equations with time-dependent nonlinearities and potentials + https://arxiv.org/abs/2411.08657 + arXiv:2411.08657v3 Announce Type: replace +Abstract: In this article, we study the Calder\'on problem for nonlocal generalizations of the semilinear Moore--Gibson--Thompson (MGT) equation and the Jordan--Moore--Gibson--Thompson (JMGT) equation of Westervelt-type. These partial differential equations are third order wave equations that appear in nonlinear acoustics, describe the propagation of high-intensity sound waves and exhibit finite speed of propagation. For semilinear MGT equations with nonlinearity $g$ and potential $q$, we show the following uniqueness properties of the Dirichlet to Neumann (DN) map $\Lambda_{q,g}$: + (i) If $g$ is a polynomial-type nonlinearity whose $m$-th order derivative is bounded, then $\Lambda_{q,g}$ uniquely determines $q$ and $(\partial^{\ell}_\tau g(x,t,0))_{2\leq \ell \leq m}$. + (ii) If $g$ is a polyhomogeneous nonlinearity of finite order $L$, then $\Lambda_{q,g}$ uniquely determines $q$ and $g$. + The uniqueness proof for polynomial-type nonlinearities is based on a higher order linearization scheme, while the proof for polyhomogeneous nonlinearities only uses a first order linearization. Finally, we demonstrate that a first linearization suffices to uniquely determine Westervelt-type nonlinearities from the related DN maps. We also remark that all the unknowns, which we wish to recover from the DN data, are allowed to depend on time. + oai:arXiv.org:2411.08657v3 math.AP - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Qifan Mao, Xinyu Wang, Xiaoping Xue + Song-Ren Fu, Yongyi Yu, Philipp Zimmermann - Some results on the $\mathfrak{g}$-stability of surfaces with boundary - https://arxiv.org/abs/2601.13077 - arXiv:2601.13077v1 Announce Type: new -Abstract: In this paper, we investigate the geometric properties associated with the $\mathfrak{g}$-stability of surfaces with boundary whose null expansion satisfies $\Theta^{+} = h \geq 0$. First, we show that a $\mathfrak{g}$-stable hypersurface with free boundary admits a metric of positive scalar curvature with minimal boundary under suitable conditions. Second, for $\mathfrak{g}$-stable surfaces with free boundary, we derive an area estimate and determine the topology of the surface. Finally, we extend our free boundary results to the case of capillary boundary. - oai:arXiv.org:2601.13077v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Sanghun Lee + Berkovich Motives + https://arxiv.org/abs/2412.03382 + arXiv:2412.03382v3 Announce Type: replace +Abstract: We construct a theory of (etale) Berkovich motives. This is closely related to Ayoub's theory of rigid-analytic motives, but works uniformly in the archimedean and nonarchimedean setting. We aim for a self-contained treatment, not relying on previous work on algebraic or analytic motives. Applying the theory to discrete fields, one still recovers the etale version of Voevodsky's theory. Two notable features of our setting which do not hold in other settings are that over any base, the cancellation theorem holds true, and under only minor assumptions on the base, the stable $\infty$-category of motivic sheaves is rigid dualizable. + oai:arXiv.org:2412.03382v3 + math.AG + math.KT + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Peter Scholze - Wasserstein geometry of nonnegative measures on finite Markov chains II: Geodesic and duality formulae - https://arxiv.org/abs/2601.13080 - arXiv:2601.13080v1 Announce Type: new -Abstract: In this paper, we investigate the geodesic structure and the associated Kantorovich-type duality for a Benamou-Brenier-type transportation metric defined on the space of nonnegative measures over a finite reversible Markov chain. The metric is introduced through a dynamic formulation that combines transport and source costs along solutions of a nonconservative continuity equation, where mass variation is constrained to occur along a fixed strictly positive reference direction. We show that geodesics associated with this metric exhibit a non-locality property: almost every time, they are supported on the whole state space, independently of the choice of endpoints. Moreover, along optimal curves, the source term displays a characteristic temporal profile, with mass creation occurring at early times and subsequent decay as the curve approaches the target measure. As an application of this property, we compare our metric with the shift-transport distance and prove that the latter is always bounded above by our metric. Finally, we establish a Kantorovich-type duality formula in terms of Hamilton-Jacobi subsolutions, which provides a characterization of the metric and highlights the role of the momentum associated with geodesic curves. - oai:arXiv.org:2601.13080v1 + On a class of Nonlinear Grushin equations + https://arxiv.org/abs/2412.08039 + arXiv:2412.08039v2 Announce Type: replace +Abstract: In this paper, we study two kinds of nonlinear degenerate elliptic equations containing the Grushin operator. First, we prove radial symmetry and a decay rate at infinity of solutions to such a Grushin equation by using the moving plane method in combination with suitable integral inequalities. Applying similar methods, we obtain nonexistence results for solutions to a second type of Grushin equation in Euclidean half space. Finally, we derive a priori estimates and the existence for positive solutions to more general types of Grushin equations by employing blow up analysis and topological degree methods, respectively. + oai:arXiv.org:2412.08039v2 math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Qifan Mao, Xinyu Wang, Xiaoping Xue + Wolfram Bauer, Yawei Wei, Xiaodong Zhou - Further progress on Wojda's conjecture - https://arxiv.org/abs/2601.13085 - arXiv:2601.13085v1 Announce Type: new -Abstract: Two digraphs of order $n$ are said to pack if they can be found as edge-disjoint subgraphs of the complete digraph of order $n$. It is well established that if the sum of the sizes of the two digraphs is at most $2n-2$, then they pack, with this bound being sharp. However, it is sufficient for the size of the smaller digraph to be only slightly below $n$ for the sum of their sizes to significantly exceed this threshold while still guaranteeing the existence of a packing. - In 1985, Wojda conjectured that for any $2 \leq m \leq n/2$, if one digraph has size at most $n - m$ and the other has size less than $2n - \lfloor n/m \rfloor$, then the two digraphs pack. It was previously known that this conjecture holds for $m = \Omega(\sqrt{n})$. In this paper, we confirm it for $m \geq 93$ and $n \geq 31m$. - oai:arXiv.org:2601.13085v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Maciej Cisi\'nski, Andrzej \.Zak + Existence and Non-Uniqueness of Ergodic Leray-Hopf Solutions to the Stochastic Power-Law Flows + https://arxiv.org/abs/2412.08622 + arXiv:2412.08622v2 Announce Type: replace +Abstract: We study long time behavior of shear-thinning fluid flows in $d \geq 3$ dimensions, driven by additive stochastic forcing of trace class, with power-law indices ranging from $1$ to $ \frac{2d}{d+2}$. We particularly focus on Leray-Hopf solutions, i.e. on analytically weak solutions satisfying energy inequality. Introducing a new kind of energy related functional into the technique of convex integration enables the construction of infinitely many such solutions that are probabilistically strong for a certain initial value. Furthermore, we provide global i time estimates which lead to the existence of infinitely many stationary and even ergodic Leray--Hopf solutions. These results represent the first construction of Leray-Hopf solutions in the framework of stochastic shear-thinning fluids within this range of power-law indices. + oai:arXiv.org:2412.08622v2 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Stefanie Elisabeth Berkemeier - Brownian Loops and the Selberg Zeta Function - https://arxiv.org/abs/2601.13086 - arXiv:2601.13086v1 Announce Type: new -Abstract: We study the Brownian loop measure on hyperbolic surfaces for Brownian motion with a constant killing rate. We compute the mass of Brownian loops with killing in a free homotopy class and then relate the total mass of loops in all essential homotopy classes to the Selberg zeta function when the surface is geometrically finite. As an application, we provide a probabilistic interpretation of different notions of regularized determinants of Laplacian, in both the compact and infinite-area cases. - oai:arXiv.org:2601.13086v1 - math.PR - math.GT - math.SP - Wed, 21 Jan 2026 00:00:00 -0500 - new + $N=1$ super Virasoro tensor categories + https://arxiv.org/abs/2412.18127 + arXiv:2412.18127v3 Announce Type: replace +Abstract: We show that the category of $C_1$-cofinite modules for the universal $N=1$ super Virasoro vertex operator superalgebra $\mathcal{S}(c,0)$ at any central charge $c$ is locally finite and admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. For central charges $c^{\mathfrak{ns}}(t)=\frac{15}{2}-3(t+t^{-1})$ with $t\notin\mathbb{Q}$, we show that this tensor category is semisimple, rigid, and slightly degenerate, and we determine its fusion rules. For central charge $c^{\mathfrak{ns}}(1)=\frac{3}{2}$, we show that this tensor category is rigid and that its simple modules have the same fusion rules as $\mathrm{Rep}\,\mathfrak{osp}(1\vert 2)$, in agreement with earlier fusion rule calculations of Milas. Finally, for the remaining central charges $c^{\mathfrak{ns}}(t)$ with $t\in \mathbb{Q}^\times$, we show that the simple $\mathcal{S}(c^{\mathfrak{ns}}(t),0)$-module $\mathcal{S}_{2,2}$ of lowest conformal weight $h^{\mathfrak{ns}}_{2,2}(t)=\frac{3(t-1)^2}{8t}$ is rigid and self-dual, except possibly when $t^{\pm 1}$ is a negative integer or when $c^{\mathfrak{ns}}(t)$ is the central charge of a rational $N=1$ superconformal minimal model. + As $\mathcal{S}_{2,2}$ is expected to generate the category of $C_1$-cofinite $\mathcal{S}(c^{\mathfrak{ns}}(t),0)$-modules under fusion, rigidity of $\mathcal{S}_{2,2}$ is the first key step to proving rigidity of this category for general $t\in\mathbb{Q}^\times$. + oai:arXiv.org:2412.18127v3 + math.QA + math-ph + math.MP + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Roman Lemonde, Jian Wang + Thomas Creutzig, Robert McRae, Florencia Orosz Hunziker, Jinwei Yang - Dynamical boundaries of affine buildings: C*-simplicity and Poisson boundaries - https://arxiv.org/abs/2601.13092 - arXiv:2601.13092v1 Announce Type: new -Abstract: We investigate a class of groups acting on possibly exotic affine buildings $X$ and possessing good proximal properties. Such groups are termed of general type, and their dynamics is analyzed through their flag limit sets in the space of chambers at infinity of $X$. For a group $G$ of general type, we prove C*-simplicity by showing that its flag limit set $\Lambda_{\mathcal F}(G)$ is topologically free, minimal, and strongly proximal. When $\Lambda_{\mathcal F}(G)$ intersects all Schubert cells relative to a limit chamber, then it is a mean proximal space, in the sense that it carries a unique proximal stationary measure for any admissible probability measure on the acting group. Lattices are established as examples of groups of general type, and their Poisson boundaries are identified. The arguments rely on constructing an equivariant barycenter map from triples of chambers in generic position to the affine building. - oai:arXiv.org:2601.13092v1 - math.GR - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Corina Ciobotaru, Corentin Le Bars + Categorical Diffusion of Weighted Lattices + https://arxiv.org/abs/2501.03890 + arXiv:2501.03890v2 Announce Type: replace +Abstract: We introduce a categorical framework for diffusion on network-structured data valued in weighted lattices, extending the Laplacian paradigm beyond the category of Hilbert spaces. Central to our approach is the Lawvere Laplacian, an endofunctor on the category of cochains of a cellular sheaf enriched in a commutative unital quantale. We establish the Tarski-Lawvere Fixed Point Theorem, generalizing Tarski's classical result to show that the suffix and prefix points of a quantale-enriched endofunctor form complete weighted lattices. Leveraging this, we prove the Hodge-Lawvere Theorem, which identifies the suffix points of the Laplacian with weighted global sections, providing a geometric characterization of equilibria. Finally, we derive a discrete-time harmonic flow that evolves data toward these sections, offering a constructive method for information aggregation in systems ranging from discrete event processes to preference dynamics. + oai:arXiv.org:2501.03890v2 + math.CT + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Robert Ghrist, Miguel Lopez, Paige Randall North, Hans Riess - Quasi-maximal ideals and ring extensions - https://arxiv.org/abs/2601.13093 - arXiv:2601.13093v1 Announce Type: new -Abstract: Alan and al. defined and studied quasi-maximal ideals. We add a comprehensive characterization of these ideals, introducing submaximal ideals. The conductor of a finite minimal extension $R\subset S$ is quasi-maximal in $S$. This allows us to give a new characterization of these extensions. We also examine the links between quasi-maximal ideals and Badawi 2-absorbing ideals. - oai:arXiv.org:2601.13093v1 - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 - new + A domain decomposition strategy for natural imposition of mixed boundary conditions in port-Hamiltonian systems + https://arxiv.org/abs/2501.06107 + arXiv:2501.06107v4 Announce Type: replace +Abstract: In this contribution, a finite element scheme to impose mixed boundary conditions without introducing Lagrange multipliers is presented for hyperbolic systems described as port-Hamiltonian systems. The strategy relies on finite element exterior calculus and domain decomposition to interconnect two systems with dual input-output behavior. The spatial domain is split into two parts by introducing an arbitrary interface. Each subdomain is discretized with a mixed finite element formulation that introduces a uniform boundary condition in a natural way as the input. In each subdomain the finite element spaces are selected from a finite element subcomplex to obtain a stable discretization. The two systems are then interconnected together by making use of a feedback interconnection. This is achieved by discretizing the boundary inputs using appropriate spaces that couple the two formulations. The final systems include all boundary conditions explicitly and do not contain any Lagrange multiplier. Time integration is performed using the implicit midpoint or St\"ormer-Verlet scheme. The method can also be applied to semilinear systems containing algebraic nonlinearities. The proposed strategy is tested on different examples: geometrically exact intrinsic beam model, the wave equation, membrane elastodynamics and the Mindlin plate. Numerical tests assess the conservation properties of the scheme, the effectiveness of the methodology and its robustness against shear locking phenomena. + oai:arXiv.org:2501.06107v4 + math.NA + cs.NA + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Gabriel Picavet, Martine Picavet-L'Hermitte + 10.1016/j.apm.2026.116775 + S. D. M. de Jong, A. Brugnoli, R. Rashad, Y. Zhang, S. Stramigioli - Equiprojective polytopes in higher dimension - https://arxiv.org/abs/2601.13095 - arXiv:2601.13095v1 Announce Type: new -Abstract: A 3-dimensional polytope is called k-equiprojective if every planar projection along a direction non-parallel to any facet is a k-gon. In this article, we generalise equiprojectivity to higher dimensions and give a lower bound on the number of combinatorial types of equiprojective polytopes. We also establish the pathwise connectedness of a subset of the Grassmannian in the case of (d-2)-dimensional spaces with conditions on the explicit path. This makes it possible to extend the Hasan--Lubiw characterisation of equiprojectivity to higher dimensions. Equiprojectivity provides cases relevant to the study of the Shadow Vertex algorithm, showing there is no hope minimising the complexity of the projection. It also offers a reverse point of view on the usual study of planar projections of polytopes as the projections have a fixed size. - oai:arXiv.org:2601.13095v1 - math.CO - math.MG - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Alice Cousaert + Geometrization of the local Langlands correspondence, motivically + https://arxiv.org/abs/2501.07944 + arXiv:2501.07944v2 Announce Type: replace +Abstract: Based on the formalism of rigid-analytic motives of Ayoub--Gallauer--Vezzani, we extend our previous work with Fargues from $\ell$-adic sheaves to motivic sheaves. In particular, we prove independence of $\ell$ of the $L$-parameters constructed there. + oai:arXiv.org:2501.07944v2 + math.AG + math.NT + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Peter Scholze - Parallel mean curvature surfaces with constant contact angle along free boundaries - https://arxiv.org/abs/2601.13101 - arXiv:2601.13101v1 Announce Type: new -Abstract: We classify branched immersed disks in space forms with non-zero parallel mean curvature vector and non-orthogonal constant contact angle along the boundary in 4-dimensional space form. For higher codimensional case, we prove a codimension reduction theorem for branched immersed bordered Riemann surfaces of higher genus with multiple boundary components under the same parallel mean curvature and constant contact angle assumptions. Furthermore, we construct a family of explicit examples of branched minimal immersions satisfying the non-orthonormal constant contact angle free boundary condition, which demonstrate the sharpness of both the classification result and the codimension reduction result. - oai:arXiv.org:2601.13101v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new + Explosive appearance of cores and bootstrap percolation on lattices + https://arxiv.org/abs/2501.18976 + arXiv:2501.18976v2 Announce Type: replace +Abstract: Consider the process where the $n$ vertices of a square $2$-dimensional torus appear consecutively in a random order. We show that typically the size of the $3$-core of the corresponding induced unit-distance graph transitions from $0$ to $n-o(n)$ within a single step. Equivalently, by infecting the vertices of the torus in a random order, under two-neighbour bootstrap percolation, the size of the infected set transitions instantaneously from $o(n)$ to $n$. This hitting time result answers a question of Benjamini. + We also study the much more challenging and general setting of bootstrap percolation on two-dimensional lattices for a variety of finite-range infection rules. In this case, powerful but fragile bootstrap percolation tools such as the rectangles process and the Aizenman-Lebowitz lemma become unavailable. We develop a new method complementing and replacing these standard techniques, thus allowing us to prove the above hitting time result for a wide family of threshold bootstrap percolation rules on the $2$-dimensional square lattice, including neighbourhoods given by large $\ell^p$ balls for $p\in[1,\infty]$. + oai:arXiv.org:2501.18976v2 + math.CO + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rui Gao, Miaomiao Zhu + Ivailo Hartarsky, Lyuben Lichev - Stochastic Gradient Descent for Nonlinear Inverse Problems in Banach Spaces - https://arxiv.org/abs/2601.13110 - arXiv:2601.13110v1 Announce Type: new -Abstract: Stochastic gradient descent (SGD) and its variants are widely used and highly effective optimization methods in machine learning, especially for neural network training. By using a single datum or a small subset of the data, selected randomly at each iteration, SGD scales well to problem size and has been shown to be effective for solving large-scale inverse problems. In this work, we investigate SGD for solving nonlinear inverse problems in Banach spaces through the lens of iterative regularization. Under general assumptions, we prove almost sure convergence of the iterates to the minimum distance solution and show the regularizing property in expectation under an a priori stopping rule. Further, we establish convergence rates under the conditional stability assumptions for both exact and noisy data. Numerical experiments on Schlieren tomography and electrical impedance tomography are presented to show distinct features of the method. - oai:arXiv.org:2601.13110v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Bangti Jin, Zeljko Kereta, Yuxin Xia + Newton-Mandelbrot set and Murase-Mandelbrot set + https://arxiv.org/abs/2502.14872 + arXiv:2502.14872v3 Announce Type: replace +Abstract: We obtain four extended Newton's methods and three extended Mandelbrot's recurrence formulas from the Wasan (Japanese mathematics in the Edo period (1603-1868)). Furthermore, two extended Newton's methods relate to one of the extended Mandelbrot's recurrence formulas. We lead four types of extended Mandelbrot recurrence formulas. Next, we show that these become the same extended Mandelbrot set, and connected, closed set. These show the originality of Wasan. + oai:arXiv.org:2502.14872v3 + math.GM + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Shunji Horiguchi - Full characterization of core for nonlinear optimization games - https://arxiv.org/abs/2601.13124 - arXiv:2601.13124v1 Announce Type: new -Abstract: We fully characterize the core of a broad class of nonlinear games by identifying a suitable relaxation for inherent nonlinearity, directly generalizing the linear frameworks in the literature. This characterization significantly expands the scope of cooperative games that can be analyzed and contributes to the literature on games induced from optimization models. We apply these insights to not only establish connections with and provide new insights on classical models but also solve new games untamed in the existing literature, including combinatorial quadratic and ratio games such as portfolio, maximum cut, matching, and assortment games. These results are further extended to more general models and also the approximate core. - oai:arXiv.org:2601.13124v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Donglei Du, Qizhi Fang, Bin Liu, Tianhang Lu, Chenchen Wu + Local geometry of high-dimensional mixture models: Effective spectral theory and dynamical transitions + https://arxiv.org/abs/2502.15655 + arXiv:2502.15655v3 Announce Type: replace +Abstract: We study the local geometry of empirical risks in high dimensions via the spectral theory of their Hessian and information matrices. We focus on settings where the data, $(Y_\ell)_{\ell =1}^n \in \mathbb{R}^d$, are i.i.d. draws of a $k$-Gaussian mixture model, and the loss depends on the projection of the data into a fixed number of vectors, namely $\mathbf{x}^\top Y$, where $\mathbf{x}\in \mathbb{R}^{d\times C}$ are the parameters, and $C$ need not equal $k$. This setting captures a broad class of problems such as classification by one and two-layer networks and regression on multi-index models. We provide exact formulas for the limits of the empirical spectral distribution and outlier eigenvalues and eigenvectors of such matrices in the proportional asymptotics limit, where the number of samples and dimension $n,d\to\infty$ and $n/d=\phi \in (0,\infty)$. These limits depend on the parameters $\mathbf{x}$ only through the summary statistic of the $(C+k)\times (C+k)$ Gram matrix of the parameters and class means, $\mathbf{G} = (\mathbf{x},\boldsymbol{\mu})^\top(\mathbf{x},\boldsymbol{\mu})$. + It is known that under general conditions, when $\mathbf{x}$ is trained by online stochastic gradient descent, the evolution of these same summary statistics along training converges to the solution of an autonomous system of ODEs, called the effective dynamics. This enables us to connect the training dynamics to the spectral theory of these matrices generated with test data. We demonstrate our general results by analyzing the effective spectrum along the effective dynamics in the case of multi-class logistic regression. In this setting, the empirical Hessian and information matrices have substantially different spectra, each with their own static and even dynamical spectral transitions. + oai:arXiv.org:2502.15655v3 + math.ST + math.PR + stat.ML + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Gerard Ben Arous, Reza Gheissari, Jiaoyang Huang, Aukosh Jagannath - An algebro-geometric perspective on the topology of moduli spaces of differentials - https://arxiv.org/abs/2601.13127 - arXiv:2601.13127v1 Announce Type: new -Abstract: Differentials on Riemann surfaces correspond to translation surfaces with conical singularities, and affine transformations acting on them preserve the orders of these singularities. This viewpoint allows the moduli spaces of differentials to appear in various guises across many areas, including algebraic geometry, dynamical systems, combinatorial enumeration, and mathematical physics. Over the past few decades, remarkable progress has been made in computing invariants of these moduli spaces, classifying linear subvarieties, understanding degenerations and compactifications, and developing intersection theory on these spaces. Despite these advances, our understanding of the topology of moduli spaces of differentials remains limited, and many fundamental questions are still open. In this survey, we aim to present, from an algebro-geometric perspective, the known results and open problems concerning the topology of moduli spaces of differentials, as well as their connections to other aspects of the field, with the hope of inspiring further developments in the coming decade. - oai:arXiv.org:2601.13127v1 - math.AG - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 - new + Integer-valued valuations + https://arxiv.org/abs/2502.21144 + arXiv:2502.21144v2 Announce Type: replace +Abstract: We obtain a complete characterization of planar monotone $\sigma$-continuous valuations taking integer values, without assuming invariance under any group of transformations. We further investigate the consequences of dropping monotonicity or $\sigma$-continuity and give a full classification of line valuations. We also introduce a construction of the product for valuations of this type. + oai:arXiv.org:2502.21144v2 + math.MG + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Dawei Chen, Fei Yu + Andrii Ilienko, Ilya Molchanov, Tommaso Vison\`a - On the splitting of Neumann eigenvalues in perforated domains - https://arxiv.org/abs/2601.13129 - arXiv:2601.13129v1 Announce Type: new -Abstract: We address the problem of splitting of eigenvalues of the Neumann Laplacian under singular domain perturbations. We consider a domain perturbed by the excision of a small spherical hole shrinking to an interior point. Our main result establishes that the splitting of multiple eigenvalues is a generic property: if the center of the hole is located outside a set of Hausdorff dimension $N-1$ and the radius is sufficiently small, multiple eigenvalues split into branches of lower multiplicity. The proof relies on the validity of an asymptotic expansion for the perturbed eigenvalues in terms of the scaling parameter. Such an asymptotic formula is of independent interest and generalizes previous results; notably, in dimension $N\geq 3$, it is valid for holes of arbitrary shape. - oai:arXiv.org:2601.13129v1 + Uniqueness of gauge covariant renormalisation of stochastic 3D Yang-Mills-Higgs + https://arxiv.org/abs/2503.03060 + arXiv:2503.03060v2 Announce Type: replace +Abstract: Local solutions to the 3D stochastic quantisation equations of Yang-Mills-Higgs were constructed in (arXiv:2201.03487), and it was shown that, in the limit of smooth mollifications, there exists a mass renormalisation of the Yang-Mills field such that the solution is gauge covariant. In this paper we prove uniqueness of the mass renormalisation that leads to gauge covariant solutions. This strengthens the main result of (arXiv:2201.03487), and is potentially important for the identification of the limit of other approximations, such as lattice dynamics. Our proof relies on systematic short-time expansions of singular stochastic PDEs and of regularised Wilson loops. We also strengthen the recently introduced state spaces to allow finer control on line integrals appearing in expansions of Wilson loops. + oai:arXiv.org:2503.03060v2 + math.PR + math-ph math.AP + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/publicdomain/zero/1.0/ + Ilya Chevyrev, Hao Shen + + + Myrvold's Results on Orthogonal Triples of $10 \times 10$ Latin Squares: A SAT Investigation + https://arxiv.org/abs/2503.10504 + arXiv:2503.10504v2 Announce Type: replace +Abstract: Ever since E. T. Parker constructed an orthogonal pair of $10\times10$ Latin squares in 1959, an orthogonal triple of $10\times10$ Latin squares has been one of the most sought-after combinatorial designs. Despite extensive work, the existence of such an orthogonal triple remains an open problem, though some negative results are known. In 1999, W. Myrvold derived some highly restrictive constraints in the special case in which one of the Latin squares in the triple contains a $4\times4$ Latin subsquare. In particular, Myrvold showed there were twenty-eight possible cases for an orthogonal pair in such a triple, twenty of which were removed from consideration. We implement a computational approach that quickly verifies all of Myrvold's nonexistence results and in the remaining eight cases finds explicit examples of orthogonal pairs -- thus explaining for the first time why Myrvold's approach left eight cases unsolved. As a consequence, the eight remaining cases cannot be removed by a strategy of focusing on the existence of an orthogonal pair; the third square in the triple must necessarily be considered as well. + Our approach uses a Boolean satisfiability (SAT) solver to derive the nonexistence of twenty of the orthogonal pair types and find explicit examples of orthogonal pairs in the eight remaining cases. To reduce the existence problem into Boolean logic we use a duality between the concepts of transversal representation and orthogonal pair and we provide a formulation of this duality in terms of a composition operation on Latin squares. Using our SAT encoding, we find transversal representations (and equivalently orthogonal pairs) in the remaining eight cases in under two hours of computing on a large computing cluster. + oai:arXiv.org:2503.10504v2 + math.CO + cs.DM + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Curtis Bright, Amadou Keita, Brett Stevens + + + An isoperimetric inequality for lower order Neumann eigenvalues in Gauss space + https://arxiv.org/abs/2503.15813 + arXiv:2503.15813v5 Announce Type: replace +Abstract: We prove a sharp isoperimetric inequality for the harmonic mean of the first $m-1$ nonzero Neumann eigenvalues for bounded Lipschitz domains symmetric about the origin in Gauss space. Our result generalizes the Szeg\"o-Weinberger type inequality in Gauss space, as proved in [8, Theorem 4.1]. + oai:arXiv.org:2503.15813v5 math.SP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Veronica Felli, Lorenzo Liverani, Roberto Ognibene + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/publicdomain/zero/1.0/ + Yi Gao, Kui Wang - Age of information cost minimization with no buffers, random arrivals and unreliable channels: A PCL-indexability analysis - https://arxiv.org/abs/2601.13130 - arXiv:2601.13130v1 Announce Type: new -Abstract: Over the last decade, the Age of Information has emerged as a key concept and metric for applications where the freshness of sensor-provided data is critical. Limited transmission capacity has motivated research on the design of tractable policies for scheduling information updates to minimize Age of Information cost based on Markov decision models, in particular on the restless multi-armed bandit problem (RMABP). This allows the use of Whittle's popular index policy, which is often nearly optimal, provided indexability (index existence) is proven, which has been recently accomplished in some models. We aim to extend the application scope of Whittle's index policy in a broader AoI scheduling model. We address a model with no buffers incorporating random packet arrivals, unreliable channels, and nondecreasing AoI costs. We use sufficient indexability conditions based on partial conservation laws previously introduced by the author to establish the model's indexability and evaluate its Whittle index in closed form under discounted and average cost criteria. We further use the index formulae to draw insights on how scheduling priority depends on model parameters. - oai:arXiv.org:2601.13130v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.3390/math11204394 - Mathematics, vol. 11, 4394 (2023) - Jos\'e Ni\~no-Mora + The rationality problem for multinorm one tori + https://arxiv.org/abs/2504.04078 + arXiv:2504.04078v3 Announce Type: replace +Abstract: In this paper, we study the rationality problem for multinorm one tori, a natural generalization of norm one tori. For multinorm one tori that split over finite Galois extensions with nilpotent Galois group, we prove that stable rationality and retract rationality are equivalent, and give a criterion for the validity of the above two conditions. This generalizes the result of Endo (2011) on the rationality problem for norm one tori. To accomplish it, we introduce a generalization of character groups of multinorm one tori. Moreover, we establish systematic reduction methods originating in work of Endo (2001) for an investigation of the rationality problem for arbitrary multinorm one tori. In addition, we provide a new example for which the multinorm principle holds. + oai:arXiv.org:2504.04078v3 + math.AG + math.NT + math.RA + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Sumito Hasegawa, Kazuki Kanai, Yasuhiro Oki - The descriptive complexity of the set of arc-connected compact subsets of the plane - https://arxiv.org/abs/2601.13135 - arXiv:2601.13135v1 Announce Type: new -Abstract: We compute the exact complexity of the set of all arc-connected compact subsets of $\boldmath R^2$, which turns out to be strictly higher than the classical $\boldmath \Sigma^1_1$ and $\boldmath \Pi^1_1$ classes of analytic and coanalytic sets, but stricly lower than the class $\boldmath \Pi^1_2$ which is the exact descriptive class of the set of all arc-connected compact subsets of $\boldmath R^3$. - oai:arXiv.org:2601.13135v1 - math.GN - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Gabriel Debs, Jean Saint Raymond + On manifolds with almost non-negative Ricci curvature and integrally-positive $k^{th}$-scalar curvature + https://arxiv.org/abs/2504.06865 + arXiv:2504.06865v3 Announce Type: replace +Abstract: We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest $k$ eigenvalues of the Ricci tensor. + If $(M^n,g)$ is a Riemannian manifold satisfying such curvature bounds for $k=2$, then we show that $M$ is contained in a neighbourhood of controlled width of an isometrically embedded $1$-dimensional sub-manifold. From this, we deduce several metric and topological consequences: $M$ has at most linear volume growth and at most two ends, it has bounded 1-Urysohn width, the first Betti number of $M$ is bounded above by $1$, and there is precise information on elements of infinite order in $\pi_1(M)$. + If $(M^n,g)$ is a Riemannian manifold satisfying such bounds for $k\geq 2$, then we show that $M$ has at most $(k-1)$-dimensional behavior at large scales. + If $k=n={\rm dim}(M)$, so that the integral lower bound is on the scalar curvature, assuming in addition that the $(n-2)$-Ricci curvature is non-negative, we prove that the dimension drop at large scales improves to $n-2$. + From the above results we deduce topological restrictions, such as upper bounds on the first Betti number. + oai:arXiv.org:2504.06865v3 + math.DG + math.MG + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Alessandro Cucinotta, Andrea Mondino - Blackwell optimality in risk-sensitive stochastic control - https://arxiv.org/abs/2601.13136 - arXiv:2601.13136v1 Announce Type: new -Abstract: In this paper, we consider a discrete-time Markov Decision Process (MDP) on a finite state-action space with a long-run risk-sensitive criterion used as the objective function. We discuss the concept of Blackwell optimality and comment on intricacies which arise when the risk-neutral expectation is replaced by the risk-sensitive entropy. Also, we show the relation between the Blackwell optimality and ultimate stationarity and provide an illustrative example that helps to better understand the structural difference between these two concepts. - oai:arXiv.org:2601.13136v1 - math.OC - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new + Morita equivalences, moduli spaces and flag varieties + https://arxiv.org/abs/2504.09293 + arXiv:2504.09293v2 Announce Type: replace +Abstract: Double Bruhat cells in a connected complex semisimple Lie group $G$ emerged as a crucial concept in the work of S. Fomin and A. Zelevinsky on total positivity and cluster algebras. These cells are special instances of a broader class of cluster varieties known as generalized double Bruhat cells, which can be studied collectively as Poisson subvarieties of $\widetilde{F}_{2n} = \mathcal{B}^{2n-1} \times G$, where $\mathcal{B}$ is the flag variety of $G$. The spaces $\widetilde{F}_{2n}$ are Poisson groupoids over $\mathcal{B}^n$ and were introduced by J.-H. Lu, V. Mouquin, and S. Yu in the study of configuration Poisson groupoids of flags. + In this work, we describe the spaces $\widetilde{F}_{2n}$ as decorated moduli spaces of flat $G$-bundles over a disc. This perspective yields the following results: (1) We explicitly integrate the Poisson groupoids $\widetilde{F}_{2n}$ to symplectic double groupoids, which are complex algebraic varieties. Furthermore, we show that these integrations are symplectically Morita equivalent for all $n$. (2) Using this construction, we integrate the Poisson subgroupoids of $\widetilde{F}_{2n}$ formed by unions of generalized double Bruhat cells to explicit symplectic double groupoids. As a corollary, we obtain integrations for the top-dimensional generalized double Bruhat cells contained therein. (3) Finally, we relate our integration to the work of P. Boalch on meromorphic connections. We lift the torus actions on $\widetilde{F}_{2n}$ to the double groupoid level and show that they correspond to the quasi-Hamiltonian actions on the fission spaces of irregular singularities. + oai:arXiv.org:2504.09293v2 + math.SG + math.AG + math.DG + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1109/CoDIT66093.2025.11321525 - Marcin Pitera, {\L}ukasz Stettner + Daniel \'Alvarez - Classical Optimal Designs for Stationary Diffusion with Multiple Phases - https://arxiv.org/abs/2601.13149 - arXiv:2601.13149v1 Announce Type: new -Abstract: We study optimal design problems for stationary diffusion involving one or more state equations and mixtures of an arbitrary number of anisotropic materials. Since such problems typically do not admit classical solutions, we adopt a homogenization-based relaxation framework. - The objective considered is the maximization of a weighted sum of the energies associated with each state equation, with particular emphasis on identifying cases in which the optimal design is classical, that is, of bang-bang type, composed solely of the original pure materials. Such cases provide valuable benchmarks for numerical methods in optimal design. - A simplified optimization problem expressed in terms of local material proportions is analyzed through a dual formulation in terms of fluxes. Using a saddle-point characterization, we establish a complete description of its optimal solutions. The proposed approach is applied in detail to spherically symmetric problems. In the case of a ball, the method yields explicit classical solutions of the homogenization-based relaxation problem. - oai:arXiv.org:2601.13149v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new + Deza graphs and regular polyhedra + https://arxiv.org/abs/2504.19204 + arXiv:2504.19204v2 Announce Type: replace +Abstract: We classify all regular polyhedra according to their type i.e., the collection of numbers of common neighbours that any pair of distinct vertices may have (polyhedra are planar, $3$-connected graphs). As an application, we recover the classification of planar Deza graphs. + Next, we focus on the class of quartic polyhedral Deza graphs, and completely characterise it in terms of medial graphs of certain specific cubic polyhedra. Furthermore, within the aforementioned class of quartic polyhedral Deza graphs, we study the extremal graphs with respect to the ratio of number of triangular faces to the total. In the maximal extreme, these notably coincide with the class of line graphs of cubic polyhedra of girth $5$. + We also fully characterise the quartic polyhedra of type $\{0,1,2,3\}$, and in particular we prove that none of them are medial graphs. + On one hand our findings fit within the novel research area of common neighbours in graphs. On the other hand, our findings imply general properties of regular planar graphs and regular polyhedra. + oai:arXiv.org:2504.19204v2 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Matko Grbac, Ivan Ivec, Marko Vrdoljak + Riccardo W. Maffucci - Factoriality of normal projective varieties - https://arxiv.org/abs/2601.13151 - arXiv:2601.13151v1 Announce Type: new -Abstract: For a normal projective variety $X$, the $\bf Q$-factoriality defect $\sigma(X)$ is defined to be the rank of the quotient of the group of Weil divisors by the subgroup of Cartier ones. We prove a slight improvement of a topological formula of S.G. Park and M. Popa asserting that $\sigma(X)=h^{2n-2}(X)-h^2(X)$ by assuming only 2-semi-rationality, that is, $R^k\pi_*{\mathcal O}_{\widetilde{X}}=0$ for $k=1,2$, instead of rational singularities for $X$, where $\pi:\widetilde{X}\to X$ is a desingularization with $h^k(X):=\dim H^k(X,{\bf Q})$ and $n:=\dim X>2$. Our proof generalizes the one by Y. Namikawa and J.H.M. Steenbrink for the case $n=3$ with isolated hypersurface singularities. We also give a proof of the assertion that $\bf Q$-factoriality implies factoriality if $X$ is a local complete intersection whose singular locus has at least codimension three. (This seems to be known to specialists in the case $X$ has only isolated hypersurface singularities with $n=3$ using Milnor's Bouquet theorem.) These imply another proof of Grothendieck's theorem in the projective case asserting that $X$ is factorial if $X$ is a local complete intersection whose singular locus has at least codimension four. We can also prove a variant with factorial and local complete intersection replaced respectively by $\bf Q$-factorial and Cohen-Macaulay, where $\bf Q$-factorial cannot be replaced by factorial. - oai:arXiv.org:2601.13151v1 + On strong Euler-homogeneity and Saito-holonomicity for complex hypersurfaces. Applications to a conjecture on free divisors + https://arxiv.org/abs/2504.21829 + arXiv:2504.21829v4 Announce Type: replace +Abstract: We first develop some criteria for a general divisor to be strongly Euler-homogeneous in terms of the Fitting ideals of certain modules. We also study new variants of Saito-holonomicity, generalizing Koszul-free type properties and characterizing them in terms of the same Fitting ideals. + Thanks to these advances, we are able to make progress in the understanding of a conjecture from 2002: a free divisor satisfying the Logarithmic Comparison Theorem (LCT) must be strongly Euler-homogeneous. Previously, it was known to be true only for ambient dimension $n \leq 3$ or assuming Koszul-freeness. We prove it in the following new cases: assuming strong Euler-homogeneity outside a discrete set of points; assuming the divisor is weakly Koszul-free; for $n=4$; for linear free divisors in $n=5$. + Finally, we refute a conjecture stating that all linear free divisors satisfy LCT, are strongly Euler-homogeneous and have $b$-functions with symmetric roots about $-1$. + oai:arXiv.org:2504.21829v4 math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - new + math.CV + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Seung-Jo Jung, Morihiko Saito + Abraham del Valle Rodr\'iguez - Character degrees in $2$-blocks of $\mathfrak{S}_n$ and $\mathfrak{A}_n$ - https://arxiv.org/abs/2601.13152 - arXiv:2601.13152v1 Announce Type: new -Abstract: Let $p$ be an odd prime. We show that for sufficiently large $n$, every $2$-block of $\mathfrak{S}_n$ and $\mathfrak{A}_n$ contains an ordinary irreducible character of degree divisible by $p$. For almost all $2$-blocks of $\mathfrak{A}_n$, we classify whether it contains a rational valued ordinary irreducible character of degree divisible by $p$. - oai:arXiv.org:2601.13152v1 - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - new + Towards the Colmez Conjecture + https://arxiv.org/abs/2505.06541 + arXiv:2505.06541v2 Announce Type: replace +Abstract: We prove a collection of results involving Colmez's periods and the Colmez Conjecture. Using Colmez's theory of periods of CM abelian varieties, we propose a definition for the height of a partial CM-type and prove that the Colmez conjecture follows from an arithmetic period formula for surfaces. We give an explicit conjecture for the form of this period formula, which relates the height of special points on a Shimura surface with special values of $L$-functions. Further, we relate the heights of periods given by Colmez to arithmetic degree of Hermitian line bundles and thus give a formulation of Colmez's full conjecture in geometric terms. + oai:arXiv.org:2505.06541v2 + math.NT + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bim Gustavsson + Roy Zhao - On the characteristic function of the asymmetric Student's $t$-distribution and an integral involving the sine function - https://arxiv.org/abs/2601.13158 - arXiv:2601.13158v1 Announce Type: new -Abstract: We obtain a new closed-form formula for the characteristic function of the asymmetric Student's $t$-distribution. As part of our analysis, we derive a new closed-form formula for the integral $\int_0^\infty \sin(ax)/(b^2+x^2)^n\,\mathrm{d}x$, for $a,b>0$, $n\in\mathbb{Z}^+$, expressed in terms of the exponential integral function. As a consequence of our integral formula, we deduce a closed-form formula for the limit $\lim_{\nu\rightarrow n} \{I_{\nu-1/2}(x)-\mathbf{L}_{1/2-\nu}(x)\}/\sin(\pi\nu)$, for $n\in\mathbb{Z}^+$, $x>0$. - oai:arXiv.org:2601.13158v1 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Robert E. Gaunt + Configurations, Tessellations and Tone Networks + https://arxiv.org/abs/2505.08752 + arXiv:2505.08752v5 Announce Type: replace +Abstract: The Eulerian tonnetz, which associates three minor chords to each major chord and three major chords to each minor chord, can be represented by a bipartite graph with twelve white vertices denoting major chords and twelve black vertices denoting minor chords. This so-called Levi graph determines a configuration of twelve points and twelve lines in $\mathbb R^2$ with the property that three points lie on each line and three lines pass through each point. Interesting features of the tonnetz, such as the existence of the four hexatonic cycles and the three octatonic cycles, crucial for the understanding of nineteenth-century harmony and voice leading, can be read off directly as properties of this configuration $\{12_3\}$ and its Levi graph. Analogous tone networks together with their Levi graphs and configurations can be constructed for pentatonic music and twelve-tone music. These and other new tonnetze offer the promise of new methods of composition. If the constraints of the Eulerian tonnetz are relaxed so as to allow movements between major and minor triads with variations at exactly two tones, the resulting bipartite graph has two components, each generating a tessellation of the plane, of a type known to Kepler, based on hexagons, squares and dodecagons. When the same combinatorial idea is applied to tetrachords of the 'Tristan' genus (dominant sevenths and half-diminished sevenths) the cycles of the resulting bipartite graph are sufficiently ample in girth to ensure the existence of a second configuration $\{12_3\}$, distinct from the Eulerian tonnetz as an incidence geometry, which can be used for a new approach to the analysis of the rich tetradic harmonies of the nineteenth century common practice. + oai:arXiv.org:2505.08752v5 + math.CO + eess.AS + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Jeffrey R. Boland, Lane P. Hughston - On the discrete logarithmic Minkowski problem in the plane - https://arxiv.org/abs/2601.13159 - arXiv:2601.13159v1 Announce Type: new -Abstract: The paper characterizes the convex hull of the closure of the cone-volume set $C_\cv(U)$, consisting of all cone-volume vectors of polygons with outer unit normals vectors contained in $U$, for any finite set $U \subseteq \R^2, \pos(U) = \R^2$. We prove that this convex hull has finitely many extreme points by providing both a vertex representation as well as a half space representation. As a consequence, we derive new necessary conditions, which depend on $U$, for the existence of solutions to the logarithmic Minkowski problem in $\R^2$. - oai:arXiv.org:2601.13159v1 - math.MG - Wed, 21 Jan 2026 00:00:00 -0500 - new + Determining $t$-motives and dual $t$-motives in Anderson's theory + https://arxiv.org/abs/2505.12779 + arXiv:2505.12779v2 Announce Type: replace +Abstract: Anderson t-modules are analogs of abelian varieties in positive characteristic. Associated to such a t-module, there are its t-motive and its dual t-motive. When dealing with these objects, several questions occur which one would like to solve algorithmically. For example, for a given t-module one would like to decide whether its t-motive is indeed finitely generated free, and determine a basis. Reversely, for a given object in the category of t-motives one would like to decide whether it is the t-motive associated to a t-module, and determine that t-module. + In this article, we positively answer such questions by providing the corresponding algorithms. + As it turned out, the main part of all these algorithms stem from a single algorithm in non-commutative algebra, and hence the first part of this article doesn't deal with Anderson's objects at all, but are results on finitely generated modules over skew polynomial rings. + oai:arXiv.org:2505.12779v2 + math.NT + math.RA + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Andreas Maurischat + + + Formalising the Bruhat-Tits Tree + https://arxiv.org/abs/2505.12933 + arXiv:2505.12933v2 Announce Type: replace +Abstract: In this article we describe the formalisation of the Bruhat-Tits tree - an important tool in modern number theory - in the Lean Theorem Prover. Motivated by the goal of connecting to ongoing research, we apply our formalisation to verify a result about harmonic cochains on the tree. + oai:arXiv.org:2505.12933v2 + math.NT + cs.LO + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Tom Baumbach + Judith Ludwig, Christian Merten - A new notion of dimension for dynamical systems and shift embeddability - https://arxiv.org/abs/2601.13161 - arXiv:2601.13161v1 Announce Type: new -Abstract: A dynamical system $(X,T)$ is \emph{shift embaddable} if $(X,T)$ embeds continuously and equivariantly in the shift over $[0,1]^d$ for some finite $d$. Refuting a major conjecture in the field, in a recent result of Dranishnikov and Levin it was shown that Gromov's mean dimension and Lebesgue covering dimension of finite orbits are not the only obstructions for shift embaddability. - We present a new notion of dimension for dynamical systems over any countable group. We show that this new notion of dimension accounts for all known obstructions for shift embaddability. - oai:arXiv.org:2601.13161v1 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tom Meyerovitch + Stack-sorting preimages and 0-1-trees + https://arxiv.org/abs/2505.18295 + arXiv:2505.18295v2 Announce Type: replace +Abstract: We define a class of partially labeled trees and use them to find simple proofs for two recent enumeration results of Colin Defant concerning stack-sorting preimages of permutation classes. + oai:arXiv.org:2505.18295v2 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Miklos Bona - Nash approximation of differentiable semialgebraic maps - https://arxiv.org/abs/2601.13164 - arXiv:2601.13164v1 Announce Type: new -Abstract: Let $T\subset{\mathbb R}^n$ be a semialgebraic set and let $\mu\ge0$ be a non-negative integer. We say that $T$ is a {\em Nash $\mu$-approximation target space} (or a $({\mathcal N},\mu)$-${\tt ats}$ for short) if it has the following universal approximation property: {\em For each $m\in{\mathbb N}$ and each locally compact semialgebraic subset $S\subset{\mathbb R}^m$, the subspace of Nash maps ${\mathcal N}(S,T)$ is dense in the space ${\mathcal S}^\mu(S,T)$ of ${\mathcal C}^\mu$ semialgebraic maps between $S$ and $T$}. A necessary condition to be a $({\mathcal N},\mu)$-${\tt ats}$ is that $T$ is locally connected by analytic paths. In this paper we show: {\em Nash manifolds with corners are $({\mathcal N},\mu)$-${\tt ats}$ for each $\mu\geq0$}. As an application of a stronger version of the previous statement, we show that if two Nash maps $f,g:S\to Q$, where $S$ is a locally compact semialgebraic set of ${\mathbb R}^m$ and $Q$ is a Nash manifold with corners, are close enough in the (strong) Whitney's semialgebraic topology of ${\mathcal S}^0(S,T)$ (and consequently they are (continuous) semialgebraically homotopic), then $f,g$ are Nash homotopic. - oai:arXiv.org:2601.13164v1 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - new + The PML method for calculating the propagative wave numbers of electromagnetic wave in periodic structures + https://arxiv.org/abs/2506.07084 + arXiv:2506.07084v2 Announce Type: replace +Abstract: When the electromagnetic wave is incident on the periodic structures, in addition to the scattering field, some guided modes that are traveling in the periodic medium could be generated. In the present paper, we study the calculation of guided modes. We formulate the problem as a nonlinear eigenvalue problem in an unbounded periodic domain. Then we use perfectly matched layers to truncate the unbounded domain, recast the problem to a quadratic eigenvalue problem, and prove the approximation property of the truncation. Finally, we formulate the quadratic eigenvalue problem to a general eigenvalue problem, use the finite element method to discrete the truncation problem, and show numerical examples to verify theoretical results. + oai:arXiv.org:2506.07084v2 + math.NA + cs.NA + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Antonio Carbone, Jos\'e F. Fernando + Lide Cai, Junqing Chen, Yanpeng Gao - PDE aspects of the dynamical optimal transport in the Lorentzian setting - https://arxiv.org/abs/2601.13167 - arXiv:2601.13167v1 Announce Type: new -Abstract: One of the crucial features of optimal transport on Riemannian manifolds is the equivalence of the `static', original, formulation of the problem and of the `dynamic' one, based on the study of the continuity equation. This furnishes the key link between Wasserstein geometry and PDEs that has found so many applications in the last 20 years. - In this paper we investigate this kind of equivalence on spacetimes. At the PDE level, this requires to transition from the continuity equation to a suitable `continuity inequality', to which we shall refer to as `causal continuity inequality'. As a direct consequence of our findings we obtain a Lorentzian version of the celebrated Benamou--Brenier formula. - oai:arXiv.org:2601.13167v1 - math.AP - math.FA - math.MG - Wed, 21 Jan 2026 00:00:00 -0500 - new + On the Markoff spectrum on the Hecke group of index six + https://arxiv.org/abs/2506.08358 + arXiv:2506.08358v2 Announce Type: replace +Abstract: The discrete part of the Markoff spectrum on the Hecke group of index 6 was determined by A.~Schmidt. In this paper, we study its Markoff and Lagrange spectra after the smallest accumulation point $4/\sqrt3$. We show that both the Markoff and Lagrange spectra below $4/\sqrt{3} + \epsilon$ have positive Hausdorff dimension for any positive $\epsilon$. We also find maximal gaps and an isolated point in the spectra. + oai:arXiv.org:2506.08358v2 + math.NT + math.DS + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Nicola Gigli, Felix Rott, Matteo Zanardini + Byungchul Cha, Dong Han Kim, Deokwon Sim - Calculating The Local Ideal Class Monoid and Gekeler Ratios - https://arxiv.org/abs/2601.13184 - arXiv:2601.13184v1 Announce Type: new -Abstract: Let $A = \mathbb{F}_q[T]$, $\mathfrak{p} \subset A$ prime, $f(x) \in A[x]$ irreducible and set $R = A[x]/f(x)$. Denote its completion by $R_\mathfrak{p}$. The ideal class monoid $\text{ICM}(R_\mathfrak{p})$ is the set of fractional $R_\mathfrak{p}$ ideals modulo the principal $R_\mathfrak{p}$ ideals. We provide an algorithm to compute $\text{ICM}(R_\mathfrak{p})$. In the process we also get algorithms to compute the overorders and weak equivalence classes of $R_\mathfrak{p}$. We then use the algorithms to compute the product of local Gekeler ratios $\prod_{\mathfrak{p} \subset A} v_\mathfrak{p}(f) = \prod_{\mathfrak{p} \subset A} \lim_{n \rightarrow \infty} \frac{|\{M \in \text{Mat}_r(A/\mathfrak{p}^n)\mid \text{charpoly}(M)=f\}}{|\text{SL}_r(A/\mathfrak{p}^n)|/|\mathfrak{p}|^{n(r-1)}}$. This provides part of an algorithm to compute the weighted size of an isogeny class of Drinfeld modules. - oai:arXiv.org:2601.13184v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Arix Eggink + Vector bundles on bielliptic surfaces: Ulrich bundles and degree of irrationality + https://arxiv.org/abs/2506.10696 + arXiv:2506.10696v2 Announce Type: replace +Abstract: This paper deals with two problems about vector bundles on bielliptic surfaces. The first is to give a classification of Ulrich bundles on such surfaces $S$, which depends on the topological type of $S$. In doing so, we study the weak Brill-Noether property for moduli spaces of sheaves with isotropic Mukai vector. Adapting an idea of Moretti, we also interpret the problem of determining the degree of irrationality of bielliptic surfaces in terms of the existence of certain stable vector bundles of rank 2, completing the work of Yoshihara. + oai:arXiv.org:2506.10696v2 + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Edoardo Mason - Radicals of Lie-solvable Novikov algebras - https://arxiv.org/abs/2601.13185 - arXiv:2601.13185v1 Announce Type: new -Abstract: We prove that in a Lie-solvable Novikov algebra, the Baer radical coincides with the set of all right-nilpotent elements, and the Andrunakievich radical coincides with the largest left-quasiregular ideal. We investigate the stability of some properties of commutative algebras with derivation after applying the Gelfand-Dorfman construction. - oai:arXiv.org:2601.13185v1 - math.RA - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fare Game: A Mean Field Model of Stochastic Intensity Control in Dynamic Ticket Pricing + https://arxiv.org/abs/2506.13088 + arXiv:2506.13088v2 Announce Type: replace +Abstract: We study the dynamic pricing of discrete goods over a finite selling horizon. One way to capture both the elastic and stochastic reaction of purchases to price is through a model where sellers control the intensity of a counting process, representing the number of sales thus far. The intensity describes the probabilistic likelihood of a sale, and is a decreasing function of the price a seller sets. A classical model for ticket pricing, which assumes a single seller and finite time horizon, is by Gallego and van Ryzin (1994) and it has been widely utilized by airlines, for instance. Extending to more realistic settings where there are multiple sellers, with finite inventories, in competition over a finite time horizon is more complicated both mathematically and computationally. We introduce a dynamic mean field game of this type, and some numerical and existence results. In particular, we analyze the associated coupled system of Hamilton-Jacobi-Bellman and Kolmogorov differential-difference equations, and we prove the existence and uniqueness results under certain conditions. Then, we demonstrate a numerical algorithm to find this solution and provide some insights into the macroeconomic market parameters. Finally, we present a qualitative comparison of our findings with airfare data. + oai:arXiv.org:2506.13088v2 + math.OC + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - A. S. Panasenko + 10.1007/s11579-026-00412-x + Aydin, B., Parmaksiz, E. & Sircar, R. Fare Game: A Mean Field Model of Stochastic Intensity Control in Dynamic Ticket Pricing. Mathematics and Financial Economics (2026) + Burak Aydin, Emre Parmaksiz, Ronnie Sircar - Onsager's Mean Field Theory of Vortex Flows with Singular Sources: Blow-Up and Concentration without Quantization - https://arxiv.org/abs/2601.13192 - arXiv:2601.13192v1 Announce Type: new -Abstract: Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we make a first step in the generalization of the mean field theory in [Caglioti, Lions, Marchioro, Pulvirenti; Comm. Math. Phys. (1995)]. On one side we prove the equivalence of statistical ensembles, on the other side we are bound to the analysis of a new blow up phenomenon, which we call "blow up and concentration without quantization", where the mass associated with the concentration is allowed to take values in a full interval of real numbers. This singular behavior may be regarded as lying between the classical blow up-concentration-quantization and the blow up without concentration phenomenon first proposed in [Lin, Tarantello; C.R. Math. Acad. Sci. Paris (2016)]. A careful analysis is needed to generalize known pointwise estimates in this non standard context, resulting in a complete description of the allowed asymptotic profiles. - oai:arXiv.org:2601.13192v1 + Asymptotic Velocity Profiles for Homoenergetic Rayleigh-Boltzmann Flows under Dominant Shear + https://arxiv.org/abs/2506.15449 + arXiv:2506.15449v2 Announce Type: replace +Abstract: In this paper, we study a particular class of solutions to the Rayleigh--Boltzmann equation, known in the nonlinear setting as \emph{homoenergetic solutions}. These solutions take the form $ g(x, v, t) = f(v - L(t)x, t),$ where the matrix $L(t)$ represents a shear flow deformation. We began our analysis in \cite{MNV}, where we rigorously proved the existence of a stationary non-equilibrium solution and established different behaviours of the solutions depending on the size of the shear parameter, for cut-off collision kernels with homogeneity parameter $0 \leq \gamma < 1$, thus including Maxwell molecules and hard potentials. In the present work, we focus on the regime in which the deformation term dominates the collision term for large times (hyperbolic-dominated regime). This scenario occurs for collision kernels with $\gamma < 0$; in particular, we focus on the range $\gamma \in (-1, 0)$. In this regime, it is challenging to obtain a clear and direct description of the long-time asymptotic behaviour of the solutions. Here we present a formal analysis of the velocity distribution's long-time asymptotics and derive for the first time the explicit form of the corresponding asymptotic profile. We also discuss the different asymptotic behaviour expected in the case of homogeneity $\gamma < -1$. In addition, we provide a probabilistic interpretation involving a stochastic process combining collisions with shear flow. The tagged particle velocity $\{v(t)\}_{t\geq 0}$ is a Markov process that arises from the combination of free flights in a shear flow along with random jumps caused by collisions. + oai:arXiv.org:2506.15449v2 math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new + math-ph + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Daniele Bartolucci, Paolo Cosentino, Lina Wu + Nicola Miele, Alessia Nota, Juan J. L. Vel\'azquez - Asymptotic Stability of Rarefaction Waves for the Hyperbolized Navier-Stokes-Fourier System - https://arxiv.org/abs/2601.13193 - arXiv:2601.13193v1 Announce Type: new -Abstract: This paper investigates the asymptotic stability of rarefaction waves for a one-dimensional compressible fluid system, where the Newton's law of viscosity and Fourier's law of heat conduction are replaced by Maxwell's law and Cattaneo's law, respectively. The system, which generalizes the classical Navier-Stokes-Fourier equations, features finite signal propagation speeds. We consider the Cauchy problem in Lagrangian coordinates with initial data connecting two different constant states via a rarefaction wave of the corresponding Euler system. Our main result proves that, provided the initial perturbation and wave strength are sufficiently small, the relaxation system admits a unique global solution. Furthermore, this solution converges uniformly to the background rarefaction wave as time approaches infinity. The proof is established through a combination of the relative entropy method and usual energy estimates. - oai:arXiv.org:2601.13193v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new + Thinning to improve two-sample discrepancy + https://arxiv.org/abs/2506.20932 + arXiv:2506.20932v2 Announce Type: replace +Abstract: The discrepancy between two independent samples \(X_1,\dots,X_n\) and \(Y_1,\dots,Y_n\) drawn from the same distribution on $\mathbb{R}^d$ typically has order \(O(\sqrt{n})\) even in one dimension. We give a simple online algorithm that reduces the discrepancy to \(O(\log^{2d} n)\) by discarding a small fraction of the points. + oai:arXiv.org:2506.20932v2 + math.PR + cs.DS + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Yuxi Hu, Mengran Yuan, Jie Zhang + Gleb Smirnov, Roman Vershynin - A Lower Bound on the Expected Number of Distinct Patterns in a Random Permutation - https://arxiv.org/abs/2601.13194 - arXiv:2601.13194v1 Announce Type: new -Abstract: Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. The authors of [2] showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ was $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$, exhibiting the fact that random permutations pack consecutive patterns near-perfectly. A conjecture was made in [11] that the same is true for non-consecutive patterns, i.e., that there are $2^n(1-o(1))$ distinct non-consecutive patterns expected in a random permutation. This conjecture is false, but, in this paper, we prove that a random permutation contains an expected number of at least $2^{n-1}(1+o(1))$ distinct permutations; this number is half of the range of the number of distinct permutations. - oai:arXiv.org:2601.13194v1 - math.CO - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new + You May Use the Same Channel Knowledge Map for Environment-Aware NLoS Sensing and Communication + https://arxiv.org/abs/2507.03589 + arXiv:2507.03589v2 Announce Type: replace +Abstract: As one of the key usage scenarios for the sixth generation (6G) wireless networks, integrated sensing and communication (ISAC) provides an efficient framework to achieve simultaneous wireless sensing and communication. However, traditional wireless sensing techniques mainly rely on the line-of-sight (LoS) assumptions, i.e., the sensing targets are directly visible to both the sensing transmitter and receiver. This hinders ISAC systems to be applied in complex environments such as the urban low-altitude airspace, which usually suffers from signal blockage and non-line-of-sight (NLoS) multi-path propagation. To address this challenge, in this paper, we propose a novel approach to enable environment-aware NLoS ISAC by leveraging the new technique called channel knowledge map (CKM), which was originally proposed for environment-aware wireless communications. One major novelty of our proposed method is that the same CKM built for wireless communication can be directly used to enable NLoS wireless sensing, thus enjoying the benefits of ``killing two birds with one stone''. To this end, the sensing targets are treated as virtual user equipment (UE), and the wireless communication channel priors are transformed into the sensing channel priors, allowing one single CKM to serve dual purposes. We illustrate our proposed framework by a specific CKM called \emph{channel angle-delay map} (CADM). Specifically, the proposed framework utilizes CADM to derive angle-delay priors of the sensing channel by exploiting the relationship between communication and sensing angle-delay distributions, enabling sensing target localization in the challenging NLoS environment. Extensive simulation results demonstrate significant performance improvements over classic geometry-based sensing methods, which is further validated by Cram\'er-Rao Lower Bound (CRLB) analysis. + oai:arXiv.org:2507.03589v2 + cs.IT + eess.SP + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ver\'onica Borr\'as-Serrano, Isabel Byrne, Anant Godbole, Nathaniel Veimau + Di Wu, Zhuoyin Dai, Yong Zeng - AI for Mathematics: Progress, Challenges, and Prospects - https://arxiv.org/abs/2601.13209 - arXiv:2601.13209v1 Announce Type: new -Abstract: AI for Mathematics (AI4Math) has emerged as a distinct field that leverages machine learning to navigate mathematical landscapes historically intractable for early symbolic systems. While mid-20th-century symbolic approaches successfully automated formal logic, they faced severe scalability limitations due to the combinatorial explosion of the search space. The recent integration of data-driven approaches has revitalized this pursuit. In this review, we provide a systematic overview of AI4Math, highlighting its primary focus on developing AI models to support mathematical research. Crucially, we emphasize that this is not merely the application of AI to mathematical activities; it also encompasses the development of stronger AI systems where the rigorous nature of mathematics serves as a premier testbed for advancing general reasoning capabilities. We categorize existing research into two complementary directions: problem-specific modeling, involving the design of specialized architectures for distinct mathematical tasks, and general-purpose modeling, focusing on foundation models capable of broader reasoning, retrieval, and exploratory workflows. We conclude by discussing key challenges and prospects, advocating for AI systems that go beyond facilitating formal correctness to enabling the discovery of meaningful results and unified theories, recognizing that the true value of a proof lies in the insights and tools it offers to the broader mathematical landscape. - oai:arXiv.org:2601.13209v1 - math.HO - Wed, 21 Jan 2026 00:00:00 -0500 - new + Direct reconstruction of general elastic inclusions + https://arxiv.org/abs/2507.04831 + arXiv:2507.04831v2 Announce Type: replace +Abstract: The inverse problem of linear elasticity is to determine the Lam\'e parameters, which characterize the mechanical properties of a domain, from pairs of pressure activations and the resulting displacements on its boundary. This work considers the specific problem of reconstructing inclusions that manifest themselves as deviations from the background Lam\'e parameters. + The monotonicity method is a direct reconstruction method that has previously been considered for domains only containing positive (or negative) inclusions with finite contrast. That is, all inclusions have previously been assumed to correspond to a finite increase (or decrease) in both Lam\'e parameters compared to their background values. We prove the general outer approach of the monotonicity method that simultaneously allows positive and negative inclusions, of both finite and extreme contrast; the latter refers to either infinitely stiff or perfectly elastic materials. + oai:arXiv.org:2507.04831v2 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Haocheng Ju, Bin Dong + Sarah Eberle-Blick, Henrik Garde, Nuutti Hyv\"onen - A Harnack-type inequality for a perturbed singular Liouville Equation - https://arxiv.org/abs/2601.13212 - arXiv:2601.13212v1 Announce Type: new -Abstract: Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we obtain a Harnack-type inequality for sequences of solutions of the following perturbed Liouville equation, \begin{equation}\nonumber - -\Delta v_n=\left({\epsilon_n^2+|x|^2}\right)^{\alpha_n}V_n(x)e^{\displaystyle v_n} \qquad\text{in} \,\,\, \Omega, \end{equation} where $\epsilon_n\to0^+$, $\alpha_n\to\alpha_\infty\in(-1,1)$, $\Omega$ is a bounded domain in $\mathbb{R}^2$ containing the origin and $V_n$ satisfies, \begin{equation}\nonumber - 0<a\leq V_n\leq b<+\infty, \,\, V_n\in C^{0}(\Omega), \,\,V_n\to V \,\, \text{locally uniformly in}\,\,{\Omega}. \end{equation} - oai:arXiv.org:2601.13212v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Daniele Bartolucci, Paolo Cosentino, Lina Wu + Extension Operators for Fractional Sobolev Spaces on Lipschitz Submanifolds + https://arxiv.org/abs/2507.04869 + arXiv:2507.04869v2 Announce Type: replace +Abstract: A well-known result is that any Lipschitz domain is an extension domain for $W^{s,p}$. This paper extends this result to Lipschitz subsets of compact Lipschitz submanifolds of $\mathbb{R}^n$. We adapt the construction of an extension operator for Lipschitz domains in arXiv:1104.4345v3 to manifolds via local coordinate charts. Furthermore, the dependence on the size of the extension domain is explicit in all estimates. This result is motivated by applications in numerical analysis, most notably geometry simplification, where the explicit dependence of the continuity constant on the domain size is essential. + oai:arXiv.org:2507.04869v2 + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Philipp Weder - An AMP-Based Asymptotic Analysis For Nonlinear One-Bit Precoding - https://arxiv.org/abs/2601.13214 - arXiv:2601.13214v1 Announce Type: new -Abstract: This paper focuses on the asymptotic analysis of a class of nonlinear one-bit precoding schemes under Rayleigh fading channels. The considered scheme employs a convex-relaxation-then-quantization (CRQ) approach to the well-known minimum mean square error (MMSE) model, which includes the classical one-bit precoder SQUID as a special case. To analyze its asymptotic behavior, we develop a novel analytical framework based on approximate message passing (AMP). We show that, the statistical properties of the considered scheme can be asymptotically characterized by a scalar ``signal plus Gaussian noise'' model. Based on this, we further derive a closed-form expression for the symbol error probability (SEP) in the large-system limit, which quantitatively characterizes the impact of both system and model parameters on SEP performance. Simulation results validate our analysis and also demonstrate that performance gains over SQUID can be achieved by appropriately tuning the parameters involved in the considered model. - oai:arXiv.org:2601.13214v1 - cs.IT - eess.SP - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new + A Unified Framework for Efficient Kernel and Polynomial Interpolation + https://arxiv.org/abs/2507.12629 + arXiv:2507.12629v3 Announce Type: replace +Abstract: We present a unified interpolation scheme that combines compactly-supported positive-definite kernels and multivariate polynomials. This unified framework generalizes interpolation with compactly-supported kernels and also classical polynomial least squares approximation. To facilitate the efficient use of this unified interpolation scheme, we present specialized numerical linear algebra procedures that leverage standard matrix factorizations. These procedures allow for efficient computation and storage of the unified interpolant. We also present a modification to the numerical linear algebra that allows us to generalize the application of the unified framework to target functions on manifolds with and without boundary. Our numerical experiments on both Euclidean domains and manifolds indicate that the unified interpolant is superior to polynomial least squares for the interpolation of target functions in settings with boundaries. + oai:arXiv.org:2507.12629v3 + math.NA + cs.NA + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zheyu Wu, Junjie Ma, Ya-Feng Liu, Bruno Clerckx + M. Belianovich, G. E. Fasshauer, A. Narayan, V. Shankar - On the Reliability of Estimation Bounds in Low-SNR Bistatic ISAC - https://arxiv.org/abs/2601.13216 - arXiv:2601.13216v1 Announce Type: new -Abstract: This paper explores a bistatic Integrated Sensing and Communication (ISAC) framework, where a base station transmits communication signal that serve both direct communication with a user and multi-target parameter estimation through reflections captured by a separate sensing receiver. We assume that the instantaneous knowledge of the transmit signal at the sensing receiver is not available, and the sensing receiver only has knowledge of the statistical properties of the received signal. Unlike prior research that focuses on power allocation or optimal beamforming design for ISAC, we emphasize the inadequacy of the Cram\'er-Rao Bound (and its variant) in low Signal-to-Noise Ratio (SNR) regimes, particularly in passive sensing scenarios. Due to severe path loss and other impairments, the received sensing SNR is often significantly lower than that of direct Line-of-Sight communication, making CRB-based performance evaluation unreliable. To address this, we adopt the Ziv-Zakai Bound (ZZB) for Angle of Arrival estimation, which provides a more meaningful lower bound on estimation error. We derive analytical expressions for the ZZB and the achievable ergodic communication rate as functions of SNR. Through numerical simulations, we analyze the pareto-front between communication and sensing performance, demonstrating why ZZB serves as a better metric in low sensing SNR ISAC where traditional CRB-based approaches fail. - oai:arXiv.org:2601.13216v1 + A Framework of Distributed Source Encryption using Mutual Information Security Criterion and the Strong Converse Theorem + https://arxiv.org/abs/2507.13294 + arXiv:2507.13294v5 Announce Type: replace +Abstract: We reinvestigate the general distributed secure source coding based on the common key cryptosystem proposed by Oohama and Santoso (ITW 2021). They proposed a framework of distributed source encryption and derived the necessary and sufficient conditions to have reliable and secure transmission. However, the bounds of the rate region, which specifies both necessary and sufficient conditions to have reliable and secure transmission under the proposed cryptosystem, were derived based on a self-tailored non-standard} security criterion. In this paper we adopt the standard security criterion, i.e., standard mutual information. We successfully establish the bounds of the rate region based on this security criterion. Information spectrum method and a variant of Birkhoff-von Neumann theorem play an important role in deriving the result. + oai:arXiv.org:2507.13294v5 cs.IT math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Ataher Sams, Besma Smida + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Yasutada Oohama, Bagus Santoso - Towards Matrix-Free Patch Smoothers for the Stokes Problem: Evaluating Local p-Multigrid Solvers - https://arxiv.org/abs/2601.13230 - arXiv:2601.13230v1 Announce Type: new -Abstract: Vertex-patch smoothers offer an effective strategy for achieving robust geometric multigrid convergence for the Stokes equations, particularly in the context of high-order finite elements. However, their practical efficiency is often limited by the computational cost of solving the local saddle-point problems, especially when explicit matrix factorizations are not feasible. We explore a fully iterative, matrix-free-compatible approach to the local patch solve using $p$-multigrid techniques. We evaluate different local solver configurations: Braess-Sarazin and block-triangular preconditioners. Our numerical experiments suggest that the Braess-Sarazin approach is particularly resilient. We find that a single iteration of the local solver yields global convergence rates comparable to those obtained with exact local solvers, even on distorted meshes and in the presence of large viscosity jumps. - oai:arXiv.org:2601.13230v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new + Rigorous dense graph limit of a model for biological transportation networks + https://arxiv.org/abs/2507.15829 + arXiv:2507.15829v2 Announce Type: replace +Abstract: We rigorously derive the dense graph limit of a discrete model describing the formation of biological transportation networks. The discrete model, defined on undirected graphs with pressure-driven flows, incorporates a convex energy functional combining pumping and metabolic costs. It is constrained by a Kirchhoff law reflecting the local mass conservation. We first rescale and reformulate the discrete energy functional as an integral `semi-discrete' functional, where the Kirchhoff law transforms into a nonlocal elliptic integral equation. Assuming that the sequence of graphs is uniformly connected and that the limiting graphon is 0-1 valued, we prove two results: (1) rigorous Gamma-convergence of the sequence of the semi-discrete functionals to a continuum limit as the number of graph nodes and edges tends to infinity; (2) convergence of global minimizers of the discrete functionals to a global minimizer of the limiting continuum functional. Our results provide a rigorous mathematical foundation for the continuum description of biological transport structures emerging from discrete networks. + oai:arXiv.org:2507.15829v2 + math.OC + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by-nc-nd/4.0/ - Micha{\l} Wichrowski + Nuno J. Alves, Jan Haskovec - Volume polynomials - https://arxiv.org/abs/2601.13249 - arXiv:2601.13249v1 Announce Type: new -Abstract: Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss applications to the combinatorics of algebraic matroids. These notes are based on lectures given at the 2025 Summer Research Institute in Algebraic Geometry at Colorado State University. - oai:arXiv.org:2601.13249v1 - math.AG - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new + Large-Amplitude Steady Electrohydrodynamic Solitary Waves with Constant Vorticity + https://arxiv.org/abs/2508.01336 + arXiv:2508.01336v2 Announce Type: replace +Abstract: This paper investigates solitary water waves propagating on the surface of a two-dimensional dielectric fluid subject to an electric field. The system is formulated as a nonlinear free boundary problem, with interfacial dynamics governed by the strong coupling between the Euler equations with constant vorticity and the electric potential equations. We aim to explore the effects of the electric field and constant vorticity on the nonlinear wave interactions in such a system, specifically examining whether large-amplitude solitary waves analogous to those in reference \cite{SVHMHW2023} exist. Although the inclusion of the electric field considerably complicates the analysis, we establish the existence of a continuous branch of large-amplitude solitary wave solutions. Moreover, along the global bifurcation curve, one of the following must occur: (i) the formation of an equilibrium stagnation point, (ii) the degeneration of the conformal mapping, (iii) the onset of flow stagnation, or (iv) an unbounded increase in the dimensionless wave speed. + oai:arXiv.org:2508.01336v2 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - June Huh + Tingting Feng, Yong Zhang, Zhitao Zhang - Inverting the Fisher information operator in non-linear models - https://arxiv.org/abs/2601.13254 - arXiv:2601.13254v1 Announce Type: new -Abstract: We consider non-linear regression models corrupted by generic noise when the regression functions form a non-linear subspace of L^2, relevant in non-linear PDE inverse problems and data assimilation. We show that when the score of the model is injective, the Fisher information operator is automatically invertible between well-identified Hilbert spaces, and we provide an operational characterization of these spaces. This allows us to construct in broad generality the efficient Gaussian involved in the classical minimax and convolution theorems to establish information lower bounds, that are typically achieved by Bayesian algorithms thus showing optimality of these methods. We illustrate our results on time-evolution PDE models for reaction-diffusion and Navier-Stokes equations. - oai:arXiv.org:2601.13254v1 - math.ST - math.AP - math.FA - math.PR - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - new + The approach of cluster symmetry to Diophantine equations + https://arxiv.org/abs/2508.02005 + arXiv:2508.02005v3 Announce Type: replace +Abstract: This paper aims to employ a cluster-theoretic approach to provide a class of Diophantine equations whose solutions can be obtained by starting from initial solutions through mutations. We establish a novel framework bridging cluster theory and Diophantine equations through the lens of cluster symmetry. On the one hand, we give the necessary and sufficient condition for Laurent polynomials to remain invariant under a given cluster symmetric map. On the other hand, we construct a discriminant algorithm to determine whether a given Laurent polynomial has cluster symmetry and whether it can be realized in a generalized cluster algebra. As applications, we solve Markov-cluster equations, describe some invariant Laurent polynomial rings, and resolve the questions posed by Gyoda and Matsushita. + oai:arXiv.org:2508.02005v3 + math.NT + math.AC + math.CO + math.RA + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by-nc-nd/4.0/ - Dimitri Konen + Leizhen Bao, Fang Li - Deep Neural networks for solving high-dimensional parabolic partial differential equations - https://arxiv.org/abs/2601.13256 - arXiv:2601.13256v1 Announce Type: new -Abstract: The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years, deep neural networks have emerged as a promising mesh free alternative, enabling the approximation of PDE solutions in tens to thousands of dimensions. This review provides a tutorial--oriented introduction to neural--network--based methods for solving high dimensional parabolic PDEs, emphasizing conceptual clarity and methodological connections. We organize the literature around three unifying paradigms: (i) PDE residual--based approaches, including physicsinformed neural networks and their high dimensional variants; (ii) stochastic methods derived from Feynman--Kac and backward stochastic differential equation formulations; and (iii) hybrid derivative--free random difference approaches designed to alleviate the computational cost of derivatives in high dimensions. For each paradigm, we outline the underlying mathematical formulation, algorithmic implementation, and practical strengths and limitations. Representative benchmark problems--including Hamilton--Jacobi--Bellman and Black--Scholes equations in up to 1000 dimensions --illustrate the scalability, effectiveness, and accuracy of the methods. The paper concludes with a discussion of open challenges and future directions for reliable and scalable solvers of high dimensional PDEs. - oai:arXiv.org:2601.13256v1 - math.NA - cs.LG - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Wenzhong Zhang, Zhenyuan Hu, Wei Cai, George EM Karniadakis + Cup product of inhomogeneous Tate cochains, and application to tori over local fields that split over cyclic extensions + https://arxiv.org/abs/2508.07288 + arXiv:2508.07288v3 Announce Type: replace +Abstract: In this note we give formulas for cup product in Tate cohomology in terms of inhomogeneous cochains. Using one of these formulas, for a torus T defined over a non-archimedean local field K and splitting over a cyclic extension of K, we compute explicit cocycles representing all cohomology classes in H^1(K,T). + oai:arXiv.org:2508.07288v3 + math.NT + math.GR + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-sa/4.0/ + Mikhail Borovoi - Entropy-Wasserstein regularization, defective local concentration and a cutoff criterion beyond non-negative curvature - https://arxiv.org/abs/2601.13259 - arXiv:2601.13259v1 Announce Type: new -Abstract: Notions of positive curvature have been shown to imply many remarkable properties for Markov processes, in terms, e.g., of regularization effects, functional inequalities, mixing time bounds and, more recently, the cutoff phenomenon. In this work, we are interested in a relaxed variant of Ollivier's coarse Ricci curvature, where a Markov kernel $P$ satisfies only a weaker Wasserstein bound $W_p(\mu P, \nu P) \leq K W_p(\mu,\nu)+M$ for constants $M\ge 0, K\in [0,1], p \ge 1$. Under appropriate additional assumptions on the one-step transition measures $\delta_x P$, we establish (i) a form of local concentration, given by a defective Talagrand inequality, and (ii) an entropy-transport regularization effect. We consider as illustrative examples the Langevin dynamics and the Proximal Sampler when the target measure is a log-Lipschitz perturbation of a log-concave measure. As an application of the above results, we derive criteria for the occurrence of the cutoff phenomenon in some negatively curved settings. - oai:arXiv.org:2601.13259v1 - math.PR - math.FA - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Francesco Pedrotti + Some generalizations of Camina pairs and orders of elements in cosets + https://arxiv.org/abs/2508.13056 + arXiv:2508.13056v2 Announce Type: replace +Abstract: In this paper, we investigate certain generalizations of Camina pairs. Let $H$ be a nontrivial proper subgroup of a finite group $G$. We first show that every nontrivial irreducible complex character of $H$ induces homogeneously to $G$ if and only if for every $x\in G\setminus H$, the element $x$ is conjugate to $xh$ for all $h\in H$. Furthermore we prove that if $xh$ is conjugate to either $x$ or $x^{-1}$ for all $h\in H$ and all $x\in G\setminus H$, then the normal closure $N$ of $H$ in $G$ also satisfies the same condition, and $N$ is nilpotent. Finally, we determine the structure of $H$ under the assumption that for every element $x\in G\setminus H$ of odd order, the coset $xH$ consists entirely of elements of odd order. + oai:arXiv.org:2508.13056v2 + math.GR + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Thu T. H. Quan, Hung P. Tong-Viet - Covariant tomography of fields - https://arxiv.org/abs/2601.13261 - arXiv:2601.13261v1 Announce Type: new -Abstract: This paper addresses the Inverse Boundary Value Problem (IBVP) for classical fields, specifically focusing on the recovery of parallelly transformed fields within a region based on known boundary data. We introduce a local solution framework, termed "covariant tomography," that uses geometric decomposition to reconstruct interior fields and currents within star-shaped open subsets. The core of our approach involves decomposing differential forms into exact and antiexact components, enabling the formulation of the parallel transport equation via a homotopy operator. We examine three primary extension techniques - radial, heat equation, and harmonic - to map boundary values into the interior, noting that the choice of extension directly influences the regularity of the resulting currents. The proposed methodology provides a systematic way to identify the realizability of boundary values and offers solutions for both current and gauge field tomography. Finally, we demonstrate the utility of this framework through illustrative examples in low-dimensional spaces and electromagnetic potential reconstruction in $\mathbb{R}^{3}$. - oai:arXiv.org:2601.13261v1 - math-ph - math.AP - math.DG - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rados{\l}aw Antoni Kycia + Generalized Symmetries From Fusion Actions + https://arxiv.org/abs/2508.13063 + arXiv:2508.13063v4 Announce Type: replace +Abstract: Let $A$ be a condensable algebra in a modular tensor category $\mathcal{C}$. We define an action of the fusion category $\mathcal{C}_A$ of $A$-modules in $\mathcal{C}$ on the morphism space $\mbox{Hom}_{\mathcal{C}}(x,A)$ for any $x$ in $\mathcal{C}$, whose characters are generalized Frobenius-Schur indicators. This fusion action can be considered on $A$, and we prove a categorical generalization of the Schur-Weyl duality for this action. For any fusion subcategory $\mathcal{B}$ of $\mathcal{C}_A$ containing all the local $A$-modules, we prove the invariant subobject $B=A^\mathcal{B}$ is a condensable subalgebra of $A$. The assignment of $\mathcal{B}$ to $A^\mathcal{B}$ defines a Galois correspondence between this kind of fusion subcategories of $\mathcal{C}_A$ and the condensable subalgebras of $A$. In the context of VOAs, we prove for any nice VOAs $U \subset A$, $U=A^{\mathcal{C}_A}$ where $\mathcal{C}=\mathcal{M}_U$ is the category of $U$-modules. In particular, if $U = A^G$ for some finite automorphism group $G$ of $A,$ the fusion action of $\mathcal{C}_A$ on $A$ is equivalent to the $G$-action on $A.$ + oai:arXiv.org:2508.13063v4 + math.QA + cond-mat.str-el + hep-th + math.CT + math.RT + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Chongying Dong, Siu-Hung Ng, Li Ren, Feng Xu - On surfaces with smooth projective models over $\mathbb{Z}$ - https://arxiv.org/abs/2601.13277 - arXiv:2601.13277v1 Announce Type: new -Abstract: In this expository article, we prove a birational classification of smooth projective models of surfaces with negative Kodaira dimension over $\mathbb{Z}$ and over more general rings of integers $\mathcal{O}_K$, depending on their arithmetic and cohomological invariants. Along the way we collect some results on smooth projective models of surfaces over Dedekind domains. - oai:arXiv.org:2601.13277v1 + GW/DT invariants and 5D BPS indices for strips from topological recursion + https://arxiv.org/abs/2508.15459 + arXiv:2508.15459v2 Announce Type: replace +Abstract: Topological string theory partition function gives rise to Gromov-Witten invariants, Donaldson-Thomas invariants and 5D BPS indices. Using the remodeling conjecture, which connects Topological Recursion with topological string theory for toric Calabi-Yau threefolds, we study a more direct connection for the subclass of strip geometries. In doing so, new developments in the theory of topological recursion are applied as its extension to Logarithmic Topological Recursion (Log-TR) and the universal $x$-$y$ duality. Through these techniques, our main result in this paper is a direct derivation of all free energies from topological recursion for general strip geometries. In analyzing the expression of free energy, we shed some light on the meaning and the influence of the $x$-$y$ duality in topological string theory and its interconnection to GW and DT invariants as well as the 5D BPS index. + oai:arXiv.org:2508.15459v2 + math-ph + hep-th math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Fabio Bernasconi, Gebhard Martin, Zsolt Patakfalvi + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Sibasish Banerjee, Alexander Hock, Olivier Marchal - On constructing topology from algebra - https://arxiv.org/abs/2601.13279 - arXiv:2601.13279v1 Announce Type: new -Abstract: In this thesis we explore natural procedures through which topological structure can be constructed from specific semigroups. We will do this in two ways: 1) we equip the semigroup object itself with a topological structure, and 2) we find a topological space for the semigroup to act on continuously. - We discuss various minimum/maximum topologies which one can define on an arbitrary semigroup (given some topological restrictions). We give explicit descriptions of each these topologies for the monoids of binary relations, partial transformations, transformations, and partial bijections on a countable set. Using similar methods we determine whether or not each of these semigroups admits a unique Polish semigroup topology. We also do this for the various other semigroups, provide a proof of Rubin's theorem, and give a description of the automorphism groups of the Brin-Thompson groups. - The thesis also contains many background results. - oai:arXiv.org:2601.13279v1 + Residual finiteness properties of some of Halls groups + https://arxiv.org/abs/2508.16452 + arXiv:2508.16452v3 Announce Type: replace +Abstract: In this article we study a class of central extensions of $\mathbb{Z}\wr\mathbb{Z}$, as first described by Hall. On the one hand, we consider groups of this type with cyclic centre, our construction yields a rich class of groups. In particular we obtain a group that is conjugacy separable with solvable word problem but unsolvable conjugacy problem, we obtain a group with large conjugacy separability growth but small conjugator length function and residual finiteness growth, and we also obtain both a class of groups that for most functions $f:\mathbb{N}\rightarrow\mathbb{N}$ larger then $n^3$, contain a group $G$ such that the conjugator length of $G$ is given by $f$, as well as a group where the conjugator length is superlinear but subquadratic. + On the other hand we also consider a different group with larger centre. This is the first example of a group where the residual finiteness growth is faster than any polynomial but slower than any exponential. + oai:arXiv.org:2508.16452v3 math.GR - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Luna Elliott + Lukas Vandeputte - Total curvature of convex hypersurfaces in Cartan-Hadamard manifolds - https://arxiv.org/abs/2601.13280 - arXiv:2601.13280v1 Announce Type: new -Abstract: We show that if the curvature of a Cartan-Hadamard $n$-manifold is constant near a convex hypersurface $\Gamma$, then the total Gauss-Kronecker curvature $\mathcal{G}(\Gamma)$ is not less than that of any convex hypersurface nested inside $\Gamma$. This extends Borb\'{e}ly's monotonicity theorem in hyperbolic space. It follows that $\mathcal{G}(\Gamma)$ is bounded below by the volume of the unit sphere in Euclidean space $\mathbf{R}^n$. - oai:arXiv.org:2601.13280v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new + Dynamics from iterated averaging + https://arxiv.org/abs/2508.21416 + arXiv:2508.21416v3 Announce Type: replace +Abstract: We prove that for a standard Lebesgue space $X$, the strong operator closure of the semigroup generated by conditional expectations on $L^\infty(X)$ contains the group of measure-preserving automorphisms. This is based on a solution to the following puzzle: given $n$ full water tanks, each containing one unit of water, and $n$ empty ones, how much water can be transferred from the full tanks to the empty ones by repeatedly equilibrating the water levels between pairs of tanks? + oai:arXiv.org:2508.21416v3 + math.DS + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mohammad Ghomi, John Ioannis Stavroulakis + Tobias Fritz, Nicol\'as Rivera - Second order periodic boundary value problems with reflection and piecewise constant arguments - https://arxiv.org/abs/2601.13291 - arXiv:2601.13291v1 Announce Type: new -Abstract: In this paper, we analyze a second-order differential equation with a piecewise constant argument and reflection coupled to periodic boundary conditions. Our main contribution is the construction of the related Green's function and a detailed analysis of its properties. In particular, we determine the region in which the Green's function has constant sign, depending on the parameters $m$ and $M$ on which it depends. In some cases, we are able to characterize these parameter values in terms of the first eigenvalue related to suitable Dirichlet problems. Building in these results, we apply the Krasnosel'skii method to establish the existence of solutions for different nonlinear problems, and prove the existence of a positive solution of a perturbed Schrodinger equation. - oai:arXiv.org:2601.13291v1 - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 - new + Moments of density-dependent branching processes and their genealogy + https://arxiv.org/abs/2509.05231 + arXiv:2509.05231v4 Announce Type: replace +Abstract: A density-dependent branching process is a particle system in which individuals reproduce independently, but in a way that depends on the current population size. This feature can model a wide range of ecological interactions at the cost of breaking the branching property. We propose a general approach for studying the genealogy of these models based on moments. Building on a recent work of Bansaye, we show how to compute recursively these moments in a similar spirit to the many-to-few formula in the theory of branching processes. These formulas enable one to deduce the convergence of the genealogy by studying the population density, for which stochastic calculus techniques are available. As a first application of these ideas, we consider a density-dependent branching process started close to a stable equilibrium of the ecological dynamics. We show that, under a finite second moment assumption, its genealogy converges to Kingman's coalescent when the carrying capacity of the population goes to infinity. + oai:arXiv.org:2509.05231v4 + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Alberto Cabada, Paula Cambeses-Franco + Mathilde Andr\'e, F\'elix Foutel-Rodier, Emmanuel Schertzer - Long-time behavior of solutions to fluid dynamic shape optimization problems via phase-field method - https://arxiv.org/abs/2601.13293 - arXiv:2601.13293v1 Announce Type: new -Abstract: We investigate the long time behavior of solutions to a shape and topology optimization problem with respect to the time-dependent Navier--Stokes equations. The sought topology is represented by a stationary phase-field that represents a smooth indicator function. The fluid equations are approximated by a porous media approach and are time-dependent. In the latter aspect, the considered problem formulation extends earlier work. - We prove that if the time horizon tends to infinity, minima of the time-dependent problem converge towards minima of the corresponding stationary problem. To do so, a convergence rate with respect to the time horizon, of the values of the objective functional, is analytically derived. This allowed us to prove that the solution to the time-dependent problem converges to a phase-field, as the time horizon goes to infinity, which is proven to be a minimizer for the stationary problem. We validate our results by numerical investigation. - oai:arXiv.org:2601.13293v1 + Time-inconsistent reinsurance and investment optimization problem with delay under random risk aversion + https://arxiv.org/abs/2509.15506 + arXiv:2509.15506v2 Announce Type: replace +Abstract: This paper considers a newly delayed reinsurance and investment optimization problem incorporating random risk aversion, in which an insurer pursues maximization of the expected certainty equivalent of her/his terminal wealth and the cumulative delayed information of the wealth over a period. Specially, the insurer's surplus dynamics are approximated using a drifted Brownian motion, while the financial market is described by the constant elasticity of variance (CEV) model. Moreover, the performance-linked capital flow feature is incorporated and the wealth process is formulated via a stochastic delay differential equation (SDDE). By adopting a game-theoretic approach, a verification theorem with rigorous proofs is established to capture the equilibrium reinsurance and investment strategy along with the equilibrium value function. Furthermore, analytical or semi-analytical equilibrium reinsurance and investment strategies, together with their equilibrium value functions, are obtained under the CEV model for the exponential utility and derived under the Black-Scholes model for both exponential and power utilities. Finally, several numerical experiments are conducted to analyze the behavioral characteristics of the freshly-derived equilibrium reinsurance and investment strategy. + oai:arXiv.org:2509.15506v2 math.OC - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Michael Hinze, Christian Kahle, John Sebastian H. Simon - - - Limit Theorems for $\theta$-expansions and the Failure of the Strong Law - https://arxiv.org/abs/2601.13296 - arXiv:2601.13296v1 Announce Type: new -Abstract: The paper presents fundamental metrical theorems for a class of continued fraction-like expansions known as $\theta$-expansions. We first prove Khinchine's Weak Law of Large Numbers for the sum of digits, followed by the Diamond-Vaaler Strong Law for the sum of digits minus the largest one. Our main result is a general theorem on the failure of the strong law, showing that no regular norming sequence can yield a finite, non-zero almost sure limit. This result extends a classical theorem of Philipp to the $\theta$-expansion setting. The proofs leverage the system's explicit invariant measure and a detailed analysis of its mixing properties. - oai:arXiv.org:2601.13296v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Andreas Rusu, Gabriela Ileana Sebe, Dan Lascu + Jian-hao Kang, Zhun Gou, Nan-jing Huang - Locally involutive semigroups - https://arxiv.org/abs/2601.13301 - arXiv:2601.13301v1 Announce Type: new -Abstract: We introduce locally involutive semigroups and embed them into the category of ordered groupoids. This embedding restricts to a correspondence between quasi-involutive semigroups and ordered groupoids with mediator, extending the classical ESN-correspondence between inverse semigroups and inductive groupoids. An important subcategory of locally involutive semigroups is formed by left involutive semigroups because the classifying topos of an inverse semigroup S is equivalent to the category of left involutive semigroups \'etale over S [4]. We recover this equivalence from a general adjointness and use the latter to determine when a left involutive semigroup \'etale over S is actually an involutive semigroup. Any left involutive semigroup \'etale over S embeds into an involutive S-algebra as we call it. The underlying semigroup of this algebra is involutive. - oai:arXiv.org:2601.13301v1 - math.GR - math.CT - Wed, 21 Jan 2026 00:00:00 -0500 - new + A non-linear Roth theorem for thick Cantor sets + https://arxiv.org/abs/2509.17880 + arXiv:2509.17880v3 Announce Type: replace +Abstract: We prove that for any function $f$ satisfying certain mild conditions and any Cantor set $K$ with Newhouse thickness greater than $1$, there exists $x\in K$ and $t>0$ such that \[ \{x-t,x,x+f(t)\}\subset K. \] This is an extension of previous work on the existence of three-term arithmetic progressions in Cantor sets to the non-linear setting. + oai:arXiv.org:2509.17880v3 + math.CA + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Clemens Berger, Jonathon Funk + Alex McDonald, Micah Nguyen - Structured eigenbases and pair state transfer on threshold graphs - https://arxiv.org/abs/2601.13318 - arXiv:2601.13318v1 Announce Type: new -Abstract: Recently, Macharete, Del-Vecchio, Teixeira and de Lima showed that a star and any threshold graph on the same number of vertices share the same eigenbasis relative to the Laplacian matrix. We use this fact to establish two main results in this paper. The first one is a characterization of threshold graphs that are \textit{simply structured}, i.e., their associated Laplacian matrices have eigenbases consisting of vectors with entries from the set $\{-1,0,1\}$. Then, we provide sufficient conditions such that a simply structured threshold graph is weakly Hadamard diagonalizable (WHD). This allows us to list all connected simply structured threshold graphs on at most 20 vertices, and identify those that are WHD. Second, we characterize Laplacian pair state transfer on threshold graphs. In particular, we show that the existence of Laplacian vertex state transfer and Laplacian pair state transfer on a threshold graph are equivalent if and only if it is not a join of a complete graph and an empty graph of certain sizes. - oai:arXiv.org:2601.13318v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Leonardo de Lima, Renata Del-Vecchio, Hermie Monterde, Heber Teixeira + Souriau-Fisher metric and Completely integrable system on Lie groups SO(2) and SO(3) + https://arxiv.org/abs/2509.20910 + arXiv:2509.20910v4 Announce Type: replace +Abstract: We study the generalize Fisher metric on SO(2) and SO(3) via the thermodynamics Lie group theories of Souriau. Then we give the effect of 2-cocycle on the integrability of gradient systems due to the Fisher metric and Souriau-Fisher metric. In addition, we show how the cocycle can locally modify the Fisher metric on a coadjoint orbit, in explicit terms of brackets and central extensions on the Lie groups SO(2) and SO(3). + oai:arXiv.org:2509.20910v4 + math.DG + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/publicdomain/zero/1.0/ + Prosper Rosaire Mama Assandje, Michel Bertrand Djiadeu Ngaha, Romain Nimpa Pefoukeu, Salomon Joseph Mbatakou - On the problem of generalized measures: an impossibility result - https://arxiv.org/abs/2601.13321 - arXiv:2601.13321v1 Announce Type: new -Abstract: This paper investigates the problem of extending measure theory to non-separable structures, from generalized descriptive set theory to a broader class of spaces beyond this framework. While various notions, such as the ideal of measure zero sets, have been generalized, the question of whether a satisfactory notion of $\lambda^+$-measure could be defined in generalized descriptive set theory has remained open. We introduce a broad class of $\lambda^+$-measures as functions taking values in arbitrary positively totally ordered monoids equipped with an infinitary sum. This definition relies on minimal assumptions and captures most natural generalizations of measures to this context. We then prove that, under certain cardinal assumptions, no continuous $\lambda^+$-measure of this kind exists on ${}^\kappa\lambda$, nor on any $\lambda^+$-Borel space or $T_0$ topological space of weight at most $\lambda$. We also show the optimality of these cardinal assumptions. - oai:arXiv.org:2601.13321v1 - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Claudio Agostini, Fernando Barrera, Vincenzo Dimonte + On the torsion-free nilpotent fundamental groups of smooth quasi-projective varieties of rank up to seven + https://arxiv.org/abs/2510.09026 + arXiv:2510.09026v2 Announce Type: replace +Abstract: Let $X$ be a smooth quasi-projective variety. Assume that the (topological) fundamental group $\pi_1(X, x)$ is torsion-free nilpotent. We show that if the first Betti number $b_1(X) \le 3$, then $\pi_1(X, x)$ is isomorphic to either $\mathbb{Z}^n$ for $n = 1, 2, 3$, a lattice in the Heisenberg group $H_3(\mathbb{R})$ or $\mathbb{R} \times H_3(\mathbb{R})$. Moreover, we prove that $\pi_1(X, x)$ is abelian or $2$-step nilpotent if its rank is less than or equal to seven. More precisely, we determine the real nilpotent Lie groups in which torsion-free nilpotent fundamental groups can be embedded as lattices for ranks up to six and seven, respectively. Our main results are a partial positive answer to a question on nilpotent (quasi-)K\"ahler groups posed by Aguilar and Campana. + oai:arXiv.org:2510.09026v2 + math.AG + math.AT + math.DG + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Taito Shimoji - Ribbon complexes for the 0-Hecke algebra - https://arxiv.org/abs/2601.13324 - arXiv:2601.13324v1 Announce Type: new -Abstract: We construct explicit tableau-level maps between indecomposable projective modules for the type A 0-Hecke algebra that assemble into canonical split short exact sequences lifting the basic ribbon product rule in NSym via concatenation and near-concatenation. Iterating these maps yields cochain complexes indexed by generalized ribbons; we prove these complexes are acyclic in positive degrees and that their zeroth cohomology is the projective module indexed by full concatenation. We apply these complexes, together with VandeBogert's ribbon Schur module criterion, to prove Koszulness for a naturally defined internally graded algebra object built from the 0-Hecke tower. Finally, we define skew projective modules whose noncommutative Frobenius characteristics realize skewing by fundamental quasisymmetric functions on NSym. - oai:arXiv.org:2601.13324v1 - math.CO - math.RA - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - new + General Casorati inequalities and implications for Riemannian maps and Riemannian submersions + https://arxiv.org/abs/2510.12760 + arXiv:2510.12760v2 Announce Type: replace +Abstract: This paper presents general forms of Casorati inequalities for Riemannian maps and Riemannian submersions between Riemannian manifolds. Using these general forms, we obtain Casorati inequalities for Riemannian maps (resp. submersions) whose target (resp. source) spaces are generalized complex and generalized Sasakian space forms. As a consequence, we give Casorati inequalities for Riemannian maps (resp. submersions) when the target (resp. source) spaces are real, complex, real K\"ahler, Sasakian, Kenmotsu, cosymplectic, and almost $C(\alpha)$ space forms. To support these general forms, in the particular cases when the target or source spaces are real, complex, Sasakian, and Kenmotsu space forms, we verify known Casorati inequalities for Riemannian maps and Riemannian submersions. Further, we give Casorati inequalities for invariant and anti-invariant Riemannian maps (resp. submersions) whose target (resp. source) spaces are generalized complex and generalized Sasakian space forms. Toward information on geometric characteristics, we discuss the equality cases. We also exemplify the general forms. + oai:arXiv.org:2510.12760v2 + math.DG + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-sa/4.0/ + 10.1016/j.jmaa.2026.130436 + Journal of Mathematical Analysis and Applications, Vol. 558, No. 1, Article 130436 (2026) + Ravindra Singh, Kiran Meena, Kapish Chand Meena + + + A Generalization of the Fox H-function + https://arxiv.org/abs/2510.15920 + arXiv:2510.15920v2 Announce Type: replace +Abstract: In this paper we present a generalization of the Fox H-function called Fox-Barnes J-function. Like the Fox H-function, it is defined as a contour integral in the complex plane, but instead of an integrand given by a ratio of products of gamma functions involving several parameters, we use a ratio of products of double gamma functions. We study the conditions for its existence and how to choose a contour of integration based on the involved parameters. We discuss how the Fox H-function appears as a particular case and prove some properties of the Fox-Barnes J-function. As an application, we show how the Laplace transform of the Kilbas-Saigo function can be conveniently written in terms of the Fox-Barnes J-function, even in cases where the usual series representation is not convergent. + oai:arXiv.org:2510.15920v2 + math.GM + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Ayah Almousa, Bryan Lu + Jayme Vaz - Domino tilings of black-and-white Temperleyan cylinders - https://arxiv.org/abs/2601.13332 - arXiv:2601.13332v1 Announce Type: new -Abstract: We consider the dimer model in cylindrical domains $\Omega_\delta$ on square grids of mesh size $\delta$ with two Temperleyan boundary components of different colors. Assuming that the $\Omega_\delta$ approximate a cylindrical domain $\Omega$ as $\delta\to 0$, we prove the convergence of height fluctuations to the Gaussian Free Field in $\Omega$ plus an independent discrete Gaussian multiple of the harmonic measure of one of the boundary components. The limit of the dimer coupling functions on $\Omega_\delta$ is holomorphic in $\Omega$ but not conformally covariant. Given this, we determine the limiting structure of height fluctuations from general principles rather than from explicit computations. In particular, our analysis justifies the inevitable appearance of the discrete Gaussian distribution in the doubly connected setup. - oai:arXiv.org:2601.13332v1 - math.PR - math-ph - math.CV - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new + Through the Grapevine: Vineyard Distance as a Measure of Topological Dissimilarity + https://arxiv.org/abs/2510.24472 + arXiv:2510.24472v2 Announce Type: replace +Abstract: We introduce a new measure of distance between datasets, based on vineyards from topological data analysis, which we call the vineyard distance. Vineyard distance measures the extent of topological change along an interpolation from one dataset to another, either along a pre-computed trajectory or via a straight-line homotopy. We demonstrate through theoretical results and experiments that vineyard distance is less sensitive than $L^p$ distance (which considers every single data value), but more sensitive than Wasserstein distance between persistence diagrams (which accounts only for shape and not location). This allows vineyard distance to reveal distinctions that the other two distance measures cannot. In our paper, we establish theoretical results for vineyard distance including as upper and lower bounds. We then demonstrate the usefulness of vineyard distance on real-world data through applications to geospatial data and to neural network training dynamics. + oai:arXiv.org:2510.24472v2 + math.AT + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Dmitry Chelkak, Zachary Deiman + Alvan Arulandu, Daniel Gottschalk, Thomas Payne, Alexander Richardson, Thomas Weighill - $D$-affinity of Quadrics Revisited - https://arxiv.org/abs/2601.13340 - arXiv:2601.13340v1 Announce Type: new -Abstract: Let $K$ be aa algebraically closed field of characteristic $p\geq3$ and let $Q_{n}\subset\mathbb{P}^{n+1}_{K}$ be a smooth quadric hypersurface. We show that if $n=2m\geq4$ then $Q_{n}$ is not $D$-affine. In particular, we show the grassmannian ${Gr}(2,4)$ is not $D$-affine, which gives an example of a non $D$-affine flag variety of minimal possible dimension in characteristic $p\geq3$. Our result complements previous work of A. Langer, who showed that if $p\geq n=2m+1$ then $Q_{n}$ is $D$-affine. - oai:arXiv.org:2601.13340v1 + Six-Functor Formalisms + https://arxiv.org/abs/2510.26269 + arXiv:2510.26269v2 Announce Type: replace +Abstract: These are lecture notes for a course in Winter 2022/23, updated and completed in October 2025. + The goal of the lectures is to present some recent developments around six-functor formalisms, in particular: the abstract theory of 6-functor formalisms; the 2-category of cohomological correspondences, and resulting simplifications in the proofs of Poincar\'e--Verdier duality results; the relation between 6-functor formalisms and ``geometric rings''; many examples of 6-functor formalisms, both old and new. + oai:arXiv.org:2510.26269v2 math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Feliks R\k{a}czka + math.AT + math.CT + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Peter Scholze - A Scalable Sequential Framework for Dynamic Inverse Problems via Model Parameter Estimation - https://arxiv.org/abs/2601.13347 - arXiv:2601.13347v1 Announce Type: new -Abstract: Large-scale dynamic inverse problems are often ill-posed due to model complexity and the high dimensionality of the unknown parameters. Regularization is commonly employed to mitigate ill-posedness by incorporating prior information and structural constraints. However, classical regularization formulations are frequently infeasible in this setting due to prohibitive memory requirements, necessitating sequential methods that process data and state information online, eliminating the need to form the full space-time problem. In this work, we propose a memory-efficient framework for reconstructing dynamic sequences of undersampled images from computerized tomography data that requires minimal hyperparameter tuning. The approach is based on a prior-informed, dimension-reduced Kalman filter with smoothing. While well suited for dynamic image reconstruction, practical deployment is challenging when the state transition model and covariance parameters must be initialized without prior knowledge and estimated in a single pass. To address these limitations, we integrate regularized motion models with expectation-maximization strategies for the estimation of state transition dynamics and error covariances within the Kalman filtering framework. We demonstrate the effectiveness of the proposed method through numerical experiments on limited-angle and single-shot computerized tomography problems, highlighting improvements in reconstruction accuracy, memory efficiency, and computational cost. - oai:arXiv.org:2601.13347v1 - math.NA - cs.NA - math.OC - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Aryeh Keating, Mirjeta Pasha + On Singular Integrals and Quantitative Rectifiability in Parabolic Space and the Heisenberg Group + https://arxiv.org/abs/2510.26934 + arXiv:2510.26934v2 Announce Type: replace +Abstract: David and Semmes proved that if all CZOs (of suitable dimension) are bounded with respect to an Ahlfors regular measure, then the measure is uniformly rectifiable. We extend this theorem to the parabolic space and the first Heisenberg group. + oai:arXiv.org:2510.26934v2 + math.AP + math.CA + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + John Hoffman, Ben Jaye - Existence and uniqueness of invariant measures for non-Feller Markov semigroups - https://arxiv.org/abs/2601.13354 - arXiv:2601.13354v1 Announce Type: new -Abstract: We study existence and uniqueness of invariant probability measures for continuous-time Markov processes on general state spaces. Existence is obtained from tightness of time averages under a weak regularity assumption inspired by quasi-Feller semigroups, allowing for discontinuous and non-Feller dynamics. - Our main contribution concerns uniqueness. Under a natural $\psi$-irreducibility assumption, we show that the normalized resolvent kernel satisfies a domination property with respect to a reference measure. As a consequence, every invariant probability measure charges this reference measure. Since distinct ergodic invariant measures are mutually singular on standard Borel spaces, this domination property implies uniqueness whenever an invariant probability measure exists. - The argument is purely measure-theoretic and does not rely on Harris recurrence, return-time estimates, or Foster--Lyapunov conditions, and applies in particular to jump processes and hybrid models with discontinuous dynamics. - oai:arXiv.org:2601.13354v1 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fundamental Lemma for Rank One Spherical Varieties of Classical Types + https://arxiv.org/abs/2511.05377 + arXiv:2511.05377v2 Announce Type: replace +Abstract: According to the relative Langlands functoriality conjecture, an admissible morphism between the $L$-groups of spherical varieties should induce a functorial transfer of the corresponding local and global automorphic spectra. Via the relative trace formula approach, two basic problems are the local transfer and the fundamental lemma on the geometric side of the relative trace formulae. In this paper, we consider the rank one spherical variety case, where the admissible morphism between the $L$-groups is the identity morphism, in which case, Y. Sakellaridis has already established the local transfer. We formulate the statement of the fundamental lemma for the general rank one spherical variety case and prove the fundamental lemma for the rank one spherical varieties of classical types. + oai:arXiv.org:2511.05377v2 + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jean-Gabriel Attali + Zhaolin Li - Center of distances of ultrametric spaces generated by labeled trees - https://arxiv.org/abs/2601.13363 - arXiv:2601.13363v1 Announce Type: new -Abstract: The center of distances of a metric space $(X,d)$ is the set $C(X) $ of all $t\in\mathbb{R}^+$ for which the equation $d(x,p)=t$ has a solution for each $p\in X$. We prove that the equalities $ C(X)=\{0\} $ or $C(X)=\{\operatorname{diam} X,0\} $ hold if $(X,d)$ is an ultrametric space generated by labeled trees. The necessary and sufficient conditions under which $\operatorname{diam} X\in C(X) $ are found. - oai:arXiv.org:2601.13363v1 - math.GN - Wed, 21 Jan 2026 00:00:00 -0500 - new + Necessary and Sufficient Conditions for Capacity-Achieving Private Information Retrieval with Adversarial Servers + https://arxiv.org/abs/2511.06003 + arXiv:2511.06003v4 Announce Type: replace +Abstract: Private information retrieval (PIR) is a mechanism for efficiently downloading messages while keeping the index of the desired message secret from the servers. PIR schemes have been extended to various scenarios with adversarial servers: PIR schemes where some servers are unresponsive or return noisy responses are called robust PIR and Byzantine PIR, respectively; PIR schemes where some servers collude to reveal the index are called colluding PIR. The information-theoretic upper bound on the download efficiency of these PIR schemes has been proved in previous studies. However, systematic ways to construct PIR schemes that achieve the upper bound are not known. In order to construct a capacity-achieving PIR schemes systematically, it is necessary to clarify the conditions that the queries should satisfy. This paper proves the necessary and sufficient conditions for capacity-achieving PIR schemes. + oai:arXiv.org:2511.06003v4 + cs.IT + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Oleksiy Dovgoshey, Olga Rovenska + Atsushi Miki, Toshiyasu Matsushima - Linear relations on star coefficients of the chromatic symmetric function - https://arxiv.org/abs/2601.13390 - arXiv:2601.13390v1 Announce Type: new -Abstract: We prove that the coefficient of the star $\mathfrak{st}_{21^{n-2}}$ in the chromatic symmetric function $X_G$ determines whether a connected graph $G$ is $2$-connected. We also prove new linear relations on other star coefficients of chromatic symmetric functions. This allows us to find new bases for certain spans of chromatic symmetric functions. Finally, we relate the coefficient of the star $\mathfrak{st}_n$ to acyclic orientations. - oai:arXiv.org:2601.13390v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new + Poncelet Triangles and Tetragons over Finite Fields + https://arxiv.org/abs/2511.06347 + arXiv:2511.06347v3 Announce Type: replace +Abstract: In the projective plane over a finite field of characteristic not equal to 2, we compute the probability that a randomly selected pair of distinct conics $(\mathscr{A},\mathscr{B})$, with $\mathscr{A}$ smooth or singular and $\mathscr{B}$ smooth, in a fixed pencil of conics will admit a triangle or a tetragon inscribed in $\mathscr{A}$ and circumscribed about $\mathscr{B}$. We do this for all pencils, classified up to projective automorphism, with at least one smooth conic; effectively allowing the case where our conic pairs intersect non-transversally. + oai:arXiv.org:2511.06347v3 + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Rosa Orellana, Foster Tom + Milena Radnovi\'c, Ruzzel Ragas - Discrete-Time Optimal Control of Species Augmentation for Predator-Prey Model - https://arxiv.org/abs/2601.13394 - arXiv:2601.13394v1 Announce Type: new -Abstract: Species augmentation is one of the methods used to promote biodiversity and prevent endangered species loss and extinction. The current work applies discrete-time optimal control theory to two models of species augmentation for predator-prey relationships. In discrete-time models, the order in which events occur can give different qualitative results. Two models representing different orders of events of optimal augmentation timing are considered. In one model, the population grows and predator-prey action occurs before the translocation of reserve species for augmentation. In the second model, the augmentation happens first and is followed by growth and then predator-prey action. - The reserve and target populations are subjected to strong Allee effects. The optimal augmentation models employed in this work aim to maximize the prey (target population) and reserve population at the final time and minimize the associated cost at each time step. Numerical simulations in the two models are conducted using the discrete version of the forward-backward sweep method and the sequential quadratic programming iterative method, respectively. The simulation results show different population levels in the two models under varying parameter scenarios. Objective functional values showing percentage increases with optimal controls are calculated for each simulation. Different optimal augmentation strategies for the two orders of events are discussed. This work represents the first optimal augmentation results for models incorporating the predator-prey relationship with discrete events. - oai:arXiv.org:2601.13394v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-sa/4.0/ - Munkaila Dasumani, Suzanne Lenhart, Gladys K. Onyambu, Stephen E. Moore + On a generalisation of Cameron's base size conjecture + https://arxiv.org/abs/2511.08705 + arXiv:2511.08705v3 Announce Type: replace +Abstract: Let $G\leqslant {\rm Sym}(\Omega)$ be a finite transitive permutation group with point stabiliser $H$. A base for $G$ is a subset of $\Omega$ whose pointwise stabiliser is trivial, and the minimal cardinality of a base is called the base size of $G$, denoted by $b(G, \Omega)$. Equivalently, $b(G, \Omega)$ is the minimal positive integer $k$ such that $G$ has a regular orbit on the Cartesian product $\Omega^k$. A well-known conjecture of Cameron from the 1990s asserts that if $G$ is an almost simple primitive group and $H$ is a so-called non-standard subgroup, then $b(G, \Omega) \leqslant 7$, with equality if and only if $G$ is the Mathieu group ${\rm M}_{24}$ in its natural action of degree $24$. This conjecture was settled in a series of papers by Burness et al. (2007-11). + In this paper, we complete the proof of a natural generalisation of Cameron's conjecture. Our main result states that if $G$ is an almost simple group and $H_1, \ldots, H_k$ are any non-standard maximal subgroups of $G$ with $k \geqslant 7$, then $G$ has a regular orbit on $G/H_1 \times \cdots \times G/H_k$, noting that Cameron's original conjecture corresponds to the special case where the $H_i$ are pairwise conjugate subgroups. In addition, we show that the same conclusion holds with $k = 6$, unless $G = {\rm M}_{24}$ and each $H_i$ is isomorphic to ${\rm M}_{23}$. For example, this means that if $G$ is a simple exceptional group of Lie type and $H_1, \ldots, H_6$ are proper subgroups of $G$, then there exist elements $g_i \in G$ such that $\bigcap_i H_i^{g_i} = 1$. By applying recent work in a joint paper with Burness, we may assume $G$ is a group of Lie type and our proof uses probabilistic methods based on fixed point ratio estimates. + oai:arXiv.org:2511.08705v3 + math.GR + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Marina Anagnostopoulou-Merkouri - Generalized Adjoint Method - https://arxiv.org/abs/2601.13395 - arXiv:2601.13395v1 Announce Type: new -Abstract: The adjoint method is an efficient way to numerically compute gradients in optimization problems with constraints, but is only formulated to differentiable cost and constraint functions on real variables. With the introduction of complex variables, which occur often in many inverse problems in electromagnetism and signal processing problems, both the cost and constraint can become non-holomorphic and hence non-differentiable in the standard definitions. Using the notion of CR-calculus, a generalized adjoint method is introduced that can compute the direction of steepest ascent for the cost function while enforcing the constraint even if both are non-holomorphic. - oai:arXiv.org:2601.13395v1 + Generalized gradient flows in Hadamard manifolds and convex optimization on entanglement polytopes + https://arxiv.org/abs/2511.12064 + arXiv:2511.12064v2 Announce Type: replace +Abstract: In this paper, we address the optimization problem of minimizing $Q(df_x)$ over a Hadamard manifold ${\cal M}$, where $f$ is a convex function on ${\cal M}$, $df_x$ is the differential of $f$ at $x \in {\cal M}$, and $Q$ is a function on the cotangent bundle of ${\cal M}$. This problem generalizes the problem of minimizing the gradient norm $\|\nabla f(x)\|$ over ${\cal M}$, studied by Hirai and Sakabe FOCS2024. We formulate a natural class of $Q$ in terms of convexity and invariance under parallel transports, and introduce a generalization of the gradient flow of $f$ that is expected to minimize $Q(df_x)$. For basic classes of manifolds, including the product of the manifolds of positive definite matrices, we prove that this gradient flow attains $\inf_{x\in {\cal M}} Q(df_x)$ in the limit, and yields a duality relation. This result is applied to the Kempf-Ness optimization for GL-actions on tensors, which is Euclidean convex optimization on the class of moment polytopes, known as the entanglement polytopes. This type of convex optimization arises from tensor-related subjects in theoretical computer science, such as quantum functional, $G$-stable rank, and noncommutative rank. + oai:arXiv.org:2511.12064v2 math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new + math.DG + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Andrew Zheng, Adam R. Stinchcombe - - - Analytic spectral perturbation theory for a high-contrast Maxwell operator - https://arxiv.org/abs/2601.13408 - arXiv:2601.13408v1 Announce Type: new -Abstract: We study analytic spectral perturbation theory for the time-harmonic Maxwell operator in a perfectly electrically conducting cavity containing a high-contrast core--shell structure. The dielectric permittivity equals $1$ in a bounded inclusion and a small complex parameter $\delta$ in the surrounding shell. The limit $\delta \to 0$ corresponds to an infinite-contrast regime and leads to a degenerate Maxwell system. Despite this degeneracy, we develop a detailed spectral theory for the limiting problem for general Lipschitz inclusions and shells. - Using a novel operator-theoretic reformulation, we prove complex-analytic dependence of the spectrum on $\delta$ in a neighborhood of $\delta = 0$. When the inclusion is a ball, we analyze the asymptotic expansion of eigenvalues and identify conditions under which the leading-order term is independent of the geometry of the surrounding shell. We also construct examples of resonances for which the leading-order asymptotics depend sensitively on the shell geometry, even in this symmetric setting. These results clarify the mechanisms underlying geometry-invariance of resonances in high-contrast Maxwell systems and explain their robustness under small complex perturbations. - oai:arXiv.org:2601.13408v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Robert V. Kohn, Raghavendra Venkatraman + Hiroshi Hirai - A uniformity principle for spatial matching - https://arxiv.org/abs/2601.13426 - arXiv:2601.13426v1 Announce Type: new -Abstract: Platforms matching spatially distributed supply to demand face a fundamental design choice: given a fixed total budget of service range, how should it be allocated across supply nodes ex ante, i.e. before supply and demand locations are realized, to maximize fulfilled demand? We model this problem using bipartite random geometric graphs where $n$ supply and $m$ demand nodes are uniformly distributed on $[0,1]^k$ ($k \ge 1$), and edges form when demand falls within a supply node's service region, the volume of which is determined by its service range. Since each supply node serves at most one demand, platform performance is determined by the expected size of a maximum matching. We establish a uniformity principle: whenever one service range allocation is more uniform than the other, the more uniform allocation yields a larger expected matching. This principle emerges from diminishing marginal returns to range expanding service range, and limited interference between supply nodes due to bounded ranges naturally fragmenting the graph. For $k=1$, we further characterize the expected matching size through a Markov chain embedding and derive closed-form expressions for special cases. Our results provide theoretical guidance for optimizing service range allocation and designing incentive structures in ride-hailing, on-demand labor markets, and drone delivery networks. - oai:arXiv.org:2601.13426v1 - math.PR - cs.DS - econ.GN - math.OC - q-fin.EC - Wed, 21 Jan 2026 00:00:00 -0500 - new + Pluripotential geometry on semi-positive effective divisors of numerical dimension one + https://arxiv.org/abs/2511.13903 + arXiv:2511.13903v2 Announce Type: replace +Abstract: We study the complex-analytic geometry of semi-positive holomorphic line bundles on compact K\"ahler manifolds. In one of our main results, for a $\mathbb{Q}$-effective line bundle satisfying a natural torsion-type assumption, we show the equivalence between semi-positivity and semi-ampleness. More generally, for an effective nef divisor of numerical dimension one, we characterize the semi-positivity of the associated line bundle in terms of the existence of a certain type of pseudoflat fundamental system of neighborhoods of the support. Furthermore, for an effective semi-positive divisor, we prove a dichotomy: either the divisor is the pull-back of a $\mathbb{Q}$-divisor by a fibration onto a Riemann surface, or the Hartogs extension phenomenon holds on the complement of its support. Our proof is based on a pluripotential method that has previously been used for studying the boundaries of pseudoconvex domains, which allows us to investigate the complex-analytic structure of neighborhoods of the support of the divisor even when the manifold is non-compact. + oai:arXiv.org:2511.13903v2 + math.CV + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Taha Ameen, Flore Sentenac, Sophie H. Yu + Takayuki Koike - Stabilization of an incompressible fluid-elastic structure system using a vacuum bubble - https://arxiv.org/abs/2601.13430 - arXiv:2601.13430v1 Announce Type: new -Abstract: We prove a priori estimates for the system of partial differential equations modeling the interaction between an elastic body and an incompressible fluid in a 3D curved domain. The fluid is governed by the incompressible Navier-Stokes equations and contains a bubble whose interior is a vacuum. The elastic body is described by a damped wave equation, and interaction with the fluid takes place along a free interface whose initial domain is curved. We show that the presence of the vacuum bubble stabilizes the system in the sense that it provides control of the average of the pressure function, and hence allows global existence and exponential decay of smooth solutions for small data. - oai:arXiv.org:2601.13430v1 + The Critical LYZ Equation in K\"ahler Geometry + https://arxiv.org/abs/2511.21492 + arXiv:2511.21492v3 Announce Type: replace +Abstract: We establish the existence of smooth solutions for the LYZ equation at the critical phase $\theta =(n-2)\frac{\pi}{2}$, thereby solving the critical case of a problem posed by Collins-Jacob-Yau and Li concerning the solvability for phase $\theta \leq (n-2)\frac{\pi}{2}$. As applications, we solve the 3D Hessian equation $\sigma_2 = 1$ and the 4D Hessian quotient equation $\sigma_3 = \sigma_1$ under weaker assumptions than previously required. + oai:arXiv.org:2511.21492v3 + math.DG math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new + math.CV + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - B. Ingimarson, I. Kukavica, W. S. O\.za\'nski + Jixiang Fu, Shing-Tung Yau, Dekai Zhang - Independence complexes of generalized Mycielskian graphs - https://arxiv.org/abs/2601.13432 - arXiv:2601.13432v1 Announce Type: new -Abstract: We show that the homotopy type of the independence complex of the generalized Mycielskian of a graph $G$ is determined by the homotopy type of the independence complex of $G$ and the homotopy type of the independence complex of the Kronecker double cover of $G$. As an application we calculate the homotopy type for paths, cycles and the categorical product of two complete graphs. - oai:arXiv.org:2601.13432v1 - math.CO - math.AT - Wed, 21 Jan 2026 00:00:00 -0500 - new + Do you precondition on the left or on the right? + https://arxiv.org/abs/2512.05160 + arXiv:2512.05160v2 Announce Type: replace +Abstract: This work is a follow-up to a poster that was presented at the DD29 conference. Participants were asked the question: ``Do you precondition on the left or on the right?''. Here we report on the results of this social experiment. We also provide context on left, right and split preconditioning, share our literature review on the topic, and analyze some of the finer points. Two examples illustrate that convergence bounds can sometimes lead to misleading conclusions. + oai:arXiv.org:2512.05160v2 + math.NA + cs.NA + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Andr\'es Carnero Bravo + Nicole Spillane (CMAP), Pierre Matalon (CMAP), Daniel B Szyld - A priori estimates and exact solvability for non-coercive stochastic control equations - https://arxiv.org/abs/2601.13444 - arXiv:2601.13444v1 Announce Type: new -Abstract: We establish, for the first time, explicit a priori and regularity estimates for solutions of the Dirichlet problem for Hamilton-Jacobi-Bellman operators from stochastic control, whose principal half-eigenvalues have opposite signs. In addition, if the negative eigenvalue is not too negative, the problem can have exactly two, one or zero solutions, depending on the valuation function. This is a novel exact multiplicity result for fully nonlinear equations, which also yields a generalization of the Ambrosetti-Prodi theorem to such equations. - oai:arXiv.org:2601.13444v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new + Geometrically vertex decomposable open neighborhood ideals + https://arxiv.org/abs/2512.12886 + arXiv:2512.12886v2 Announce Type: replace +Abstract: In this paper, we prove that the open neighborhood ideal of a TD-unmixed tree is geometrically vertex decomposable. This result implies that the associated Stanley-Reisner complex is vertex decomposable. We further demonstrate that Cohen-Macaulay open neighborhood ideals of trees are special cases of Cohen-Macaulay facet ideals of simplicial trees. Finally, we investigate open neighborhood ideals of chordal graphs and establish that almost all square-free monomial ideal can be realized as the open neighborhood ideal of a chordal graph. + oai:arXiv.org:2512.12886v2 + math.AC + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Maria Lu\'isa Pasinato, Boyan Sirakov + Jounglag Lim - Monge-Ampere type equations on compact Hermitian manifolds with bounded mass property - https://arxiv.org/abs/2601.13446 - arXiv:2601.13446v1 Announce Type: new -Abstract: In this paper, we study possibly non-closed big (1, 1)-forms on a compact Hermitian manifold satisfying the bounded mass property. We propose several criteria for the existence of rooftop envelopes. As applications, we establish the existence of solutions to complex Monge-Ampere type equations with prescribed singularities, allowing for non-pluripolar measures on the right-hand side. We also obtain stability results when singularity types vary, by extending the Darvas-Di Nezza-Lu distance to the Hermitian context. - oai:arXiv.org:2601.13446v1 - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 - new + A characterization of the local structure of two-dimensional sets with positive reach + https://arxiv.org/abs/2512.17606 + arXiv:2512.17606v2 Announce Type: replace +Abstract: The main result of the article is a complete characterization of the local structure of two-dimensional sets with positive reach in $R^d$. We also present a more elementary proof of a recent result of A. Lytchak which describes for $k\leq d$ the local structure of $k$-dimensional sets with positive reach $A$ in $R^d$ at points where the tangent cone of $A$ is $k$-dimensional. As an easy corollary of our and Lytchak's results we obtain a characterization of compact two-dimensional sets with positive reach in $R^d$. Our method also shows that, for any set $A\subset R^d$ with positive reach, the set of points at which the tangent cone of $A$ is $k$-dimensional is locally contained in a $k$-dimensional $C^{1,1}$ surface. As a consequence we obtain that if $1\leq k<d$, and $A$ is $k$-dimensional, it can be covered by countably many $k$-dimensional $C^{1,1}$ surfaces. + oai:arXiv.org:2512.17606v2 + math.MG + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xuan Li + Jan Rataj, Ludek Zajicek - From multiplicative to additive geometry: Deformation theory and 2D TQFT - https://arxiv.org/abs/2601.13455 - arXiv:2601.13455v1 Announce Type: new -Abstract: In this paper, we present a theory of Poisson deformation of Hamiltonian quasi-Poisson manifolds to Hamiltonian Poisson manifolds that include degenerate cases. More significantly, this theory extends to singular cases arising from symplectic implosion: we introduce a generalized Hamiltonian deformation theory and we show that the imploded cross section of the double $D(G)_\imp$ deforms to the implosion of the cotangent bundle $T^*G_\imp$ with applications to the master moduli space of $G$-flat connections.\\ In parallel, we construct a topological quantum field theory $\N: \text{Cob}_{2}\to \mathbf{QHam}$, where $\mathbf{QHam}$ is the category of quasi-Hamiltonian manifolds. To each cobordism $\Sigma$, we associate a quasi-Hamiltonian space $\N(\Sigma)$ built from the fusion product of copies of the double $D(G).$ We show that these spaces are invariant under the \emph{quiver homotopy} and that the composition of cobordisms corresponds to a quasi-Hamiltonian reduction. This provides a multiplicative version of the 2D Hamiltonian TQFT of Maiza-Mayrand. - oai:arXiv.org:2601.13455v1 - math.SG - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Mohamed Moussadek Maiza - - - Attached Submanifolds Beyond Symmetric Spaces - https://arxiv.org/abs/2601.13461 - arXiv:2601.13461v1 Announce Type: new -Abstract: We study submanifold geometry in the presence of symmetry, focusing on submanifolds of solvmanifolds with an unusual property relative to Ricci curvature. We generalize work of H. Tamaru \cite{tamaru-11} in which he explores the geometry of submanifolds of symmetric spaces of noncompact type constructed from parabolic subgroups of the isometry group. He calls these attached submanifolds. The Ricci curvatures of attached submanifolds coincide with the restrictions of the Ricci curvatures of ambient symmetric spaces. - We broaden Tamaru's construction by weakening the hypotheses on the ambient space, allowing a pseudo-Riemannian scalar product, and defining attached submanifolds in terms of root spaces. We demonstrate that in this setting, the Ricci curvature restriction property for attached submanifolds holds if and only if the submanifold satisfies an algebraic criterion that we call the Jacobi Star Condition. Like attached submanifolds of symmetric spaces, our attached submanifolds are minimal, and are only totally geodesic under hypotheses analogous to hypotheses in the symmetric space case. - Finally, we give an example of a solvmanifold that has an attached submanifold and is not a symmetric space, demonstrating that attached submanifolds are not unique to symmetric spaces. - oai:arXiv.org:2601.13461v1 + Holomorphic Deformations of Hyperbolicity Notions on Compact Complex Manifolds + https://arxiv.org/abs/2512.19284 + arXiv:2512.19284v3 Announce Type: replace +Abstract: We investigate deformation properties of balanced hyperbolicity, with particular emphasis on degenerate balanced manifolds and their behavior under modifications. + In this context, we introduce two new notions of hyperbolicity for compact non-K\"ahler manifolds $X$ of complex dimension $\dim_{\mathbb{C}}X=n$ in degree $1 \leq p \leq n-1$, inspired by the work of F. Haggui and S. Marouani on $p$-K\"ahler hyperbolicity. The first notion, called \emph{p-SKT hyperbolicity}, generalizes the notions of SKT hyperbolicity and Gauduchon hyperbolicity introduced by S. Marouani. The second notion, called \emph{p-HS hyperbolicity}, extends the notion of sG hyperbolicity defined by Y. Ma. + We investigate the relationship between these notions of analytic nature and their geometric counterparts, namely Kobayashi hyperbolicity and \emph{p-cyclic hyperbolicity} for $2 \leq p \leq n-1$, and we examine the openness under holomorphic deformations of both $p$-HS hyperbolicity and $p$-K\"ahler hyperbolicity. + oai:arXiv.org:2512.19284v3 + math.CV + math.AG math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Megan M. Kerr, Tracy L. Payne - - - Fiber-preserving and orientation-reversing involutions of Seifert fibered 3-manifolds - https://arxiv.org/abs/2601.13469 - arXiv:2601.13469v1 Announce Type: new -Abstract: We consider fiber-preserving, orientation-reversing involutions on orientable Seifert fibered 3-manifolds and the conditions on a manifold for admissibility of such involutions. We construct a class $\Psi$ of fiber-preserving, orientation-reversing involutions that act trivially on the base. Each element of $\Psi$ is obtained by extending a product involution across Seifert pieces of type $V(2,2;-1)$ - a solid torus with three fibers filled according to Seifert invariants $(2,1)$, $(2,1)$, and $(1,-1)$. We show that $\Psi$ forms a single conjugacy class under fiber-preserving diffeomorphisms. Our main result establishes that any fiber-preserving, orientation-reversing involution factors as $\psi\circ g$, where $g$ is fiber-preserving and orientation-preserving and $\psi\in\Psi$, thus reducing the problem to the previously known orientation-preserving case. Through the orientable base-space double covering, we further extend the classification to manifolds with non-orientable base orbifold. - oai:arXiv.org:2601.13469v1 - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Benjamin Peet - - - Directional Ballistic Transport in Quantum Waveguides - https://arxiv.org/abs/2601.13471 - arXiv:2601.13471v1 Announce Type: new -Abstract: We study the transport properties of Schr\"odinger operators on $\mathbb{R}^d$ with potentials that are periodic in some directions and compactly supported in the others. Such systems are known to produce surface states that are weakly confined near the support of the potential. We show that a natural set of surface states exhibits directional ballistic transport, characterized by ballistic transport in the periodic directions and its absence in the others. To prove this, we develop a Floquet theory that captures the analytic variation of surface states. The main idea consists of reformulating the eigenvalue problem for surface states as a Fredholm problem via the Dirichlet-to-Neumann map. - oai:arXiv.org:2601.13471v1 - math.SP - math-ph - math.AP - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Adam Black, David Damanik, Peter Kuchment, Tal Malinovitch, Giorgio Young - - - Elias-type Bounds for Codes in the Symmetric Limited-Magnitude Error Channel - https://arxiv.org/abs/2601.13477 - arXiv:2601.13477v1 Announce Type: new -Abstract: We study perfect error-correcting codes in $\mathbb{Z}^n$ for the symmetric limited-magnitude error channel, where at most $e$ coordinates of an integer vector may be altered by a value whose magnitude is at most $s$. Geometrically, such codes correspond to tilings of $\mathbb{Z}^n$ by the symmetric limited-magnitude error ball $\mathcal{B}(n,e,s,s)$. Given $n$ and $s$, we adapt the geometric ideas underlying the Elias bound for the Hamming metric to the distance $d_s$ tailed for this channel, and derive new necessary conditions on $e$ for the existence of perfect codes / tilings, without assuming any lattice structure. Our main results identify two distinct regimes depending on the error magnitude. For small error magnitudes ($s \in \{1, 2\}$), we prove that if the number of correctable errors does not exceed a certain fraction of $n$, then it is asymptotically bounded by $e = \mathcal{O}(\sqrt{n \log n})$. In contrast, for larger magnitudes ($s \geq 3$), we establish a significantly sharper bound of $e < \sqrt{12.36n}$, which holds without any restriction on $e$ being below a given fraction of $n$. Finally, by extending our method to non-perfect codes, we derive an upper bound on packing density, showing that for codes correcting a linear or $\Omega(\sqrt{n})$ number of errors, the density is bounded by a factor inversely proportional to the error magnitude $s$. - oai:arXiv.org:2601.13477v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Zhihao Guan, Hengjia Wei + Abdelouahab Khelifati - iCanonical basis arising from quasi-split rank one iquantum group - https://arxiv.org/abs/2601.13482 - arXiv:2601.13482v1 Announce Type: new -Abstract: We compute icanonical basis of the quasi-split rank one modified iquantum group, by obtaining explicit transition matrices among the icanonical basis, monomial basis, and standardized canonical basis; all these bases can be naturally categorified. These transition matrices follow from their counterparts computed in this paper among the icanonical basis, monomial basis, and canonical basis on simple finite-dimensional modules of quantum $\mathfrak{sl}_3$. - oai:arXiv.org:2601.13482v1 - math.QA - Wed, 21 Jan 2026 00:00:00 -0500 - new + Erd\H{o}s-Wintner theorem for linear recurrent bases + https://arxiv.org/abs/2512.20882 + arXiv:2512.20882v2 Announce Type: replace +Abstract: Let $(G_n)_{n\geqslant 0}$ be a linear recurrence sequence defining a numeration system and satisfying mild structural hypotheses. For real-valued G-additive functions (additive in the greedy G-digits), we establish an Erd\H{o}s-Wintner-type theorem: convergence of two canonical series (a first-moment series and a quadratic digit-energy series) is necessary and sufficient for the existence of a limiting distribution along initial segments of the integers. In that case, the limiting characteristic function admits an explicit infinite-product factorization whose local factors depend only on the underlying digit system. We also indicate conditional extensions of this two-series criterion to Ostrowski numeration systems with bounded partial quotients and to Parry $\beta$-expansions with Pisot-Vijayaraghavan base $\beta$. + oai:arXiv.org:2512.20882v2 + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ziming Chen + Johann Verwee - Gorenstein Special Fiber Rings of Ladder Determinantal Modules - https://arxiv.org/abs/2601.13483 - arXiv:2601.13483v1 Announce Type: new -Abstract: A ladder determinantal module is an arbitrary direct sum of ideals of maximal minors of a generic ladder matrix. In this article, we give necessary and sufficient conditions for the special fiber ring of such modules to be Gorenstein. These conditions are expressed in terms of data obtained from the underlying matrix. - oai:arXiv.org:2601.13483v1 - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 - new + Piecewise Smooth Dynamical Systems Regularized by Convolution + https://arxiv.org/abs/2601.00697 + arXiv:2601.00697v2 Announce Type: replace +Abstract: We present a general regularization procedure for piecewise smooth vector fields whose discontinuity locus is a variety of normal crossings type. We show that such regularization can be smoothed through a finite sequence of blowings-up, thereby reducing the problem to study of the dynamics of a smooth vector field in a manifold with corners. The procedure will be illustrated in the cases of piecewise smooth vector fields on $\mathbb{R}^2$ with discontinuity locus $x=0$ or $xy=0$, and on $\mathbb{R}^3$ with discontinuity locus $xyz=0$. We will see that some unexpected dynamical phenomena may arise even in the case of piecewise constant vector fields. + oai:arXiv.org:2601.00697v2 + math.DS + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Louiza Fouli, Kuei-Nuan Lin, Haydee Lindo, Maral Mostafazadehfard + Claudio A. Buzzi, Daniel Panazzolo, Paulo R. da Silva - Self-Supervised Learning of Parametric Approximation for Security-Constrained DC-OPF - https://arxiv.org/abs/2601.13486 - arXiv:2601.13486v1 Announce Type: new -Abstract: This paper introduces a self-supervised learning framework for approximating the Security-Constrained DC Optimal Power Flow (SC-DCOPF) problem using a parametric linear model. The approach preserves the physical structure of the DC-OPF while incorporating demand-dependent tunable parameters that scale transmission line limits. These parameters are predicted via a Graph Neural Network and optimized through differentiable layers, enabling direct training from contingency costs without requiring labeled data. The framework integrates pre- and post-contingency optimization layers into an implicit loss function. Numerical experiments on benchmark systems demonstrate that the proposed method achieves high dispatch accuracy, low cost approximation error, and strong data efficiency, outperforming semi-supervised and end-to-end baselines. This scalable and interpretable approach offers a promising solution for real-time secure power system operations. - oai:arXiv.org:2601.13486v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new + Polygons in Polygons with a Twist + https://arxiv.org/abs/2601.00899 + arXiv:2601.00899v4 Announce Type: replace +Abstract: This is a study of the construction of particular regular sub-n-gons T in regular n-gons P using a special system of chords of P. In particular, some of these sub-n-gons have areas which are integer divisors of the area of the given n-gon P. Initially, the study will concentrate on chords which are from a vertex to special points of one of the opposite sides of P. Several examples are explored. However, it will become apparent that a much more general situation exists. Dynamic Geometry software is the key to investigating this new relationship. + oai:arXiv.org:2601.00899v4 + math.HO + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Anderson Anrrango, Andr\'e Quisaguano, Gonzalo E. Constante-Flores, Can Li + James M Parks - A note on "Higher order linear differential equations for unitary matrix integrals: applications and generalisations" - https://arxiv.org/abs/2601.13488 - arXiv:2601.13488v1 Announce Type: new -Abstract: In this note, we briefly introduce the background and motivation of the collaborative work [arXiv:2508.20797], and provide an outline of the main results. The latter relates to matrix and higher order scalar differential equations satisfied by certain Hankel and Toeplitz determinants involving I-Bessel functions, or equivalently certain unitary matrix integrals, and moreover puts this property in a broader context. We also investigate large gaps between zeros of the derivatives of the Hardy $\mathsf{Z}$-function, assuming the validity of a certain joint moments conjecture in random matrix theory. - oai:arXiv.org:2601.13488v1 - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new + Two algebraic proofs of the transcendence of $\mathrm{e}$ based on formal power series + https://arxiv.org/abs/2601.01019 + arXiv:2601.01019v5 Announce Type: replace +Abstract: We remind the classical analytical proof of the transcendence of $\mathrm{e}$ due to Hilbert. Then, using formal power series, we give two algebraic semiformal proofs of this result. The first proof is a specialization of the proof of the Lindemann-Weierstrass theorem found by Beukers, B\'ezivin and Robba [2]. The second proof uses improper integrals of formal power series and is due to this author. We explain what ``semiformal'' means. + oai:arXiv.org:2601.01019v5 + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Peter J. Forrester, Fei Wei + Martin Klazar - Noncommutative Minkowski integral inequality and a unitary categorification criterion for fusion rings - https://arxiv.org/abs/2601.13490 - arXiv:2601.13490v1 Announce Type: new -Abstract: We prove a noncommutative analogue of Minkowski's integral inequality for commuting squares of tracial von Neumann algebras. The inequality implies a necessary condition for a quadruple of graphs to be realized as inclusion graphs of a commuting square of multi-matrix algebras. As a corollary, we obtain a unitary categorification criterion for based rings, in particular, fusion rings. - oai:arXiv.org:2601.13490v1 - math.OA - math.CT - math.FA - math.QA - Wed, 21 Jan 2026 00:00:00 -0500 - new + A countable-support symmetric iteration separating PP from AC + https://arxiv.org/abs/2601.01855 + arXiv:2601.01855v4 Announce Type: replace +Abstract: We construct, from a ground model of $ZFC$, a transitive symmetric model $M$ satisfying $ZF + DC + PP + AC_{wo} + \neg AC$. The construction starts with a Cohen symmetric seed model $N$ over $Add(\omega,\omega_1)$ and performs an Ord-length countable-support symmetric iteration. For fixed parameters $S:=A^\omega$ and $T:=PowerSet(S)$ (as computed in $N$), successor stages add orbit-symmetrized packages which force the localized splitting principle $PP^{\mathrm{split}}\!\restriction T$ (hence $PP\restriction T$) and the choice principle $AC_{wo}$, while preserving $DC$ and keeping $A$ non-well-orderable. A diagonal-lift/diagonal-cancellation scheme produces $\omega_1$-complete normal limit filters. A persistence argument yields $SVC^+(T)$ in M, and Ryan--Smith localization then upgrades $PP\restriction T$ and $AC_{wo}$ to $PP$. + oai:arXiv.org:2601.01855v4 + math.LO + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Junhwi Lim + Frank Gilson - LQ Mean Field Games with Common Noise in Hilbert Spaces: Small and Arbitrary Finite Time Horizons - https://arxiv.org/abs/2601.13493 - arXiv:2601.13493v1 Announce Type: new -Abstract: We extend the results of (Liu and Firoozi, 2025), which develops the theory of linear-quadratic (LQ) mean field games in Hilbert spaces, by incorporating a common noise. This common noise is an infinite-dimensional Wiener process affecting the dynamics of all agents. In the presence of common noise, the mean-field consistency condition is characterized by a system of coupled forward-backward stochastic evolution equations (FBSEEs) in Hilbert spaces, whereas in its absence, it is represented by forward-backward deterministic evolution equations. We establish the existence and uniqueness of solutions to the coupled linear FBSEEs associated with the LQ MFG setting for small time horizons and prove the $\epsilon$-Nash property of the resulting equilibrium strategy. Furthermore, for the first time in the literature, we develop an analysis that establishes the well-posedness of these coupled linear FBSEEs in Hilbert spaces, for which only mild solutions exist, over arbitrary finite time horizons. - oai:arXiv.org:2601.13493v1 - math.OC - math.FA + Distributional Limits for Eigenvalues of Graphon Kernel Matrices + https://arxiv.org/abs/2601.04584 + arXiv:2601.04584v2 Announce Type: replace +Abstract: We study the fluctuation behavior of individual eigenvalues of kernel matrices arising from dense graphon-based random graphs. Under minimal integrability and boundedness assumptions on the graphon, we establish distributional limits for simple, well-separated eigenvalues of the associated integral operator. A sharp probabilistic dichotomy emerges: in the non-degenerate regime, the properly normalized empirical eigenvalue satisfies a central limit theorem with an explicit variance, whereas in the degenerate regime the leading stochastic term vanishes and the centered eigenvalue converges to a weighted chi-square law determined by the operator spectrum. + The analysis requires no smoothness or Lipschitz conditions on the kernel. Prior work under comparable assumptions established only operator convergence and eigenspace consistency; the present results characterize the full distributional behavior of individual eigenvalues, extending fluctuation theory beyond the reach of classical operator-level arguments. The proofs combine second-order perturbation expansions, concentration bounds for kernel matrices, and Hoeffding decompositions for symmetric statistics, revealing that at the $\sqrt{n}$ scale the dominant randomness arises from latent-position sampling rather than Bernoulli edge noise. + oai:arXiv.org:2601.04584v2 math.PR - q-fin.MF - Wed, 21 Jan 2026 00:00:00 -0500 - new + math.ST + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hanchao Liu, Dena Firoozi + Behzad Aalipur - Double Hall-Littlewood symmetric polynomials - https://arxiv.org/abs/2601.13497 - arXiv:2601.13497v1 Announce Type: new -Abstract: We establish a ring isomorphism between the derived Hall algebra of the Jordan quiver and the ring of double symmetric functions (i.e., the ring of symmetric polynomials in two sets of countably many variables, invariant under the respective actions of their symmetric groups) with a parameter $t$. This isomorphism maps the derived Hall basis (the natural basis of the derived Hall algebra) to a class of double Hall-Littlewood (HL) symmetric functions, which are formulated via raising and lowering operators. These double HL functions are parameterized by bipartitions; they reduce to the classical HL functions when one of the partitions is empty, and specialize to Schur Laurent symmetric functions at $t = 0$. We also derive the Pieri rules for these double HL functions. Additionally, we obtain several natural generating functions for the derived Hall algebra as well as their transition relations, which can be transferred to the ring of double symmetric functions via the established ring isomorphism. - oai:arXiv.org:2601.13497v1 - math.QA - math.CO - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - new + On the effects of protection zone and directed population flux in prey-predator dynamics + https://arxiv.org/abs/2601.05054 + arXiv:2601.05054v2 Announce Type: replace +Abstract: We study a spatial predator-prey model in which prey can enter a protection zone (refuge) inaccessible to predators, while predators exhibit directed movement toward prey-rich regions. The directed movement is modeled by a far-sighted population flux motivated by classical movement rules, in contrast to the more commonly analyzed near-sighted chemotaxis-type mechanisms. We first establish local-in-time well-posedness for the corresponding nonstationary problem under Neumann boundary conditions, despite the discontinuity induced by the refuge interface. We then investigate the stationary problem, focusing on how the coexistence states emerge and organize globally in parameter space. In particular, we identify the bifurcation threshold for positive steady states from semitrivial predator-only equilibria, and describe the global continuation of the resulting branches. Our analysis reveals that strong directed movement can induce turning-point structures and multiplicity of coexistence steady states, highlighting a nontrivial interplay between spatial protection and predator movement behavior. + oai:arXiv.org:2601.05054v2 + math.AP + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Jiayi Chen, Ming Lu, Shiquan Ruan + Kousuke Kuto, Kazuhiro Oeda - Group Relative Policy Optimization for Robust Blind Interference Alignment with Fluid Antennas - https://arxiv.org/abs/2601.13506 - arXiv:2601.13506v1 Announce Type: new -Abstract: Fluid antenna system (FAS) leverages dynamic reconfigurability to unlock spatial degrees of freedom and reshape wireless channels. This paper proposes, for the first time, a robust fluid antenna-driven blind interference alignment (BIA) framework for a K-user MISO downlink under imperfect channel state information (CSI). We formulate a robust sum-rate maximization problem through optimizing fluid antenna positions. To solve this challenging non-convex problem, we employ group relative policy optimization (GRPO), a novel deep reinforcement learning algorithm that eliminates the critic network. This robust design reduces model size and floating point operations (FLOPs) by nearly half compared to proximal policy optimization (PPO) while significantly enhancing performance through group-based exploration that escapes bad local optima. Simulation results demonstrate that GRPO outperforms PPO by 4.17%, and a 100K-step pre-trained PPO by 30.29%. Due to error distribution learning, GRPO exceeds heuristic MaximumGain and RandomGain by 200.78% and 465.38%, respectively. - oai:arXiv.org:2601.13506v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new + Foundations and Fundamental Properties of a Two-Parameter Memory-Weighted Velocity Operator + https://arxiv.org/abs/2601.05122 + arXiv:2601.05122v2 Announce Type: replace +Abstract: We introduce and analyze a **memory-weighted velocity operator** \(\mathscr{V}_{\alpha,\beta}\) as a mathematical framework for describing rates of change in systems with time-varying, power-law memory. The operator employs two independent continuous exponents \(\alpha(t)\) and \(\beta(t)\) that separately weight past state increments and elapsed time scaling, motivated by physical systems where these memory aspects may evolve differently -- such as viscoelastic materials with stress-dependent relaxation or anomalous transport with history-dependent characteristics. + We establish the operator's foundational properties: an explicit integral representation, linearity, and **continuous dependence** on the memory exponents with respect to uniform convergence. Central to the analysis are **weighted pointwise estimates** revealing how the exponent difference \(\beta(t)-\alpha(t)\) modulates \(\mathscr{V}_{\alpha,\beta}[x](t)\), leading to conditions under which \(\mathscr{V}_{\alpha,\beta}\) defines a bounded linear operator between standard function spaces. These estimates exhibit a natural compensation mechanism between the two memory weightings. + For the uniform-memory case \(\alpha=\beta\equiv1\), we prove that \(\mathscr{V}_{\alpha,\beta}[x](t)\) **asymptotically recovers** the classical derivative \(\dot{x}(0)\) as \(t\to 0^{+}\), ensuring consistency with local calculus. The mathematical framework is supported by self-contained technical appendices. By decoupling the memory weighting of state increments from that of elapsed time, \(\mathscr{V}_{\alpha,\beta}\) provides a structured approach to modeling systems with independently evolving memory characteristics, offering potential utility in formulating evolution equations for complex physical processes with non-stationary memory. + oai:arXiv.org:2601.05122v2 + math-ph + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jianqiu Peng, Tong Zhang, Shuai Wang, Mingjie Shao, Hao Xu, Rui Wang + Jiahao Jiang - Hidden convexity of quadratic systems and its application to quadratic programming - https://arxiv.org/abs/2601.13511 - arXiv:2601.13511v1 Announce Type: new -Abstract: In this paper, we present sufficient conditions ensuring that the sum of the image of quadratic functions and the nonnegative orthant is convex. The hidden convexity of the trust-region problem with linear inequality constraints is established under a newly proposed assumption, which is compared with the previous one in [{\it Math. Program. 147, 171--206, 2014}]. We also provide a complete proof of the hidden convexity of a system of two quadratic functions in [{\it J. Glob. Optim. 56, 1045--1072, 2013}]. Furthermore, necessary and sufficient conditions for the S-lemma concerning systems of quadratic inequalities are investigated. Finally, we derive necessary and sufficient global optimality conditions and strong duality results for quadratic programming. - oai:arXiv.org:2601.13511v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new + Homogenization of L\'evy-type operators: operator estimates with correctors + https://arxiv.org/abs/2601.06832 + arXiv:2601.06832v2 Announce Type: replace +Abstract: The goal of the paper is to study in $L_2(\R^d)$ a self-adjoint operator ${\mathbb A}_\eps$, $\eps >0$, of the form $$ ({\mathbb A}_\eps u) (\x) = \int_{\R^d} \mu(\x/\eps, \y/\eps) \frac{\left( u(\x) - u(\y) \right)}{|\x - \y|^{d+\alpha}}\,d\y $$ with $1< \alpha < 2$; + here the function + $\mu(\x,\y)$ is $\Z^d$-periodic in the both variables, satisfies the symmetry relation $\mu(\x,\y) = \mu(\y,\x)$ and + the estimates $0< \mu_- \leqslant \mu(\x,\y) \leqslant \mu_+< \infty$. The rigorous definition of the operator ${\mathbb A}_\eps$ is given in terms of the corresponding quadratic form. In the previous work of the authors it was shown that the resolvent $({\mathbb A}_\eps + I)^{-1}$ converges, as $\eps\to0$, in the operator norm in $L_2(\mathbb R^d)$ to the resolvent of the effective operator $A^0$, and the estimate $\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} \| = O(\eps^{2-\alpha})$ holds. In the present work we achieve a more accurate approximation of the resolvent of ${\mathbb A}_\eps$ which takes into account the correctors. Namely, for $N\in\mathbb N$ such that $2-1/N < \alpha \le 2-1/(N+1)$, we obtain $$ \bigl\|({\mathbb A}_\eps + I)^{-1} - (\A^0 + I)^{-1} - \sum_{m=1}^N \eps^{m(2-\alpha)} \mathbb{K}_m \bigr\| = O(\eps). $$ + oai:arXiv.org:2601.06832v2 + math.AP + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Nguyen Quang Huy, Nguyen Huy Hung, Tran Van Nghi, Hoang Ngoc Tuan, Nguyen Van Tuyen + Andrey Piatnitski, Vladimir Sloushch, Tatiana Suslina, Elena Zhizhina - Categorical Entropies of Hilbert Schemes of Points on Surfaces and Hyperk\"ahler Manifolds - https://arxiv.org/abs/2601.13526 - arXiv:2601.13526v1 Announce Type: new -Abstract: This paper studies the categorical entropy of autoequivalences of derived categories of Hilbert schemes of points on surfaces and hyperk\"ahler manifolds. One of the central questions about categorical entropy is whether it satisfies a Gromov-Yomdin type formula $h_{\mathrm{cat}}(\Phi) = \log\rho(\Phi)$. We say that $X$ has the Gromov-Yomdin (GY) property if this formula holds. We prove that if a surface $S$ fails to satisfy the (GY) property (e.g., K3 surfaces), then so does $\mathrm{Hilb}^n(S)$. Moreover, we show that no hyperk\"ahler or Enriques manifold satisfies the (GY) property by constructing an explicit autoequivalence with positive categorical entropy but unipotent action on the cohomology ring. - oai:arXiv.org:2601.13526v1 - math.AG - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - new + $A_3$-formality for Demushkin groups at odd primes + https://arxiv.org/abs/2601.07551 + arXiv:2601.07551v2 Announce Type: replace +Abstract: We study a weak form of formality for differential graded algebras, called $A_3$-formality, for the cohomology of pro-p Demushkin groups at odd primes p. We show that the differential graded $\mathbb{F}_p$-algebras of continuous cochains of Demushkin groups with q-invariant not equal 3 are $A_3$-formal, whereas Demushkin groups with q-invariant 3 are not $A_3$-formal. We prove these results by an explicit computation of the Benson-Krause-Schwede canonical class in Hochschild cohomology. + oai:arXiv.org:2601.07551v2 + math.GR + math.AT + math.KT + math.NT + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tomoki Yoshida + Ambrus P\'al, Gereon Quick - A construction of smooth varieties admitting small contractions - https://arxiv.org/abs/2601.13527 - arXiv:2601.13527v1 Announce Type: new -Abstract: We construct smooth varieties admitting small contractions from arbitrary smooth projective varieties. This construction generalizes Kawamata's four-dimensional example. We also give sufficient conditions for divisors on these varieties to be nef. As an application, we obtain weak Fano fourfolds from products of two del Pezzo surfaces. - oai:arXiv.org:2601.13527v1 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yuto Masamura, Tomoki Yoshida + Computational Evidence Against Quadratic-Cubic Factorization for the Second Cuboid Quintic + https://arxiv.org/abs/2601.07899 + arXiv:2601.07899v2 Announce Type: replace +Abstract: Let $Q_{p,q}(t)\in\mathbb{Z}[t]$ be Sharipov's even monic degree-$10$ second cuboid polynomial depending on coprime integers $p\neq q>0$. Writing $Q_{p,q}(t)$ as a quintic in $t^{2}$ produces an associated monic quintic polynomial. After the weighted normalization $r=p/q$ and $s=r^{2}$ we obtain a one-parameter family $P_s(x)\in\mathbb{Q}[x]$ such that \[ Q_{p,q}(t)=q^{20}\,P_s\!\left(\frac{t^{2}}{q^{4}}\right)\qquad\text{with}\qquad s=\left(\frac{p}{q}\right)^{2}. \] Assuming a quadratic divisor $x^{2}+ax+b$ with $a,b\in\mathbb{Q}$, we reduce divisibility of $P_s(x)$ to the vanishing of an explicit remainder \[ R(x)=R_{1}(s,a,b)\,x+R_{0}(s,a,b). \] A key structural observation is that $R_1$ and $R_0$ are quadratic in $b$ and that, on the equation $R_1=0$, the second condition becomes linear in $b$. This yields a one-direction elimination to a plane obstruction curve $F(s,a)=0$ with $F\in\mathbb{Z}[s,a]$, without any lifting-back issues: when the linear coefficient is nonzero, the parameter $b$ is forced to be the rational value $b=C/L$. We isolate the degenerate locus $L=C=0$ and show it produces only $s=\pm 1$ (hence only $s=1$ in the cuboid domain $s>0$). Let $\overline{C}\subset\mathbb{P}^{2}$ be the projective closure of $F(s,a)=0$. Using Magma we perform a height-bounded search for rational points on $\overline{C}$. With bound $H=10^{9}$, the search returns $8$ rational points, whose affine part has $s\in\{-1,0,1\}$. In particular, no affine rational point with $s>0$ and $s\neq 1$ is found up to this bound. This provides strong computational evidence that for rational $s>0$, $s\neq 1$, the quintic $P_s(x)$ admits no quadratic factor over $\mathbb{Q}$ (equivalently, no $2+3$ (quadratic-cubic) factorization over $\mathbb{Q}$), and yields a conditional exclusion assuming completeness of the rational-point enumeration on $\overline{C}$. + oai:arXiv.org:2601.07899v2 + math.GM + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Valery Asiryan, Randall L. Rathbun - Sparse Identification of Nonlinear Distributed-Delay Dynamics via the Linear Chain Trick - https://arxiv.org/abs/2601.13536 - arXiv:2601.13536v1 Announce Type: new -Abstract: The Sparse Identification of Nonlinear Dynamics (SINDy) framework has been frequently used to discover parsimonious differential equations governing natural and physical systems. This includes recent extensions to SINDy that enable the recovery of discrete delay differential equations, where delay terms are represented explicitly in the candidate library. However, such formulations cannot capture the distributed delays that naturally arise in biological, physical, and engineering systems. In the present work, we extend SINDy to identify distributed-delay differential equations by incorporating the Linear Chain Trick (LCT), which provides a finite-dimensional ordinary differential equation representing the distributed memory effects. Hence, SINDy can operate in an augmented state space using conventional sparse regression while preserving a clear interpretation of delayed influences via the chain trick. From time-series data, the proposed method jointly infers the governing equations, the mean delay, and the dispersion of the underlying delay distribution. We numerically verify the method on several models with distributed delay, including the logistic growth model and a Hes1--mRNA gene regulatory network model. We show that the proposed method accurately reconstructs distributed delay dynamics, remains robust under noise and sparse sampling, and provides a transparent, data-driven approach for discovering nonlinear systems with distributed-delay. - oai:arXiv.org:2601.13536v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new + A proof of Alexander's conjecture on an inequality of Cassels + https://arxiv.org/abs/2601.10411 + arXiv:2601.10411v2 Announce Type: replace +Abstract: Let $z_1,\dots,z_n$ be complex numbers with $|z_j|\le \rho$, where $\rho>1$. Cassels proved that, under an additional restriction on $\rho$, the inequality \[ \prod_{j\ne k}\bigl|1-\overline{z_j}z_k\bigr| \le \left(\frac{\rho^{2n}-1}{\rho^2-1}\right)^{\!n} \] holds. In a subsequent note, Alexander conjectured that this inequality is in fact valid without any restriction on $\rho$. In this paper, we confirm Alexander's conjecture. + oai:arXiv.org:2601.10411v2 + math.CV + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mohammed Alanazi, Majid Bani-Yaghoub + Myriam Ouna\"ies - A hybrid numerical method for a microscopic and macroscopic traffic flow model - https://arxiv.org/abs/2601.13541 - arXiv:2601.13541v1 Announce Type: new -Abstract: In this paper, we introduce a traffic flow model based on a microscopic follow-the-leader model, while enforcing maximal constraints on the density and velocity of the flow. The related macroscopic model can be represented in conservative formulation. By introducing an advected variable up with the flow, where p is the velocity offset, and u is the relative velocity, we reformulate the classical Aw-Rascle-Zhang (ARZ) model and the modified Aw-Rascle model to describe a realistic fundamental diagrams. The elementary waves are derived, and the Riemann problem is solved to validate the model's theoretical consistency. We further extend to a two-dimensional model. Numerical simulations are given for both one-and two-dimensional case by using the hybrid Godunov-Glimm scheme to verify the model's performance. - oai:arXiv.org:2601.13541v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yuanhong Wu, Shuzhi Liu, Qinglong Zhang + Error-Correcting Codes for Two Bursts of t1-Deletion-t2-Insertion with Low Computational Complexity + https://arxiv.org/abs/2601.10540 + arXiv:2601.10540v2 Announce Type: replace +Abstract: Burst errors involving simultaneous insertions, deletions, and substitutions occur in practical scenarios, including DNA data storage and document synchronization, motivating developments of channel codes that can correct such errors. In this paper, we address the problem of constructing error-correcting codes (ECCs) capable of handling multiple bursts of $t_1$-deletion-$t_2$-insertion ($(t_1,t_2)$-DI) errors, where each burst consists of $t_1$ deletions followed by $t_2$ insertions in a binary sequence. We make three key contributions: Firstly, we establish the fundamental equivalence of (1) two bursts of $(t_1,t_2)$-DI ECCs, (2) two bursts of $(t_2,t_1)$-DI ECCs, and (3) one burst each of $(t_1,t_2)$-DI and $(t_2,t_1)$-DI ECCs. Then, we derive lower and upper bounds on the code size of two bursts of $(t_1,t_2)$-DI ECCs, which can naturally be extended to the case of multiple bursts. Finally, we present constructions of two bursts of $(t_1,t_2)$-DI ECCs. Compared to the codes obtained by the syndrome compression technique, the resulting codes achieve significantly lower computational complexity. + oai:arXiv.org:2601.10540v2 + cs.IT + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Yajuan Liu, Tolga M. Duman - Dean's conjecture and cycles modulo k - https://arxiv.org/abs/2601.13552 - arXiv:2601.13552v1 Announce Type: new -Abstract: Dean conjectured three decades ago that every graph with minimum degree at least $k\ge 3$ contains a cycle whose length is divisible by $k$. While the conjecture has been verified for $k\in \{3,4\}$, it remains open for $k\ge 5$. A weaker version, also proposed by Dean, asserting that every $k$-connected graph contains a cycle of length divisible by $k$, was resolved by Gao, Huo, Liu, and Ma using the notion of admissible cycles. - In this paper, we resolve Dean's conjecture for all $k\ge 6$. In fact, we prove a stronger result by showing that every graph with minimum degree at least $k$ contains cycles of length $r \pmod k$ for every even integer $r$, unless every end-block belongs to a specific family of exceptional graphs, which fail only to contain cycles of length $2 \pmod k$. We also establish a strengthened result on the existence of admissible cycles. Our proof introduces two sparse graph families, called trigonal graphs and tetragonal graphs, which provide a flexible framework for studying path and cycle lengths and may be of independent interest. - oai:arXiv.org:2601.13552v1 + The inducibility of Tur\'an graphs + https://arxiv.org/abs/2601.10548 + arXiv:2601.10548v2 Announce Type: replace +Abstract: Let $I(F,n)$ denote the maximum number of induced copies of a graph $F$ in an $n$-vertex graph. The inducibility of $F$, defined as $i(F)=\lim_{n\to \infty} I(F,n)/\binom{n}{v(F)}$, is a central problem in extremal graph theory. In this work, we investigate the inducibility of Tur\'an graphs $F$. This topic has been extensively studied in the literature, including works of Pippenger--Golumbic, Brown--Sidorenko, Bollob\'as--Egawa--Harris--Jin, Mubayi, Reiher, and the first author, and Yuster. Broadly speaking, these results resolve or asymptotically resolve the problem when the part sizes of $F$ are either sufficiently large or sufficiently small (at most four). + We complete this picture by proving that for every Tur\'an graph $F$ and sufficiently large $n$, the value $I(F,n)$ is attained uniquely by the $m$-partite Tur\'an graph on $n$ vertices, where $m$ is given explicitly in terms of the number of parts and vertices of $F$. This confirms a conjecture of Bollob\'as--Egawa--Harris--Jin from 1995, and we also establish the corresponding stability theorem. Moreover, we prove an asymptotic analogue for $I_{k+1}(F,n)$, the maximum number of induced copies of $F$ in an $n$-vertex $K_{k+1}$-free graph, thereby completely resolving a recent problem of Yuster. Finally, our results extend to a broader class of complete multipartite graphs in which the largest and smallest part sizes differ by at most on the order of the square root of the smallest part size. + oai:arXiv.org:2601.10548v2 math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yufan Luo, Jie Ma, Ziyuan Zhao - - - Universality of the Basilica - https://arxiv.org/abs/2601.13553 - arXiv:2601.13553v1 Announce Type: new -Abstract: We establish universality of the fat Basilica Julia set $J(z^2-\frac34)$ in conformal dynamics in the following sense: $J(z^2-\frac34)$ is quasiconformally equivalent to the fat Basilica Julia set of any polynomial as well as to the limit set of any geometrically finite closed surface Bers boundary group. We thus obtain the first example of a connected rational Julia set, not homeomorphic to the circle or the sphere, that is quasiconformally equivalent to a Kleinian limit set. It follows that any geometrically finite Bers boundary limit set is conformally removable. Other consequences of this universality result include quasi-symmetric uniformization of polynomial fat Basilicas by round Basilicas, and the existence of infinitely many non-commensurable uniformly quasi-symmetric surface subgroups of the Basilica quasi-symmetry group. We apply our techniques to cuspidal Basilica Julia sets arising from Schwarz reflections and cubic polynomials, yielding further universality classes. We also show that the standard Basilica Julia set $J(z^2-1)$ is the archbasilica in the David hierarchy. - oai:arXiv.org:2601.13553v1 - math.DS - math.CV - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yusheng Luo, Mahan Mj, Sabyasachi Mukherjee + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Xizhi Liu, Jie Ma, Tianming Zhu - Mirror construction of Hecke correspondence between Nakajima quiver varieties - https://arxiv.org/abs/2601.13555 - arXiv:2601.13555v1 Announce Type: new -Abstract: Nakajima constructed geometric representations of a deformed Kac-Moody Lie algebra using Hecke correspondences between quiver varieties. In this paper, we show that Hecke correspondences, which are holomorphic Lagrangians in products of Nakajima quiver varieties, can be obtained by applying the localized mirror construction to the morphism spaces between families of framed Lagrangian branes supported on the core of a plumbing of two-spheres. Moreover, for a non-ADE quiver, we show that the localized mirror functor is fully-faithful. - oai:arXiv.org:2601.13555v1 - math.SG - math.AG - math.RA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Siu-Cheong Lau, Ju Tan + A note on strong similarity and the Connes embedding problem + https://arxiv.org/abs/2601.10654 + arXiv:2601.10654v2 Announce Type: replace +Abstract: We show that there exists a completely bounded (c.b. in short) homomorphism $u$ from a $C^*$-algebra $C$ with the lifting property (in short LP) into a QWEP von Neumann algebra $N$ that is not strongly similar to a $*$-homomorphism, i.e. the similarities that ``orthogonalize" $u$ (which exist since $u$ is c.b.) cannot belong to the von Neumann algebra $N$. Moreover, the map $u$ does not admit any c.b. lifting up into the WEP $C^*$-algebra of which $N$ is a quotient. We can take $C=C^*(G)$ (full $C^*$-algebra) where $G$ is any nonabelian free group and $N= B(H)\bar \otimes M$ where $M$ is the von Neumann algebra generated by the reduced $C^*$-algebra of $G$. + oai:arXiv.org:2601.10654v2 + math.OA + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://creativecommons.org/licenses/by/4.0/ + Gilles Pisier - On the radius of analyticity and Gevrey regularity for the Boltzmann equation - https://arxiv.org/abs/2601.13560 - arXiv:2601.13560v1 Announce Type: new -Abstract: This paper investigates the non-cutoff Boltzmann equation for hard potentials in a perturbative setting. We first establish a sharp short-time estimate on the radius of analyticity and Gevrey regularity of mild solutions. Furthermore, we obtain a global-in-time radius estimate in Gevrey space. The proof combines hypoelliptic estimates with the macro-micro decomposition. - oai:arXiv.org:2601.13560v1 + Boundary Delocalization and Spectral Packets for Dirichlet Eigenfunctions + https://arxiv.org/abs/2601.11605 + arXiv:2601.11605v3 Announce Type: replace +Abstract: We establish a boundary delocalization principle for high-frequency Dirichlet eigenfunctions on smooth strictly convex domains. The main result excludes persistent boundary concentration at the level of individual eigenmodes when compared to short spectral packets of sublinear length. Quantitatively, we compare boundary energies of single eigenfunctions to packet sums over frequency windows of size N_k = o(k), without asserting any asymptotic gain in magnitude. The main mode-to-packet estimate relies only on the Rellich identity. For the multi-mode bias exclusion we additionally use the boundary local Weyl law to obtain a packet zero-mean cancellation estimate. This mode-to-packet comparison is independent of eigenvalue monotonicity and is stable under eigenvalue crossings. + oai:arXiv.org:2601.11605v3 + math.SP math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Wei-Xi Li, Lvqiao Liu, Hao Wang - - - Logarithmic geometry and Infinitesimal Hodge Theory - https://arxiv.org/abs/2601.13568 - arXiv:2601.13568v1 Announce Type: new -Abstract: This paper develops a systematic approach to infinitesimal variations of Hodge structure for singular and equisingular families by means of logarithmic geometry and residue theory. The central idea is that logarithmic vector fields encode precisely those deformation directions that preserve singularities and act trivially on Hodge structures, while the effective variation is entirely governed by residue calculus. This viewpoint provides a conceptual reinterpretation of classical results of Griffiths, Green, and Voisin, and extends them to settings involving singular varieties and equisingular deformations. The resulting framework yields a geometric explanation for the appearance of Jacobian rings in infinitesimal Hodge theory and clarifies the structure of deformation spaces underlying Severi varieties and related moduli problems. - oai:arXiv.org:2601.13568v1 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - new + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mounir Nisse - - - Control policies for a two-stage queueing system with parallel and single server options - https://arxiv.org/abs/2601.13576 - arXiv:2601.13576v1 Announce Type: new -Abstract: We study a two-stage tandem service queue attended by two servers. Each job-server pair must complete both service phases together, with the server unable to begin a new job until the current one is fully processed after two stages. Immediately after the first phase of service, the server decides whether to send the job/customer to a downstream station that allows parallel processing or to a single-service facility that offers faster or higher-quality service but handles only one job at a time. This choice determines whether the second phase commences immediately or (potentially) after waiting in a queue for the single-service facility to become available. - The decision-making scenario is modeled via a Markov decision process formulation, of a clearing system with holding costs at each station. We fully characterize the structural properties of an optimal control policy based on the relationship between the service rates at the downstream stations. A numerical study highlights the significance of optimal control by comparing its performance against several natural heuristic policies. - oai:arXiv.org:2601.13576v1 - math.OC - cs.SY - eess.SY - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Shuwen Lu, Jamol Pender, Mark E. Lewis - - - Nonlinear fractional-periodic boundary value problems with Hilfer fractional derivative: existence and numerical approximations of solutions - https://arxiv.org/abs/2601.13584 - arXiv:2601.13584v1 Announce Type: new -Abstract: We prove conditions for existence of analytical solutions for boundary value problems with the Hilfer fractional derivative, generalizing the commonly used Riemann-Liouville and Caputo operators. The boundary values, referred to in this paper as fractional-periodic, are fractional integral conditions generalizing recurrent solution values for the non-Caputo case of the Hilfer fractional derivative. Analytical solutions to the studied problem are obtained using a perturbation of the corresponding initial value problem with enforced boundary conditions. In general, solutions to the boundary value problem are singular for $t\downarrow 0$. To overcome this singularity, we construct a sequence of converging solutions in a weighted continuous function space. We present a Bernstein splines-based implementation to numerically approximate solutions. We prove convergence of the numerical method, providing convergence criteria and asymptotic convergence rates. Numerical examples show empirical convergence results corresponding with the theoretical bounds. Moreover, the method is able to approximate the singular behavior of solutions and is demonstrated to converge for nonlinear problems. Finally, we apply a grid search to obtain correspondence to the original, non-perturbed system. - oai:arXiv.org:2601.13584v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Niels Goedegebure, Kateryna Marynets - - - Balancing Independent and Collaborative Service - https://arxiv.org/abs/2601.13586 - arXiv:2601.13586v1 Announce Type: new -Abstract: We study a two-type server queueing system where flexible Type-I servers, upon their initial interaction with jobs, decide in real time whether to process them independently or in collaboration with dedicated Type-II servers. Independent processing begins immediately, as does collaborative service if a Type-II server is available. Otherwise, the job and its paired Type-I server wait in queue for collaboration. Type-I servers are non-preemptive and cannot engage with new jobs until their current job is completed. - We provide a complete characterization of the structural properties of the optimal policy for the clearing system. In particular, an optimal control is shown to follow a threshold structure based on the number of jobs in the queue before a Type-I first interaction and on the number of jobs in either independent or collaborative service. - We propose simple threshold heuristics, based on linear approximations, for real-time decision-making. In much of the parameter and state spaces, we establish theoretical bounds that compare the thresholds proposed by our heuristics to those of optimal policies and identify parameter configurations where these bounds are attained. Outside of these regions, the optimal thresholds are infinite. Numerical experiments further demonstrate the accuracy and robustness of our heuristics, particularly when the initial queue length is high. Our proposed heuristics achieve costs within 0.5% of the optimal policy on average and significantly outperform benchmark policies that exhibit extreme sensitivity to system parameters, sometimes incurring costs exceeding 100% of the optimal. - oai:arXiv.org:2601.13586v1 - math.OC - cs.SY - eess.SY - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Shuwen Lu, Mark E. Lewis, Jamol Pender - - - An Elementary Approach to Scheduling in Generative Diffusion Models - https://arxiv.org/abs/2601.13602 - arXiv:2601.13602v1 Announce Type: new -Abstract: An elementary approach to characterizing the impact of noise scheduling and time discretization in generative diffusion models is developed. Considering a simplified model where the source distribution is multivariate Gaussian with a given covariance matrix, the explicit closed-form evolution trajectory of the distributions across reverse sampling steps is derived, and consequently, the Kullback-Leibler (KL) divergence between the source distribution and the reverse sampling output is obtained. The effect of the number of time discretization steps on the convergence of this KL divergence is studied via the Euler-Maclaurin expansion. An optimization problem is formulated, and its solution noise schedule is obtained via calculus of variations, shown to follow a tangent law whose coefficient is determined by the eigenvalues of the source covariance matrix. For an alternative scenario, more realistic in practice, where pretrained models have been obtained for some given noise schedules, the KL divergence also provides a measure to compare different time discretization strategies in reverse sampling. Experiments across different datasets and pretrained models demonstrate that the time discretization strategy selected by our approach consistently outperforms baseline and search-based strategies, particularly when the budget on the number of function evaluations is very tight. - oai:arXiv.org:2601.13602v1 - cs.IT - cs.LG - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Qiang Sun, H. Vincent Poor, Wenyi Zhang - - - Optimizing Parallel Schemes with Lyapunov Exponents and kNN-LLE Estimation - https://arxiv.org/abs/2601.13604 - arXiv:2601.13604v1 Announce Type: new -Abstract: Inverse parallel schemes remain indispensable tools for computing the roots of nonlinear systems, yet their dynamical behavior can be unexpectedly rich, ranging from strong contraction to oscillatory or chaotic transients depending on the choice of algorithmic parameters and initial states. A unified analytical-data-driven methodology for identifying, measuring, and reducing such instabilities in a family of uni-parametric inverse parallel solvers is presented in this study. On the theoretical side, we derive stability and bifurcation characterizations of the underlying iterative maps, identifying parameter regions associated with periodic or chaotic behavior. On the computational side, we introduce a micro-series pipeline based on kNN-driven estimation of the local largest Lyapunov exponent (LLE), applied to scalar time series derived from solver trajectories. The resulting sliding-window Lyapunov profiles provide fine-grained, real-time diagnostics of contractive or unstable phases and reveal transient behaviors not captured by coarse linearized analysis. Leveraging this correspondence, we introduce a Lyapunov-informed parameter selection strategy that identifies solver settings associated with stable behavior, particularly when the estimated LLE indicates persistent instability. Comprehensive experiments on ensembles of perturbed initial guesses demonstrate close agreement between the theoretical stability diagrams and empirical Lyapunov profiles, and show that the proposed adaptive mechanism significantly improves robustness. The study establishes micro-series Lyapunov analysis as a practical, interpretable tool for constructing self-stabilizing root-finding schemes and opens avenues for extending such diagnostics to higher-dimensional or noise-contaminated problems. - oai:arXiv.org:2601.13604v1 - math.NA - cs.LG - cs.NA - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mudassir Shams, Andrei Velichko, Bruno Carpentieri - - - Quasi-periodic Dynamics for Multi-dimensional Quasi-linear Schr\"{o}dinger Equations via Resonant Mode Control - https://arxiv.org/abs/2601.13611 - arXiv:2601.13611v1 Announce Type: new -Abstract: This paper focuses on the problem of quasi-periodic solutions for multi-dimensional quasi-linear Schr\"odinger equation. To address the challenge of unbounded perturbations caused by quasi-linear terms in the equation, we define the resonant mode set $\mathcal{K}$ to control nonlinear resonant effects. Combining KAM (Kolmogorov-Arnold-Moser) ( or Nash-Moser ) theory and Fourier analysis methods, we prove that there are plenty of quasi-periodic solutions of the equation. We also present the Fourier expansion form of the solutions and the estimation of frequency shifts. - oai:arXiv.org:2601.13611v1 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zuhong You, Xiaoping Yuan - - - Reflections over the Sea: Reconfigurable Intelligent Surface for Maritime Self-Powered Communications - https://arxiv.org/abs/2601.13618 - arXiv:2601.13618v1 Announce Type: new -Abstract: Maritime communication is becoming a vital component of 6G networks, driven by the rapid expansion of the maritime economy. However, existing technologies face critical challenges in signal coverage, availability, and robustness, especially under harsh sea conditions. This paper proposes a novel framework for the maritime Internet-of-Things (IoT) communications that leverages the reconfigurable intelligent surface (RIS) mounted on offshore infrastructures, such as wind turbines, to enhance coverage and reliability. To capture dynamic maritime environment, a near-ocean-surface channel model is developed considering the impact of sea waves. In addition, a wave energy harvesting (EH) system is designed to self-power IoT sensors for data acquisition, processing, and transmission. To support real-time adaptation, channel state information is continuously measured to optimize RIS reflection parameters and maximize multi-user communication rates. Simulation results show that the proposed system significantly improves IoT communication performance by over 20%, under harsh sea conditions. - oai:arXiv.org:2601.13618v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Qianqian Zhang, Long Wang, Ben Wu, Jia Mi - - - Locally analytic vectors in the completed cohomology of quaternionic Shimura curves - https://arxiv.org/abs/2601.13625 - arXiv:2601.13625v1 Announce Type: new -Abstract: We use the methods introduced by Lue Pan to study the locally analytic vectors of the completed cohomology of Shimura curves associated to an indefinite quaternion algebra $D$ which is ramified at a prime number $p$. Let $D_p^{\times}$ be the group of units of $D$ at $p$. Using $p$-adic uniformization of the quaternionic Shimura curves, we compute the Hecke eigenspace of the completed cohomology with the Hecke eigenvalues associated to a classical automorphic form on another quaternion algebra $\bar D$ (switching invariants of $D$ at $p,\infty$). We present this locally analytic $D_p^\times$-representation using the de Rham complex of the Lubin-Tate tower of dimension $1$. This is analogous to the Breuil-Strauch conjecture for the group $\mathrm{GL}_2(\mathbb{Q}_p)$. We show that the locally analytic $D_p^{\times}$-representation does not detect the Hodge filtration of the local de Rham Galois representation at $p$ in the crystalline case, and also give applications for the locally analytic Jacquet--Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$ and $D_p^\times$. - oai:arXiv.org:2601.13625v1 - math.NT - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Zhenghui Li, Benchao Su, Zhixiang Wu - - - Symmetric multiple Eisenstein series - https://arxiv.org/abs/2601.13626 - arXiv:2601.13626v1 Announce Type: new -Abstract: In this paper, we introduce the symmetric multiple Eisenstein series, a variant of the multiple Eisenstein series. As a fundamental result, we show that they satisfy the linear shuffle relation. As a case study, we investigate the vector space spanned by symmetric double Eisenstein series of weight $k$. When $k$ is even, it coincides with the space spanned by modular forms of weight $k$ and the derivative of the Eisenstein series of weight $k-2$. For $k$ odd, we prove that its dimension equals $\lfloor k/3\rfloor$. We further provide an explicit correspondence between the linear shuffle relation and the Fay-shuffle relation satisfied by elliptic double zeta values, which may be of independent interest. In connection with modular forms, we prove that every modular form can be expressed as a linear combination of symmetric triple Eisenstein series. This will serve as a first step toward understanding modular phenomena for symmeric multiple zeta values observed by Kaneko and Zagier. - oai:arXiv.org:2601.13626v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Takashi Hara, Kenji Sakugawa, Koji Tasaka - - - Exact solution of the (2+1)-dimensional damping forcing coupled Burgers equation by using Darboux transformation - https://arxiv.org/abs/2601.13634 - arXiv:2601.13634v1 Announce Type: new -Abstract: In this article, we investigate the (2+1)-dimensional damping forcing coupled Burgers equation, which is obtain by adding damping and forcing terms from couple Burgers equation. The Lax pair of the (2+1)-dimensional damping forcing coupled Burgers equation is established. With the help of Lax pair, we derive the $N$-fold Darboux transformation of (2+1)-dimensional damping forcing coupled Burgers equation. Using one fold and two fold Darboux transformation, we demonstrated some wave solutions including solitary wave solution and periodic wave solution. The impact of damping and forcing terms in solitary wave solution and periodic solution is graphically demonstrated. - oai:arXiv.org:2601.13634v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Prasanta Chatterjee, Nanda Kanan Pal, Dipan Saha, Santanu Raut - - - Direct Finite-Time Contraction (Step-Log) Profiling--Driven Optimization of Parallel Schemes for Nonlinear Problems on Multicore Architectures - https://arxiv.org/abs/2601.13637 - arXiv:2601.13637v1 Announce Type: new -Abstract: Efficient computation of all distinct solutions of nonlinear problems is essential in many scientific and engineering applications. Although high-order parallel iterative schemes offer fast convergence, their practical performance is often limited by sensitivity to internal parameters and the lack of reproducible tuning procedures. Classical parameter selection tools based on analytical conditions and dynamical-system diagnostics can be problem-dependent and computationally demanding, which motivates lightweight data-driven alternatives. - In this study, we propose a parameterized single-step bi-parametric parallel Weierstrass-type scheme with third-order convergence together with a training-free tuning framework based on Direct finite-time contraction (step-log) profiling. The approach extracts Lyapunov-like finite-time contraction information directly from solver trajectories via step norms and step-log ratios, aggregates the resulting profiles over micro-launch ensembles, and ranks parameter candidates using two compact scores: the stability minimum S_min and the stability moment S_mom. Numerical results demonstrate consistent improvements in convergence rate, stability, and robustness across diverse nonlinear test problems, establishing the proposed profiling-based strategy as an efficient and reproducible alternative to classical parameter tuning methods. - oai:arXiv.org:2601.13637v1 - math.NA - cs.NA - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mudassir Shams, Andrei Velichko, Bruno Carpentieri - - - Borcherds products approximating Gersten complex - https://arxiv.org/abs/2601.13643 - arXiv:2601.13643v1 Announce Type: new -Abstract: For an orthogonal modular variety, we construct a complex which is defined in terms of lattices and elliptic modular forms, which resembles the Gersten complex in Milnor K-theory, and which has a morphism to the Gersten complex of the modular variety by the Borcherds lifting. This provides a formalism for approaching the higher Chow groups of the modular variety by special cycles and Borcherds products. The construction is an incorporation of the theory of Borcherds products and ideas from Milnor K-theory. - oai:arXiv.org:2601.13643v1 - math.AG - math.KT - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Shouhei Ma - - - Gromov-Hausdorff stability of global attractors of damped wave equations under perturbations of the domain - https://arxiv.org/abs/2601.13650 - arXiv:2601.13650v1 Announce Type: new -Abstract: In this paper, we will make use of the Gromov-Hausdorff distance between compact metric spaces to establish the continuous dependence and the Gromov-Hausdorff stability of global attractors for damped wave equations under perturbations of the domain. - oai:arXiv.org:2601.13650v1 - math.DS - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Ngoctu Bui, Jihoon Lee - - - Kenmotsu Contact Geometry Through the Lens of $\ast-\boldsymbol{\kappa}$-Ricci-Bourguignon Almost Solitons - https://arxiv.org/abs/2601.13661 - arXiv:2601.13661v1 Announce Type: new -Abstract: This paper focuses on the study of the newly introduced $\ast-\boldsymbol{\kappa}$-Ricci-Bourguignon almost soliton pertaining to Kenmotsu structure manifolds. Our analysis concerns the characteristics of this soliton and derive the scalar curvature for a Kenmotsu manifold admitting such a structure. Further, we formulate the corresponding vector fields under the assumption that the manifold supports a $\ast-\boldsymbol{\kappa}-$Ricci-Bourguignon soliton. Additionally, we explore applications involving torse-forming vector fields within the framework of the $\ast-\boldsymbol{\kappa}-$Ricci-Bourguignon almost soliton on Kenmotsu structure manifolds. To support the theoretical findings, we provide a concrete illustration belonging to a $\ast-\boldsymbol{\kappa}-$Ricci-Bourguignon almost soliton in a 5D Kenmotsu structure manifold. - oai:arXiv.org:2601.13661v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lavanya Kumar, Soumendu Roy - - - The norm of the Hilbert matrix operator on Bergman spaces - https://arxiv.org/abs/2601.13672 - arXiv:2601.13672v1 Announce Type: new -Abstract: Karapetrovi\'c conjectured that the norm of the Hilbert matrix operator on the Bergman space $A^p_\alpha$ is equal to $\pi/\sin((2+\alpha)\pi/p)$ when $-1<\alpha<p-2$. In this paper, we provide a proof of this conjecture for $0\leq \alpha \leq \frac{6p^3-29p^2+17p-2+2p\sqrt{6p^2-11p+4}}{(3p-1)^2}$, and this range of $\alpha$ improves the best known result when $\alpha>\frac{1}{47}$ and $\alpha \not=1$. - oai:arXiv.org:2601.13672v1 - math.CV - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Guanlong Bao, Liu Tian, Hasi Wulan - - - The spectral measures of random Jacobi matrices related to beta ensembles at high temperature and Dirichlet processes - https://arxiv.org/abs/2601.13674 - arXiv:2601.13674v1 Announce Type: new -Abstract: In a high temperature regime where $\beta N \to 2c$, the empirical distribution of the eigenvalues of Gaussian beta ensembles, beta Laguerre ensembles and beta Jacobi ensembles converges to a limiting measure which is related to associated Hermite polynomials, associated Laguerre polynomials and associated Jacobi polynomials, respectively. Here $\beta$ is the inverse temperature parameter, $N$ is the system size and $c>0$ is a given constant. This paper studies the spectral measure of the random tridiagonal matrix model of the three classical beta ensembles. We show that in the high temperature regime, the spectral measure converges in distribution to a Dirichlet process with base distribution being the limiting distribution, and scaling parameter $c$. Consequently, the spectral measure of a related semi-infinite Jacobi matrix coincides with that Dirichlet process, which provides examples of random Jacobi matrices with explicit spectral measures. - oai:arXiv.org:2601.13674v1 - math-ph - math.MP - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Fumihiko Nakano, Hoang Dung Trinh, Khanh Duy Trinh - - - Distributed Coverage Control on Poriferous Surface via Poly-Annulus Conformal Mapping - https://arxiv.org/abs/2601.13688 - arXiv:2601.13688v1 Announce Type: new -Abstract: The inherent non-convexity of poriferous surfaces typically entraps agents in local minima and complicates workload distribution. To resolve this, we propose a distributed diffeomorphic coverage control framework for the multi-agent system (MAS) in such surfaces. First, we establish a distributed poly-annulus conformal mapping that transforms arbitrary poriferous surfaces into a multi-hole disk. Leveraging this topological equivalence, a collision-free sectorial partition mechanism is designed in the multi-hole disk, which rigorously induces strictly connected subregions and workload balance on the poriferous surfaces. This mechanism utilizes a buffer-based sequence mechanism to ensure strict topological safety when bypassing obstacles. Furthermore, a pull-back Riemannian metric is constructed to define the length metric that encodes safety constraints. Based on this metric, a distributed gradient-based control law is synthesized to drive agents toward optimal configurations, ensuring simultaneous obstacle avoidance and coverage optimization. Theoretical analyses guarantee the Input-to-State Stability (ISS) of the partition dynamics and the asymptotic convergence of the closed-loop system. Numerical simulations confirm the reachability and robustness of the proposed coverage algorithm, offering a scalable solution for distributed coverage in poriferous surfaces. - oai:arXiv.org:2601.13688v1 - math.OC - cs.SY - eess.SY - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xun Feng, Chao Zhai - - - Nonlinear compressive reduced basis approximation : when Taylor meets Kolmogorov - https://arxiv.org/abs/2601.13712 - arXiv:2601.13712v1 Announce Type: new -Abstract: This paper investigates model reduction methods for efficiently approximating the solution of parameter-dependent PDEs with a multi-parameter vector $\vec{\mu} \in \mathbb{R}^p$. In cases where the Kolmogorov $N$-width decays fast enough, it is effective to approximate the solution as a sum of $N$ separable terms, each being the product of a parameter-dependent coefficient and a space-dependent function. This leads to reduced-order models with $N$ degrees of freedom and complexity of order ${\mathcal O}(N^3)$. - However, when the $N$-width decays slowly, $N$ must be large to achieve acceptable accuracy, making cubic complexity prohibitive. The linear complexity measure in terms of Kolmogorov width must be replaced by the Gelfand width, with its associated sensing number. Recent nonlinear approaches based on this notion decompose the $N$ coordinates into two groups: $n$ free variables and $\overline{n}$ dependent variables, where the latter are nonlinear functions of the former ($N= n+\overline n$). Several works have focused on cases where these $\overline{n}$ functions are homogeneous quadratic forms of the $n$ variables, with optimization strategies for choosing $n$ given a target accuracy. - A rigorous analysis of the local sensing number is carried out, showing that $n = p$ is optimal and appropriate, at least locally, around a reference point. In practical scenarios involving wide parameter ranges, the condition $p\le n \le p + k$ (with $k$ small) is valid and more robust from continuity arguments. Additionally, the assumption of a quadratic mapping, while justified in a local sense, becomes insufficient. More expressive nonlinear mappings-including those using machine learning-become necessary. This work contributes a theoretical foundation for such strategies and highlights the need for further investigations to push back the Kolmogorov Barrier. - oai:arXiv.org:2601.13712v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Joubine Aghili, Hassan Ballout, Yvon Maday, Christophe Prud'homme - - - On the Birkhoff Spectrum for Hyperbolic Dynamics - https://arxiv.org/abs/2601.13720 - arXiv:2601.13720v1 Announce Type: new -Abstract: In this paper, we study the structure of Birkhoff spectra for hyperbolic dynamical systems. For a H\"older observable $f$ on a basic set $\Lambda$, we prove that if the Birkhoff sums take both positive and negative values, then the Birkhoff spectrum $\mathcal{B}(f,\varphi,\Lambda)$ is dense in $\mathbb{R}$. This extends the density result of Gan, Shi, and Xia \cite{Shaobo} from transitive Anosov diffeomorphisms in infranilmanifolds to general basic sets, and yields new density theorems for Axiom~A systems, including Anosov diffeos.\\ Conversely, when the spectrum is not dense, we characterize \emph{concentrated} observables, whose spectrum is confined to one side of zero. For these, we establish two rigidity results: (i) boundedness of the spectrum is equivalent to the function being cohomologous to a zero, which constitutes an extension of the Liv\v sic theorem; (ii) if the spectrum exhibits an arithmetic structure, the function is cohomologous to a constant. - Finally, we extend the results to continuous time. For Anosov flows$-$ including geodesic flows on Anosov manifolds$-$we obtain analogous density results for Birkhoff integrals along closed orbits. In particular, we generalize a theorem of Dairbekov--Sharafutdinov (\cite{Dairbekov}) by showing that a bounded or ``arithmetically sparse'' spectrum forces a smooth function to vanish or be constant, respectively. - oai:arXiv.org:2601.13720v1 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sergio Roma\~na - - - Central Limit Theorems in Multiplicative Diophantine Approximation - https://arxiv.org/abs/2601.13726 - arXiv:2601.13726v1 Announce Type: new -Abstract: We investigate the number of integer solutions to a multiplicative Diophantine approximation problem and show that the associated counting function converges in distribution to a normal law. Our approach relies on the analysis of correlations of measures on homogeneous spaces, together with estimates for Siegel transforms restricted to subspaces. - oai:arXiv.org:2601.13726v1 - math.NT - math.DS - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Michael Bj\"orklund, Reynold Fregoli, Alexander Gorodnik - - - Some Consequences of the Grunewald-O'Halloran Conjecture for Pseudoquonic Operators - https://arxiv.org/abs/2601.13736 - arXiv:2601.13736v1 Announce Type: new -Abstract: Investigating a recent positive solution of a conjecture of Grunewald and O'Halloran for complex finite dimensional nilpotent Lie algebras, we are in the position to find results of existence and uniqueness for the construction of complex nilpotent Lie algebras of arbitrary dimension via pseudobosonic operators. We involve the so-called theory of the deformation of Lie algebras of Gerstenhaber, in order to prove our main results. There isn't a generalized version of the Grunewald-O'Halloran Conjecture when we consider pseudoquonic operators, which specialize to pseudobosonic operators in many cirumstances. Therefore we prove a result of existence (and a direct construction) of pseudobosonic $O^*$-algebras of operators, but leave open the problem of the uniqueness of the construction. - oai:arXiv.org:2601.13736v1 - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Fabio Bagarello, Yanga Bavuma, Francesco G. Russo - - - Second Order Asymptotics for the Hard Wall Probability of the 2D Harmonic Crystal - https://arxiv.org/abs/2601.13738 - arXiv:2601.13738v1 Announce Type: new -Abstract: We estimate the probability that the discrete Gaussian free field on a planar domain with Dirichlet boundary conditions stays positive in the bulk. Improving upon the result by Bolthausen, Deuschel and Giacomin from 2001, we derive the order of the subleading term of this probability when a sequence of discretized scale-ups of given domain and compactly included smooth bulk are considered. A main ingredient in the proof is the double exponential decay of the right tail of the centered minimum of the field in the bulk, conditioned on a certain weighted average of its values to be zero. - oai:arXiv.org:2601.13738v1 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Maximilian Fels, Oren Louidor, Tianqi Wu - - - Abstract maximal hypoellipticity and applications - https://arxiv.org/abs/2601.13741 - arXiv:2601.13741v1 Announce Type: new -Abstract: We prove an abstract theorem of maximal hypoellipticy showing that in an abstract calculus under some natural assumptions, an operator is maximally hypoelliptic if and only if its principal symbol is left invertible. We then show that our theorem implies various known results in the literature like regularity theorem for elliptic operators, Helffer and Nourrigat's resolution of the Rockland conjecture, Rodino's theorem on regularity of operators on products of manifolds, and our resolution of the Helffer-Nourrigat conjecture. Other examples like our resolution of the microlocal Helffer-Nourrigat conjecture will be given in a sequel to this paper. - Our arguments are based on the theory of $C^*$-algebras of Type I. - oai:arXiv.org:2601.13741v1 - math.OA - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Omar Mohsen - - - A Note on k-NN Gating in RAG - https://arxiv.org/abs/2601.13744 - arXiv:2601.13744v1 Announce Type: new -Abstract: We develop a statistical proxy framework for retrieval-augmented generation (RAG), designed to formalize how a language model (LM) should balance its own predictions with retrieved evidence. For each query x, the system combines a frozen base model q0 ($\times$ x) with a k-nearest neighbor retriever r (k ) ($\times$ x) through a measurable gate k(x). A retrieval-trust weight wfact (x) quantifies the geometric reliability of the retrieved neighborhood and penalizes retrieval in low-trust regions. We derive the Bayes-optimal per-query gate and analyze its effect on a discordance-based hallucination criterion that captures disagreements between LM predictions and retrieved evidence. We further show that this discordance admits a deterministic asymptotic limit governed solely by the structural agreement (or disagreement) between the Bayes rule and the LM. To account for distribution mismatch between queries and memory, we introduce a hybrid geometric-semantic model combining covariate deformation and label corruption. Overall, this note provides a principled statistical foundation for factuality-oriented RAG systems. - oai:arXiv.org:2601.13744v1 - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - G\'erard Biau (SU, IUF, MEGAVOLT), Claire Boyer (IUF, CELESTE) - - - Hamiltonian hydrodynamic reductions of one-dimensional Vlasov equations - https://arxiv.org/abs/2601.13746 - arXiv:2601.13746v1 Announce Type: new -Abstract: We investigate Hamiltonian fluid reductions of the one-dimensional Vlasov-Poisson equation. Our approach utilizes the hydrodynamic Poisson bracket framework, which allows us to systematically identify fundamental normal variables derived from the analysis of the Casimir invariants of the resulting Poisson bracket. This framework is then applied to analyze several well-established Hamiltonian closures of the onedimensional Vlasov equation, including the multi-delta distribution and the waterbag models. Our key finding is that all of these seemingly distinct closures consistently lead to the formulation of a unified form of parametric closures: When expressed in terms of the identified normal variables, the parameterization across all these closures is revealed to be polynomial and of the same degree. All these parametric closures are uniquely generated from one of the moments, called $\mu$2, a cubic polynomial in the normal variables. This result establishes a structural connection between these different physical models, offering a path toward a more unified and simplified description of the one-dimensional Vlasov-Poisson dynamics through its reduced hydrodynamic forms with an arbitrary number of fluid variables. - oai:arXiv.org:2601.13746v1 - math-ph - math.MP - nlin.CD - physics.plasm-ph - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rayan Oufar (I2M), Cristel Chandre (I2M) - - - closed $\mathrm{G}_2$-structures with $\mathbb{T}^3$-symmetry and hypersymplectic structures - https://arxiv.org/abs/2601.13747 - arXiv:2601.13747v1 Announce Type: new -Abstract: Closed $\mathrm{G}_2$-structures $\varphi$ with an effective $\mathbb{T}^3$-symmetry on connected manifolds are roughly classified into three types according to the evaluation of $\varphi$ on the principal orbits. Type 1: if there is neither associative nor isotropic orbit, then the action is free and $\varphi$ reduces to a hypersymplectic structure on the quotient manifold admitting three linearly independent closed 1-forms; in particular, it is diffeomorphic to $\mathbb{T}^4$ if the manifold is compact. Type 2: if some orbit is associative, then the action is almost-free and $\varphi$ reduces to a good hypersymplectic orbifold with cyclic isotropic groups. Type 3: if some orbit is isotropic, then the action is locally multi-Hamiltonian for $\varphi$. Moreover, the open and dense subset of principal orbits is foliated by $\mathbb{T}^3$-invariant hypersymplectic manifolds. If $\varphi$ is torsion-free and complete, then the hypersymplectic manifold is flat and $\varphi$ is flat for Type 1; the good hypersymplectic orbifold is good hyperk\"ahler orbifold for Type 2; $\varphi$ is locally toric for Type 3. As shown, hypersymplectic structures have intimate link with closed $\mathrm{G}_2$-structure with effective $\mathbb{T}^3$-symmetry. - oai:arXiv.org:2601.13747v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chengjian Yao, Ziyi Zhou - - - Sharp Quantitative Forms of the Hardy Inequality on Cartan-Hadamard Manifolds via Sobolev-Lorentz Embeddings - https://arxiv.org/abs/2601.13750 - arXiv:2601.13750v1 Announce Type: new -Abstract: In this article, we investigate the quantitative form of the classical Hardy inequality. In our first result, we prove the following quantitative bound under the assumption that the $\mathbb{M}^N$ is a Riemannian model satisfying the centered isoperimetric inequality: We prove that $$ \|\nabla_g u\|^2_{L^{2}(\mathbb{M}^N)} - \frac{(N-2)^2}{4}\left\|\frac{u}{r(x)}\right\|^2_{L^2(\mathbb{M}^N)} \geq C [\mbox{dist}(u, Z)]^{\frac{4N}{N-2}}\left\|\frac{u}{r(x)}\right\|^2_{L^2(\mathbb{M}^N)},$$ for every real-valued weakly differentiable function $u$ on $\mathbb{M}^N$ such that $|\nabla_g u| \in L^2(\mathbb{M}^N)$ and $u$ decays to zero at infinity. Here $r(x) = d_g(x,x_0)$ denotes the geodesic distance from a fixed pole $x_0,$ the set $Z$ represents the family of virtual extremals, and the distance is understood in an appropriate generalized Lorentz-type space. Our approach is built on the symmetrization technique on manifolds, combined with a novel Jacobian-type transformation that provides a precise way for comparing volume growth, level sets, and gradient terms across the two geometries of Euclidean and manifold settings. When coupled with symmetrization, this framework yields sharp control over the relevant functionals and reveals how the underlying curvature influences extremal behavior. Our result generalizes the seminal result of Cianchi-Ferone [Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire 25 (2008)] to the curved spaces. Moreover, building upon this transformation, we succeed in extending Sobolev-Lorentz embedding-classically formulated in the Euclidean setting to the broader framework of Cartan-Hadamard models and we establish an optimal Sobolev-Lorentz embedding in this geometric setting. Finally, we establish a quantitative correspondence between the Hardy deficit on the manifold and an appropriate weighted Hardy deficit in Euclidean space, showing that each controls the other. - oai:arXiv.org:2601.13750v1 - math.AP - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Avas Banerjee, Debdip Ganguly, Prasun Roychowdhury - - - A turnpike property in an eigenvalue optimization problem - https://arxiv.org/abs/2601.13756 - arXiv:2601.13756v1 Announce Type: new -Abstract: We consider a constrained eigenvalue optimization problem that arises in an important nonlinear dynamical model for mRNA translation in the cell. We prove that the ordered list of optimal parameters admits a turnpike property, namely, it includes three parts with the first and third part relatively short, and the values in the middle part are all approximately equal. Turnpike properties have attracted considerable attention in econometrics and optimal control theory, but to the best of our knowledge this is the first rigorous proof of such a structure in an eigenvalue optimization problem. - oai:arXiv.org:2601.13756v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Adam Kaminer, Thomas Kriecherbauer, Lars Gr\"une, Michael Margaliot - - - Quantum Entanglement Geometry on Severi-Brauer Schemes: Subsystem Reductions of Azumaya Algebras - https://arxiv.org/abs/2601.13764 - arXiv:2601.13764v1 Announce Type: new -Abstract: We formulate pure-state entanglement in families as a geometric obstruction. In standard quantum information, entanglement is defined relative to a chosen tensor-product factorization of a fixed Hilbert space. In contrast, for a twisted family of pure-state spaces, which can be described by Azumaya algebras $A$ of degree $n$ on $X$ and their Severi-Brauer schemes \[ SB(A)=P\times^{PGL_n}\mathbb{P}^{n-1}\to X, \] such a subsystem choice may fail to globalize. We formalize this algebro-geometrically: fixing a factorization type $\mathbf d=(d_1,\dots,d_s)$ with $n=\prod_i d_i$, the existence of a global product-state locus of type $\mathbf d$ is equivalent to a reduction of the underlying $PGL_n$-torsor $P\to X$ to the stabilizer $G_{\mathbf d}\subset PGL_n$. Thus, entanglement is the obstruction to the existence of a relative Segre subscheme inside $SB(A)$. - Writing $\Sigma_{\mathbf d}\subset \mathbb{P}^{n-1}$ for the Segre variety, we call a reduction to $G_{\mathbf d}$ a $\mathbf d$-subsystem structure. Our first main result identifies the moduli of $\mathbf d$-subsystem structures with the quotient $P/G_{\mathbf d}$. Moreover, we realize naturally $P/G_{\mathbf d}$ as a locally closed subscheme of the relative Hilbert scheme, \[ \text{Hilb}^{\Sigma_{\mathbf d}}\!\bigl(SB(A)/X\bigr)\ \subset\ \text{Hilb}\bigl(SB(A)/X\bigr), \] parametrizing relative closed subschemes fppf-locally isomorphic to $\Sigma_{\mathbf d}\times X$. - oai:arXiv.org:2601.13764v1 - math.AG - hep-th - math-ph - math.MP - math.RA - quant-ph - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kazuki Ikeda - - - The Harnack inequality without convexity for curve shortening flow - https://arxiv.org/abs/2601.13767 - arXiv:2601.13767v1 Announce Type: new -Abstract: In 1995, Hamilton introduced a Harnack inequality for convex solutions of the mean curvature flow. In this paper we prove an alternative Harnack inequality for curve shortening flow, i.e. one-dimensional mean curvature flow, that does not require any assumption of convexity. For an initial proper curve in the plane whose ends are radial lines but which is otherwise arbitrarily wild, we use the Harnack inequality to give an explicit time by which the curve shortening flow evolution must become graphical. This gives a new instance of delayed parabolic regularity. The Harnack inequality also gives estimates describing how a polar graphical flow with radial ends settles down to an expanding solution. Finally, we relate our Harnack inequality to Hamilton's by identifying a pointwise curvature estimate implied by both Harnack inequalities in the special case of convex flows. - oai:arXiv.org:2601.13767v1 - math.DG - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Arjun Sobnack, Peter M. Topping - - - Existence and regularity of minimizers for a variational problem of species population density - https://arxiv.org/abs/2601.13771 - arXiv:2601.13771v1 Announce Type: new -Abstract: We study a variational problem motivated by models of species population density in a nonhomogeneous environment. We first analyze local minimizers and the structure of the saturated region (where the population attains its maximal density) from a free boundary perspective. By comparing the original problem with a radially symmetric minimization problem and studying its properties, we then establish the existence and structure of a global solution. Analytic examples of radially symmetric solutions and numerical simulations illustrate the theoretical results and provide insight into spatial saturation patterns in population models. We further highlight an unresolved question regarding the quasiconcavity of minimizers. - oai:arXiv.org:2601.13771v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pu-Zhao Kow, Masato Kimura, Hiroshi Ohtsuka - - - Bialgebraic structures on boolean functions - https://arxiv.org/abs/2601.13773 - arXiv:2601.13773v1 Announce Type: new -Abstract: We study several bialgebraic structures on boolean functions, that is to say maps defined on the set of subsets of a finite set $X$, taking the value $0$ on $\emptyset$. Examples of boolean functions are given by the indicator function of the hyperedges of a given hypergraph, or the rank function of a matroid. We give the species of boolean functions a two-parameters family of products and a coproduct, and this defines a two-parameters family of twisted bialgebras. We then try to define a second coproduct on boolean functions, based on contractions, in order to obtain a double bialgebra. We show that this is not possible on the whole species of boolean functions, but that there exists a maximal subspecies where this is possible. This subspecies being rather mysterious, we introduce rigid boolean functions and show that this subspecies has indeed a second coproduct, as wished, and that it contains rank functions of matroids and indicator functions associated to hypergraphs. As a consequence, we obtain a unique polynomial invariant on rigid boolean functions, which is a generalization of the chromatic polynomial of graphs. - oai:arXiv.org:2601.13773v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lo\"ic Foissy (LMPA) - - - Extension of the Fundamental Theorem of Algebra to Polynomial Matrix Equations over $Q$-Circulant Matrices - https://arxiv.org/abs/2601.13775 - arXiv:2601.13775v1 Announce Type: new -Abstract: In this paper, we establish an analogue of the Fundamental Theorem of Algebra for polynomial matrix equations, where both the coefficient matrices and the unknown matrix are $Q$-circulant matrices. This result generalizes Abramov's result for circulant matrices. - oai:arXiv.org:2601.13775v1 - math.RA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hongjian Li - - - Maximum spanning trees in normed planes - https://arxiv.org/abs/2601.13779 - arXiv:2601.13779v1 Announce Type: new -Abstract: Extending some properties from the Euclidean plane to any normed plane, we show the validity of the Monma-Paterson-Suri-Yao algorithm for finding the maximum-weighted spanning tree of a set of $n$ points, where the weight of an edge is the distance between the end points measured by the norm and there are not repeated distances. For strictly convex normed planes, we expose an strategy for moving slightly the points of the set in order to obtain distinct distances. - oai:arXiv.org:2601.13779v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Javier Alonso, Pedro Mart\'in - - - Moving Least Squares without Quasi-Uniformity: A Stochastic Approach - https://arxiv.org/abs/2601.13782 - arXiv:2601.13782v1 Announce Type: new -Abstract: Local Polynomial Regression (LPR) and Moving Least Squares (MLS) are closely related nonparametric estimation methods, developed independently in statistics and approximation theory. While statistical LPR analysis focuses on overcoming sampling noise under probabilistic assumptions, the deterministic MLS theory studies smoothness properties and convergence rates with respect to the \textit{fill-distance} (a resolution parameter). Despite this similarity, the deterministic assumptions underlying MLS fail to hold under random sampling. We begin by quantifying the probabilistic behavior of the fill-distance $h_n$ and \textit{separation} $\delta_n$ of an i.i.d. random sample. That is, for a distribution satisfying a mild regularity condition, $h_n\propto n^{-1/d}\log^{1/d} (n)$ and $\delta_n \propto n^{-1/d}$. We then prove that, for MLS of degree $k\!-\!1$, the approximation error associated with a differential operator $Q$ of order $|m|\le k-1$ decays as $h_n^{\,k-|m|}$ up to logarithmic factors, establishing stochastic analogues of the classical MLS estimates. Additionally, We show that the MLS approximant is smooth with high probability. Finally, we apply the stochastic MLS theory to manifold estimation. Assuming that the sampled Manifold is $k$-times smooth, we show that the Hausdorff distance between the true manifold and its MLS reconstruction decays as $h_n^k$, extending the deterministic Manifold-MLS guarantees to random samples. This work provides the first unified stochastic analysis of MLS, demonstrating that -- despite the failure of deterministic sampling assumptions -- the classical convergence and smoothness properties persist under natural probabilistic models - oai:arXiv.org:2601.13782v1 - math.ST - cs.NA - math.NA - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Shir Tapiro-Moshe, Yariv Aizenbud, Barak Sober - - - The Genus-Decreasing Property of Mean Curvature Flow, I - https://arxiv.org/abs/2601.13787 - arXiv:2601.13787v1 Announce Type: new -Abstract: This paper proves that, in mean curvature flow of a compact surface in a complete $3$-manifold with Ricci curvature bounded below, the genus of the regular set is a decreasing function of time as long as the only singularities are given by shrinking sphere and shrinking cylinder tangent flows. The paper also proves some local versions of that fact. - oai:arXiv.org:2601.13787v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Brian White - - - Characterizations of a class of Musielak--Orlicz BMO spaces via commutators of Riesz potential operators - https://arxiv.org/abs/2601.13788 - arXiv:2601.13788v1 Announce Type: new -Abstract: The fractional integral operators $I_\alpha$ can be used to characterize the Musielak--Orlicz Hardy spaces. This paper shows that for $b\in \rm BMO(\mathbb R^n)$, the commutators $[b,I_\alpha]$ generated by fractional integral operators $I_\alpha$ with $b$ are bounded from the Musielak--Orlicz Hardy spaces $H^{\varphi_1}(\mathbb R^n)$ to the Musielak--Orlicz spaces $L^{\varphi_2}(\mathbb R^n)$ (where $1<u<\infty$ and $\varphi_1$, $\varphi_2$ are growth functions) if and only if $b\in \mathcal {BMO}_{\varphi_1,u}(\mathbb R^n)$, which are a class of non-trivial subspaces of $\rm BMO(\mathbb R^n)$. Additionally, we obtain the boundedness of the commutator $[b,I_\alpha]$ from $H^{\varphi_1}(\mathbb R^n)$ to $H^{\varphi_2}(\mathbb R^n)$. The corresponding results are also provided for commutators of fractional integrals associated with general homogeneous kernels. - oai:arXiv.org:2601.13788v1 - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yanyan Han, Hongwei Huang, Jinghan Shao, Huoxiong Wu - - - Nijenhuis BiHom-Lie bialgebras and differential Lie bialgebras - https://arxiv.org/abs/2601.13791 - arXiv:2601.13791v1 Announce Type: new -Abstract: In this paper, we first introduce the concept of Nijenhuis BiHom-Lie algebras. We then establish the equivalence relations between the Manin triples of Nijenhuis BiHom-Lie algebras, Nijenhuis BiHom-Lie bialgebras, and matched pairs of Nijenhuis BiHom-Lie algebras. Furthermore, we show that such an equivalence also holds for differential Lie bialgebras, together with their associated Manin triples and corresponding matched pairs. - oai:arXiv.org:2601.13791v1 - math.RA - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jiaqi Liu, Lin Gao, Yuanyuan Zhang - - - Asymptotic Properties of Filtrations of Ideals - https://arxiv.org/abs/2601.13794 - arXiv:2601.13794v1 Announce Type: new -Abstract: We introduce a unified framework for studying persistence phenomena in commutative algebra via filtrations of ideals. For a filtration $\mathcal{F} = \{I_i\}_{i \in \mathbb{N}}$, we define $\mathcal{F}$-persistence and $\mathcal{F}$-strong persistence, extending the classical notions for ordinary and symbolic powers of ideals. We show that if $\mathcal{F}$ is strongly persistent, then $\mathcal{F}_{\mathrm{sym}}$ is strongly persistent, where $\mathcal{F}_{\mathrm{sym}}$ denotes the symbolic filtration associated with the filtration $\mathcal{F}$. In addition, we prove that if $\mathcal{F}$ is strongly persistent, then $\mathcal{F}$ is persistent. - oai:arXiv.org:2601.13794v1 - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Mehrdad Nasernejad, Jonathan Toledo - - - A Hybridizable Discontinuous Galerkin Method for the non--local Camassa--Holm--Kadomtsev--Petviashvili equation - https://arxiv.org/abs/2601.13800 - arXiv:2601.13800v1 Announce Type: new -Abstract: This paper develops a hybridizable discontinuous Galerkin method for the two-dimensional Camassa--Holm--Kadomtsev--Petviashvili equation. The method employs Cartesian meshes with tensor-product polynomial spaces, enabling separate treatment of \(x\) and \(y\) derivatives. The non-local operator \(\partial_{x}^{-1}u_{y}\) is localized through an auxiliary variable \(v\) satisfying \(v_x = u_y\), allowing efficient element-by-element computations. We prove energy stability of the semi-discrete scheme and derive \(\mathcal{O}(h^{k+1/2})\) convergence in space. Numerical experiments validate the theoretical results and demonstrate the method's capability to accurately resolve smooth solutions and peaked solitary waves (peakons). - oai:arXiv.org:2601.13800v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/publicdomain/zero/1.0/ - Mukul Dwivedi, Ruben Gutendorf, Andreas Rupp - - - On rough ideal convergence - https://arxiv.org/abs/2601.13805 - arXiv:2601.13805v1 Announce Type: new -Abstract: We continue the study of ideal convergence for sequences $(x_n)$ with values in a topological space $X$ with respect to a family $\{F_\eta:\eta\in X\}$ of subsets of $X$ with $\eta\in F_\eta$, where each $F_\eta$ measures the allowed ``roughness'' of convergence toward $\eta$. - More precisely, after introducing the corresponding notions of cluster and limit points, we prove several inclusion and invariance properties, discuss their structural properties, and give examples showing that the rough notions are genuinely different from the classical ideal ones. - oai:arXiv.org:2601.13805v1 - math.GN - math.CA - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Paolo Leonetti - - - Non-finitely generated $(\mathbb{Z}_2)^k$-equivariant bordism ring - https://arxiv.org/abs/2601.13807 - arXiv:2601.13807v1 Announce Type: new -Abstract: In 1998, Mukherjee and Sankaran posed two problems concerning the algebraic structure of the equivariant bordism ring of smooth closed $(\mathbb{Z}_2)^k$-manifolds with only isolated fixed points. One is the property of being finitely generated as a $\mathbb{Z}_2$-algebra, and the other is the existence of indecomposable elements. This paper definitively resolves both problems for the fully effective case. Specifically, let $\mathcal{Z}_*((\mathbb{Z}_2)^k)$ denote the equivariant bordism ring of smooth closed manifolds equipped with fully effective smooth $(\mathbb{Z}_2)^k$-actions having only isolated fixed points. We prove that $\mathcal{Z}_*((\mathbb{Z}_2)^k)$ is not finitely generated as a $\mathbb{Z}_2$-algebra for all $k\geqslant 3$. Moreover, the proof explicitly constructs an infinite family of indecomposable elements with unbounded degrees, thereby settling the second problem simultaneously. - oai:arXiv.org:2601.13807v1 - math.AT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Yuanxin Guan, Zhi L\"u - - - Multi-Trace M\"uller Boundary Integral Equation for Electromagnetic Scattering by Composite Objects - https://arxiv.org/abs/2601.13823 - arXiv:2601.13823v1 Announce Type: new -Abstract: This paper introduces a boundary integral equation for time-harmonic electromagnetic scattering by composite dielectric objects. The formulation extends the classical M\"uller equation to composite structures through the global multi-trace method. The key ingredient enabling this extension is the use of the Stratton-Chu representation in complementary region, also known as the extinction property, which augments the off-diagonal blocks of the interior representation operator. The resulting block system is composed entirely of second-kind operators. A Petrov-Galerkin (mixed) discretization using Rao-Wilton-Glisson trial functions and Buffa-Christiansen test functions is employed, yielding linear systems that remain well conditioned on dense meshes and at low frequencies without the need for additional stabilization. This reduces computational costs associated with matrix-vector multiplications and iterative solving. Numerical experiments demonstrate the accuracy of the method in computing field traces and derived quantities. - oai:arXiv.org:2601.13823v1 - math.NA - cs.NA - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Van Chien Le, Kristof Cools - - - Dimensional Constraints from SU(2) Representation Theory in Graph-Based Quantum Systems - https://arxiv.org/abs/2601.13828 - arXiv:2601.13828v1 Announce Type: new -Abstract: We investigate dimensional constraints arising from representation theory when abstract graph edges possess internal degrees of freedom but lack geometric properties. We prove that such internal degrees of freedom can only encode directional information, necessitating quantum states in $\mathbb{C}^2$ (qubits) as the minimal representation. Any geometrically consistent projection of these states maps necessarily to $\mathbb{R}^3$ via the Bloch sphere. This dimensional constraint $d=3$ emerges through self-consistency: edges without intrinsic geometry force directional encoding ($\mathbb{C}^2$), whose natural symmetry group $SU(2)$ has three-dimensional Lie algebra, yielding emergent geometry that validates the hypothesis via Bloch sphere correspondence ($S^2 \subset \mathbb{R}^3$). We establish uniqueness (SU($N>2$) yields $d>3$) and robustness (dimensional saturation under graph topology changes). The Euclidean metric emerges canonically from the Killing form on $\mathfrak{su}(2)$. A global gauge consistency axiom is justified via principal bundle trivialization for finite graphs. Numerical simulations verify theoretical predictions. This result demonstrates how dimensional structure can be derived from information-theoretic constraints, with potential relevance to quantum information theory, discrete geometry, and quantum foundations. - oai:arXiv.org:2601.13828v1 - math-ph - math.MP - quant-ph - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jo\~ao P. da Cruz - - - Analytic description of the moving moisture front in soils - https://arxiv.org/abs/2601.13833 - arXiv:2601.13833v1 Announce Type: new -Abstract: The fact that moisture propagates in soils at a finite speed is confirmed by natural everyday experience as well as by controlled laboratory tests. In this text, we rigorously derive analytical upper bounds for the speed of moisture front propagation under gravity for the solution to the Richards equation with compactly supported initial data. The main result is an explicit criterion describing a competition between gravity and capillarity, where the dominant effect is determined by the characteristics of the soil. If capillarity prevails, the initially wet regions remain wet for all times, while if gravity is dominant, moisture travels downward at a speed that is asymptotically bounded from below and above. As a by-product, we prove the existence and uniqueness of a solution to an initial value problem for the degenerate Richards equation on the whole space. Numerical simulations based on the proposed model confirm the theoretical predictions, with results that closely match experimental observations. - oai:arXiv.org:2601.13833v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bettina Detmann, Chiara Gavioli, Pavel Krej\v{c}\'i, Yanyan Zhang - - - Derivative free data-driven stabilization of continuous-time linear systems from input-output data - https://arxiv.org/abs/2601.13848 - arXiv:2601.13848v1 Announce Type: new -Abstract: This letter presents a data-driven framework for the design of stabilizing controllers from input-output data in the continuous-time, linear, and time-invariant domain. Rather than relying on measurements or reliable estimates of input and output time derivatives, the proposed approach uses filters to derive a parameterization of the system dynamics. This parameterization is amenable to the application of linear matrix inequalities enabling the design of stabilizing output feedback controllers from input-output data and the knowledge of the order of the system. - oai:arXiv.org:2601.13848v1 - math.OC - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Corrado Possieri - - - Basic Albanese maps of regular Riemannian foliations - https://arxiv.org/abs/2601.13853 - arXiv:2601.13853v1 Announce Type: new -Abstract: In the paper we introduce the notion of basic Albanese map which we define for foliated Riemannian manifolds using basic 1-forms. We relate this mapping to the classical Albanese map for the ambient manifold. The study of general properties is supplemented with the description of several important examples. - oai:arXiv.org:2601.13853v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Kinga S{\l}owik, Robert Wolak - - - Jacob's ladders, point of contact of the remained in the prime-number law with the Fermat-Wiles theorem and multiplicative puzzles on some sets of integrals - https://arxiv.org/abs/2601.13855 - arXiv:2601.13855v1 Announce Type: new -Abstract: In this paper we prove, on the Riemann hypothesis, the existence of such increments of the Ingham integral (1932) that generate new functionals together with corresponding new $P\zeta$-equivalents of the Fermat-Wiles theorem. We obtain also new results in this direction. - oai:arXiv.org:2601.13855v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Jan Moser - - - Patterns and Tracks - https://arxiv.org/abs/2601.13861 - arXiv:2601.13861v1 Announce Type: new -Abstract: Patterns in triangulated $2$-spheres and $3$-spheres are investigated. A new proof of a lemma in Abigail Thompson's proof of the Recognition Algorithm for $3$-spheres is obtained. - oai:arXiv.org:2601.13861v1 - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - M. J. Dunwoody - - - Constrained MARL for Coexisting TN-NTN Resource Allocation: Scalability and Flexibility - https://arxiv.org/abs/2601.13883 - arXiv:2601.13883v1 Announce Type: new -Abstract: This paper considers the joint TN-NTN constrained resource allocation, where terrestrial base stations and non-terrestrial base stations coexist in the spectrum. We focus on large-scale and practical scenarios characterized by large numbers of transmission channels and users, alongside highly dynamic user behaviors. As common learning solutions fail to address these challenges, we propose a decomposition solution based on the special properties of the cross-segment interference, and then tackle the original problem via solving subproblems in a sequential learning manner. Furthermore, to enhance the flexibility of the learned policies, we design a stochastic training environment that captures the key characteristics of real-world systems. Simulation results tested on the full 20MHz bandwidth with various numerologies show that our solution significantly improves scalability compared to existing solutions and remains robust in highly dynamic scenarios. - oai:arXiv.org:2601.13883v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Cuong Le, Thang X. Vu, Stefano Andrenacci, Symeon Chatzinotas - - - Optimizing the Geometry of an L-Shaped Building to Enhance Energy Efficiency and Sustainability - https://arxiv.org/abs/2601.13884 - arXiv:2601.13884v1 Announce Type: new -Abstract: The geometric form of a building strongly influences its material use, heat losses, and energy efficiency. This paper presents an analytical optimization of L-shaped residential buildings aimed at minimizing the external surface area for a prescribed volume. Both symmetric and asymmetric configurations are examined under realistic design constraints, including fixed or bounded wing aspect ratios and fixed building height. Using explicit optimization methods and Karush-Kuhn-Tucker conditions, closed-form expressions for the optimal geometric parameters and minimal envelope area are derived. The results show that unconstrained optimization leads to degenerate cuboid shapes, highlighting the importance of geometric constraints to preserve the L-shaped form. The obtained results provide practical design guidelines for architects and engineers, supporting informed early stage decisions that balance functional requirements, regulatory constraints, architectural intent, and energy performance. Case studies of existing houses demonstrate that the proposed approach can reduce external surface area or confirm near-optimality of practical designs, supporting energy-efficient early-stage architectural decisions. - oai:arXiv.org:2601.13884v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ewa Rokita-Magdziarz, Barbara Gronostajska, Marcin Magdziarz - - - From geometry to sustainability: Optimal shapes of hip roof houses - https://arxiv.org/abs/2601.13896 - arXiv:2601.13896v1 Announce Type: new -Abstract: In this paper, we develop a rigorous mathematical framework for the optimization of hip roof house geometry, with the primary goal of minimizing the external surface of the building envelope for a given set of design constraints. Five optimization scenarios are systematically analyzed: fixed volume, fixed footprint ratio, fixed slenderness ratio, fixed floor area, and constrained height. For each case, explicit formulas for the optimal dimensions are derived, offering architects and engineers practical guidelines for improving material efficiency, reducing construction costs, and enhancing energy performance. To illustrate the practical relevance of the theoretical results, case studies of real-world hip roof houses are presented, revealing both inefficiencies in common practice and near-optimal examples. Furthermore, a freely available software application has been developed to support designers in applying the optimization methods directly to architectural projects. The findings confirm that square-based footprints combined with balanced slenderness ratios yield the most efficient forms, while deviations toward elongated or flattened proportions significantly increase energy and material demands. This work demonstrates how mathematical modeling and architectural design can be integrated to support sustainable architecture, providing both theoretical insight and practical tools for shaping energy-efficient, cost-effective, and aesthetically coherent residential buildings. - oai:arXiv.org:2601.13896v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ewa Rokita-Magdziarz, Barbara Gronostajska, Marcin Magdziarz - - - Homogeneous substructures in random ordered uniform matchings - https://arxiv.org/abs/2601.13906 - arXiv:2601.13906v1 Announce Type: new -Abstract: An ordered $r$-uniform matching of size $n$ is a collection of $n$ pairwise disjoint $r$-subsets of a linearly ordered set of $rn$ vertices. For $n=2$, such a matching is called an $r$-pattern, as it represents one of $\tfrac12\binom{2r}r$ ways two disjoint edges may intertwine. Given a set $\mathcal{P}$ of $r$-patterns, a $\mathcal{P}$-clique is a matching with all pairs of edges belonging to $\mathcal{P}$. In this paper we determine the order of magnitude of the size of a largest $\mathcal{P}$-clique in a random ordered $r$-uniform matching for several sets $\mathcal{P}$, including all sets of size $|\mathcal{P}|\le2$ and the set $\mathcal{R}^{(r)}$ of all $2^{r-1}$ $r$-partite $r$-patterns. - oai:arXiv.org:2601.13906v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Andrzej Dudek, Jaros{\l}aw Grytczuk, Jakub Przyby{\l}o, Andrzej Ruci\'nski - - - Improving the local solution of the DG predictor of the ADER-DG method for solving systems of ordinary differential equations and its applicability to systems of differential-algebraic equations - https://arxiv.org/abs/2601.13908 - arXiv:2601.13908v1 Announce Type: new -Abstract: Improved local numerical solution for the ADER-DG numerical method with a local DG predictor for solving the initial value problem for a first-order ODE system is proposed. The improved local numerical solution demonstrates convergence orders of one higher than the convergence order of the local numerical solution of the original ADER-DG numerical method and has the property of continuity at grid nodes. Rigorous proofs of the approximation orders of the local numerical solution and the improved local numerical solution are presented. Obtaining the proposed improved local numerical solution does not require significant changes to the structure of the ADER-DG numerical method. Therefore, all conclusions regarding the convergence orders of the numerical solution at grid nodes, the resulting superconvergence, and the high stability of the ADER-DG numerical method remain unchanged. A wide range of applications of the ADER-DG numerical method is presented for solving specific initial value problems for ODE systems for a wide range of polynomial degrees. The obtained results provide strong confirmation for the developed rigorous theory. The improved local numerical solution is shown to exhibit both higher accuracy and improved smoothness and point-wise comparability. Empirical convergence orders of all individual numerical solutions were calculated for a wide range of error norms, which well agree with the expected convergence orders. The rigorous proof, based on the $\epsilon$-embedding method, of the applicability of the ADER-DG numerical method with a local DG predictor to solving DAE systems is presents. - oai:arXiv.org:2601.13908v1 - math.NA - cs.NA - math.FA - physics.app-ph - physics.comp-ph - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - I. S. Popov - - - Designing sustainable barn-type houses: Optimal shapes for minimal envelope and energy use - https://arxiv.org/abs/2601.13911 - arXiv:2601.13911v1 Announce Type: new -Abstract: Barn-type houses have become one of the most popular single-family housing typologies in Poland and across Europe due to their simplicity, functionality, and potential for energy efficiency. Despite their widespread use, systematic methods for optimizing their geometry in terms of envelope area and energy performance remain limited. This paper develops a rigorous mathematical framework for determining the optimal proportions of barn-type houses with respect to minimizing the external surface area while satisfying constraints of either fixed volume or fixed floor area. Closed-form solutions for the optimal width, length, and height are derived as explicit functions of the roof slope, together with formulas for the minimal achievable surface. A recently introduced dimensionless compactness measure is also calculated, allowing quantitative assessment of how far a given design deviates from the theoretical optimum. The methodology is applied to case studies of three existing houses, showing that while some designs deviate significantly from optimal compactness, others already closely approximate it. The results confirm that theoretical optimization can lead to meaningful reductions in construction costs and energy demand. To support practical implementation, two original freely available software tools were developed, enabling architects and engineers to perform optimization analyses. - oai:arXiv.org:2601.13911v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ewa Rokita-Magdziarz, Barbara Gronostajska, Marcin Magdziarz - - - Geometry-Driven Conditioning of Multivariate Vandermonde Matrices in High-Degree Regimes - https://arxiv.org/abs/2601.13915 - arXiv:2601.13915v1 Announce Type: new -Abstract: We study multivariate monomial Vandermonde matrices $V_N(Z)$ with arbitrary distinct nodes $Z=\{z_1,\dots,z_s\}\subset B_2^n$ in the high-degree regime $N\ge s-1$. Introducing a projection-based geometric statistic -- the \emph{max-min projection separation} $\rho(Z,j)$ and its minimum $\kappa(Z)=\min_j\rho(Z,j)$ -- we construct Lagrange polynomials $Q_j\in\mathcal P_N^n$ with explicit coefficient bounds $$ \|Q_j\|_\infty \lesssim s\Bigl(\frac{4n}{\rho(Z,j)}\Bigr)^{s-1}. $$ These polynomials yield quantitative distance-to-span estimates for the rows of $V_N(Z)$ and, as consequences, $$ \sigma_{\min}(V_N(Z)) \gtrsim \frac{\kappa(Z)^{s-1}}{(4n)^{s-1} s\sqrt{s \nu(n,N)}}, \quad \nu(n,N)={N+n\choose N}, $$ and an explicit right inverse $V_N(Z)^+$ with operator-norm control $$ \|V_N(Z)^+\| \lesssim s^{3/2}\sqrt{\nu(n,N)}\Bigl(\frac{4n}{\kappa(Z)}\Bigr)^{s-1}. $$ Our estimates are dimension-explicit and expressed directly in terms of the local geometry parameter $\kappa(Z)$; they apply to \emph{every} distinct node set $Z\subset B_2^n$ without any \emph{a priori} separation assumptions. In particular, $V_N(Z)$ has full row rank whenever $N\ge s-1$. The results complement the Fourier-type theory (on the complex unit circle/torus), where lower bounds for $\sigma_{\min}$ hinge on uniform separation or cluster structure; here stability is quantified instead via high polynomial degree and the projection geometry of $Z$. - oai:arXiv.org:2601.13915v1 - math.CA - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Omer Friedland, Yosef Yomdin - - - Wiener Algebras Methods for Liouville Theorems on the Stationary Navier-Stokes System - https://arxiv.org/abs/2601.13916 - arXiv:2601.13916v1 Announce Type: new -Abstract: We prove some Liouville theorems for the stationary Navier-Stokes system for incompressible fluids. We provide some sufficient conditions on the low frequency part of the solution, using some properties of classical singular integrals with respect to Wiener algebras. - oai:arXiv.org:2601.13916v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Nicolas Lerner - - - A finiteness result on representations of Nori's fundamental group scheme - https://arxiv.org/abs/2601.13917 - arXiv:2601.13917v1 Announce Type: new -Abstract: Let $(X,x)$ be a pointed geometrically connected smooth projective variety over a sub-$p$-adic field $K$. For any given rank $n$, we prove that there are only finitely many isomorphism classes of representations $\pi_{1}^{EF}(X,x)\rightarrow \mathrm{GL}_{n}$, where $\pi_{1}^{EF}(X,x)$ is Nori's fundamental group of essentially finite bundles. Equivalently, there are only finitely many isomorphism classes of essentially finite bundles of rank $n$. This answers a question from C.Gasbarri. - oai:arXiv.org:2601.13917v1 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xiaodong Yi - - - The properad of quadratic Poisson structures is Koszul - https://arxiv.org/abs/2601.13921 - arXiv:2601.13921v1 Announce Type: new -Abstract: In this paper, we suggest a sufficient condition on the properadic envelope of a quadratic dioperad to be Koszul in terms of twisted associative algebras. - As a particular new example, we show that the properad of quadratic Poisson structures is Koszul. - oai:arXiv.org:2601.13921v1 - math.QA - math-ph - math.KT - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Anton Khoroshkin - - - Proactive Coded Caching Scheme for D2D Networks - https://arxiv.org/abs/2601.13929 - arXiv:2601.13929v1 Announce Type: new -Abstract: Coded caching and device-to-device (D2D) communication are two effective techniques for alleviating network traffic. Secure transmission and file privacy have also become critical concerns in these domains. However, prevailing coded caching schemes typically assume that a user's cached content is inaccessible to others, overlooking the risk of file privacy leakage due to attacks targeting the cache itself. In this paper, we propose a secure coded caching scheme for D2D networks that guarantees both file privacy and secure delivery. We demonstrate that the proposed scheme achieves order-optimal performance when the file size is sufficiently large and the cache memory is ample. - oai:arXiv.org:2601.13929v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Qiaoling Zhang, Changlu Lin, Minquan Cheng - - - On spectral clustering under non-isotropic Gaussian mixture models - https://arxiv.org/abs/2601.13930 - arXiv:2601.13930v1 Announce Type: new -Abstract: We evaluate the misclustering probability of a spectral clustering algorithm under a Gaussian mixture model with a general covariance structure. The algorithm partitions the data into two groups based on the sign of the first principal component score. As a corollary of the main result, the clustering procedure is shown to be consistent in a high-dimensional regime. - oai:arXiv.org:2601.13930v1 - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Kohei Kawamoto, Yuichi Goto, Koji Tsukuda - - - Packing minima of convex bodies - https://arxiv.org/abs/2601.13937 - arXiv:2601.13937v1 Announce Type: new -Abstract: In 2021, Henk, Schymura and Xue introduced packing minima, associated with a convex body and a lattice, as packing counterparts to the covering minima of Kannan and Lov\'asz. Motivated by conjectures on the volume inequalities for the successive minima, we generalized the definition of the packing minima to the class of all convex bodies that contain the origin in their interior. For these packing minima, we presented several novel volume inequalities and calculated the specific values of the packing minima for several special convex bodies. - oai:arXiv.org:2601.13937v1 - math.MG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Mei Han, Martin Henk, Fei Xue - - - Gallai-Ramsey Numbers for $\ell$-Connected Graphs - https://arxiv.org/abs/2601.13944 - arXiv:2601.13944v1 Announce Type: new -Abstract: Given a nonempty graph $G$, a collection of nonempty graphs $\cal{H}$, and a positive integer $k$, the Gallai-Ramsey number $\mathrm{gr}_k(G:\mathcal{H})$ is defined to be the minimum positive integer $n$ such that every exact $k$-edge-coloring of a complete graph $K_n$ contains either a rainbow copy of $G$ or a monochromatic copy of some element in $\mathcal{H}$. In this paper, we obtain some exact values and general lower and upper bounds for $\mathrm{gr}_k(G:\mathcal{F}^\ell)$, where $\mathcal{F}^\ell$ is the set of $\ell$-connected graphs and $G\in\{P_5, K_{1,3}\}$. - oai:arXiv.org:2601.13944v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Zhao Wang, Lanyanni Zhang, Meiqin Wei, Mark Budden - - - Topological Criteria for Hypothesis Testing with Finite-Precision Measurements - https://arxiv.org/abs/2601.13946 - arXiv:2601.13946v1 Announce Type: new -Abstract: We establish topological necessary and sufficient conditions under which a pair of statistical hypotheses can be consistently distinguished when i.i.d. observations are recorded only to finite precision. Requiring the test's decision regions to be open in the sample-space topology to accommodate finite-precision data, we show that a pair of null- and alternative hypotheses $H_0$ and $H_1$ admits a consistent test if and only if they are $F_\sigma$ in the weak topology on the space of probability measures $W := H_0\cup H_1$. Additionally, the hypotheses admit uniform error control under $H_0$ and/or $H_1$ if and only if $H_0$ and/or $H_1$ are closed in $W$. Under compactness assumptions, uniform consistency is characterised by $H_0$ and $H_1$ having disjoint closures in the ambient space of probability measures. These criteria imply that - without regularity assumptions - conditional independence is not consistently testable. We introduce a Lipschitz-continuity assumption on the family of conditional distributions under which we recover testability of conditional independence with uniform error control under the null, with testable smoothness constraints. - oai:arXiv.org:2601.13946v1 - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Philip Boeken, Eduardo Skapinakis, Konstantin Genin, Joris M. Mooij - - - Wold-type decomposition for doubly twisted left-invertible covariant representations - https://arxiv.org/abs/2601.13950 - arXiv:2601.13950v1 Announce Type: new -Abstract: We will introduce the notion of a near-isometric covariant representation of a $C^*$-correspondence and prove its Wold-type decomposition. Wold-type decomposition for doubly twisted left-invertible covariant representations of a product system is also obtained. - oai:arXiv.org:2601.13950v1 - math.OA - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Niraj Kumar, Azad Rohilla, Harsh Trivedi - - - Hypercube subgroups of (outer) reduced Weyl groups of the Cuntz algebras - https://arxiv.org/abs/2601.13952 - arXiv:2601.13952v1 Announce Type: new -Abstract: We develop some tools, of an algebraic and combinatorial nature, which enable us to obtain a detailed description of certain quadratic subgroups of the (outer) reduced Weyl group of the Cuntz algebra ${\mathcal O}_n$. In particular, for $n=4$ our findings give a self-contained theoretical interpretation of the groups tabulated in [AJS18], which were obtained with the help of a computer. For each of these groups we provide a set of generators. A prominent role in our analysis is played by a certain family of subgroups of the symmetric group of a discrete square which we call bicompatible. - oai:arXiv.org:2601.13952v1 - math.OA - math-ph - math.CO - math.FA - math.GR - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Francesco Brenti, Roberto Conti, Gleb Nenashev - - - Uniform Consistency of Generalized Cross-Validation for Ridge Regression in High-Dimensional Misspecified Linear Models - https://arxiv.org/abs/2601.13955 - arXiv:2601.13955v1 Announce Type: new -Abstract: This study examines generalized cross-validation for the tuning parameter selection for ridge regression in high-dimensional misspecified linear models. The set of candidates for the tuning parameter includes not only positive values but also zero and negative values. We demonstrate that if the second moment of the specification error converges to zero, generalized cross-validation is still a uniformly consistent estimator of the out-of-sample prediction risk. This implies that generalized cross-validation selects the tuning parameter for which ridge regression asymptotically achieves the smallest prediction risk among the candidates if the degree of misspecification for the regression function is small. Our simulation studies show that ridge regression tuned by generalized cross-validation exhibits a prediction performance similar to that of optimally tuned ridge regression and outperforms the Lasso under correct and incorrect model specifications. - oai:arXiv.org:2601.13955v1 - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Akira Shinkyu - - - A Bregman Regularized Proximal Point Method for Solving Equilibrium Problems on Hadamard Manifolds - https://arxiv.org/abs/2601.13959 - arXiv:2601.13959v1 Announce Type: new -Abstract: In this paper we develop a Bregman regularized proximal point algorithm for solving monotone equilibrium problems on Hadamard manifolds. It has been shown that the regularization term induced by a Bregman function is, in general, nonconvex on Hadamard manifolds unless the curvature is zero. Nevertheless, we prove that the proposed Bregman regularization scheme does converge to a solution of the equilibrium problem on Hadamard manifolds in the presence of a strong assumption on the convexity of the set formed by the regularization term. Moreover, we employ a coercivity condition on the Bregman function which is weaker than those typically assumed in the existing literature on Bregman regularization. Numerical experiments on illustrative examples demonstrate the practical effectiveness of our proposed method. - oai:arXiv.org:2601.13959v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Shikher Sharma, Simeon Reich - - - Information-Theoretic and Computational Limits of Correlation Detection under Graph Sampling - https://arxiv.org/abs/2601.13966 - arXiv:2601.13966v1 Announce Type: new -Abstract: Correlation analysis is a fundamental problem in statistics. In this paper, we consider the correlation detection problem between a pair of Erdos-Renyi graphs. Specifically, the problem is formulated as a hypothesis testing problem: under the null hypothesis, the two graphs are independent; under the alternative hypothesis, the two graphs are edge-correlated through a latent permutation. We focus on the scenario where only two induced subgraphs are sampled, and characterize the sample size threshold for detection. At the information-theoretic level, we establish the sample complexity rates that are optimal up to constant factors over most parameter regimes, and the remaining gap is bounded by a subpolynomial factor. On the algorithmic side, we propose polynomial-time tests based on counting trees and bounded degree motifs, and identify the regimes where they succeed. Moreover, leveraging the low-degree conjecture, we provide evidence of computational hardness that matches our achievable guarantees, showing that the proposed polynomial-time tests are rate-optimal. Together, these results reveal a statistical--computational gap in the sample size required for correlation detection. Finally, we validate the proposed algorithms on synthetic data and a real coauthor network, demonstrating strong empirical performance. - oai:arXiv.org:2601.13966v1 - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Dong Huang, Pengkun Yang - - - Dispersive estimate for quasi-periodic Klein-Gordon equation on 1-d lattices - https://arxiv.org/abs/2601.13967 - arXiv:2601.13967v1 Announce Type: new -Abstract: The dispersive estimate plays a pivotal role in establishing the long-term behavior of solutions to the nonlinear equation, thereby being crucial for investigating the well-posedness of the equation.In this work we prove that the solutions to Klein-Gordon equation on 1-d lattices follow the dispersive estimate provided that potential is quasi-periodic with Diophantine frequencies and closed to positive constants. - oai:arXiv.org:2601.13967v1 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Hongyu Cheng - - - Toric Euler-Jacobi vanishing theorem and zeros at infinity - https://arxiv.org/abs/2601.13977 - arXiv:2601.13977v1 Announce Type: new -Abstract: Residues appear naturally in various questions in complex and algebraic geometry: interpolation, duality, representation problems, and obstructions. The first global vanishing result in the projective plane, known as the Euler-Jacobi theorem, was established by Jacobi in 1835. In the toric case, the input is a system of n Laurent sparse polynomials with fixed Newton polytopes, and the first version of the Euler-Jacobi toric vanishing theorem for residues in the n-torus is due to Khovanskii in 1978, under restrictive genericity assumptions. In this paper, we provide geometric conditions on the input Newton polytopes to ensure that this global vanishing is equivalent to the existence of zeros at infinity in the associated compact toric variety. We relate these conditions to the dimension at the toric critical degree of the quotient of the Cox ring by the ideal generated by the (multi)homogenizations of the input polynomials. We also relate the existence of zeros at infinity to interpolation questions. - oai:arXiv.org:2601.13977v1 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Carlos D'Andrea, Alicia Dickenstein - - - Spectral Gaps on Large Hyperbolic Surfaces - https://arxiv.org/abs/2601.13988 - arXiv:2601.13988v1 Announce Type: new -Abstract: In this expository paper, we review the history and the recent breakthroughs in the spectral theory of large volume hyperbolic surfaces. More precisely, we focus mostly on the investigation of the first non-trivial eigenvalue $\lambda_1$ and its possible behaviour in the large volume regime. - oai:arXiv.org:2601.13988v1 - math.SP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Laura Monk, Fr\'ed\'eric Naud - - - Eigensets of switching dynamical systems - https://arxiv.org/abs/2601.13990 - arXiv:2601.13990v1 Announce Type: new -Abstract: Reachability sets of linear switching dynamical systems (systems of ODE with time-dependent matrices that take values from a given compact set) are analysed. An eigenset is a non-trivial compact set M that possesses the following property: the closure of the set of points reachable by trajectories starting in M in time t is equal to exp(at)M. This concept introduced in a recent paper of E.Viscovini is an analogue of an eigenvector for compact sets of matrices. We prove the existence of eigensets, analyse their structure and properties, and find ``eigenvalues'' a for an arbitrary system. The question which compact sets, in particular, which convex sets and polyhedra, can be presented as eigensets of suitable systems, is studied. - oai:arXiv.org:2601.13990v1 - math.DS - math.MG - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Vladimir Protasov - - - Computing Crystalline Cohomology and p-Divisible Groups for Curves over Finite Fields - https://arxiv.org/abs/2601.14006 - arXiv:2601.14006v1 Announce Type: new -Abstract: Let $X$ be a smooth projective curve over a finite field of characteristic $p$. We describe and implement a practical algorithm for computing the $p$-divisible group $Jac(X)[p^\infty]$ via computing its Dieudonn\'{e} module, or equivalently computing the Frobenius and Verschiebung operators on the first crystalline cohomology of $X$. We build on Tuitman's $p$-adic point counting algorithm, which computes the rigid cohomology of $X$ and requires a ``nice'' lift of $X$ to be provided. - oai:arXiv.org:2601.14006v1 - math.NT - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jeremy Booher - - - Numerical solution of Smoluchowski coagulation equation combined with Ostwald ripening - https://arxiv.org/abs/2601.14011 - arXiv:2601.14011v1 Announce Type: new -Abstract: The processes of simultaneous coagulation and Ostwald ripening of particles in the concluding stage of phase transformation are considered. We solve the integro-differential system of Smoluchowski-type kinetic and mass balance equations using a computationally efficient numerical algorithm based on low-rank matrices. We compare our numerical solutions for different initial particle-volume distributions with the universal distribution function for combined coagulation and Ostwald ripening. Our calculations confirm the tendency of a particulate ensemble to the universal particle-volume distribution to be approached asymptotically after a sufficiently long time, no matter what the initial particle-volume distribution might be. - oai:arXiv.org:2601.14011v1 - math.NA - cond-mat.soft - cond-mat.stat-mech - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Robert T. Zaks, Sergey A. Matveev, Margarita A. Nikishina, Dmitri V. Alexandrov - - - Robustness for free: asymptotic size and power of max-tests in high dimensions - https://arxiv.org/abs/2601.14013 - arXiv:2601.14013v1 Announce Type: new -Abstract: Consider testing a zero restriction on the mean of a $d$-dimensional random vector based on an i.i.d. sample of size $n$. Suppose further that the coordinates are only assumed to possess $m>2$ moments. Then, max-tests based on arithmetic means and critical values derived from Gaussian approximations are not guaranteed to be asymptotically valid unless $d$ is relatively small compared to $n$, because said approximation faces a polynomial growth barrier of $d=o(n^{m/2-1})$. - We propose a max-test based on winsorized means, and show that it holds the desired asymptotic size even when $d$ grows at an exponential rate in $n$ and the data are adversarially contaminated. Our characterization of its asymptotic power function shows that these benefits do not come at the cost of reduced asymptotic power: the robustified max-test has identical asymptotic power to that based on arithmetic means whenever the stronger assumptions underlying the latter are satisfied. - We also investigate when -- and when not -- data-driven (bootstrap) critical values can strictly increase asymptotic power of the robustified max-test. - oai:arXiv.org:2601.14013v1 - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Anders Bredahl Kock, David Preinerstorfer - - - Non-linear traces of Choquet type on AF algebras - https://arxiv.org/abs/2601.14016 - arXiv:2601.14016v1 Announce Type: new -Abstract: We study non-linear traces of Choquet type on AF algebras. Building on the characterization of Choquet traces on matrix algebras due to Nagisa--Watatani, we generalize the construction to arbitrary unital AF algebras. We show that there is a one-to-one correspondence between such traces and increasing functions on the dimension scale, and we obtain explicit Choquet formulas in terms of the spectrum and ranks of spectral projections along a fixed AF filtration. - oai:arXiv.org:2601.14016v1 - math.OA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ryota Ninomiya - - - Principal $p-$frequency estimates on non-compact manifolds with negative Ricci curvature - https://arxiv.org/abs/2601.14018 - arXiv:2601.14018v1 Announce Type: new -Abstract: We establish a lower bound for the principal $p-$frequency $\lambda_{1,p}(\Omega)$ on a bounded domain $\Omega$ in a non-compact Riemannian manifold of dimension $n.$ Under the assumption that the Ricci curvature satisfies $\operatorname{Ric} \geq (n-1)K$ with $K<0,$ we prove that $\lambda_{1,p}(\Omega) > \bar{\lambda}_{D,K,n}$, where $D$ is the diameter of $\Omega$ and $\bar{\lambda}_{D,K,n}$ is explicitly defined as the first eigenvalue of an associated one-dimensional ordinary differential equation model that incorporates both $D$ and $K.$ Moreover, the estimate is sharp. This work extends previous results for the case $K=0$ to the geometrically more complex setting of negative Ricci curvature, and providing a new quantitative connection between the eigenvalue, the diameter of domains, and the curvature lower bound. - oai:arXiv.org:2601.14018v1 - math.DG - math.SP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Xiaoshang Jin, Zhiwei L\"u - - - Tensor Abelian geometry of VI-modules - https://arxiv.org/abs/2601.14020 - arXiv:2601.14020v1 Announce Type: new -Abstract: In this short note, we study the spectrum of prime Serre ideals of global represen tations for noetherian families. In particular, we prove that the spectrum of prime Serre ideals of finitely generated VI-modules is homeomorphic to N^{*}, the one-point compactification of N, which differs from the Balmer spectrum of derived VI-modules. Our method could also be applied to the category of finitely generated FI-modules and the category of global representations for the family of cyclic p-groups. - oai:arXiv.org:2601.14020v1 - math.CT - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Peng Xu - - - Some Results on Causal Modalities in General Spacetimes - https://arxiv.org/abs/2601.14029 - arXiv:2601.14029v1 Announce Type: new -Abstract: Causality is one of the fundamental structures of spacetimes, it determines the possible behaviour and propagation of physical information through different relations. Causal structure can be analysed through the various modal logics it induces. The modal logics for the standard chronological and causal relations of the archetypal Minkowski spacetime have been classified. However only partial results have been achieved for the strict variant of the causal relation, also known as the after relation. - The present work continues this analysis towards arbitrary spacetimes. By utilizing the definition of the causal relations through causal paths, we can lift known results about the modal logics of Minkowski spacetime to general spacetimes. In particular, for the after relation, we show that a previously studied formula within the logics of Minkowski spacetime holds in arbitrary spacetimes. We introduce a related modal formula that demonstrates that the logic of two-dimensional spacetimes are more expressive than higher dimensional ones. Lastly, we study the interrelation between the logical properties and physical properties along the causal ladder, a classification of causal structures according to a hierarchy of physically relevant properties. - oai:arXiv.org:2601.14029v1 - math.LO - gr-qc - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Marco Lewis, Nesta van der Schaaf - - - Classification of invariant tight contact structures on the 3-space, -ball and -sphere - https://arxiv.org/abs/2601.14040 - arXiv:2601.14040v1 Announce Type: new -Abstract: We prove some classification results for tight contact structure in the 3-space, -ball and -sphere that are invariant with respect to some arbitrary involution, that is conjugated to the standard rotation around the x-axis. Unlike the classical scenario, a new integral torsion appears, dictating a splitting between equivalence classes. These tools could be useful fur future classification results regarding strongly invertible Legendrian knots. - oai:arXiv.org:2601.14040v1 - math.GT - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Mirko Torresani - - - On the Diophantine Equation Involving Elementary Symmetric Polynomials and the Decomposition of Unity - https://arxiv.org/abs/2601.14057 - arXiv:2601.14057v1 Announce Type: new -Abstract: We consider the equality of the values of the $n$th and $k$th elementary symmetric polynomials of $n$ not necessarily distinct positive integers. For $k < n$, we prove that this equation always has a solution, but only finitely many solutions. Furthermore, we consider the equality of the values of the $n$th and $(n-2)$th elementary symmetric polynomials of $n$ not necessarily distinct positive integers. In particular, we show that the number of solutions of this equation tends to infinity if $n$ tends to infinity. - oai:arXiv.org:2601.14057v1 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - S\'andor Z. Kiss, Csaba S\'andor, Maciej Zakarczemny - - - A Splitting Theorem for non-positively curved Lorentzian spaces - https://arxiv.org/abs/2601.14058 - arXiv:2601.14058v1 Announce Type: new -Abstract: We prove a splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature. Additionally, we extend the first variation formula to spaces with any timelike curvature bound, either from above or below, and different from 0. - oai:arXiv.org:2601.14058v1 - math.DG - math-ph - math.MG - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Joe Barton, Tobias Beran, Mauricio Che, Sebastian Gieger, Jona R\"ohrig, Felix Rott - - - Frostman dimension of Furstenberg measure for $\mathrm{SL}(2,\mathbb{R})$ random matrix products - https://arxiv.org/abs/2601.14061 - arXiv:2601.14061v1 Announce Type: new -Abstract: For compactly supported $\mu \in \mathcal{P}(\mathrm{SL}(2,\mathbb{R}))$ satisfying strong irreducibility and proximality, we obtain a formula for the Frostman dimension of the associated Furstenberg measure. We also describe the left neighbourhood of 0 for which the classical transfer operators defined by Le Page have a spectral gap on H\"older spaces in this setting. - oai:arXiv.org:2601.14061v1 - math.DS - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tom Rush - - - Time-dependent metrics and connections - https://arxiv.org/abs/2601.14064 - arXiv:2601.14064v1 Announce Type: new -Abstract: Time-dependent structures often appear in differential geometry, particularly in the study of non-autonomous differential equations on manifolds. One may study the geodesics associated with a time-dependent Riemannian metric by extremizing the corresponding energy functional, but also through the introduction of a more general concept of time-dependent covariant derivative operator. This relies on the examination of connections on the product manifold $\mathbb{R}\times M$. For these time-dependent covariant derivatives we explore the notions of parallel transport, geodesics and torsion. We also define the derivative of a one-parameter family of connections. - oai:arXiv.org:2601.14064v1 - math.DG - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Xavier Gr\`acia, Xavier Rivas, Daniel Torres - - - On the optimal shape parameter for kernel methods: Sharp direct and inverse statements - https://arxiv.org/abs/2601.14070 - arXiv:2601.14070v1 Announce Type: new -Abstract: The search for the optimal shape parameter for Radial Basis Function (RBF) kernel approximation has been an outstanding research problem for decades. In this work, we establish a theoretical framework for this problem by leveraging a recently established theory on sharp direct, inverse and saturation statements for kernel based approximation. In particular, we link the search for the optimal shape parameter to superconvergence phenomena. Our analysis is carried out for finitely smooth Sobolev kernels, thereby covering large classes of radial kernels used in practice, including those emerging from current machine-learning methodologies. Our results elucidate how approximation regimes, kernel regularity, and parameter choices interact, thereby clarifying a question that has remained unresolved for decades. - oai:arXiv.org:2601.14070v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tizian Wenzel, Gabriele Santin - - - LU-type factorizations for birth--death processes and their Darboux transformations - https://arxiv.org/abs/2601.14074 - arXiv:2601.14074v1 Announce Type: new -Abstract: We study LU-type factorizations of the infinitesimal generator of a birth--death process on $\mathbb{N}_0$. Our goal is to characterize those factorizations whose Darboux transformations (that is, inverting the order of the factors) yield new infinitesimal generators of birth--death processes. Two types are considered: lower--upper (LU), which is unique and upper--lower (UL), which involves a free parameter. For both cases, we determine the conditions under which such factorizations can occur, derive explicit formulas for their coefficients, and provide a probabilistic interpretation of the factors. The spectral properties and associated orthogonal polynomials of the Darboux transformations are also analyzed. Finally, the general results are applied to classical examples such as the $M/M/1$ and $M/M/\infty$ queues and to different cases of linear birth--death processes. - oai:arXiv.org:2601.14074v1 - math.PR - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jos\'e Arcia-Manoleskos, Manuel Dom\'inguez de la Iglesia - - - Utilizing the Perceived Age to Maximize Freshness in Query-Based Update Systems - https://arxiv.org/abs/2601.14075 - arXiv:2601.14075v1 Announce Type: new -Abstract: Query-based sampling has become an increasingly popular technique for monitoring Markov sources in pull-based update systems. However, most of the contemporary literature on this assumes an exponential distribution for query delay and often relies on the assumption that the feedback or replies to the queries are instantaneous. In this work, we relax both of these assumptions and find optimal sampling policies for monitoring continuous-time Markov chains (CTMC) under generic delay distributions. In particular, we show that one can obtain significant gains in terms of mean binary freshness (MBF) by employing a waiting based strategy for query-based sampling. - oai:arXiv.org:2601.14075v1 - cs.IT - cs.SY - eess.SY - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sahan Liyanaarachchi, Sennur Ulukus, Nail Akar - - - On large periodic traveling wave solutions to the free boundary Stokes and Navier-Stokes equations - https://arxiv.org/abs/2601.14085 - arXiv:2601.14085v1 Announce Type: new -Abstract: We study the free boundary problem for a finite-depth layer of viscous incompressible fluid in arbitrary dimension, modeled by the Stokes or Navier-Stokes equations. In addition to the gravitational field acting in the bulk, the free boundary is acted upon by surface tension and an external stress tensor posited to be in traveling wave form. We prove that for any isotropic stress tensor with periodic profile, there exists a locally unique periodic traveling wave solution, which can have large amplitude. Moreover, we prove that the constructed traveling wave solutions are asymptotically stable for the dynamic free boundary Stokes equations. Our proofs rest on the analysis of the nonlocal normal-stress to normal-Dirichlet operators for the Stokes and Navier-Stokes equations in domains of Sobolev regularity. - oai:arXiv.org:2601.14085v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Seyed Abdolhamid Banihashemi, Huy Q. Nguyen - - - Near Optimal Code Construction for the Adversarial Torn Paper Channel With Edit Errors - https://arxiv.org/abs/2601.14088 - arXiv:2601.14088v1 Announce Type: new -Abstract: Motivated by DNA storage systems and 3D fingerprinting, this work studies the adversarial torn paper channel with edit errors. This channel first applies at most $t_e$ edit errors (i.e., insertions, deletions, and substitutions) to the transmitted word and then breaks it into $t+1$ fragments at arbitrary positions. In this paper, we construct a near optimal error correcting code for this channel, which will be referred to as a $t$-breaks $t_e$-edit-errors resilient code. This code enables reconstructing the transmitted codeword from the $t+1$ noisy fragments. Moreover, we study list decoding of the torn paper channel by deriving bounds on the size of the list (of codewords) obtained from cutting a codeword of a $t$-breaks resilient code $t'$ times, where $t' > t$. - oai:arXiv.org:2601.14088v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Maria Abu-Sini, Reinhard Heckel - - - Period collapse of Markov triangles - https://arxiv.org/abs/2601.14090 - arXiv:2601.14090v1 Announce Type: new -Abstract: Cristofaro-Gardiner and Kleinman showed the complete period collapse of the Ehrhart quasipolynomial of Fibonacci triangles and their irrational limits, by studying the Fourier-Dedekind sums involved in the Ehrhart function of right-angled rational triangles. We generalize this result using integral affine geometrical methods to all Markov triangles, as defined by Vianna. In particular, we show new occurrences of strong period collapse, namely by constructing for each Markov number $p$ a two-sided sequence of rational triangles and two irrational limits with quasipolynomial Ehrhart function of period $p$. - oai:arXiv.org:2601.14090v1 - math.CO - math.SG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Marc Fares - - - Basis Number and Pathwidth - https://arxiv.org/abs/2601.14095 - arXiv:2601.14095v1 Announce Type: new -Abstract: We prove two results relating the basis number of a graph $G$ to path decompositions of $G$. Our first result shows that the basis number of a graph is at most four times its pathwidth. Our second result shows that, if a graph $G$ has a path decomposition with adhesions of size at most $k$ in which the graph induced by each bag has basis number at most $b$, then $G$ has basis number at most $b+O(k\log^2 k)$. The first result, combined with recent work of Geniet and Giocanti shows that the basis number of a graph is bounded by a polynomial function of its treewidth. The second result (also combined with the work of Geniet and Giocanti) shows that every $K_t$-minor-free graph has a basis number bounded by a polynomial function of $t$. - oai:arXiv.org:2601.14095v1 - math.CO - cs.DM - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Babak Miraftab, Pat Morin, Yelena Yuditsky - - - Simple subquotients of crossed products by abelian groups and twisted group algebras - https://arxiv.org/abs/2601.14097 - arXiv:2601.14097v1 Announce Type: new -Abstract: Motivated by work of Poguntke we study the question under what conditions simple subquotients of crossed products $A\rtimes_{\alpha}G$ by (twisted) actions of abelian groups $G$ are isomorphic to simple twisted group algebras of abelian groups. As a consequence, we recover a theorem of Poguntke's saying that the simple subquotients of group $C^*$-algebras of connected groups are either stably isomorphic to $\mathbb C$ or they are stably isomorphic to simple non-commutative tori. - oai:arXiv.org:2601.14097v1 - math.OA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Siegfried Echterhoff - - - Angular pair-of-pants decompositions of complex varieties - https://arxiv.org/abs/2601.14116 - arXiv:2601.14116v1 Announce Type: new -Abstract: We define the notion of torically hyperbolic varieties and we construct pair-of-pants decompositions for these in terms of angle sets of essential projective hyperplane complements. This construction generalizes the classical pair-of-pants decomposition for hyperbolic Riemann surfaces. In our first main theorem, we prove that the natural angle map associated to an essential projective hyperplane complement is a homotopy equivalence, extending earlier work of Salvetti and Bj\"orner-Ziegler. By a topological argument, we further show that the angle map for a finite Kummer covering of an essential projective hyperplane complement is likewise a homotopy equivalence. We then explain how these local building blocks can be glued along the dual intersection complex of a semistable degeneration. Using the theory of Kato-Nakayama spaces, we prove that the resulting space is homotopy equivalent to the original algebraic variety. We make this explicit for complete intersections in projective space using techniques from tropical geometry. - oai:arXiv.org:2601.14116v1 - math.AG - math.AT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Yassine Elmaazouz, Paul Alexander Helminck - - - Scalar-rigid submersions are Riemannian products - https://arxiv.org/abs/2601.14117 - arXiv:2601.14117v1 Announce Type: new -Abstract: Scalar-rigid maps are Riemannian submersions by works of Llarull, Goette--Semmelmann, and the second named author. In this article we show that they are essentially Riemannian products of the base manifold with a Ricci-flat fiber. As an application we obtain a Llarull-type theorem for non-zero degree maps onto products of manifolds of non-negative curvature operator and positive Ricci curvature with some enlargeable manifold. The proof is based on spin geometry for Dirac operators and an analysis connecting Clifford multiplication with the representation theory of the curvature operator. - oai:arXiv.org:2601.14117v1 - math.DG - math.KT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Oskar Riedler, Thomas Tony - - - Universal Chord Theorem and a Topological Analysis - https://arxiv.org/abs/2601.14120 - arXiv:2601.14120v1 Announce Type: new -Abstract: We study the set of chords of a real-valued continuous function on [0,1] with f(0)=f(1)=0. We describe which chords may appear as isolated points and provide examples illustrating our characterization. Maximal Hopf sets are introduced and analyzed. - oai:arXiv.org:2601.14120v1 - math.GN - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-sa/4.0/ - Ion Ciudin, Eugen J. Ionascu - - - $Q_p$-weighted zero-sum constants - https://arxiv.org/abs/2601.14122 - arXiv:2601.14122v1 Announce Type: new -Abstract: A sequence $S=(x_1,\ldots, x_k)$ in $\mathbb Z_p$ is called a $(Q_p,\mathbf 1)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in Q_p$ such that $a_1x_1+\cdots+a_kx_k=0$ and $a_1+\cdots+a_k=0$. The constant $E_{Q_p,\mathbf 1}$ is defined to be the smallest positive integer $k$ such that every sequence of length $k$ in $\mathbb Z_p$ has a $(Q_p,\mathbf 1)$-weighted zero-sum subsequence of length $p$. We determine the constant $E_{Q_p,\mathbf 1}$ and the related constants $C_{Q_p,\mathbf 1}$ and $D_{Q_p,\mathbf 1}$. We also study some $(Q_p,B)$-weighted zero-sum constants where $B$ is a subset of $Q_p$. - oai:arXiv.org:2601.14122v1 - math.NT - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Krishnendu Paul, Shameek Paul - - - Flexible curves and Hausdorff dimension - https://arxiv.org/abs/2601.14125 - arXiv:2601.14125v1 Announce Type: new -Abstract: We show that given a log-singular circle homeomorphism $h$ and given any $s\in[1,2]$, there is a flexible curve of Hausdorff dimension $s$ with welding $h$. We also see that there is another curve with welding $h$ and positive area. In particular, this implies that given a flexible curve $\Gamma$, there is a homeomorphism of the plane $\phi\colon\mathbb{C}\to\mathbb{C}$, conformal off $\Gamma$, so that $\phi(\Gamma)$ has positive area. This answers a particular case of the corresponding conjecture for general non-conformally removable sets, for a class of curves that is residual in the space of all Jordan curves. - oai:arXiv.org:2601.14125v1 - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alex Rodriguez - - - Structural properties of graphs and the Universal Difference Property - https://arxiv.org/abs/2601.14126 - arXiv:2601.14126v1 Announce Type: new -Abstract: We study the Universal Difference Property (UDP) introduced by Alt{\i}nok, Anders, Arreola, Asencio, Ireland, Sar{\i}o\u{g}lan, and Smith, focusing on the relationship between the structural properties of a graph and UDP. We present condtions for when UDP must hold on unicyclic graphs. We then prove that if UDP does not hold on an edge-labeled graph, then it cannot hold on any subdivision of that graph. Additionally, we show that if an edge-labeled graph satisfies the pairwise edge-disjoint path property, then the graph satisfies UDP. Lastly, we explore the relationship between UDP and subgraphs and prove that trees and cycles are the only two families of connected graphs for which UDP must hold for any edge-labeling over any ring. - oai:arXiv.org:2601.14126v1 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Katie Anders, Able Martinez, Patrick McHugh, Jenna Rogers, Remi Salinas Schmeis - - - Scheme theory for commutative semirings - https://arxiv.org/abs/2601.14136 - arXiv:2601.14136v1 Announce Type: new -Abstract: In this survey, we describe two different approaches to constructing affine schemes for commutative semirings: one based on prime ideals, and another based on prime kernels (also called subtractive ideals). We then explain how these two approaches are related through the theory of universal valuations. - oai:arXiv.org:2601.14136v1 - math.AG - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Roberto Gualdi, Arne Kuhrs, Mayo Mayo Garcia, Xavier Xarles - - - A global stochastic maximum principle for delayed forward-backward stochastic control systems - https://arxiv.org/abs/2601.14138 - arXiv:2601.14138v1 Announce Type: new -Abstract: In this paper, we study a delayed forward-backward stochastic control system in which all the coefficients depend on the state and control terms, and the control domain is not necessarily convex. A global stochastic maximum principle is obtained by using a new method. More precisely, this method introduces first-order and second-order auxiliary equations and offers a novel approach to deriving the adjoint equations as well as the variational equation for $y^\e - y^*$. - oai:arXiv.org:2601.14138v1 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Feng Li - - - Symmetry Breaking and Phase Transitions in Random Non-Commutative Geometries and Related Random-Matrix Ensembles - https://arxiv.org/abs/2601.14141 - arXiv:2601.14141v1 Announce Type: new -Abstract: Ensembles of random fuzzy non-commutative geometries - may be described in terms of finite (\(N^2\)-dimensional) - Dirac operators and a probability measure. - Dirac operators of type \((p,q)\) are defined in - terms of commutators and anti-commutators of \(2^{p+q-1}\) - hermitian matrices \(H_k\) - and tensor products with a representation of a Clifford algebra. - Ensembles based on this idea have recently been - used as a toy model for - quantum gravity, - and they are interesting random-matrix - ensembles in their own right. - We provide - a complete theoretical picture of crossovers, - phase transitions, and symmetry breaking - in the \(N \to \infty \) limit of - 1-parameter families of quartic Barrett-Glaser ensembles - in the one-matrix cases \((1,0)\) and \((0,1)\) that - depend on one coupling constant \(g\). - Our theoretical results are in full agreement with previous - and new Monte-Carlo simulations. - oai:arXiv.org:2601.14141v1 - math-ph - gr-qc - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Mauro D'Arcangelo, Sven Gnutzmann - - - Vector Coded Caching Multiplicatively Boosts MU-MIMO Systems Under Practical Considerations - https://arxiv.org/abs/2601.14142 - arXiv:2601.14142v1 Announce Type: new -Abstract: This work presents a first comprehensive analysis of the impact of vector coded caching (VCC) in multi-user multiple-input multiple-output (MU-MIMO) systems with multiple receive antennas and variable pathloss -- two key factors that critically influence systems with inherent MU unicasting behavior. We investigate two widely adopted precoding strategies: (i) blockdiagonalization (BD) at the transmitter combined with maximal ratio combining (MRC) at the receivers, and (ii) zero-forcing (ZF) precoding. Our analysis explicitly accounts for practical considerations such as channel fading, channel state information (CSI) acquisition overhead, and fairness-oriented power allocation. - Our contributions span both analytical and simulation-based fronts. On the analytical side, we derive analytical expressions for the achievable throughput under BD-MRC and ZF, highlighting the performance benefits of equipping multi-antenna users with cache-aided interference management. Specifically, we develop a low-complexity BD-MRC optimization method that leverages matrix structure to significantly reduce the dimensionality involved in precoding computation, followed by solving the associated maxmin fairness problem through an efficient one-dimensional search. In the massive MIMO regime, an asymptotic expression for the achievable throughput over Rayleigh fading channels is also derived. Simulations validate our theoretical results, confirming that VCC delivers substantial performance gains over optimized cacheless MU-MIMO systems. For example, with 32 transmit antennas and 2 receive antennas per user, VCC yields throughput improvements exceeding 300%. These gains are further amplified under imperfect CSI at the transmitter, where VCC's ability to offload interference mitigation to the receivers ensures robust performance even in the face of degraded CSI quality and elevated acquisition costs. - oai:arXiv.org:2601.14142v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hui Zhao, Petros Elia - - - On the $p$-adic deformation problem for the $K$-theory of semistable schemes - https://arxiv.org/abs/2601.14146 - arXiv:2601.14146v1 Announce Type: new -Abstract: We establish a semistable generalization of the Beilinson-Bloch-Esnault-Kerz fiber square, relating the algebraic K-theory of a semistable scheme to its logarithmic topological cyclic homology. We prove that the obstruction to lifting K-theory classes is governed by the Hyodo-Kato Chern character. This answers the $p$-adic deformation problem for continuous K-theory in the semistable case, extending the work of Antieau-Mathew-Morrow-Nikolaus. As an application, we provide a purely K-theoretic proof of Yamashita's semistable $p$-adic Lefschetz $(1,1)$-theorem. - oai:arXiv.org:2601.14146v1 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Federico Binda, Tommy Lundemo, Alberto Merici, Doosung Park - - - Gradient Flow for Finding E-optimal Designs - https://arxiv.org/abs/2601.14147 - arXiv:2601.14147v1 Announce Type: new -Abstract: We investigate the use of Wasserstein gradient flows for finding an $E$-optimal design for a regression model. Unlike the commonly used $D$- and $L$-optimality criteria, the $E$-criterion finds a design that maximizes the smallest eigenvalue of the information matrix, and so it is a non-differentiable criterion unless the minimum eigenvalue has geometric multiplicity equals to one. Such maximin design problems abound in statistical applications and present unique theoretical and computational challenges. Building on the differential structure of the $2$-Wasserstein space, we derive explicit formulas for the Wasserstein gradient of the $E$-optimality criterion in the simple-eigenvalue case. For higher multiplicities, we propose a Wasserstein steepest ascent direction and show that it can be computed exactly via a semidefinite programming (SDP) relaxation. We develop particle approximations that connect infinite-dimensional flows with finite-dimensional optimization, and provide approximation guarantees for empirical measures. Our framework extends naturally to constrained designs via projected Wasserstein gradient flows. Numerical experiments demonstrate that the proposed methods successfully recover $E$-optimal designs for both linear and nonlinear regression models, with competitive accuracy and scalability compared to existing heuristic approaches. This work highlights the potential of optimal transport-based dynamics as a unifying tool for studying challenging optimal design problems. - oai:arXiv.org:2601.14147v1 - math.OC - stat.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jieling Shi, Kim-Chuan Toh, Xin T. Tong, Weng Kee Wong - - - A Natural Representation of Volumes Yields a Remarkable Affine Consequence - https://arxiv.org/abs/2601.14149 - arXiv:2601.14149v1 Announce Type: new -Abstract: At the beginning of the 20th Century there was a growing interest for the investigation of the action of linear groups on the geometry of surfaces. In that context of ideas, the quest for a connection between curvature and the behaviour of linear groups rose naturally. Pursuing the original thought, we investigate how the geometric meaning of this idea is intimately related to the concept of volume of parallelepiped boxes. We show how the ratio of the Gaussian curvature divided by the fourth power of a certain distance of interest in the geometry of surfaces can be represented as a function of volumes. This geometric description explores the profound meaning of a quantity considered by {\c{T}}i{\c{t}}eica in 1907, in a work that sparked a growing interest in affine differential geometry, as an illustration of Felix Klein's Erlangen Program, in which the quest for geometric invariants was the main point of inquiry. - oai:arXiv.org:2601.14149v1 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Wladimir G. Boskoff, Bogdan D. Suceav\u{a} - - - Achievable Burning Densities of Growing Grids - https://arxiv.org/abs/2601.14151 - arXiv:2601.14151v1 Announce Type: new -Abstract: Graph burning is a discrete-time process on graphs where vertices are sequentially activated and burning vertices cause their neighbours to burn over time. In this work, we focus on a dynamic setting in which the graph grows over time, and at each step we burn vertices in the growing grid $G_n = [-f(n),f(n)]^2$. We investigate the set of achievable burning densities for functions of the form $f(n)=\lceil cn^\alpha\rceil$, where $\alpha \ge 1$ and $c>0$. We show that for $\alpha=1$, the set of achievable densities is $[1/(2c^2),1]$, for $1<\alpha<3/2$, every density in $[0,1]$ is achievable, and for $\alpha=3/2$, the set of achievable densities is $[0,(1+\sqrt{6}c)^{-2}]$. - oai:arXiv.org:2601.14151v1 - math.CO - cs.DM - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Jordan Barrett, Karen Gunderson, JD Nir, Pawel Pralat - - - Poisson-Dirichlet graphons and permutons - https://arxiv.org/abs/2601.14166 - arXiv:2601.14166v1 Announce Type: new -Abstract: We introduce classes of supergraphs and superpermutations with novel universal graphon and permuton limiting objects whose construction involves the two-parameter Poisson-Dirichlet process introduced by Pitman and Yor (1997). We demonstrate the universality of these limiting objects through general invariance principles in a heavy-tailed regime and establish a comprehensive phase diagram for the asymptotic shape of superstructures. - oai:arXiv.org:2601.14166v1 - math.PR - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Benedikt Stufler - - - Heights on toric varieties for singular metrics: Local theory - https://arxiv.org/abs/2601.14167 - arXiv:2601.14167v1 Announce Type: new -Abstract: We show that the (toric) local height of a toric variety with respect to a semipositive torus-invariant singular metric is given by the integral of a concave function over a compact convex set. This generalizes a result of Burgos, Philippon, and Sombra for the case of continuous metrics and answers a question raised by Burgos, Kramer, and K\"uhn in 2016. - oai:arXiv.org:2601.14167v1 - math.AG - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gari Y. Peralta Alvarez - - - The 2-categorical S-matrix of a braided fusion 1-category is a character table - https://arxiv.org/abs/2601.14168 - arXiv:2601.14168v1 Announce Type: new -Abstract: The semisimple module categories over a braided fusion category $\mathcal{C}$ form a connected fusion 2-category $\text{Mod}(\mathcal{C})$. Its Drinfeld center $\mathcal{Z}(\text{Mod}(\mathcal{C}))$ is a braided fusion 2-category. To any braided fusion 2-category, Johnson-Freyd and Reutter arXiv:2105.15167v3 [math.QA] have associated a matrix-valued invariant, the 2-categorical $S$-matrix. In this short note we investigate this matrix of $\mathcal{Z}(\text{Mod}(\mathcal{C}))$ as an invariant for the braided fusion 1-category $\mathcal{C}$ and show that it reduces to the character table of the M\"uger center of $\mathcal{C}$. - oai:arXiv.org:2601.14168v1 - math.QA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Alea Hofstetter, Christoph Schweigert - - - Chaos propagation in genetic algorithms: An optimal transport approach - https://arxiv.org/abs/2601.14169 - arXiv:2601.14169v1 Announce Type: new -Abstract: Genetic algorithms are high-level heuristic optimization methods which enjoy great popularity thanks to their intuitive description, flexibility, and, of course, effectiveness. The optimization procedure is based on the evolution of possible solutions following three mechanisms: selection, mutation, and crossover. In this paper, we look at the algorithm as an interacting particle system and show that it is described by a Boltzmann-type equation in the many particles limit. Specifically, we prove a propagation of chaos result with a novel technique that leverages the optimal transport formulation of the Kantorovich-Rubinstein norm and naturally incorporates the crossover mechanism into the analysis. The convergence rate is sharp with respect to the number of particles. - oai:arXiv.org:2601.14169v1 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Giacomo Borghi - - - Wasserstein distances between ERGMs and Erd\H{o}s-R\'enyi models - https://arxiv.org/abs/2601.14170 - arXiv:2601.14170v1 Announce Type: new -Abstract: Ferromagnetic exponential random graph models (ERGMs) are random graph models under which the presence of certain small structures (such as triangles) is encouraged; they can be constructed by tilting an Erd\H{o}s--R\'enyi model by the exponential of a particular nonlinear Hamiltonian. These models are mixtures of metastable wells which each behave macroscopically like an Erd\H{o}s--R\'enyi model, exhibiting the same laws of large numbers for subgraph counts [CD13]. However, on the microscopic scale these metastable wells are very different from Erd\H{o}s--R\'enyi models, with the total variation distance between the two measures tending to 1 [MX23]. In this article we clarify this situation by providing a sharp (up to constants) bound on the Hamming-Wasserstein distance between the two models, which is the average number of edges at which they differ, under the coupling which minimizes this average. In particular, we show that this distance is $\Theta(n^{3/2})$, quantifying exactly how these models differ. - An upper bound of this form has appeared in the past [RR19], but this was restricted to the subcritical (high-temperature) regime of parameters. We extend this bound, using a new proof technique, to the supercritical (low-temperature) regime, and prove a matching lower bound which has only previously appeared in the subcritical regime of special cases of ERGMs satisfying a "triangle-free" condition [DF25]. To prove the lower bound in the presence of triangles, we introduce an approximation of the discrete derivative of the Hamiltonian, which controls the dynamical properties of the ERGM, in terms of local counts of triangles and wedges (two-stars) near an edge. This approximation is the main technical and conceptual contribution of the article, and we expect it will be useful in a variety of other contexts as well. Along the way, we also prove a bound on the marginal edge probability under the ERGM via a new bootstrapping argument. Such a bound has already appeared [FLSW25], but again only in the subcritical regime and using a different proof strategy. - oai:arXiv.org:2601.14170v1 - math.PR - cond-mat.stat-mech - cs.DM - math.CO - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Vilas Winstein - - - Wavelet-Packet Content for Positive Operators - https://arxiv.org/abs/2601.14174 - arXiv:2601.14174v1 Announce Type: new -Abstract: We give a simple way to attach ``content" to the nodes of a wavelet packet tree when a positive operator is given. At a fixed packet depth, the packet projections split the operator into positive pieces, and this decomposition induces a boundary measure on the packet path space, together with vector-dependent densities that show how energy is distributed across the tree. We then study a sequential extraction procedure and two depth-fixed greedy rules for choosing packet blocks, one based on trace weights and one based on Hilbert-Schmidt weights. The main results are explicit geometric decay estimates for the remainder under these greedy removals. In the Hilbert-Schmidt case we also isolate a coherence quantity that measures how close the operator is to being block-diagonal in the packet partition. We close with a concrete patch-based denoising procedure for images, where packet blocks are selected by these content weights computed from an empirical second-moment operator; the construction ensures that both the approximants and the remainders stay positive at every step. - oai:arXiv.org:2601.14174v1 - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Myung-Sin Song, James F. Tian - - - Quantum mixing on large Schreier graphs - https://arxiv.org/abs/2601.14182 - arXiv:2601.14182v1 Announce Type: new -Abstract: Quantum ergodicity describes the delocalization of most eigenfunctions of Laplace-type operators on graphs or manifolds exhibiting chaotic classical dynamics. Quantum mixing is a stronger notion, additionally controlling correlations between eigenfunctions at different energy levels. In this work, we study families of finite Schreier graphs that converge to an infinite Cayley graph and establish quantum mixing under the assumption that the limiting Cayley graph has absolutely continuous spectrum. The convergence of Schreier graphs is understood in the Benjamini-Schramm sense or in the sense of strong convergence in distribution. Our proofs rely on a new approach to quantum ergodicity, based on trace computations, resolvent approximations and representation theory. We illustrate our assumptions on several examples and provide applications to Schreier graphs associated with free products of groups and right-angled Coxeter groups. - oai:arXiv.org:2601.14182v1 - math.SP - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Charles Bordenave, Cyril Letrouit, Mostafa Sabri - - - The nonlinear Steklov problem in outward cuspidal domains - https://arxiv.org/abs/2601.14186 - arXiv:2601.14186v1 Announce Type: new -Abstract: In this article, we consider the nonlinear Steklov eigenvalue problem in outward cuspidal domains. Using the compactness of the weighted trace embedding we obtain the variational characterization of the first non-trivial eigenvalue and prove the existence of a corresponding weak solution. - oai:arXiv.org:2601.14186v1 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pier Domenico Lamberti, Alexander Ukhlov - - - From big q-Jacobi and Chebyshev polynomials to exponential-reproducing subdivision: new identities - https://arxiv.org/abs/2601.14189 - arXiv:2601.14189v1 Announce Type: new -Abstract: In this paper we derive new identities satisfied by Chebyshev polynomials of the first kind and big q-Jacobi polynomials. An immediate benefit of the derived identities is the achievement of closed-form expressions for the Laurent polynomials that identify minimum-support interpolating subdivision schemes reproducing finite sets of integer powers of exponentials. - oai:arXiv.org:2601.14189v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Leonard Peter Bos, Lucia Romani, Alberto Viscardi - - - Translation invariant curvature measures of convex bodies - https://arxiv.org/abs/2601.14193 - arXiv:2601.14193v1 Announce Type: new -Abstract: In a series of papers, Weil initiated the investigation of translation invariant curvature measures of convex bodies, which include as prime examples Federer's curvature measures. In this paper, we continue this line of research by introducing new tools to study curvature measures. Our main results suggest that the space of curvature measures, which is graded by degree and parity, is highly structured: We conjecture that each graded component has length at most $2$ as a representation of the general linear group, and we prove this in degrees $0$ and $n-2$. Beyond this conjectural picture, our methods yield a characterization of Federer's curvature measures under weaker assumptions. - oai:arXiv.org:2601.14193v1 - math.DG - math.MG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jakob Schuhmacher, Thomas Wannerer - - - Local electrical impedance tomography via projections - https://arxiv.org/abs/2601.14198 - arXiv:2601.14198v1 Announce Type: new -Abstract: This paper introduces a method for approximately eliminating the effect that conductivity changes outside the region of interest have in electrical impedance tomography, allowing to form a local reconstruction in the region of interest only. The method considers the Jacobian matrix of the forward map, i.e., of the map that sends the discretized conductivity to the electrode measurements, at an initial guess for the conductivity. The Jacobian matrix is divided columnwise into two parts: one corresponding to the region of interest and a nuisance Jacobian corresponding to the rest of the domain. The leading idea is to project both the electrode measurements and the forward map onto the orthogonal complement of the span of a number of left-hand singular vectors for a suitably weighted nuisance Jacobian. The weighting can, e.g., account for the element sizes in a finite element discretization or to prior information on the conductivity outside the region of interest. The inverse problem is then solved by considering the projected relation between the measurements and the forward map, only reconstructing the conductivity in the region of interest. The functionality of the method is demonstrated by applying a reconstruction algorithm that combines lagged diffusivity iteration and total variation regularization to experimental data. In particular, data from a head-shaped water tank is considered, with the conductivity change in the region of interest mimicking growth of a hemorrhagic stroke and the changes outside the region of interest imitating physiological variations in the conductivity of the scalp. - oai:arXiv.org:2601.14198v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - A. J\"a\"askel\"ainen, A. Vavilov, J. Toivanen, A. H\"anninen, V. Kolehmainen, N. Hyv\"onen - - - Convergence analysis and a novel Lagrange multiplier partitioned method for fluid-poroelastic interaction - https://arxiv.org/abs/2601.14201 - arXiv:2601.14201v1 Announce Type: new -Abstract: We propose a partitioned method for the monolithic formulation of the Stokes-Biot system that incorporates Lagrange multipliers enforcing the interface conditions. The monolithic system is discretized using finite elements, and we establish convergence of the resulting approximation. A Schur complement based algorithm is developed together with an efficient preconditioner, enabling the fluid and poroelastic structure subproblems to be decoupled and solved independently at each time step. The Lagrange multipliers approximate the interface fluxes and act as Neumann boundary conditions for the subproblems, yielding parallel solution of the Stokes and Biot equations. Numerical experiments demonstrate the effectiveness of the proposed algorithm and validate the theoretical error estimate. - oai:arXiv.org:2601.14201v1 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Amy de Castro, Hyesuk Lee - - - Storage-Rate Trade-off in A-XPIR - https://arxiv.org/abs/2601.14202 - arXiv:2601.14202v1 Announce Type: new -Abstract: We consider the storage problem in an asymmetric $X$-secure private information retrieval (A-XPIR) setting. The A-XPIR setting considers the $X$-secure PIR problem (XPIR) when a given arbitrary set of servers is communicating. We focus on the trade-off region between the average storage at the servers and the average download cost. In the case of $N=4$ servers and two non-overlapping sets of communicating servers with $K=2$ messages, we characterize the achievable region and show that the three main inequalities compared to the no-security case collapse to two inequalities in the asymmetric security case. In the general case, we derive bounds that need to be satisfied for the general achievable region for an arbitrary number of servers and messages. In addition, we provide the storage and retrieval scheme for the case of $N=4$ servers with $K=2$ messages and two non-overlapping sets of communicating servers, such that the messages are not replicated (in the sense of a coded version of each symbol) and at the same time achieve the optimal achievable rate for the case of replication. Finally, we derive the exact capacity for the case of asymmetric security and asymmetric collusion for $N=4$ servers, with the communication links $\{1,2\}$ and $\{3,4\}$, which splits the servers into two groups, i.e., $g=2$, and with the collusion links $\{1,3\}$, $\{2,4\}$, as $C=\frac{1}{3}$. More generally, we derive a capacity result for a certain family of asymmetric collusion and asymmetric security cases. - oai:arXiv.org:2601.14202v1 - cs.IT - cs.CR - cs.NI - eess.SP - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mohamed Nomeir, Sennur Ulukus - - - Tropical Methods for Counting Plane Curves -- Complex, Real and Quadratically Enriched - https://arxiv.org/abs/2601.14216 - arXiv:2601.14216v1 Announce Type: new -Abstract: Since the first famous correspondence theorem by Mikhalkin appeared in 2005, tropical geometry has allowed a parallel treatment of real and complex counting problems. A prime example are the genus 0 Gromov-Witten invariants of the plane which count rational plane curves of degree d satisfying point conditions and their real counterpart, the Welschinger invariants, which both can be determined using tropical methods. Remarkably, the tropical computation of the two types of invariants works entirely in parallel. Recently, quadratically enriched enumerative geometry enables us to combine such real and complex counts under one roof, providing a simultaneous approach which can also be used for counts over other fields. Tropical geometry is a successful tool for the study and computation of such quadratically enriched enumerative invariants, too. In this survey, we provide an overview of tropical methods for plane curve counting problems over the real and complex numbers, and the new quadratically enriched counts. - oai:arXiv.org:2601.14216v1 - math.AG - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Andr\'es Jaramillo Puentes, Hannah Markwig, Sabrina Pauli, Felix R\"ohrle - - - Symmetry Testing in Time Series using Ordinal Patterns: A U-Statistic Approach - https://arxiv.org/abs/2601.14223 - arXiv:2601.14223v1 Announce Type: new -Abstract: We introduce a general framework for testing temporal symmetries in time series based on the distribution of ordinal patterns. While previous approaches have focused on specific forms of asymmetry, such as time reversal, our method provides a unified framework applicable to arbitrary symmetry tests. We establish asymptotic results for the resulting test statistics under a broad class of stationary processes. Comprehensive experiments on both synthetic and real data demonstrate that the proposed test achieves high sensitivity to structural asymmetries while remaining fully data-driven and computationally efficient. - oai:arXiv.org:2601.14223v1 - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Annika Betken, Giorgio Micali, Manuel Ruiz Mar\'in - - - New Topological Restrictions For Spaces With Nonnegative Ricci Curvature - https://arxiv.org/abs/2601.14231 - arXiv:2601.14231v1 Announce Type: new -Abstract: We obtain new topological restrictions for complete Riemannian manifolds with nonnegative Ricci curvature and RCD(0,n) spaces. Our main results are a Betti number rigidity theorem which answers a question open since work of M.-T. Anderson in 1990, and a vanishing theorem for the simplicial volume generalizing a theorem of M. Gromov from 1982. Combining such results we obtain a new proof of the classification of noncompact 3-manifolds with nonnegative Ricci curvature, originally due to G. Liu in 2011, which extends to the synthetic setting. - oai:arXiv.org:2601.14231v1 - math.DG - math.GT - math.MG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Alessandro Cucinotta, Mattia Magnabosco, Daniele Semola - - - Stabilizer-Assisted Inactivation Decoding of Quantum Error-Correcting Codes with Erasures - https://arxiv.org/abs/2601.14236 - arXiv:2601.14236v1 Announce Type: new -Abstract: In this work, we develop a reduced complexity maximum likelihood (ML) decoder for quantum low-density parity-check (QLDPC) codes over erasures. Our decoder combines classical inactivation decoding, which integrates peeling with symbolic guessing, with a new dual peeling procedure. In the dual peeling stage, we perform row operations on the stabilizer matrix to efficiently reveal stabilizer generators and their linear combinations whose support lies entirely on the erased set. Each such stabilizer identified allows us to freely fix a bit in its support without affecting the logical state of the decoded result. This removes one degree of freedom that would otherwise require a symbolic guess, reducing the number of inactivated variables and decreasing the size of the final linear system that must be solved. We further show that dual peeling combined with standard peeling alone, without inactivation, is sufficient to achieve ML for erasure decoding of surface codes. Simulations across several QLDPC code families confirm that our decoder matches ML logical failure performance while significantly reducing the complexity of inactivation decoding, including more than a 20% reduction in symbolic guesses for the B1 lifted product code at high erasure rates. - oai:arXiv.org:2601.14236v1 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Giulio Pech, Mert G\"okduman, Hanwen Yao, Henry D. Pfister - - - Partial Linearity in Categories - https://arxiv.org/abs/2601.14237 - arXiv:2601.14237v1 Announce Type: new -Abstract: In this paper we study partial linearity in a category by replacing isomorphism between coproducts and products in a linear category with isomorphism between suitable monoidal structures on a category. The main results a coherence theorem and a generalization of the theory of central morphisms from unital categories to our context of partial linearity - oai:arXiv.org:2601.14237v1 - math.CT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Roy Ferguson, Zurab Janelidze - - - Conformal dimension and its attainment on self-similar Laakso-type fractal spaces - https://arxiv.org/abs/2601.14241 - arXiv:2601.14241v1 Announce Type: new -Abstract: A general construction of Laakso-type fractal spaces was recently introduced by the first two authors. In this paper, we establish a simple condition characterizing when the Ahlfors regular conformal dimension of a symmetric Laakso-type fractal space is attained. The attaining metrics are constructed explicitly. This gives new examples of attainment and clarifies the possible obstructions. - oai:arXiv.org:2601.14241v1 - math.MG - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Riku Anttila, Sylvester Eriksson-Bique, Lassi Rainio - - - Detecting Limit Tori in Non-Smooth Systems: An Analytic Approach with Applications to 3D Piecewise Linear Systems - https://arxiv.org/abs/2601.14247 - arXiv:2601.14247v1 Announce Type: new -Abstract: This work investigates a class of non-autonomous $T$-periodic piecewise smooth differential systems and their associated time-$T$ maps. Our main result provides an analytical approach for detecting, within this class of piecewise differential systems, isolated invariant tori associated with normally hyperbolic invariant closed curves of the time-$T$ map. To achieve this, we derive sufficient conditions under which smooth near-identity maps undergo a Neimark--Sacker bifurcation. As an application of our main result, we present a family of 3D piecewise linear differential systems exhibiting attracting and repelling isolated invariant tori which, moreover, persist under small perturbations. To the best of our knowledge, this family provides the first examples in which limit tori are analytically detected in piecewise linear systems. - oai:arXiv.org:2601.14247v1 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Murilo R. C\^andido, Douglas D. Novaes, Joan S. G. Rivera - - - Identification capacity and rate-query tradeoffs in classification systems - https://arxiv.org/abs/2601.14252 - arXiv:2601.14252v1 Announce Type: new -Abstract: We study a one-shot identification analogue of rate-distortion for discrete classification under three resources: tag rate L (bits of side information stored per entity), identification cost W (attribute-membership queries per identification, excluding global preprocessing and amortized caching), and distortion D (misclassification probability). The question is to characterize achievable triples (L,W,D) when a decoder must recover an entity's class from limited observations. Zero-error barrier. If two distinct classes induce the same attribute profile, then the observation pi(V) is identical for both and no decoder can identify the class from attribute queries alone. Thus, if the profile map pi is not injective on classes, zero-error identification without tags is impossible (a zero-error feasibility threshold). Achievability and converse at D=0. With k classes, nominal tags of L = ceil(log2 k) bits enable O(1) identification cost with D=0. Conversely, any scheme with D=0 must satisfy L >= log2 k bits (tight). Without tags (L=0), identification requires Omega(n) queries in the worst case and may incur D>0. Combinatorial structure. Minimal sufficient query families form the bases of a matroid; the induced distinguishing dimension is well-defined and links to zero-error source coding via graph entropy. We illustrate implications for type systems, databases, and biological taxonomy. All results are mechanized in Lean4 (6000+ lines, 0 sorry). - oai:arXiv.org:2601.14252v1 - cs.IT - cs.PL - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Tristan Simas - - - Post-Hoc Uncertainty Quantification in Pre-Trained Neural Networks via Activation-Level Gaussian Processes - https://arxiv.org/abs/2502.20966 - arXiv:2502.20966v1 Announce Type: cross -Abstract: Uncertainty quantification in neural networks through methods such as Dropout, Bayesian neural networks and Laplace approximations is either prone to underfitting or computationally demanding, rendering these approaches impractical for large-scale datasets. In this work, we address these shortcomings by shifting the focus from uncertainty in the weight space to uncertainty at the activation level, via Gaussian processes. More specifically, we introduce the Gaussian Process Activation function (GAPA) to capture neuron-level uncertainties. Our approach operates in a post-hoc manner, preserving the original mean predictions of the pre-trained neural network and thereby avoiding the underfitting issues commonly encountered in previous methods. We propose two methods. The first, GAPA-Free, employs empirical kernel learning from the training data for the hyperparameters and is highly efficient during training. The second, GAPA-Variational, learns the hyperparameters via gradient descent on the kernels, thus affording greater flexibility. Empirical results demonstrate that GAPA-Variational outperforms the Laplace approximation on most datasets in at least one of the uncertainty quantification metrics. - oai:arXiv.org:2502.20966v1 - stat.ML - cs.LG - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Richard Bergna, Stefan Depeweg, Sergio Calvo Ordonez, Jonathan Plenk, Alvaro Cartea, Jose Miguel Hernandez-Lobato - - - Embeddings and intersections of adelic groups - https://arxiv.org/abs/2510.22408 - arXiv:2510.22408v1 Announce Type: cross -Abstract: We prove the embeddings of adelic groups $\mathbb{A}_I(X, \mathcal{F})\hookrightarrow \mathbb{A}_J(X, \mathcal{F})$ on an excellent scheme $X$ of special type, where $I\subset J$ and $\mathcal{F}$ is a flat quasicoherent sheaf on $X$. We prove the equality $\mathbb{A}_I(X, \mathcal{F})\cap \mathbb{A}_J(X, \mathcal{F}) = \mathbb{A}_{I\setminus 0}(X, \mathcal{F})$ for a normal excellent scheme of special type $X$ and a flat quasicoherent sheaf $\mathcal{F}$ on it, where $I\cap J = I\setminus 0$. We show that the limit of restrictions of global sections of a locally free sheaf on a Cohen-Macaulay projective scheme to power thickenings of integral subschemes is equal to the group of global sections of this sheaf. Using this result, we prove the equality $\mathbb{A}_{\dim X - 1}(X, \mathcal{F})\cap\mathbb{A}_{\dim X}(X, \mathcal{F}) = H^0(X, \mathcal{F})$ for a Cohen-Macaulay projective variety $X$ and a locally free sheaf $\mathcal{F}$ on it. Hence we deduce a theorem on intersections of adelic groups for the case of a normal projective surface. We also compute cohomology groups for a particular case of a curtailed adelic complex. Hence we show that on a three-dimensional regular projective variety over a countable field $X$ there is an equality $\mathbb{A}_I(X, \mathcal{F})\cap \mathbb{A}_J(X, \mathcal{F}) = \mathbb{A}_{I\cap J}(X, \mathcal{F})$ for any $I, J\subset \{0, 1, 2, 3\}$ and any locally free sheaf $\mathcal{F}$ on $X$. - oai:arXiv.org:2510.22408v1 - math.AG - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Dmitry Badulin - - - Phase Space Formulation of S-matrix - https://arxiv.org/abs/2512.23100 - arXiv:2512.23100v1 Announce Type: cross -Abstract: We establish an exact relation between the S-symplectomorphism and the S-matrix by means of the phase space formulation of quantum mechanics. The adjoint action of the S-matrix defines a fuzzy diffeomorphism on phase space whose classical limit is the S-symplectomorphism. The relation between classical and quantum eikonals is immediate via $\hbar$-deformation of each Poisson bracket in the Magnus formula. Diagrammatic computation of quantum eikonal is illustrated for quantizations in both symmetric and normal orderings. - oai:arXiv.org:2512.23100v1 - hep-th - math-ph - math.CO - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Joon-Hwi Kim - - - Tight bounds on recurrence time in closed quantum systems - https://arxiv.org/abs/2601.10409 - arXiv:2601.10409v1 Announce Type: cross -Abstract: The evolution of an isolated quantum system inevitably exhibits recurrence: the state returns to the vicinity of its initial condition after finite time. Despite its fundamental nature, a rigorous quantitative understanding of recurrence has been lacking. We establish upper bounds on the recurrence time, $t_{\mathrm{rec}} \lesssim t_{\mathrm{exit}}(\epsilon)(1/\epsilon)^d$, where $d$ is the Hilbert-space dimension, $\epsilon$ the neighborhood size, and $t_{\mathrm{exit}}(\epsilon)$ the escape time from this neighborhood. For pure states evolving under a Hamiltonian $H$, estimating $t_{\mathrm{exit}}$ is equivalent to an inverse quantum speed limit problem: finding upper bounds on the time a time-evolved state $\psi_t$ needs to depart from the $\epsilon$-vicinity of the initial state $\psi_0$. We provide a partial solution, showing that under mild assumptions $t_{\mathrm{exit}}(\epsilon) \approx \epsilon /\sqrt{ \Delta(H^2)}$, with $\Delta(H^2)$ the Hamiltonian variance in $\psi_0$. We show that our upper bound on $t_{\mathrm{rec}}$ is generically saturated for random Hamiltonians. Finally, we analyze the impact of coherence of the initial state in the eigenbasis of $H$ on recurrence behavior. - oai:arXiv.org:2601.10409v1 - quant-ph - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Marcin Kotowski, Micha{\l} Oszmaniec - - - Umbral theory and the algebra of formal power series - https://arxiv.org/abs/2601.10443 - arXiv:2601.10443v1 Announce Type: cross -Abstract: Umbral theory, formulated in its modern version by S. Roman and G.~C. Rota, has been reconsidered in more recent times by G. Dattoli and collaborators with the aim of devising a working computational tool in the framework of special function theory. Concepts like umbral image and umbral vacuum have been introduced as pivotal elements of the discussion, which, albeit effective, lacks of generality. - This article is directed towards endowing the formalism with a rigorous formulation within the context of the formal power series with complex coefficients $(\mathbb{C}[[ t ]], \partial)$. The new formulation is founded on the definition of the umbral operator $\operatorname{\mathfrak{u}}$ as a functional in the "umbral ground state" subalgebra of analytically convergent formal series $\varphi \in \mathbb{C}\{t\}$. - We consider in detail some specific classes of umbral ground states $\varphi$ and analyse the conditions for analytic convergence of the corresponding umbral identities, defined as formal series resulting from the action on $\varphi$ of operators of the form $f(\zeta \operatorname{\mathfrak{u}}^\mu)$ with $f \in \mathbb{C}\{t\}$ and $\mu, \zeta \in \mathbb{C}$. For these umbral states, we exploit the Gevrey classification of formal power series to establish a connection with the theory of Borel-Laplace resummation, enabling to make rigorous sense of a large class of -- even divergent -- umbral identities. - As an application of the proposed theoretical framework, we introduce and investigate the properties of new umbral images for the Gaussian trigonometric functions, which emphasise the trigonometric-like nature of these functions and enable to define the concept of "Gaussian Fourier transform", a potentially powerful tool for applications. - oai:arXiv.org:2601.10443v1 - math.CA - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Roberto Ricci (ENEA, Nuclear Department NUC-DTT, Frascati Research Center, Via E. Fermi 45, 00044 Frascati RM Italy) - - - A new class of special functions arising in plasma linear susceptibility tensor calculations - https://arxiv.org/abs/2601.11276 - arXiv:2601.11276v1 Announce Type: cross -Abstract: We investigate some fundamental properties of a peculiar class of special functions strictly related to Bessel, Anger and Weber functions, whose introduction was originally motivated by linear susceptibility tensor calculations in a hot, magnetised plasma. We show that these functions are solutions of an inhomogeneous Bessel ODE, with specified initial conditions and a distinct right-hand-side term fulfilling the Nielsen's requirement. Beside deriving recurrence relations and an alternative representation involving incomplete Anger-Weber functions, we show that these functions admit a simple series expansion in terms of Bessel functions of integer order, obtained by resorting to the Jacobi-Anger formula. In plasma applications this eventually leads to expressions involving infinite sums of products of Bessel functions, not particularly apt to numerical evaluation ought to their slow convergence rate when the particle's gyro-radius is larger than the wavelength. By exploiting the previously determined recurrence properties of the new class of functions we present a particularly simple derivation of the linear susceptibility tensor that enables to avoid this inconvenience. - oai:arXiv.org:2601.11276v1 - physics.plasm-ph - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Roberto Ricci - - - From HNSW to Information-Theoretic Binarization: Rethinking the Architecture of Scalable Vector Search - https://arxiv.org/abs/2601.11557 - arXiv:2601.11557v1 Announce Type: cross -Abstract: Modern semantic search and retrieval-augmented generation (RAG) systems rely predominantly on in-memory approximate nearest neighbor (ANN) indexes over high-precision floating-point vectors, resulting in escalating operational cost and inherent trade-offs between latency, throughput, and retrieval accuracy. This paper analyzes the architectural limitations of the dominant "HNSW + float32 + cosine similarity" stack and evaluates existing cost-reduction strategies, including storage disaggregation and lossy vector quantization, which inevitably sacrifice either performance or accuracy. We introduce and empirically evaluate an alternative information-theoretic architecture based on maximally informative binarization (MIB), efficient bitwise distance metrics, and an information-theoretic scoring (ITS) mechanism. Unlike conventional ANN systems, this approach enables exhaustive search over compact binary representations, allowing deterministic retrieval and eliminating accuracy degradation under high query concurrency. Using the MAIR benchmark across 14 datasets and 10,038 queries, we compare this architecture against Elasticsearch, Pinecone, PGVector, and Qdrant. Results demonstrate retrieval quality comparable to full-precision systems, while achieving substantially lower latency and maintaining constant throughput at high request rates. We show that this architectural shift enables a truly serverless, cost-per-query deployment model, challenging the necessity of large in-memory ANN indexes for high-quality semantic search. - oai:arXiv.org:2601.11557v1 - cs.DB - cs.IR - cs.IT - cs.PF - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-sa/4.0/ - Seyed Moein Abtahi, Majid Fekri, Tara Khani, Akramul Azim - - - On Analyzing the Conditions for Stability of Opportunistic Supply Chains Under Network Growth - https://arxiv.org/abs/2601.11566 - arXiv:2601.11566v1 Announce Type: cross -Abstract: Even large firms such as Walmart, Apple, and Coca-Cola face persistent fluctuations in costs, demand, and raw material availability. These are not \textit{rare events} and cannot be evaluated using traditional disruption models focused on infrequent events. Instead, sustained volatility induces opportunistic behavior, as firms repeatedly reconfigure partners in absence of long-term contracts, often due to trust deficits. The resulting web of transient relationships forms opportunistic supply chains (OSCs). To capture OSC evolution, we develop an integrated mathematical framework combining a Geometric Brownian Motion (GBM) model to represent stochastic price volatility, a Bayesian learning model to describe adaptive belief updates regarding partner reliability, and a Latent Order Logistic (LOLOG) network model for endogenous changes in network structure. This framework is implemented in an agent-based simulation to examine how volatility, trust, and network structure jointly shape SC resilience. Our modeling approach identifies critical volatility threshold; a tipping point beyond which the network shifts from a stable, link-preserving regime to a fragmented regime marked by rapid relationship dissolution. We analytically establish monotonic effects of volatility on profitability, trust, and link activation; derive formal stability conditions and volatility-driven phase transitions, and show how these mechanisms shape node importance and procurement behavior. These theoretical mechanisms are illustrated through computational experiments reflecting industry behaviors in fast fashion, electronics, and perishables. Overall, our contribution is to develop an integrated GBM-Bayesian-LOLOG framework to analyze OSC stability and our model can be extended to other OSCs including humanitarian, pharmaceutical, and poultry networks. - oai:arXiv.org:2601.11566v1 - econ.GN - math.OC - q-fin.EC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Gurkirat Wadhwa, Priyank Sinha - - - Privacy-Preserving Black-Box Optimization (PBBO): Theory and the Model-Based Algorithm DFOp - https://arxiv.org/abs/2601.11570 - arXiv:2601.11570v1 Announce Type: cross -Abstract: This paper focuses on solving unconstrained privacy-preserving black-box optimization (PBBO), its corresponding least Frobenius norm updating of quadratic models, and the differentially privacy mechanisms for PBBO. Optimization problems with transformed/encrypted objective functions aim to minimize F(x), which is encrypted/transformed/encrypted to F_k(x) as the output at the k-th iteration. A new derivative-free solver named DFOp, with its implementation, is proposed in this paper, which has a new updating formula for the quadratic model functions. The convergence of DFOp for solving problems with transformed/encrypted objective functions is given. Other analyses, including the new model updating formula and the analysis of the transformation's impact to model functions are presented. We propose two differentially private noise-adding mechanisms for privacy-preserving black-box optimization. Numerical results show that DFOp performs better than compared algorithms. To the best of our knowledge, DFOp is the first derivative-free solver that can solve black-box optimization problems with step-encryption and privacy-preserving black-box problems exactly, which also tries to answer the open question about the combination of derivative-free optimization and privacy. - oai:arXiv.org:2601.11570v1 - cs.CR - cs.NA - math.NA - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pengcheng Xie - - - Level set-based topology optimization of micropolar solids under thermo-mechanical loading - https://arxiv.org/abs/2601.11607 - arXiv:2601.11607v1 Announce Type: cross -Abstract: We propose a novel level set-based topology optimization for micropolar solids subjected to thermo-mechanical loading. To capture the size effects, we have incorporated the microstructural length-scale information into the level set-based topology optimization method by adopting a micropolar theory. The proposed non-local topology optimization method can provide accurate topology optimization for size-dependent solids under thermo-mechanical loading. We have demonstrated the effectiveness of the proposed method through a few representative two-dimensional benchmark problems. The numerical results reveal the substantial influence of underlying micro-structures, incorporated in the model through micropolar parameters, and temperature on topology optimization, highlighting the necessity of the proposed thermo-mechanical micropolar formulation for materials with pronounced non-local effects. For the numerical implementation of the proposed model, we have used open-source finite element libraries, \texttt{Gridap.jl}, and \texttt{GridapTopOpt.jl}, available in Julia, to ensure transparency and reproducibility of the reported computational results. - oai:arXiv.org:2601.11607v1 - physics.comp-ph - cs.NA - math.NA - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mayank Shekhar, Ayyappan Unnikrishna Pillai, Subhayan De, Mohammad Masiur Rahaman - - - Exact Computation of the Catalan Number $C(2,050,572,903)$ - https://arxiv.org/abs/2601.11621 - arXiv:2601.11621v1 Announce Type: cross -Abstract: This paper presents a two-phase algorithm for computing exact Catalan numbers at an unprecedented scale. The method is demonstrated by computing $C(n)$ for $n = 2,050,572,903$ yielding a result with a targeted $1,234,567,890$ decimal digits. To circumvent the memory limitations associated with evaluating large factorials, the algorithm operates exclusively in the prime-exponent domain. Phase 1 employs a parallel segmented sieve to enumerate primes up to $2n$ and applies Legendre's formula to determine the precise prime factorization of $C(n)$. The primes are grouped by exponent and serialized to disk. Phase 2 reconstructs the final integer using a memory-efficient balanced product tree with chunking. The algorithm runs on a time complexity of $\Theta(n(\log n)^2)$ bit-operations and a space complexity of $\Theta(n \log n)$ bits. This result represents the largest exact Catalan number computed to date. Performance statistics for a single-machine execution are reported, and verification strategies -- including modular checks and SHA-256 hash validation -- are discussed. The source code and factorization data are provided to ensure reproducibility. - oai:arXiv.org:2601.11621v1 - cs.DS - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Mahesh Ramani - - - Weyl Mutations in Quiver Yangians - https://arxiv.org/abs/2601.11736 - arXiv:2601.11736v1 Announce Type: cross -Abstract: The problem of solving non-linear equations would be considerably simplified by a possibility to convert known solutions into the new ones. This could seem an element of art, but in the context of ADHM-like equations describing quiver varieties there is a systematic approach. In this note we study moduli spaces and dualities of quiver gauge theories associated to effective dynamics of D-branes compactified on Calabi-Yau resolutions. We concentrate on a subfamily of quivers $\mathfrak{Q}_{\mathfrak{g}}$ covering Dynkin diagrams for simple Lie algebras $\mathfrak{g}$, where the respective BPS algebra is expected to be the Yangian algebra $Y(\mathfrak{g})$. For Yangians labeled by quivers their representations are described by solutions of ADHM-like equations. As quivers substitute Dynkin diagrams a generalization of the Weyl group $\mathcal{W}_{\mathfrak{g}}$ acts on the ADHM solutions. Here we work with the case $\mathfrak{g}=\mathfrak{sl}_{n+1}$ and treat this group as a group of electro-magnetic Seiberg-like dualities (we call them Weyl mutations) on the respective quiver gauge theories. We lift it to the case of higher representations associated to rectangular Young diagrams. An action of Weyl mutations on the BPS Yangian algebra is also discussed. - oai:arXiv.org:2601.11736v1 - hep-th - math-ph - math.AG - math.MP - math.QA - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Dmitry Galakhov, Alexei Gavshin, Alexei Morozov - - - On Nonasymptotic Confidence Intervals for Treatment Effects in Randomized Experiments - https://arxiv.org/abs/2601.11744 - arXiv:2601.11744v1 Announce Type: cross -Abstract: We study nonasymptotic (finite-sample) confidence intervals for treatment effects in randomized experiments. In the existing literature, the effective sample sizes of nonasymptotic confidence intervals tend to be looser than the corresponding central-limit-theorem-based confidence intervals by a factor depending on the square root of the propensity score. We show that this performance gap can be closed, designing nonasymptotic confidence intervals that have the same effective sample size as their asymptotic counterparts. Our approach involves systematic exploitation of negative dependence or variance adaptivity (or both). We also show that the nonasymptotic rates that we achieve are unimprovable in an information-theoretic sense. - oai:arXiv.org:2601.11744v1 - stat.ME - math.ST - stat.AP - stat.ML - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ricardo J. Sandoval, Sivaraman Balakrishnan, Avi Feller, Michael I. Jordan, Ian Waudby-Smith - - - Generalized Shiraishi--Mori construction is exhaustive for ferromagnetic quantum many-body scars - https://arxiv.org/abs/2601.11806 - arXiv:2601.11806v1 Announce Type: cross -Abstract: Quantum many-body scars (QMBS) constitute a subtle violation of ergodicity through a set of non-thermal eigenstates, referred to as scar states, which are embedded in an otherwise thermal spectrum. In a broad class of known examples, these scar states admit a simple interpretation: they are magnon excitations of fixed momentum on top of a ferromagnetic background. In this paper we prove that any Hamiltonian hosting such ``ferromagnetic scar states'' necessarily admits a structural decomposition into a Zeeman term and an ``annihilator'' that annihilates the entire scar manifold. Moreover, we show that this annihilator must itself decompose into a sum of terms built from local projectors that locally annihilate the scar states. This architecture is closely related to the Shiraishi--Mori construction, and our main theorem establishes that an appropriate generalization of that construction is in fact essentially exhaustive for this class of scar states. - oai:arXiv.org:2601.11806v1 - quant-ph - cond-mat.str-el - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Keita Omiya - - - Analysis of a Random Local Search Algorithm for Dominating Set - https://arxiv.org/abs/2601.11841 - arXiv:2601.11841v1 Announce Type: cross -Abstract: Dominating Set is a well-known combinatorial optimization problem which finds application in computational biology or mobile communication. Because of its $\mathrm{NP}$-hardness, one often turns to heuristics for good solutions. Many such heuristics have been empirically tested and perform rather well. However, it is not well understood why their results are so good or even what guarantees they can offer regarding their runtime or the quality of their results. For this, a strong theoretical foundation has to be established. We contribute to this by rigorously analyzing a Random Local Search (RLS) algorithm that aims to find a minimum dominating set on a graph. We consider its performance on cycle graphs with $n$ vertices. We prove an upper bound for the expected runtime until an optimum is found of $\mathcal{O}\left(n^4\log^2(n)\right)$. In doing so, we introduce several models to represent dominating sets on cycles that help us understand how RLS explores the search space to find an optimum. For our proof we use techniques which are already quite popular for the analysis of randomized algorithms. We further apply a special method to analyze a reversible Markov Chain, which arises as a result of our modeling. This method has not yet found wide application in this kind of runtime analysis. - oai:arXiv.org:2601.11841v1 - cs.DS - math.CO - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Hendrik Higl - - - Necessity of Cooperative Transmissions for Wireless MapReduce - https://arxiv.org/abs/2601.11844 - arXiv:2601.11844v1 Announce Type: cross -Abstract: The paper presents an improved upper bound (achievability result) on the optimal tradeoff between Normalized Delivery Time (NDT) and computation load for distributed computing MapReduce systems in certain ranges of the parameters. The upper bound is based on interference alignment combined with zero-forcing. The paper further provides a lower bound (converse) on the optimal NDT-computation tradeoff that can be achieved when IVAs are partitioned into sub-IVAs, and these sub-IVAs are then transmitted (in an arbitrary form) by a single node, without cooperation among nodes. For appropriate linear functions (e.g., XORs), such non-cooperative schemes can achieve some of the best NDT-computation tradeoff points so far obtained in the literature. However, as our lower bound shows, any non-cooperative scheme achieves a worse NDT-computation tradeoff than our new proposed scheme for certain parameters, thus proving the necessity of cooperative schemes like zero-forcing to attain the optimal NDT-computation tradeoff. - oai:arXiv.org:2601.11844v1 - eess.SP - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Yue Bi, Mich\`ele Wigger - - - Least-Squares Multi-Step Koopman Operator Learning for Model Predictive Control - https://arxiv.org/abs/2601.11901 - arXiv:2601.11901v1 Announce Type: cross -Abstract: MPC is widely used in real-time applications, but practical implementations are typically restricted to convex QP formulations to ensure fast and certified execution. Koopman-based MPC enables QP-based control of nonlinear systems by lifting the dynamics to a higher-dimensional linear representation. However, existing approaches rely on single-step EDMD. Consequently, prediction errors may accumulate over long horizons when the EDMD operator is applied recursively. Moreover, the multi-step prediction loss is nonconvex with respect to the single-step EDMD operator, making long-horizon model identification particularly challenging. This paper proposes a multi-step EDMD framework that directly learns the condensed multi-step state-control mapping required for Koopman-MPC, thereby bypassing explicit identification of the lifted system matrices and subsequent model condensation. The resulting identification problem admits a convex least-squares formulation. We further show that the problem decomposes across prediction horizons and state coordinates, enabling parallel computation and row-wise $\ell_1$-regularization for automatic dictionary pruning. A non-asymptotic finite-sample analysis demonstrates that, unlike one-step EDMD, the proposed method avoids error compounding and yields error bounds that depend only on the target multi-step mapping. Numerical examples validate improved long-horizon prediction accuracy and closed-loop performance. - oai:arXiv.org:2601.11901v1 - eess.SY - cs.SY - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Liang Wu, Wallace Gian Yion Tan, Leqi Zhou, Richard D. Braatz, Jan Drgona - - - LIBRA: Language Model Informed Bandit Recourse Algorithm for Personalized Treatment Planning - https://arxiv.org/abs/2601.11905 - arXiv:2601.11905v1 Announce Type: cross -Abstract: We introduce a unified framework that seamlessly integrates algorithmic recourse, contextual bandits, and large language models (LLMs) to support sequential decision-making in high-stakes settings such as personalized medicine. We first introduce the recourse bandit problem, where a decision-maker must select both a treatment action and a feasible, minimal modification to mutable patient features. To address this problem, we develop the Generalized Linear Recourse Bandit (GLRB) algorithm. Building on this foundation, we propose LIBRA, a Language Model-Informed Bandit Recourse Algorithm that strategically combines domain knowledge from LLMs with the statistical rigor of bandit learning. LIBRA offers three key guarantees: (i) a warm-start guarantee, showing that LIBRA significantly reduces initial regret when LLM recommendations are near-optimal; (ii) an LLM-effort guarantee, proving that the algorithm consults the LLM only $O(\log^2 T)$ times, where $T$ is the time horizon, ensuring long-term autonomy; and (iii) a robustness guarantee, showing that LIBRA never performs worse than a pure bandit algorithm even when the LLM is unreliable. We further establish matching lower bounds that characterize the fundamental difficulty of the recourse bandit problem and demonstrate the near-optimality of our algorithms. Experiments on synthetic environments and a real hypertension-management case study confirm that GLRB and LIBRA improve regret, treatment quality, and sample efficiency compared with standard contextual bandits and LLM-only benchmarks. Our results highlight the promise of recourse-aware, LLM-assisted bandit algorithms for trustworthy LLM-bandits collaboration in personalized high-stakes decision-making. - oai:arXiv.org:2601.11905v1 - cs.AI - cs.LG - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Junyu Cao, Ruijiang Gao, Esmaeil Keyvanshokooh, Jianhao Ma - - - A Constraint Programming Model for the Super-Agile Earth Observation Satellite Imaging Scheduling Problem - https://arxiv.org/abs/2601.11967 - arXiv:2601.11967v1 Announce Type: cross -Abstract: As the dependence on satellite imaging continues to grow, modern satellites have become increasingly agile, with the new generation, namely super-agile Earth observation satellites (SAEOS), providing unprecedented imaging flexibility. The highly dynamic capabilities of these satellites introduce additional challenges to the scheduling of observation tasks, as existing approaches for conventional agile satellites do not account for variable observation durations and multiple imaging directions. Although some efforts have been made in this regard, the SAEOS imaging scheduling problem (SAEOS-ISP) remains largely unexplored, and no exact approaches have yet been proposed. In this context, this study presents the first exact Constraint Programming formulation for the SAEOS-ISP, considering flexible observation windows, multiple pointing directions and sequence-dependent transition times across multiple satellites. Computational experiments on a newly generated benchmark set demonstrate that the model can be solved efficiently and within very short computational times. Moreover, the results also show that the proposed approach has the potential to achieve higher computational performance compared to the non-exact approaches that are currently considered state-of-the-art. - oai:arXiv.org:2601.11967v1 - eess.SY - cs.CV - cs.SY - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Margarida Caleiras, Samuel Moniz, Paulo Nascimento - - - Parameterized Complexity of Scheduling Problems in Robotic Process Automation - https://arxiv.org/abs/2601.11984 - arXiv:2601.11984v1 Announce Type: cross -Abstract: This paper studies the growing domain of Robotic Process Automation (RPA) problems. Motivated by scheduling problems arising in RPA, we study the parameterized complexity of the single-machine problem $1|\text{prec},r_j,d_j|*$. We focus on parameters naturally linked to RPA systems, including chain-like precedences, the number of distinct processing times, and the structure of the time windows. We show that the problem is W[2]-hard parameterized by the number of chains, even with only two prescribed processing times and two distinct time-window lengths. This hardness remains even for distinct processing times and time windows under prec-consistent time windows. On the positive side, we obtain polynomial-time algorithm when all jobs share a single time-window length and FPT when the processing times, release times and deadlines are chain-uniform. We also show that the problem lies in XP when parameterized by the width of the precedence relation. - oai:arXiv.org:2601.11984v1 - cs.DS - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Michal Dvo\v{r}\'ak, Anton\'in Nov\'ak, P\v{r}emysl \v{S}\r{u}cha, Du\v{s}an Knop, Claire Hanen - - - Observing rurality of a geographical area from road graph geometry -- a qualitative study - https://arxiv.org/abs/2601.12006 - arXiv:2601.12006v1 Announce Type: cross -Abstract: In this paper we analyze the Finnish road network as a graph in order to measure whether the "rurality" or "urbanity" of an area correlates with local geometrical properties of the graph. Our primary motivation is the observation that the road systems in rural areas look similar to hyperbolic graphs, while in large cities they resemble more the Cayley graph of $\mathbb{Z}^2$. We do not aim for a comprehensive analysis, but rather wish to demonstrate that this observation can be measured and analyzed through looking at various "hyperbolicity measures" of randomly sampled geodesic triangles in the road graph. - oai:arXiv.org:2601.12006v1 - physics.soc-ph - math.HO - math.MG - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Rami Luisto - - - Stability of equilibrium points in modified elliptic restricted three-body problem with various perturbation sources - https://arxiv.org/abs/2601.12026 - arXiv:2601.12026v1 Announce Type: cross -Abstract: This study examines the dynamics of the third body in an elliptic restricted three-body problem (ERTBP) framework, taking into account perturbations from radiation pressure, oblateness, and elongation of the primary bodies, as well as disk-like structures. The objectives are to determine the positions and stability of the equilibrium points, asses how these points shift under the influence of perturbations, and evaluate the dependence of their stability on the orbital eccentricity and perturbation parameters. The ERTBP model is modified to include a radiating, oblate primary body and an elongated secondary body modeled as a finite straight segment, alongside perturbations from a surrounding disk. The system's equations of motion are numerically solved using parameters from perturbed and classical cases. Equilibrium positions are computed over a range of eccentricities and perturbation values, and stability is analyzed using linearized equations and eigenvalue methods. In all cases, we have found three collinear ($L_1$, $L_2$, $L_3$) and two non-collinear ($L_4$, $L_5$) equilibrium points solutions. The inclusion of radiations, oblateness, elongation using a finite straight segment, and disk perturbation systematically displaces each equilibrium point from its classical location, with the magnitude and direction of the displacement varying with the perturbation parameter. Stability analysis confirms that the collinear points remain linearly stable under all tested conditions. Meanwhile, non-collinear points are stable under a specific condition. We investigate the stability boundary of these points as a function of orbital eccentricity and we found there is a critical range of eccentricity values within which stability is preserved. - oai:arXiv.org:2601.12026v1 - astro-ph.EP - math.DS - physics.space-ph - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - M. B. Saputra, H. S. Ramadhan, Ibnu N. Huda, Leonardus B. Putra - - - sangkuriang: A pseudo-spectral Python library for Korteweg-de Vries soliton simulation - https://arxiv.org/abs/2601.12029 - arXiv:2601.12029v1 Announce Type: cross -Abstract: The Korteweg-de Vries (KdV) equation serves as a foundational model in nonlinear wave physics, describing the balance between dispersive spreading and nonlinear steepening that gives rise to solitons. This article introduces sangkuriang, an open-source Python library for solving this equation using Fourier pseudo-spectral spatial discretization coupled with adaptive high-order time integration. The implementation leverages just-in-time (JIT) compilation for computational efficiency while maintaining accessibility for instructional purposes. Validation encompasses progressively complex scenarios including isolated soliton propagation, symmetric two-wave configurations, overtaking collisions between waves of differing amplitudes, and three-body interactions. Conservation of the classical invariants is monitored throughout, with deviations remaining small across all test cases. Measured soliton velocities conform closely to theoretical predictions based on the amplitude-velocity relationship characteristic of integrable systems. Complementary diagnostics drawn from information theory and recurrence analysis confirm that computed solutions preserve the regular phase-space structure expected for completely integrable dynamics. The solver outputs data in standard scientific formats compatible with common analysis tools and generates visualizations of spatiotemporal wave evolution. By combining numerical accuracy with practical accessibility on modest computational resources, sangkuriang offers a platform suitable for both classroom demonstrations of nonlinear wave phenomena and exploratory research into soliton dynamics. - oai:arXiv.org:2601.12029v1 - nlin.PS - cs.NA - math.NA - physics.ao-ph - physics.comp-ph - physics.ed-ph - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sandy H. S. Herho, Faruq Khadami, Iwan P. Anwar, Dasapta E. Irawan - - - A method for optimizing the structure of the software and hardware complex of a distributed process control system for large industrial enterprises - https://arxiv.org/abs/2601.12070 - arXiv:2601.12070v1 Announce Type: cross -Abstract: The article proposes a method for optimizing the structure of the software and hardware complex of an automated control system for continuous technological processes for large industrial enterprises. General information is given on the relevance of the problem of choosing the structure of a system built on the basis of serially produced components, a formal description of the optimization problem is given, the criterion and limitations are highlighted. A solution method using the metaheuristic algorithm of ant colonies is described. A numerical example of the solution is given, the results of the algorithm are analyzed, and directions for further research are determined. - oai:arXiv.org:2601.12070v1 - eess.SY - cs.SY - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1109/MLSD65526.2025.11220659 - Ruslan Zakirzyanov - - - Peres-type Criterion of Einstein-Podolsky-Rosen Steering for Two Qubits - https://arxiv.org/abs/2601.12085 - arXiv:2601.12085v1 Announce Type: cross -Abstract: Quantum nonlocality manifests in multipartite systems through entanglement, Bell's nonlocality, and Einstein-Podolsky-Rosen (EPR) steering. While Peres's positive-partial-transpose criterion provides a simple and powerful test for entanglement, a comparably elegant spectral criterion for detecting EPR steering remains an open challenge. In this work, we systematically explore whether a Peres-type criterion can be established for EPR steering in the two-qubit system. Focusing on rank-2 (including rank-1) states and the two-qubit Werner state, we analyze the eigenvalues of their partially transposed density matrices and construct a significant steering criterion based on symmetric combinations of these eigenvalues. We prove that this criterion serves as a necessary and sufficient condition for steerability for the Werner state, all two-qubit pure states, all two-qubit rank-2 states. Furthermore, we validate the criterion for higher-rank states (rank-3 and rank-4) and show that the results align with known steering inequalities. Our findings suggest a more unified framework for detecting quantum nonlocality via partial transposition and open avenues for further theoretical and numerical investigations into steering detection. - oai:arXiv.org:2601.12085v1 - quant-ph - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Yu-Xuan Zhang, Jing-Ling Chen - - - Tensionless spinning string: emergence of world-sheet torsion and covariant density of energy-momentum tensor - https://arxiv.org/abs/2601.12086 - arXiv:2601.12086v1 Announce Type: cross -Abstract: The action of tensionless spinning string invariant under reparametrizions, both local supersymmetry and dilatations, is considered. The density of energy-momentum tensor is constructed and vanishing of its covariant divergence is proved. This result arises from mutual cancellation of the bosonic and fermionic contributions. Differences in the geometry of worldsheets swept by tensionless and tensionfull spinning strings are analyzed. Shown is emergence of covariant trace of a torsion tensor on w-s of the tensionless spinning string. It is derived from the condition for the fermionic scalar density to be a composite one including the 2-dim. w-s density simulating the 4-dim. Rarita-Schwinger field. The said condition is accompanied with the Noether condition for covariant divergence of the vector metric density to vanish. - oai:arXiv.org:2601.12086v1 - hep-th - gr-qc - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - A. A/ Zheltukhin - - - Solvability of The Output Corridor Control Problem by Pulse-Modulated Feedback - https://arxiv.org/abs/2601.12210 - arXiv:2601.12210v1 Announce Type: cross -Abstract: The problem of maintaining the output of a positive time-invariant single-input single-output system within a predefined corridor of values is treated. For third-order plants possessing a certain structure, it is proven that the problem is always solvable under stationary conditions by means of pulse-modulated feedback. The obtained result is utilized to assess the feasibility of patient-specific pharmacokinetic-pharmacodynamic models with respect to patient safety. A population of Wiener models capturing the dynamics of a neuromuscular blockade agent is studied to investigate whether or not they can be driven into the desired output corridor by clinically acceptable sequential drug doses (boluses). It is demonstrated that low values of a parameter in the nonlinear pharmacodynamic part lie behind the detected model infeasibility. - oai:arXiv.org:2601.12210v1 - eess.SY - cs.SY - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Alexander Medvedev, Anton V. Proskurnikov - - - One-Sided Matrix Completion from Ultra-Sparse Samples - https://arxiv.org/abs/2601.12213 - arXiv:2601.12213v1 Announce Type: cross -Abstract: Matrix completion is a classical problem that has received recurring interest across a wide range of fields. In this paper, we revisit this problem in an ultra-sparse sampling regime, where each entry of an unknown, $n\times d$ matrix $M$ (with $n \ge d$) is observed independently with probability $p = C / d$, for a fixed integer $C \ge 2$. This setting is motivated by applications involving large, sparse panel datasets, where the number of rows far exceeds the number of columns. When each row contains only $C$ entries -- fewer than the rank of $M$ -- accurate imputation of $M$ is impossible. Instead, we estimate the row span of $M$ or the averaged second-moment matrix $T = M^{\top} M / n$. - The empirical second-moment matrix computed from observed entries exhibits non-random and sparse missingness. We propose an unbiased estimator that normalizes each nonzero entry of the second moment by its observed frequency, followed by gradient descent to impute the missing entries of $T$. The normalization divides a weighted sum of $n$ binomial random variables by the total number of ones. We show that the estimator is unbiased for any $p$ and enjoys low variance. When the row vectors of $M$ are drawn uniformly from a rank-$r$ factor model satisfying an incoherence condition, we prove that if $n \ge O({d r^5 \epsilon^{-2} C^{-2} \log d})$, any local minimum of the gradient-descent objective is approximately global and recovers $T$ with error at most $\epsilon^2$. - Experiments on both synthetic and real-world data validate our approach. On three MovieLens datasets, our algorithm reduces bias by $88\%$ relative to baseline estimators. We also empirically validate the linear sampling complexity of $n$ relative to $d$ on synthetic data. On an Amazon reviews dataset with sparsity $10^{-7}$, our method reduces the recovery error of $T$ by $59\%$ and $M$ by $38\%$ compared to baseline methods. - oai:arXiv.org:2601.12213v1 - cs.LG - math.OC - stat.ML - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Trans. Mach. Learn. Res. 2026 - Hongyang R. Zhang, Zhenshuo Zhang, Huy L. Nguyen, Guanghui Lan - - - Rotational flow underlying coupled surface and internal waves. I: Eulerian perspective - https://arxiv.org/abs/2601.12229 - arXiv:2601.12229v1 Announce Type: cross -Abstract: In this paper we examine the flow generated by coupled surface and internal small-amplitude water waves in a two-fluid layer model, where we take the upper layer to be rotational (constant vorticity) and the lower layer to be irrotational. The presence of vorticity greatly complicates the underlying analysis, yet it generates a rich array of otherwise unobservable phenomena such as the presence of critical layers, and stagnation points, in the fluid interior. We employ a phase-plane analysis to elucidate the qualitative behaviour of streamlines for a range of different coupled-wave, and vorticity, regimes. Although the water waves considered are linear in the fluid dynamics sense, the dynamical systems which govern their motion are nonlinear. - oai:arXiv.org:2601.12229v1 - physics.flu-dyn - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - 10.1016/j.jde.2025.113871 - Journal of Differential Equations, Volume 453, Part 4, (2026) 113871 - David Henry, Rossen I. Ivanov, Zisis N. Sakellaris - - - On the Provable Suboptimality of Momentum SGD in Nonstationary Stochastic Optimization - https://arxiv.org/abs/2601.12238 - arXiv:2601.12238v1 Announce Type: cross -Abstract: While momentum-based acceleration has been studied extensively in deterministic optimization problems, its behavior in nonstationary environments -- where the data distribution and optimal parameters drift over time -- remains underexplored. We analyze the tracking performance of Stochastic Gradient Descent (SGD) and its momentum variants (Polyak heavy-ball and Nesterov) under uniform strong convexity and smoothness in varying stepsize regimes. We derive finite-time bounds in expectation and with high probability for the tracking error, establishing a sharp decomposition into three components: a transient initialization term, a noise-induced variance term, and a drift-induced tracking lag. Crucially, our analysis uncovers a fundamental trade-off: while momentum can suppress gradient noise, it incurs an explicit penalty on the tracking capability. We show that momentum can substantially amplify drift-induced tracking error, with amplification that becomes unbounded as the momentum parameter approaches one, formalizing the intuition that using 'stale' gradients hinders adaptation to rapid regime shifts. Complementing these upper bounds, we establish minimax lower bounds for dynamic regret under gradient-variation constraints. These lower bounds prove that the inertia-induced penalty is not an artifact of analysis but an information-theoretic barrier: in drift-dominated regimes, momentum creates an unavoidable 'inertia window' that fundamentally degrades performance. Collectively, these results provide a definitive theoretical grounding for the empirical instability of momentum in dynamic environments and delineate the precise regime boundaries where SGD provably outperforms its accelerated counterparts. - oai:arXiv.org:2601.12238v1 - stat.ML - cs.LG - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Sharan Sahu, Cameron J. Hogan, Martin T. Wells - - - DeepRAHT: Learning Predictive RAHT for Point Cloud Attribute Compression - https://arxiv.org/abs/2601.12255 - arXiv:2601.12255v1 Announce Type: cross -Abstract: Regional Adaptive Hierarchical Transform (RAHT) is an effective point cloud attribute compression (PCAC) method. However, its application in deep learning lacks research. In this paper, we propose an end-to-end RAHT framework for lossy PCAC based on the sparse tensor, called DeepRAHT. The RAHT transform is performed within the learning reconstruction process, without requiring manual RAHT for preprocessing. We also introduce the predictive RAHT to reduce bitrates and design a learning-based prediction model to enhance performance. Moreover, we devise a bitrate proxy that applies run-length coding to entropy model, achieving seamless variable-rate coding and improving robustness. DeepRAHT is a reversible and distortion-controllable framework, ensuring its lower bound performance and offering significant application potential. The experiments demonstrate that DeepRAHT is a high-performance, faster, and more robust solution than the baseline methods. Project Page: https://github.com/zb12138/DeepRAHT. - oai:arXiv.org:2601.12255v1 - eess.IV - cs.CV - cs.IT - cs.MM - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chunyang Fu, Tai Qin, Shiqi Wang, Zhu Li - - - Opportunistic Scheduling for Optimal Spot Instance Savings in the Cloud - https://arxiv.org/abs/2601.12266 - arXiv:2601.12266v1 Announce Type: cross -Abstract: We study the problem of scheduling delay-sensitive jobs over spot and on-demand cloud instances to minimize average cost while meeting an average delay constraint. Jobs arrive as a general stochastic process, and incur different costs based on the instance type. This work provides the first analytical treatment of this problem using tools from queuing theory, stochastic processes, and optimization. We derive cost expressions for general policies, prove queue length one is optimal for low target delays, and characterize the optimal wait-time distribution. For high target delays, we identify a knapsack structure and design a scheduling policy that exploits it. An adaptive algorithm is proposed to fully utilize the allowed delay, and empirical results confirm its near-optimality. - oai:arXiv.org:2601.12266v1 - cs.DC - cs.NI - cs.PF - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Neelkamal Bhuyan, Randeep Bhatia, Murali Kodialam, TV Lakshman - - - Analyzing Collection Strategies: A Computational Perspective on the Coupon Collector Problem - https://arxiv.org/abs/2601.12351 - arXiv:2601.12351v1 Announce Type: cross -Abstract: The Coupon Collector Problem (CCP) is a well-known combinatorial problem that seeks to estimate the number of random draws required to complete a collection of $n$ distinct coupon types. Various generalizations of this problem have been applied in numerous engineering domains. However, practical applications are often hindered by the computational challenges associated with deriving numerical results for moments and distributions. In this work, we present three algorithms for solving the most general form of the CCP, where coupons are collected under any arbitrary drawing probability, with the objective of obtaining $t$ copies of a subset of $k$ coupons from a total of $n$. The First algorithm provides the base model to compute the expectation, variance, and the second moment of the collection process. The second algorithm utilizes the construction of the base model and computes the same values in polynomial time with respect to $n$ under the uniform drawing distribution, and the third algorithm extends to any general drawing distribution. All algorithms leverage Markov models specifically designed to address computational challenges, ensuring exact computation of the expectation and variance of the collection process. Their implementation uses a dynamic programming approach that follows from the Markov models framework, and their time complexity is analyzed accordingly. - oai:arXiv.org:2601.12351v1 - cs.DS - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hadas Abraham, Ido Feldman, Eitan Yaakobi - - - Topological quantum color code model on infinite lattice - https://arxiv.org/abs/2601.12409 - arXiv:2601.12409v1 Announce Type: cross -Abstract: The color code model is a crucial instance of a Calderbank--Shor--Steane (CSS)-type topological quantum error-correcting code, which notably supports transversal implementation of the full Clifford group. Its robustness against local noise is rooted in the structure of its topological excitations. From the perspective of quantum phases of matter, it is essential to understand these excitations in the thermodynamic limit. In this work, we analyze the color code model on an infinite lattice within the quasi-local $C^{*}$-algebra framework, using a cone-localized Doplicher-Haag-Roberts (DHR) analysis. We classify its irreducible anyon superselection sectors and construct explicit string operators that generate anyonic excitations from the ground state. We further examine the fusion and braiding properties of these excitations. Our results show that the topological order of the color code is described by $\mathsf{Rep}(D(\mathbb{Z}_2 \times \mathbb{Z}_2)) \simeq \mathsf{Rep}(D(\mathbb{Z}_2)) \boxtimes \mathsf{Rep}(D(\mathbb{Z}_2))$, which is equivalent to a double layer of the toric code and consistent with established analyses on finite lattices. - oai:arXiv.org:2601.12409v1 - quant-ph - hep-th - math-ph - math.MP - math.QA - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Shiyu Cao, Zhian Jia, Sheng Tan - - - HOT-POT: Optimal Transport for Sparse Stereo Matching - https://arxiv.org/abs/2601.12423 - arXiv:2601.12423v1 Announce Type: cross -Abstract: Stereo vision between images faces a range of challenges, including occlusions, motion, and camera distortions, across applications in autonomous driving, robotics, and face analysis. Due to parameter sensitivity, further complications arise for stereo matching with sparse features, such as facial landmarks. To overcome this ill-posedness and enable unsupervised sparse matching, we consider line constraints of the camera geometry from an optimal transport (OT) viewpoint. Formulating camera-projected points as (half)lines, we propose the use of the classical epipolar distance as well as a 3D ray distance to quantify matching quality. Employing these distances as a cost function of a (partial) OT problem, we arrive at efficiently solvable assignment problems. Moreover, we extend our approach to unsupervised object matching by formulating it as a hierarchical OT problem. The resulting algorithms allow for efficient feature and object matching, as demonstrated in our numerical experiments. Here, we focus on applications in facial analysis, where we aim to match distinct landmarking conventions. - oai:arXiv.org:2601.12423v1 - cs.CV - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Antonin Clerc, Michael Quellmalz, Moritz Piening, Philipp Flotho, Gregor Kornhardt, Gabriele Steidl - - - A Mixture of Experts Vision Transformer for High-Fidelity Surface Code Decoding - https://arxiv.org/abs/2601.12483 - arXiv:2601.12483v1 Announce Type: cross -Abstract: Quantum error correction is a key ingredient for large scale quantum computation, protecting logical information from physical noise by encoding it into many physical qubits. Topological stabilizer codes are particularly appealing due to their geometric locality and practical relevance. In these codes, stabilizer measurements yield a syndrome that must be decoded into a recovery operation, making decoding a central bottleneck for scalable real time operation. Existing decoders are commonly classified into two categories. Classical algorithmic decoders provide strong and well established baselines, but may incur substantial computational overhead at large code distances or under stringent latency constraints. Machine learning based decoders offer fast GPU inference and flexible function approximation, yet many approaches do not explicitly exploit the lattice geometry and local structure of topological codes, which can limit performance. In this work, we propose QuantumSMoE, a quantum vision transformer based decoder that incorporates code structure through plus shaped embeddings and adaptive masking to capture local interactions and lattice connectivity, and improves scalability via a mixture of experts layer with a novel auxiliary loss. Experiments on the toric code demonstrate that QuantumSMoE outperforms state-of-the-art machine learning decoders as well as widely used classical baselines. - oai:arXiv.org:2601.12483v1 - quant-ph - cs.IT - cs.LG - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hoang Viet Nguyen, Manh Hung Nguyen, Hoang Ta, Van Khu Vu, Yeow Meng Chee - - - Semidefinite Programming for Quantum Channel Learning - https://arxiv.org/abs/2601.12502 - arXiv:2601.12502v1 Announce Type: cross -Abstract: The problem of reconstructing a quantum channel from a sample of classical data is considered. When the total fidelity can be represented as a ratio of two quadratic forms (e.g., in the case of mapping a mixed state to a pure state, projective operators, unitary learning, and others), Semidefinite Programming (SDP) can be applied to solve the fidelity optimization problem with respect to the Choi matrix. A remarkable feature of SDP is that the optimization is convex, which allows the problem to be efficiently solved by a variety of numerical algorithms. We have tested several commercially available SDP solvers, all of which allowed for the reconstruction of quantum channels of different forms. A notable feature is that the Kraus rank of the obtained quantum channel typically comprises less than a few percent of its maximal possible value. This suggests that a relatively small Kraus rank quantum channel is typically sufficient to describe experimentally observed classical data. The theory was also applied to the problem of reconstructing projective operators from data. Finally, we discuss a classical computational model based on quantum channel transformation, performed and calculated on a classical computer, possibly hardware-optimized. - oai:arXiv.org:2601.12502v1 - cs.LG - cs.NA - math.NA - quant-ph - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Mikhail Gennadievich Belov, Victor Victorovich Dubov, Vadim Konstantinovich Ivanov, Alexander Yurievich Maslov, Olga Vladimirovna Proshina, Vladislav Gennadievich Malyshkin - - - Rerandomization for quantile treatment effects - https://arxiv.org/abs/2601.12540 - arXiv:2601.12540v1 Announce Type: cross -Abstract: Although complete randomization is widely regarded as the gold standard for causal inference, covariate imbalance can still arise by chance in finite samples. Rerandomization has emerged as an effective tool to improve covariate balance across treatment groups and enhance the precision of causal effect estimation. While existing work focuses on average treatment effects, quantile treatment effects (QTEs) provide a richer characterization of treatment heterogeneity by capturing distributional shifts in outcomes, which is crucial for policy evaluation and equity-oriented research. In this article, we establish the asymptotic properties of the QTE estimator under rerandomization within a finite-population framework, without imposing any distributional or modeling assumptions on the covariates or outcomes.The estimator exhibits a non-Gaussian asymptotic distribution, represented as a linear combination of Gaussian and truncated Gaussian random variables. To facilitate inference, we propose a conservative variance estimator and construct corresponding confidence interval. Our theoretical analysis demonstrates that rerandomization improves efficiency over complete randomization under mild regularity conditions. Simulation studies further support the theoretical findings and illustrate the practical advantages of rerandomization for QTE estimation. - oai:arXiv.org:2601.12540v1 - stat.ME - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Tingxuan Han, Yuhao Wang - - - Quantum Filtering for Squeezed Noise Inputs - https://arxiv.org/abs/2601.12564 - arXiv:2601.12564v1 Announce Type: cross -Abstract: We derive the quantum filter for a quantum open system undergoing quadrature measurements (homodyning) where the input field is in a general quasi-free state. This extends previous work for thermal input noise and allows for squeezed inputs. We introduce a convenient class of Bogoliubov transformations which we refer to as balanced and formulate the quantum stochastic model with squeezed noise as an Araki-Woods type representation. We make an essential use of the Tomita-Takesaki theory to construct the commutant of the C*-algebra describing the inputs and obtain the filtering equations using the quantum reference probability technique. The derived quantum filter must be independent of the choice of representation and this is achieved by fixing an independent quadrature in the commutant algebra. - oai:arXiv.org:2601.12564v1 - quant-ph - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - John Gough, Dylon Rees - - - Partial Identification under Stratified Randomization - https://arxiv.org/abs/2601.12566 - arXiv:2601.12566v1 Announce Type: cross -Abstract: This paper develops a unified framework for partial identification and inference in stratified experiments with attrition, accommodating both equal and heterogeneous treatment shares across strata. For equal-share designs, we apply recent theory for finely stratified experiments to Lee bounds, yielding closed-form, design-consistent variance estimators and properly sized confidence intervals. Simulations show that the conventional formula can overstate uncertainty, while our approach delivers tighter intervals. When treatment shares differ across strata, we propose a new strategy, which combines inverse probability weighting and global trimming to construct valid bounds even when strata are small or unbalanced. We establish identification, introduce a moment estimator, and extend existing inference results to stratified designs with heterogeneous shares, covering a broad class of moment-based estimators which includes the one we formulate. We also generalize our results to designs in which strata are defined solely by observed labels. - oai:arXiv.org:2601.12566v1 - econ.EM - math.ST - stat.ME - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bruno Ferman, Davi Siqueira, Vitor Possebom - - - Explicit Almost-Optimal $\varepsilon$-Balanced Codes via Free Expander Walks - https://arxiv.org/abs/2601.12606 - arXiv:2601.12606v1 Announce Type: cross -Abstract: We study the problem of constructing explicit codes whose rate and distance match the Gilbert-Varshamov bound in the low-rate, high-distance regime. In 2017, Ta-Shma gave an explicit family of codes where every pair of codewords has relative distance $\frac{1-\varepsilon}{2}$, with rate $\Omega(\varepsilon^{2+o(1)})$, matching the Gilbert-Varshamov bound up to a factor of $\varepsilon^{o(1)}$. Ta-Shma's construction was based on starting with a good code and amplifying its bias with walks arising from the $s$-wide-replacement product. - In this work, we give an arguably simpler almost-optimal construction, based on what we call free expander walks: ordinary expander walks where each step is taken on a distinct expander from a carefully chosen sequence. This sequence of expanders is derived from the construction of near-$X$-Ramanujan graphs due to O'Donnell and Wu. - oai:arXiv.org:2601.12606v1 - cs.CC - cs.DM - cs.DS - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jun-Ting Hsieh, Sidhanth Mohanty, Rachel Yun Zhang - - - An efficient numerical method for simulating two-dimensional non-periodic metasurfaces - https://arxiv.org/abs/2601.12674 - arXiv:2601.12674v1 Announce Type: cross -Abstract: Metasurfaces are extremely useful for controlling and manipulating electromagnetic waves. Full-wave numerical simulation is highly desired for their design and optimization, but it is notoriously difficult, even for two-dimensional metasurfaces, when they comprise a huge number of subwavelength elements. This paper focuses on two-dimensional non-periodic metasurfaces that contain only a relatively small number of distinct subwavelength elements. We develop an efficient numerical method based on Neumann-to-Dirichlet operators, the finite element method and local function expansions. Our method drastically reduces the total number of unknowns and is capable of simulating two-dimensional metasurfaces with $10^{5}$ subwavelength elements on a personal computer. Numerical examples demonstrate that the method maintains high accuracy while offering significant advantages in both computational time and memory usage compared to the classical full-domain finite element method, making it particularly suited for the analysis of large metasurfaces. - oai:arXiv.org:2601.12674v1 - physics.optics - cs.NA - math.NA - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Fuhao Liu, Ya Yan Lu - - - An Introduction to Razborov's Flag Algebra as a Proof System for Extremal Graph Theory - https://arxiv.org/abs/2601.12741 - arXiv:2601.12741v1 Announce Type: cross -Abstract: Razborov's flag algebra forms a powerful framework for deriving asymptotic inequalities between induced subgraph densities, underpinning many advances in extremal graph theory. This survey introduces flag algebra to computer scientists working in logic, programming languages, automated verification, and formal methods. We take a logical perspective on flag algebra and present it in terms of syntax, semantics, and proof strategies, in a style closer to formal logic. One popular proof strategy derives valid inequalities by first proving inequalities in a labelled variant of flag algebra and then transferring them to the original unlabelled setting using the so-called downward operator. We explain this strategy in detail and highlight that its transfer mechanism relies on the notion of what we call an adjoint pair, reminiscent of Galois connections and categorical adjunctions, which appear frequently in work on automated verification and programming languages. Along the way, we work through representative examples, including Mantel's theorem and Goodman's bound on Ramsey multiplicity, to illustrate how mathematical arguments can be carried out symbolically in the flag algebra framework. - oai:arXiv.org:2601.12741v1 - cs.PL - cs.LO - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Gyeongwon Jeong, Seonghun Park, Hongseok Yang - - - Approximation Schemes for Sequential Hiring Problems - https://arxiv.org/abs/2601.12750 - arXiv:2601.12750v1 Announce Type: cross -Abstract: The main contribution of this paper resides in providing novel algorithmic advances and analytical insights for the sequential hiring problem, a recently introduced dynamic optimization model where a firm adaptively fills a limited number of positions from a pool of applicants with known values and acceptance probabilities. While earlier research established a strong foundation -- notably an LP-based $(1 - \frac{e^{-k}k^k}{k!})$-approximation by Epstein and Ma (Operations Research, 2024) -- the attainability of superior approximation guarantees has remained a central open question. - Our work addresses this challenge by establishing the first polynomial-time approximation scheme for sequential hiring, proposing an $O(n^{O(1)} \cdot T^{2^{\tilde{O}(1/\epsilon^{2})}})$-time construction of semi-adaptive policies whose expected reward is within factor $1 - \epsilon$ of optimal. To overcome the constant-factor optimality loss inherent to earlier literature, and to circumvent intrinsic representational barriers of adaptive policies, our approach is driven by the following innovations: - -- The block-responsive paradigm: We introduce block-responsive policies, a new class of decision-making strategies, selecting ordered sets (blocks) of applicants rather than single individuals, while still allowing for internal reactivity. - -- Adaptivity and efficiency: We prove that these policies can nearly match the performance of general adaptive policies while utilizing polynomially-sized decision trees. - -- Efficient construction: By developing a recursive enumeration-based framework, we resolve the problematic ``few-positions'' regime, bypassing a fundamental hurdle that hindered previous approaches. - oai:arXiv.org:2601.12750v1 - cs.DS - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Danny Segev, Uri Stein - - - Sensing-Limited Control of Noiseless Linear Systems Under Nonlinear Observations - https://arxiv.org/abs/2601.12782 - arXiv:2601.12782v1 Announce Type: cross -Abstract: This paper investigates the fundamental information-theoretic limits for the control and sensing of noiseless linear dynamical systems subject to a broad class of nonlinear observations. We analyze the interactions between the control and sensing components by characterizing the minimum information flow required for stability. Specifically, we derive necessary conditions for mean-square observability and stabilizability, demonstrating that the average directed information rate from the state to the observations must exceed the intrinsic expansion rate of the unstable dynamics. Furthermore, to address the challenges posed by non-Gaussian distributions inherent to nonlinear observation channels, we establish sufficient conditions by imposing regularity assumptions, specifically log-concavity, on the system's probabilistic components. We show that under these conditions, the divergence of differential entropy implies the convergence of the estimation error, thereby closing the gap between information-theoretic bounds and estimation performance. By establishing these results, we unveil the fundamental performance limits imposed by the sensing layer, extending classical data-rate constraints to the more challenging regime of nonlinear observation models. - oai:arXiv.org:2601.12782v1 - eess.SY - cs.IT - cs.SY - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ming Li, Fan Liu, Yifeng Xiong, Jie Xu, Tao Liu - - - Pollutant-induced changes in fish pigmentation and spatial patterns - https://arxiv.org/abs/2601.12801 - arXiv:2601.12801v1 Announce Type: cross -Abstract: Pigmentation abnormalities, ranging from hypo- to hyperpigmentation, can serve as biomarkers of developmental disruption in fish exposed to environmental contaminants. However, the mechanistic pathways underlying these alterations remain poorly understood. Studies have shown that pattern formation in fish development requires specific pigment cell interactions. Motivated by experimental observations of pigmentation alterations following contaminant exposure, we investigate how pollutants influence pigment cell self-organization using a continuum reaction-diffusion-advection framework. The model incorporates nonlocal Morse-type kernels to describe short- and long-range interactions among melanophores and xanthophores. Our results show that perturbations to the strengths of adhesion or repulsion can drive transitions between stripes, spots, and mixed patterns, reproducing phenotypes characteristic of fish pigmentation mutants. In particular, homotypic interactions are sensitive to contamination, leading to pronounced changes in melanophore density and resulting pigmentation patterns. Time-dependent simulations indicate that pigment changes from early short-term contaminant exposure may be recoverable, whereas prolonged exposure can lead to sustained pigment loss. In a growing fish, contaminant-induced changes in cell-cell interactions directly influence stripe formation rate, stripe number, and pigmentation levels. Overall, our study provides insight into the mechanistic link between experimentally observed pigmentation alterations and the changes in spatial patterns of adult fish. - oai:arXiv.org:2601.12801v1 - q-bio.QM - math.AP - physics.bio-ph - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Pranali Roy Chowdhury, Tian Xu Wang, Abbey MacDonald, Keith B. Tierney, Hao Wang - - - Angular Sensing by Highly Reconfigurable Pixel Antennas with Joint Radiating Aperture and Feeding Ports Reconfiguration - https://arxiv.org/abs/2601.12867 - arXiv:2601.12867v1 Announce Type: cross -Abstract: Angular sensing capability is realized using highly reconfigurable pixel antenna (HRPA) with joint radiating aperture and feeding ports reconfiguration. Pixel antennas represent a general class of reconfigurable antenna designs in which the radiating surface, regardless of its shape or size, is divided into sub-wavelength elements called pixels. Each pixel is connected to its neighboring elements through radio frequency switches. By controlling pixel connections, the pixel antenna topology can be flexibly adjusted so that the resulting radiation pattern can be reconfigured. However, conventional pixel antennas have only a single, fixed-position feeding port, which is not efficient for angular sensing. Therefore, in this work, we further extend the reconfigurability of pixel antennas by introducing the HRPA, which enables both geometry control of the pixel antenna and switching of its feeding ports. The model of the proposed HRPA, including both circuit and radiation parameters, is derived. A codebook is then defined, consisting of pixel connection states and feeding port positions for each sensing area. Based on this codebook, an efficient optimization approach is developed to minimize the Cram\acute{\mathrm{\mathbf{e}}}r-Rao lower bound (CRLB) and obtain the optimal HRPA geometries for angular sensing within a given area. Numerical results show that the HRPA reduces the angle estimation error by more than 50% across the full three-dimensional sphere when compared with a conventional uniform planar array of the same size. This demonstrates the effectiveness of the proposed approach and highlights the potential of HRPA for integrated sensing and communication systems. - oai:arXiv.org:2601.12867v1 - eess.SP - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zixiang Han, Hanning Wang, Shiwen Tang, Yujie Zhang - - - Transverse modulation in electrovac Brinkmann pp-waves: Maxwell consistency and curvature universality - https://arxiv.org/abs/2601.12949 - arXiv:2601.12949v1 Announce Type: cross -Abstract: Electrovac pp--waves in Brinkmann form provide exact Einstein--Maxwell solutions for co--propagating null radiation. Motivated by lensing or scattering, one often ``modulates'' a plane electromagnetic wave by a weak transverse envelope $1+\gamma f(x,y)$. We show that, within the aligned null pp--wave ansatz ($A_v=0$, no $v$--dependence, $F_{xy}=0$) and enforcing the source--free Maxwell equations to $\mathcal O(\gamma)$, a generic profile $f(x,y)$ is incompatible with Maxwell: the transverse field $F_{ui}$ must be both divergence--free and curl--free on the transverse plane, hence $F_{ui}=\partial_i\Phi$ with $\Delta_\perp\Phi=0$. - We give a minimal, polarization--agnostic gauge completion of the modulated potential and prove a cancellation theorem: under standard decay/regularity (or zero--mode) conditions that exclude additional harmonic transverse modes, all $\mathcal O(\gamma)$ dependence on $f$ drops out of $F_{ui}$ and therefore out of the electrovac source $T_{uu}$. Consequently, the electromagnetic contribution to the Brinkmann profile is universal at $\mathcal O(\gamma)$: the familiar cycle--averaged isotropic $r^2$ term plus an isotropic oscillatory correction at frequency $2\omega$, present only for non-circular polarisation. We isolate the residual Maxwell--admissible freedom as harmonic (holomorphic) transverse data and, by Kerr--Schild linearity, superpose an arbitrary co--propagating vacuum gravitational pp--wave, relating TT--gauge strain to Brinkmann amplitudes. Modelling genuinely localised beams, therefore, requires currents, non-null components, or more general Kundt/gyraton geometries. - oai:arXiv.org:2601.12949v1 - gr-qc - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Galin S. Valchev - - - When Is Distributed Nonlinear Aggregation Private? Optimality and Information-Theoretical Bounds - https://arxiv.org/abs/2601.13001 - arXiv:2601.13001v1 Announce Type: cross -Abstract: Nonlinear aggregation is central to modern distributed systems, yet its privacy behavior is far less understood than that of linear aggregation. Unlike linear aggregation where mature mechanisms can often suppress information leakage, nonlinear operators impose inherent structural limits on what privacy guarantees are theoretically achievable when the aggregate must be computed exactly. This paper develops a unified information-theoretic framework to characterize privacy leakage in distributed nonlinear aggregation under a joint adversary that combines passive (honest-but-curious) corruption and eavesdropping over communication channels. - We cover two broad classes of nonlinear aggregates: order-based operators (maximum/minimum and top-$K$) and robust aggregation (median/quantiles and trimmed mean). We first derive fundamental lower bounds on leakage that hold without sacrificing accuracy, thereby identifying the minimum unavoidable information revealed by the computation and the transcript. We then propose simple yet effective privacy-preserving distributed algorithms, and show that with appropriate randomized initialization and parameter choices, our proposed approaches can attach the derived optimal bounds for the considered operators. Extensive experiments validate the tightness of the bounds and demonstrate that network topology and key algorithmic parameters (including the stepsize) govern the observed leakage in line with the theoretical analysis, yielding actionable guidelines for privacy-preserving nonlinear aggregation. - oai:arXiv.org:2601.13001v1 - eess.SP - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Wenrui Yu, Jaron Skovsted Gundersen, Richard Heusdens, Qiongxiu Li - - - On nonlinear self-duality in $4p$ dimensions - https://arxiv.org/abs/2601.13022 - arXiv:2601.13022v1 Announce Type: cross -Abstract: We demonstrate that every model for self-dual nonlinear electrodynamics in four dimensions has a $\mathsf{U}(1)$ duality-invariant extension to $4p>4$ dimensions. - oai:arXiv.org:2601.13022v1 - hep-th - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sergei M. Kuzenko - - - Post-Quantum Secure Aggregation via Code-Based Homomorphic Encryption - https://arxiv.org/abs/2601.13031 - arXiv:2601.13031v1 Announce Type: cross -Abstract: Secure aggregation enables aggregation of inputs from multiple parties without revealing individual contributions to the server or other clients. Existing post-quantum approaches based on homomorphic encryption offer practical efficiency but predominantly rely on lattice-based hardness assumptions. We present a code-based alternative for secure aggregation by instantiating a general framework based on key- and message-additive homomorphic encryption under the Learning Parity with Noise (LPN) assumption. Our construction employs a committee-based decryptor realized via secret sharing and incorporates a Chinese Remainder Theorem (CRT)-based optimization to reduce the communication costs of LPN-based instantiations. We analyze the security of the proposed scheme under a new Hint-LPN assumption and show that it is equivalent to standard LPN for suitable parameters. Finally, we evaluate performance and identify regimes in which our approach outperforms information-theoretically secure aggregation protocols. - oai:arXiv.org:2601.13031v1 - cs.CR - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sebastian Bitzer, Maximilian Egger, Mumin Liu, Antonia Wachter-Zeh - - - Failure of the mean-field Hartree approximation for a bosonic many-body system with non-Hermitian Hamiltonian - https://arxiv.org/abs/2601.13038 - arXiv:2601.13038v1 Announce Type: cross -Abstract: Mean-field Hartree theory is a central tool for reducing interacting many-body dynamics to an effective nonlinear one-particle evolution. This approximation has been employed also when the Hamiltonian that governs the many-body dynamics is not Hermitian. Indeed, non-Hermitian Hamiltonians model particle gain/loss or the evolution of open quantum systems between consecutive quantum jumps. Furthermore, the validity of the Hartree approximation for generic non-Hermitian Hamiltonians lies at the basis of a quantum algorithm for nonlinear differential equations. In this work, we show that this approximation can fail. We analytically solve a model of $N$ bosonic qubits with two-body interactions generated by a purely anti-Hermitian Hamiltonian, determine an analytic expression for the limit for $N\to\infty$ of the one-particle marginal state and show that such a limit does not agree with the solution of the non-Hermitian Hartree evolution equation. We further show that there exists an initial condition such that the exact one-particle marginal state undergoes a finite-time transition to a mixed state, a phenomenon that is completely absent in the case of Hermitian Hamiltonians. Our findings challenge the validity of the mean-field Hartree approximation for non-Hermitian Hamiltonians, and call for additional conditions for the validity of the mean-field regime to model the dynamics of particle gain and loss and the open-system dynamics in bosonic many-body systems. - oai:arXiv.org:2601.13038v1 - quant-ph - cond-mat.stat-mech - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Matias Ginzburg, Giacomo De Palma, Simone Rademacher - - - Approximate full conformal prediction in RKHS - https://arxiv.org/abs/2601.13102 - arXiv:2601.13102v1 Announce Type: cross -Abstract: Full conformal prediction is a framework that implicitly formulates distribution-free confidence prediction regions for a wide range of estimators. However, a classical limitation of the full conformal framework is the computation of the confidence prediction regions, which is usually impossible since it requires training infinitely many estimators (for real-valued prediction for instance). The main purpose of the present work is to describe a generic strategy for designing a tight approximation to the full conformal prediction region that can be efficiently computed. Along with this approximate confidence region, a theoretical quantification of the tightness of this approximation is developed, depending on the smoothness assumptions on the loss and score functions. The new notion of thickness is introduced for quantifying the discrepancy between the approximate confidence region and the full conformal one. - oai:arXiv.org:2601.13102v1 - stat.ML - cs.LG - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Davidson Lova Razafindrakoto, Alain Celisse, J\'er\^ome Lacaille - - - xBound: Join Size Lower Bounds - https://arxiv.org/abs/2601.13117 - arXiv:2601.13117v1 Announce Type: cross -Abstract: Cloud database vendors invest substantial resources into their query optimizers, and for good reason. Cardinality estimation, a cornerstone of the optimizer, is critical for the selection of efficient query plans, as well as downstream tasks such as resource allocation and query scheduling. Yet, as many practitioners and researchers have noted, it is also the optimizer's Achilles heel. Prior studies on a number of industrial-strength databases show substantial cardinality estimation errors on all tested systems, with a far greater tendency to underestimate than to overestimate. Unfortunately, cardinality underestimation is more problematic than overestimation, as it misleads the optimizer to choose plans designed for small data, leading to underprovisioned CPU and memory. - While previous work on pessimistic cardinality estimation has proposed provable join size upper bounds, such methods can only correct overestimation, leaving the more harmful problem of underestimation unaddressed. To fill this critical gap, we introduce xBound, the very first framework for deriving provable join size lower bounds. xBound successfully reduces underestimation in real systems: On the JOBlight benchmark, it corrects 17.5% of subexpression underestimates in DuckDB and 8.7% in PostgreSQL, while on a Microsoft enterprise workload, it fixes 36.1% of Fabric Data Warehouse's underestimates, demonstrating a significant step towards solving this long-standing problem. - oai:arXiv.org:2601.13117v1 - cs.DB - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-sa/4.0/ - Mihail Stoian, Tiemo Bang, Hangdong Zhao, Jes\'us Camacho-Rodr\'iguez, Yuanyuan Tian, Andreas Kipf - - - Global stability of a Hebbian/anti-Hebbian network for principal subspace learning - https://arxiv.org/abs/2601.13170 - arXiv:2601.13170v1 Announce Type: cross -Abstract: Biological neural networks self-organize according to local synaptic modifications to produce stable computations. How modifications at the synaptic level give rise to such computations at the network level remains an open question. Pehlevan et al. [Neur. Comp. 27 (2015), 1461--1495] proposed a model of a self-organizing neural network with Hebbian and anti-Hebbian synaptic updates that implements an algorithm for principal subspace analysis; however, global stability of the nonlinear synaptic dynamics has not been established. Here, for the case that the feedforward and recurrent weights evolve at the same timescale, we prove global stability of the continuum limit of the synaptic dynamics and show that the dynamics evolve in two phases. In the first phase, the synaptic weights converge to an invariant manifold where the `neural filters' are orthonormal. In the second phase, the synaptic dynamics follow the gradient flow of a non-convex potential function whose minima correspond to neural filters that span the principal subspace of the input data. - oai:arXiv.org:2601.13170v1 - q-bio.NC - cs.NE - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - David Lipshutz, Robert J. Lipshutz - - - Decentralized Cooperative Beamforming for BDRIS-Assisted Cell-Free MIMO OFDM Systems - https://arxiv.org/abs/2601.13201 - arXiv:2601.13201v1 Announce Type: cross -Abstract: In this paper, a wideband cell-free multi-stream multi-user Multiple-Input Multiple-Output (MIMO) Orthogonal Frequency Division Multiplexing (OFDM) system is considered operating within a smart wireless environment enabled by multiple Beyond Diagonal Reconfigurable Intelligent Surfaces (BDRISs). A novel decentralized active and passive beamforming framework, robust to imperfect channel state availability and with minimal cooperation among the system's multiple Base Stations (BSs) for deciding the final configurations of the shared BDRISs, is proposed, which aims to substantially reduce the overhead inherent in centralized solutions necessitating a central processing unit of high computational power. By considering a Dynamic Group-Connected (DGC) BDRIS architecture with frequency-selective responses per unit element, we formulate the system's sum-rate maximization problem with respect to the tunable capacitances and permutation matrices of the BDRISs as well as the precoding matrices of the BSs, which is solved via successive concave approximation and alternating projections as well as consensus-based updates for the BDRISs' design. Through extensive simulation results, it is showcased that the proposed robust decentralized cooperative approach with diverse BDRIS architectures outperforms non-cooperation benchmarks. It is also demonstrated that the considered DGC BDRIS architecture is able to provide sum-rate performance gains sufficiently close to the more complex fully-connected BDRIS structure. - oai:arXiv.org:2601.13201v1 - eess.SP - cs.ET - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Konstantinos D. Katsanos, George C. Alexandropoulos - - - Emissions and cost tradeoffs of time-matched clean electricity procurement under inter-annual weather variability: case study of hydrogen production - https://arxiv.org/abs/2601.13202 - arXiv:2601.13202v1 Announce Type: cross -Abstract: Time-matching requirements (TMRs) for clean electricity procurement are increasingly adopted in voluntary corporate sustainability initiatives and regulatory frameworks. While prior research has evaluated cost and emissions impacts of hourly vs. annual TMR, these studies typically rely on single-year weather scenarios that do not capture inter-annual variability in variable renewable energy (VRE) generation. We use a capacity expansion model to assess how inter-annual weather variability affects procurement-driven infrastructure investments, costs, and emissions for a grid-connected hydrogen producer under both annual and hourly time-matching strategies. Using a Texas case study, we compare deterministic (single weather scenario) and stochastic (nine weather scenarios) modeling approaches. Both procurement investments and cost and emissions outcomes are sensitive to weather scenario, with annual matching exhibiting greater sensitivity than hourly matching. Stochastic modeling finds higher cost premiums for hourly versus annual matching compared to deterministic modeling, though emissions trends remain directionally consistent. Demand flexibility through H2 storage is critical for lowering hourly matching cost premiums under weather-driven VRE variability. Partial hourly matching (e.g., 80-90% compliance) can modestly reduce costs while maintaining minimal emissions impacts. Finally, we examine how grid-level renewable portfolio standards (RPS) affect additionality and emissions. When stringent additionality is achieved via binding RPS constraints on non-H2 electricity demand, annual matching can produce emissions reductions comparable to hourly matching at lower cost. - oai:arXiv.org:2601.13202v1 - eess.SY - cs.SY - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Michael Giovanniello, Dharik S. Mallapragada - - - Bihamiltonian tests for integrable systems associated to rank-$1$ F-CohFTs - https://arxiv.org/abs/2601.13203 - arXiv:2601.13203v1 Announce Type: cross -Abstract: Double ramification (DR) hierarchies associated to rank-$1$ F-CohFTs are important integrable perturbations of the Riemann--Hopf hierarchy. In this paper, we perform bihamiltonian tests for these DR hierarchies, and conjecture that the ones that are bihamiltonian form a $2$-parameter family. Remarkably, our computations suggest that there is a $1$-parameter subfamily of the rank-$1$ F-CohFTs, where the corresponding DR hierarchy is conjecturally Miura equivalent to the Camassa--Holm hierarchy. We also prove a conjecture regarding bihamiltonian Hodge hierarchies. Finally, we systematically study Miura invariants, and for another $1$-parameter subfamily propose a conjectural relation to the Degasperis--Procesi hierarchy. - oai:arXiv.org:2601.13203v1 - nlin.SI - math-ph - math.AG - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexandr Buryak, Jianghao Xu, Di Yang - - - Hierarchical Sparse Vector Transmission for Ultra Reliable and Low Latency Communications - https://arxiv.org/abs/2601.13204 - arXiv:2601.13204v1 Announce Type: cross -Abstract: Sparse vector transmission (SVT) is a promising candidate technology for achieving ultra-reliable low-latency communication (URLLC). In this paper, a hierarchical SVT scheme is proposed for multi-user URLLC scenarios. The hierarchical SVT scheme partitions the transmitted bits into common and private parts. The common information is conveyed by the indices of non-zero sections in a sparse vector, while each user's private information is embedded into non-zero blocks with specific block lengths. At the receiver, the common bits are first recovered from the detected non-zero sections, followed by user-specific private bits decoding based on the corresponding non-zero block indices. Simulation results show the proposed scheme outperforms state-of-the-art SVT schemes in terms of block error rate. - oai:arXiv.org:2601.13204v1 - eess.SP - cs.IT - eess.IV - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yanfeng Zhang, Xi'an Fan, Jinkai Zheng, Xiaoye Jing, Weiwei Yang, Xu Zhu - - - Discover the GLM theory on four pages - https://arxiv.org/abs/2601.13237 - arXiv:2601.13237v1 Announce Type: cross -Abstract: The General Lagrangian Mean (GLM) theory uses a version of the averaged equations of fluid dynamics, designed to examine interactions between small-amplitude waves and mean flows. These equations are formulated in coordinates following the fluid's average velocity and are often referred to as `pseudo-Lagrangian'. This paper focuses on the principles for deriving the GLM equations, using an inviscid, incompressible, homogeneous fluid as a demonstration case. Our exposition methodically differs from others and is aimed at the learners of this theory. - oai:arXiv.org:2601.13237v1 - physics.flu-dyn - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/publicdomain/zero/1.0/ - V. A. Vladimirov - - - The table maker's quantum search - https://arxiv.org/abs/2601.13306 - arXiv:2601.13306v1 Announce Type: cross -Abstract: We show that quantum search can be used to compute the hardness to round an elementary function, that is, to determine the minimum working precision required to compute the values of an elementary function correctly rounded to a target precision of $n$ digits for all possible precision-$n$ floating-point inputs in a given interval. For elementary functions $f$ related to the exponential function, quantum search takes time $\tilde O(2^{n/2} \log (1/\delta))$ to return, with probability $1-\delta$, the hardness to round $f$ over all $n$-bit floating-point inputs in a given binade. For periodic elementary functions in large binades, standalone quantum search yields an asymptotic speedup over the best known classical algorithms and heuristics. - oai:arXiv.org:2601.13306v1 - quant-ph - cs.NA - math.NA - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Stefanos Kourtis - - - Renewal theory for Brownian motion across a stochastically gated interface - https://arxiv.org/abs/2601.13414 - arXiv:2601.13414v1 Announce Type: cross -Abstract: Stochastically gated interfaces play an important role in a variety of cellular diffusion processes. Examples include intracellular transport via stochastically gated ion channels and pores in the plasma membrane of a cell, intercellular transport between cells coupled by stochastically gated gap junctions, and oxygen transport in insect respiration. Most studies of stochastically-gated interfaces are based on macroscopic models that track the particle concentration averaged with respect to different realisations of the gate dynamics. In this paper we use renewal theory to develop a probabilistic model of single-particle Brownian motion (BM) through a stochastically gated interface. We proceed by constructing a renewal equation for 1D BM with an interface at the origin, which effectively sews together a sequence of BMs on the half-line with a totally absorbing boundary at $x=0$. Each time the particle is absorbed, the stochastic process is immediately restarted according to the following rule: if the gate is closed then BM restarts on the same side of the interface, whereas if the gate is open then BM restarts on either side of the interface with equal probability. In order to ensure that diffusion restarts in a state that avoids immediate re-absorption. we assume that whenever the particle reaches the interface it is instantaneously shifted a distance $\epsilon$ from the origin. We explicitly solve the renewal equation for $\epsilon>0$ and show how the solution of a corresponding forward Kolmogorov equation is recovered in the limit $\epsilon\rightarrow 0$. However, the renewal equation provides a more general mathematical framework by explicitly separating the first passage time problem of detecting the gated interface (absorption) and the subsequent rule for restarting BM. We conclude by extending the theory to higher-dimensional interfaces. - oai:arXiv.org:2601.13414v1 - cond-mat.stat-mech - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Paul C Bressloff - - - Optimal estimation of generalized causal effects in cluster-randomized trials with multiple outcomes - https://arxiv.org/abs/2601.13428 - arXiv:2601.13428v1 Announce Type: cross -Abstract: Cluster-randomized trials (CRTs) are widely used to evaluate group-level interventions and increasingly collect multiple outcomes capturing complementary dimensions of benefit and risk. Investigators often seek a single global summary of treatment effect, yet existing methods largely focus on single-outcome estimands or rely on model-based procedures with unclear causal interpretation or limited robustness. We develop a unified potential outcomes framework for generalized treatment effects with multiple outcomes in CRTs, accommodating both non-prioritized and prioritized outcome settings. The proposed cluster-pair and individual-pair causal estimands are defined through flexible pairwise contrast functions and explicitly account for potentially informative cluster sizes. We establish nonparametric estimation via weighted clustered U-statistics and derive efficient influence functions to construct covariate-adjusted estimators that integrate debiased machine learning with U-statistics. The resulting estimators are consistent and asymptotically normal, attain the semiparametric efficiency bounds under mild regularity conditions, and have analytically tractable variance estimators that are proven to be consistent under cross-fitting. Simulations and an application to a CRT for chronic pain management illustrate the practical utility of the proposed methods. - oai:arXiv.org:2601.13428v1 - stat.ME - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Xinyuan Chen, Fan Li - - - Distribution-Free Confidence Ellipsoids for Ridge Regression with PAC Bounds - https://arxiv.org/abs/2601.13436 - arXiv:2601.13436v1 Announce Type: cross -Abstract: Linearly parametrized models are widely used in control and signal processing, with the least-squares (LS) estimate being the archetypical solution. When the input is insufficiently exciting, the LS problem may be unsolvable or numerically unstable. This issue can be resolved through regularization, typically with ridge regression. Although regularized estimators reduce the variance error, it remains important to quantify their estimation uncertainty. A possible approach for linear regression is to construct confidence ellipsoids with the Sign-Perturbed Sums (SPS) ellipsoidal outer approximation (EOA) algorithm. The SPS EOA builds non-asymptotic confidence ellipsoids under the assumption that the noises are independent and symmetric about zero. This paper introduces an extension of the SPS EOA algorithm to ridge regression, and derives probably approximately correct (PAC) upper bounds for the resulting region sizes. Compared with previous analyses, our result explicitly show how the regularization parameter affects the region sizes, and provide tighter bounds under weaker excitation assumptions. Finally, the practical effect of regularization is also demonstrated via simulation experiments. - oai:arXiv.org:2601.13436v1 - stat.ML - cs.LG - cs.SY - eess.SP - eess.SY - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Szabolcs Szentp\'eteri, Bal\'azs Csan\'ad Cs\'aji - - - Fairness-informed Pareto Optimization : An Efficient Bilevel Framework - https://arxiv.org/abs/2601.13448 - arXiv:2601.13448v1 Announce Type: cross -Abstract: Despite their promise, fair machine learning methods often yield Pareto-inefficient models, in which the performance of certain groups can be improved without degrading that of others. This issue arises frequently in traditional in-processing approaches such as fairness-through-regularization. In contrast, existing Pareto-efficient approaches are biased towards a certain perspective on fairness and fail to adapt to the broad range of fairness metrics studied in the literature. In this paper, we present BADR, a simple framework to recover the optimal Pareto-efficient model for any fairness metric. Our framework recovers its models through a Bilevel Adaptive Rescalarisation procedure. The lower level is a weighted empirical risk minimization task where the weights are a convex combination of the groups, while the upper level optimizes the chosen fairness objective. We equip our framework with two novel large-scale, single-loop algorithms, BADR-GD and BADR-SGD, and establish their convergence guarantees. We release badr, an open-source Python toolbox implementing our framework for a variety of learning tasks and fairness metrics. Finally, we conduct extensive numerical experiments demonstrating the advantages of BADR over existing Pareto-efficient approaches to fairness. - oai:arXiv.org:2601.13448v1 - cs.LG - math.OC - stat.ML - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Sofiane Tanji, Samuel Vaiter, Yassine Laguel - - - Labels or Preferences? Budget-Constrained Learning with Human Judgments over AI-Generated Outputs - https://arxiv.org/abs/2601.13458 - arXiv:2601.13458v1 Announce Type: cross -Abstract: The increasing reliance on human preference feedback to judge AI-generated pseudo labels has created a pressing need for principled, budget-conscious data acquisition strategies. We address the crucial question of how to optimally allocate a fixed annotation budget between ground-truth labels and pairwise preferences in AI. Our solution, grounded in semi-parametric inference, casts the budget allocation problem as a monotone missing data framework. Building on this formulation, we introduce Preference-Calibrated Active Learning (PCAL), a novel method that learns the optimal data acquisition strategy and develops a statistically efficient estimator for functionals of the data distribution. Theoretically, we prove the asymptotic optimality of our PCAL estimator and establish a key robustness guarantee that ensures robust performance even with poorly estimated nuisance models. Our flexible framework applies to a general class of problems, by directly optimizing the estimator's variance instead of requiring a closed-form solution. This work provides a principled and statistically efficient approach for budget-constrained learning in modern AI. Simulations and real-data analysis demonstrate the practical benefits and superior performance of our proposed method. - oai:arXiv.org:2601.13458v1 - stat.ML - cs.AI - cs.LG - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zihan Dong, Ruijia Wu, Linjun Zhang - - - Quantum Entanglement, Stratified Spaces, and Topological Matter: Towards an Entanglement-Sensitive Langlands Correspondence - https://arxiv.org/abs/2601.13467 - arXiv:2601.13467v1 Announce Type: cross -Abstract: Recently, quantum entanglement has been presented as a cohomological obstruction to reconstructing a global quantum state from locally compatible information, where sheafification provides a functor that is forgetful with regards to global-from-local signatures while acting faithfully with respect to within-patch multipartite structures. Nontrivial connections to Hecke modifications and the geometric Langlands program are explored in the process. The aim of this work is to validate and extend a number of the claims made in [arXiv:2511.04326] through both theoretical analysis and numerical simulations, employing concrete perspectives from condensed matter physics. - oai:arXiv.org:2601.13467v1 - quant-ph - cond-mat.mes-hall - math-ph - math.MP - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kazuki Ikeda, Steven Rayan - - - Resampling-free Inference for Time Series via RKHS Embedding - https://arxiv.org/abs/2601.13468 - arXiv:2601.13468v1 Announce Type: cross -Abstract: In this article, we study nonparametric inference problems in the context of multivariate or functional time series, including testing for goodness-of-fit, the presence of a change point in the marginal distribution, and the independence of two time series, among others. Most methodologies available in the existing literature address these problems by employing a bandwidth-dependent bootstrap or subsampling approach, which can be computationally expensive and/or sensitive to the choice of bandwidth. To address these limitations, we propose a novel class of kernel-based tests by embedding the data into a reproducing kernel Hilbert space, and construct test statistics using sample splitting, projection, and self-normalization (SN) techniques. Through a new conditioning technique, we demonstrate that our test statistics have pivotal limiting null distributions under absolute regularity and mild moment assumptions. We also analyze the limiting power of our tests under local alternatives. Finally, we showcase the superior size accuracy and computational efficiency of our methods as compared to some existing ones. - oai:arXiv.org:2601.13468v1 - stat.ME - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Deep Ghoshal, Xiaofeng Shao - - - Preconditioning Benefits of Spectral Orthogonalization in Muon - https://arxiv.org/abs/2601.13474 - arXiv:2601.13474v1 Announce Type: cross -Abstract: The Muon optimizer, a matrix-structured algorithm that leverages spectral orthogonalization of gradients, is a milestone in the pretraining of large language models. However, the underlying mechanisms of Muon -- particularly the role of gradient orthogonalization -- remain poorly understood, with very few works providing end-to-end analyses that rigorously explain its advantages in concrete applications. We take a step by studying the effectiveness of a simplified variant of Muon through two case studies: matrix factorization, and in-context learning of linear transformers. For both problems, we prove that simplified Muon converges linearly with iteration complexities independent of the relevant condition number, provably outperforming gradient descent and Adam. Our analysis reveals that the Muon dynamics decouple into a collection of independent scalar sequences in the spectral domain, each exhibiting similar convergence behavior. Our theory formalizes the preconditioning effect induced by spectral orthogonalization, offering insight into Muon's effectiveness in these matrix optimization problems and potentially beyond. - oai:arXiv.org:2601.13474v1 - cs.LG - cs.AI - math.OC - stat.ML - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jianhao Ma, Yu Huang, Yuejie Chi, Yuxin Chen - - - Small Gradient Norm Regret for Online Convex Optimization - https://arxiv.org/abs/2601.13519 - arXiv:2601.13519v1 Announce Type: cross -Abstract: This paper introduces a new problem-dependent regret measure for online convex optimization with smooth losses. The notion, which we call the $G^\star$ regret, depends on the cumulative squared gradient norm evaluated at the decision in hindsight $\sum_{t=1}^T \|\nabla \ell(x^\star)\|^2$. We show that the $G^\star$ regret strictly refines the existing $L^\star$ (small loss) regret, and that it can be arbitrarily sharper when the losses have vanishing curvature around the hindsight decision. We establish upper and lower bounds on the $G^\star$ regret and extend our results to dynamic regret and bandit settings. As a byproduct, we refine the existing convergence analysis of stochastic optimization algorithms in the interpolation regime. Some experiments validate our theoretical findings. - oai:arXiv.org:2601.13519v1 - stat.ML - cs.LG - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Wenzhi Gao, Chang He, Madeleine Udell - - - StoTAM: Stochastic Alternating Minimization for Tucker-Structured Tensor Sensing - https://arxiv.org/abs/2601.13522 - arXiv:2601.13522v1 Announce Type: cross -Abstract: Low-rank tensor sensing is a fundamental problem with broad applications in signal processing and machine learning. Among various tensor models, low-Tucker-rank tensors are particularly attractive for capturing multi-mode subspace structures in high-dimensional data. Existing recovery methods either operate on the full tensor variable with expensive tensor projections, or adopt factorized formulations that still rely on full-gradient computations, while most stochastic factorized approaches are restricted to tensor decomposition settings. In this work, we propose a stochastic alternating minimization algorithm that operates directly on the core tensor and factor matrices under a Tucker factorization. The proposed method avoids repeated tensor projections and enables efficient mini-batch updates on low-dimensional tensor factors. Numerical experiments on synthetic tensor sensing demonstrate that the proposed algorithm exhibits favorable convergence behavior in wall-clock time compared with representative stochastic tensor recovery baselines. - oai:arXiv.org:2601.13522v1 - cs.LG - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/publicdomain/zero/1.0/ - Shuang Li - - - Near-field Physical Layer Security: Robust Beamforming under Location Uncertainty - https://arxiv.org/abs/2601.13549 - arXiv:2601.13549v1 Announce Type: cross -Abstract: In this paper, we study robust beamforming design for near-field physical-layer-security (PLS) systems, where a base station (BS) equipped with an extremely large-scale array (XL-array) serves multiple near-field legitimate users (Bobs) in the presence of multiple near-field eavesdroppers (Eves). Unlike existing works that mostly assume perfect channel state information (CSI) or location information of Eves, we consider a more practical and challenging scenario, where the locations of Bobs are perfectly known, while only imperfect location information of Eves is available at the BS. We first formulate a robust optimization problem to maximize the sum-rate of Bobs while guaranteeing a worst-case limit on the eavesdropping rate under location uncertainty. By transforming Cartesian position errors into the polar domain, we reveal an important near-field angular-error amplification effect: for the same location error, the closer the Eve, the larger the angle error, severely degrading the performance of conventional robust beamforming methods based on imperfect channel state information. To address this issue, we first establish the conditions for which the first-order Taylor approximation of the near-field channel steering vector under location uncertainty is largely accurate. Then, we propose a two-stage robust beamforming method, which first partitions the uncertainty region into multiple fan-shaped sub-regions, followed by the second stage to formulate and solve a refined linear-matrix-inequality (LMI)-based robust beamforming optimization problem. In addition, the proposed method is further extended to scenarios with multiple Bobs and multiple Eves. Finally, numerical results validate that the proposed method achieves a superior trade-off between rate performance and secrecy robustness, hence significantly outperforming existing benchmarks under Eve location uncertainty. - oai:arXiv.org:2601.13549v1 - eess.SP - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chao Zhou, Changsheng You, Cong Zhou, Chengwen Xing, Jianhua Zhang - - - Additive-Functional Approach to Transport in Periodic and Tilted Periodic Potentials - https://arxiv.org/abs/2601.13561 - arXiv:2601.13561v1 Announce Type: cross -Abstract: We present a unified theoretical framework for effective transport in periodic and tilted periodic potentials based on additive functionals of stochastic processes. By systematically combining the Poisson equation, corrector construction, and martingale decomposition, we show that both the long-time drift and diffusion of overdamped Brownian motion can be derived within a single and transparent scheme. In the absence of external tilt, the formalism naturally recovers the classical Lifson-Jackson formula for the effective diffusion coefficient. When a constant bias is applied, breaking detailed balance and inducing a finite stationary current, the same approach yields the Stratonovich expressions for the effective drift and diffusion in tilted periodic potentials. Beyond one dimension, we demonstrate that the same additive-functional structure extends directly to two-dimensional and general N dimensional periodic diffusions, leading to the standard homogenized drift and diffusion tensor expressed in terms of vector-valued correctors. Our derivation highlights the central role of additive functionals in separating bounded microscopic corrections from unbounded macroscopic transport and clarifies the connection between reversible and nonequilibrium steady states. This work provides a conceptually unified and mathematically controlled route to transport coefficients in periodic media, with direct relevance to stochastic transport, soft matter, and nonequilibrium statistical physics. - oai:arXiv.org:2601.13561v1 - cond-mat.stat-mech - math.PR - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sang Yang, Zhixin Peng - - - On the stability, complexity, and distribution of similarity classes of the longest edge bisection process for triangles - https://arxiv.org/abs/2601.13663 - arXiv:2601.13663v1 Announce Type: cross -Abstract: The Longest Edge Bisection (LEB) of a triangle is performed by joining the midpoint of its longest edge to the opposite vertex. Applying this procedure iteratively produces an infinite family of triangles. Surprisingly, a classical result of Adler (1983) shows that for any initial triangle, this infinite family falls into finitely many similarity classes. - While the set of classes is finite, we show that a far smaller, stable subset of ``fat'' triangles, called {\bf terminal quadruples}, effectively dominates the final mesh structure. We prove the following asymptotic area distribution result: for every initial triangle, the portion of area occupied by terminal quadruples tends to one, with the convergence occurring at an exponential rate. In fact, we provide the precise distribution of triangles in every step. We introduce the {\bf bisection graph} and use spectral methods to establish this result. - Given this dominance, we provide a complete characterization of triangles possessing a single terminal quadruple, while conversely exhibiting a sequence of triangles with an unbounded number of terminal quadruples. Furthermore, we reveal several fundamental geometric properties of the points of a terminal quadruple, laying the groundwork for studying the geometric distribution of the entire orbit. Our analysis leverages the hyperbolic geometry framework of Perdomo and Plaza (2014) and refines their techniques. - oai:arXiv.org:2601.13663v1 - cs.CG - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Daniel Kalmanovich, Yaar Solomon - - - Does Privacy Always Harm Fairness? Data-Dependent Trade-offs via Chernoff Information Neural Estimation - https://arxiv.org/abs/2601.13698 - arXiv:2601.13698v1 Announce Type: cross -Abstract: Fairness and privacy are two vital pillars of trustworthy machine learning. Despite extensive research on these individual topics, the relationship between fairness and privacy has received significantly less attention. In this paper, we utilize the information-theoretic measure Chernoff Information to highlight the data-dependent nature of the relationship among the triad of fairness, privacy, and accuracy. We first define Noisy Chernoff Difference, a tool that allows us to analyze the relationship among the triad simultaneously. We then show that for synthetic data, this value behaves in 3 distinct ways (depending on the distribution of the data). We highlight the data distributions involved in these cases and explore their fairness and privacy implications. Additionally, we show that Noisy Chernoff Difference acts as a proxy for the steepness of the fairness-accuracy curves. Finally, we propose a method for estimating Chernoff Information on data from unknown distributions and utilize this framework to examine the triad dynamic on real datasets. This work builds towards a unified understanding of the fairness-privacy-accuracy relationship and highlights its data-dependent nature. - oai:arXiv.org:2601.13698v1 - cs.LG - cs.AI - cs.IT - math.IT - stat.ML - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Arjun Nichani (Richard), Hsiang Hsu (Richard), Chun-Fu (Richard), Chen, Haewon Jeong - - - Cost-Effectiveness of Adult Hepatitis A Vaccination Strategies in Korea Under an Aging Susceptibility Profile - https://arxiv.org/abs/2601.13714 - arXiv:2601.13714v1 Announce Type: cross -Abstract: Hepatitis A severity increases sharply with age, while Korea is experiencing a cohort shift in which low seroprevalence adult cohorts are aging into older, higher fatality age groups. This demographic and immunological transition creates an urgent policy question regarding how adult vaccination should be prioritized under resource constraints. We evaluated three adult vaccination scenarios targeting low seroprevalence age groups (S1) 20 to 39 years, (S2) 40 to 59 years, and (S3) 20 to 59 years. Using an age structured dynamic transmission model calibrated to Korean data, we derived dynamically feasible vaccination allocation trajectories under realistic capacity constraints using an optimal control framework and linked these trajectories to long term transmission model simulations. We conducted DALY based cost effectiveness analyses over a lifetime horizon from both healthcare system and societal perspectives, and characterized uncertainty using probabilistic sensitivity analysis (PSA) and cost effectiveness acceptability curves (CEACs). Robustness was examined using one way sensitivity analyses. In the base case, S2 consistently yields the most favorable and robust cost effectiveness profile under both perspectives, with the lowest ICER. S3 achieved the largest reduction in DALYs but requires substantially higher incremental costs, resulting in a higher ICER than S2. S1 produces the smallest DALY reduction and is the least efficient strategy. PSA and CEACs confirm that S2 remains the preferred option across most willingness to pay ranges. S2 offers the most balanced and robustly cost effective strategy in Korea, capturing substantial mortality reduction while limiting additional program costs. S3 may be justified when higher budgets or willingness to pay thresholds are acceptable, but S2 provides the clearest value for money under epidemiological and economic conditions. - oai:arXiv.org:2601.13714v1 - q-bio.PE - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Yuna Lim, Gerardo Chowell, Eunok Jung - - - Area-universality in Outerplanar Graphs - https://arxiv.org/abs/2601.13781 - arXiv:2601.13781v1 Announce Type: cross -Abstract: A rectangular floorplan is a partition of a rectangle into smaller rectangles such that no four rectangles meet at a single point. Rectangular floorplans arise naturally in a variety of applications, including VLSI design, architectural layout, and cartography, where efficient and flexible spatial subdivisions are required. A central concept in this domain is that of area-universality: a floorplan (or more generally, a rectangular layout) is area-universal if, for any assignment of target areas to its constituent rectangles, there exists a combinatorially equivalent layout that realizes these areas. - In this paper, we investigate the structural conditions under which an outerplanar graph admits an area-universal rectangular layout. We establish a necessary and sufficient condition for area-universality in this setting, thereby providing a complete characterization of admissible outerplanar graphs. Furthermore, we present an algorithmic construction that guarantees that the resulting layout is always area-universal. - oai:arXiv.org:2601.13781v1 - cs.CG - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ravi Suthar, Raveena, Krishnendra Shekhawat - - - Soliton dynamics in the ABS nonlinear spinor model with external fields - https://arxiv.org/abs/2601.13783 - arXiv:2601.13783v1 Announce Type: cross -Abstract: We consider the novel nonlinear model in (1 + 1)-dimensions for Dirac spinors recently introduced by Alexeeva, Barashenkov, and Saxena [1] (ABS model), which admits an exact explicit solitary-wave (soliton for short) solution. The charge, the momentum, and the energy of this solution are conserved. We investigate the dynamics of the soliton subjected to several potentials: a ramp, a harmonic, and a periodic potential. We develop a Collective Coordinates Theory by making an ansatz for a moving soliton where the position, rapidity, and momentum, are functions of time. We insert the ansatz into the Lagrangian density of the model, integrate over space and obtain a Lagrangian as a function of the collective coordinates. This Lagrangian differs only in the charge and mass with the Lagrangian of a collective coordinates theory for the Gross-Neveu equation. Thus the soliton dynamics in the ABS spinor model is qualitatively the same as in the Gross-Neveu equation, but quantitatively it differs. These results of the collective coordinates theory are confirmed by simulations, i.e., by numerical solutions for solitons of the ABS spinor model, subjected to the above potentials. - oai:arXiv.org:2601.13783v1 - nlin.PS - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - 10.1088/1751-8121/ac190b - J. Phys. A: Math. Theor. 54 405702 (2021) - Franz G. Mertens, Bernardo S\'anchez-Rey, Niurka R. Quintero - - - Zero-free regions and concentration inequalities for hypergraph colorings in the local lemma regime - https://arxiv.org/abs/2601.13796 - arXiv:2601.13796v1 Announce Type: cross -Abstract: We show that for $q$-colorings in $k$-uniform hypergraphs with maximum degree $\Delta$, if $k\ge 50$ and $q\ge 700\Delta^{\frac{5}{k-10}}$, there is a "Lee-Yang" zero-free strip around the interval $[0,1]$ of the partition function, which includes the special case of uniform enumeration of hypergraph colorings. As an immediate consequence, we obtain Berry-Esseen type inequalities for hypergraph $q$-colorings under such conditions, demonstrating the asymptotic normality for the size of any color class in a uniformly random coloring. Our framework also extends to the study of "Fisher zeros", leading to deterministic algorithms for approximating the partition function in the zero-free region. - Our approach is based on extending the recent work of [Liu, Wang, Yin, Yu, STOC 2025] to general constraint satisfaction problems (CSP). We focus on partition functions defined for CSPs by introducing external fields to the variables. A key component in our approach is a projection-lifting scheme, which enables us to essentially lift information percolation type analysis for Markov chains from the real line to the complex plane. Last but not least, we also show a Chebyshev-type inequality under the sampling LLL condition for atomic CSPs. - oai:arXiv.org:2601.13796v1 - cs.DS - cs.DM - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jingcheng Liu, Yixiao Yu - - - Composing $p$-adic qubits: from representations of SO(3)$_p$ to entanglement and universal quantum logic gates - https://arxiv.org/abs/2601.13808 - arXiv:2601.13808v1 Announce Type: cross -Abstract: In the context of $p$-adic quantum mechanics, we investigate composite systems of $p$-adic qubits and $p$-adically controlled quantum logic gates. We build on the notion of a single $p$-adic qubit as a two-dimensional irreducible representation of the compact $p$-adic special orthogonal group SO(3)$_p$. We show that the classification of these representations reduces to the finite case, as they all factorise through some finite quotient SO(3)$_p$ mod $p^k$. Then, we tackle the problem of $p$-adic qubit composition and entanglement, fundamental for a $p$-adic formulation of quantum information processing. We classify the representations of SO(3)$_p$ mod $p$, and analyse tensor products of two $p$-adic qubit representations lifted from SO(3)$_p$ mod $p$. We solve the Clebsch-Gordan problem for such systems, revealing that the coupled bases decompose into singlet and doublet states. We further study entanglement arising from those stable subsystems. For $p=3$, we construct a set of gates from $4$-dimensional irreducible representations of SO(3)$_p$ mod $p$ that we prove to be universal for quantum computation. - oai:arXiv.org:2601.13808v1 - quant-ph - math-ph - math.MP - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Ilaria Svampa, Sonia L'Innocente, Stefano Mancini, Andreas Winter - - - Two-dimensional FrBD friction models for rolling contact: extension to linear viscoelasticity - https://arxiv.org/abs/2601.13818 - arXiv:2601.13818v1 Announce Type: cross -Abstract: This paper extends the distributed rolling contact FrBD framework to linear viscoelasticity by considering classic derivative Generalised Maxwell and Kelvin-Voigt rheological representations of the bristle element. With this modelling approach, the dynamics of the bristle, generated friction forces, and internal deformation states are described by a system of 2(n+1) hyperbolic partial differential equations (PDEs), which can capture complex relaxation phenomena originating from viscoelastic behaviours. By appropriately specifying the analytical expressions for the transport and rigid relative velocity, three distributed formulations of increasing complexity are introduced, which account for different levels of spin excitation. For the linear variants, well-posedness and passivity are analysed rigorously, showing that these properties hold for any physically meaningful parametrisation. Numerical experiments complement the theoretical results by illustrating steady-state characteristics and transient relaxation effects. The findings of this paper substantially advance the FrBD paradigm by enabling a unified and systematic treatment of linear viscoelasticity. - oai:arXiv.org:2601.13818v1 - physics.app-ph - cs.NA - math.NA - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Luigi Romano - - - Rigid Body Dynamics in Ambient Fluids - https://arxiv.org/abs/2601.13971 - arXiv:2601.13971v1 Announce Type: cross -Abstract: We present a novel framework for rigid body dynamics in ambient media, such as air or water, enabling accurate motion prediction of objects without requiring computational fluid dynamics simulations. Our method computes the added mass of the fluid and replaces heuristic models for shape-dependent lift and drag with a generalized estimate of flow separation and dynamic pressure. Our method is the first within the rigid body dynamics context to reproduce the full range of falling plate behaviors: fluttering, tumbling, chaotic and steady modes, as well as phenomena such as the Magnus effect and the flight dynamics of an American football (tight spiral pass paradox). The resulting algorithm is simple to implement, robust, does not rely on specialized integrators and incorporates seamlessly into existing physics engines for real-time simulation. - oai:arXiv.org:2601.13971v1 - physics.flu-dyn - cs.NA - math.NA - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Marcel Padilla, Aviv Segall, Olga Sorkine-Hornung - - - Achieving Full Multipath Diversity by Random Constellation Rotation: a Theoretical Perspective - https://arxiv.org/abs/2601.13997 - arXiv:2601.13997v1 Announce Type: cross -Abstract: Diversity is an essential concept associated with communication reliability in multipath channels since it determines the slope of bit error rate performance in the medium to high signal-to-noise ratio regions. However, most of the existing analytical frameworks were developed for specific modulation schemes while the efficient validation of full multipath diversity for general modulation schemes remains an open problem. To fill this research gap, we propose to utilize random constellation rotation to ease the conditions for full-diversity modulation designs. For linearly precoded cyclic-prefix orthogonal frequency division multiplexing (OFDM) systems, we prove that maximum multipath diversity can be attained as long as the spread matrix does not have zero entries, which is a sufficient but easily satisfied condition. Furthermore, we derive the sufficient and necessary condition for general modulation schemes, whose verification can be divided into validation tasks for each column of the modulation matrix. Based on the proposed conditions, maximum diversity order can be attained with the probability of 1 by enabling a randomly generated rotation pattern for both time and doubly dispersive channels. The theoretical analysis in this paper also demonstrates that the diversity evaluation can be concentrated on the pairwise error probability when the number of error symbols is one, which reduces the complexity of diversity-driven design and performance analysis for novel modulation schemes significantly in both time and doubly dispersive channels. Finally, numerical results for various modulation schemes confirm that the theoretical analysis holds in both time and doubly dispersive channels. Furthermore, when employing practical detectors, the random constellation rotation technique consistently enhance the transmission reliability for both coded and uncoded systems. - oai:arXiv.org:2601.13997v1 - eess.SP - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xuehan Wang, Jinhong Yuan, Jintao Wang, Kehan Huang - - - The rate of purification of quantum trajectories - https://arxiv.org/abs/2601.14023 - arXiv:2601.14023v1 Announce Type: cross -Abstract: We investigate the behavior of quantum trajectories conditioned on measurement outcomes. Under a condition related to the absence of so-called dark subspaces, K\"{u}mmerer and Maassen had shown that such trajectories almost surely purify in the long run. In this article, we first present a simple alternative proof of this result using Lyapunov methods. We then strengthen the conclusion by proving that purification actually occurs at an exponential rate in expectation, again using a Lyapunov approach. Furthermore, we address the quantum state estimation problem by propagating two trajectories under the same measurement record--one from the true initial state and the other from an arbitrary initial guess--and show that the estimated trajectory converges exponentially fast to the true one, thus quantifying the rate at which information is progressively revealed through the measurement process. - oai:arXiv.org:2601.14023v1 - quant-ph - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ma\"el Bompais, Nina H. Amini, Juan P. Garrahan, M\u{a}d\u{a}lin Gu\c{t}\u{a} - - - Performance enhancing of hybrid quantum-classical Benders approach for MILP optimization - https://arxiv.org/abs/2601.14024 - arXiv:2601.14024v1 Announce Type: cross -Abstract: Mixed-integer linear programming problems are extensively used in industry for a wide range of optimization tasks. However, as they get larger, they present computational challenges for classical solvers within practical time limits. Quantum annealers can, in principle, accelerate the solution of problems formulated as quadratic unconstrained binary optimization instances, but their limited scale currently prevents achieving practical speedups. Quantum-classical algorithms have been proposed to take advantage of both paradigms and to allow current quantum computers to be used in larger problems. In this work, a hardware-agnostic Benders' decomposition algorithm and a series of enhancements with the goal of taking the most advantage of quantum computing are presented. The decomposition consists of a master problem with integer variables, which is reformulated as a quadratic unconstrained binary optimization problem and solved with a quantum annealer, and a linear subproblem solved by a classical computer. The enhancements consist, among others, of different embedding processes that substantially reduce the pre-processing time of the embedding computation without compromising solution quality, a conservative handling of cut constraints, and a stopping criterion that accounts for the limited size of current quantum computers and their heuristic nature. The proposed algorithm is benchmarked against classical approaches using a D-Wave quantum annealer for a scalable family of transmission network expansion planning problems. - oai:arXiv.org:2601.14024v1 - quant-ph - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Sergio L\'opez-Ba\~nos, Elisabeth Lobe, Ontje L\"unsdorf, Oriol Ravent\'os - - - Universal Approximation Theorem for Input-Connected Multilayer Perceptrons - https://arxiv.org/abs/2601.14026 - arXiv:2601.14026v1 Announce Type: cross -Abstract: We introduce the Input-Connected Multilayer Perceptron (IC-MLP), a feedforward neural network architecture in which each hidden neuron receives, in addition to the outputs of the preceding layer, a direct affine connection from the raw input. We first study this architecture in the univariate setting and give an explicit and systematic description of IC-MLPs with an arbitrary finite number of hidden layers, including iterated formulas for the network functions. In this setting, we prove a universal approximation theorem showing that deep IC-MLPs can approximate any continuous function on a closed interval of the real line if and only if the activation function is nonlinear. We then extend the analysis to vector-valued inputs and establish a corresponding universal approximation theorem for continuous functions on compact subsets of $\mathbb{R}^n$. - oai:arXiv.org:2601.14026v1 - cs.LG - cs.NE - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Vugar Ismailov - - - Generalised contextuality of continuous variable quantum theory can be revealed with a single projective measurement - https://arxiv.org/abs/2601.14067 - arXiv:2601.14067v1 Announce Type: cross -Abstract: Generalized contextuality is a possible indicator of non-classical behaviour in quantum information theory. In finite-dimensional systems, this is justified by the fact that noncontextual theories can be embedded into some simplex, i.e. into a classical theory. We show that a direct application of the standard definition of generalized contextuality to continuous variable systems does not envelope the statistics of some basic measurements, such as the position observable. In other words, we construct families of fully classical, i.e. commuting, measurements that nevertheless can be used to show contextuality of quantum theory. To overcome the apparent disagreement between the two notions of classicality, that is commutativity and noncontextuality, we propose a modified definition of generalised contextuality for continuous-variable systems. The modified definition is based on a physically-motivated approximation procedure, that uses only finite sets of measurement effects. We prove that in the limiting case this definition corresponds exactly to an extension of noncontextual models that benefits from non-constructive response functions. In the process, we discuss the extension of a known connection between contextuality and no-broadcasting to the continuous-variable scenario, and prove structural results regarding fixed points of infinite-dimensional entanglement breaking channels. - oai:arXiv.org:2601.14067v1 - quant-ph - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pauli Jokinen, Mirjam Weilenmann, Martin Pl\'avala, Juha-Pekka Pellonp\"a\"a, Jukka Kiukas, Roope Uola - - - Sharp Inequalities for Schur-Convex Functionals of Partial Traces over Unitary Orbits - https://arxiv.org/abs/2601.14158 - arXiv:2601.14158v1 Announce Type: cross -Abstract: While many bounds have been proved for partial trace inequalities over the last decades for a large variety of quantities, recent problems in quantum information theory demand sharper bounds. In this work, we study optimal bounds for partial trace quantities in terms of the spectrum; equivalently, we determine the best bounds attainable over unitary orbits of matrices. We solve this question for Schur-convex functionals acting on a single partial trace in terms of eigenvalues for self-adjoint matrices and then we extend these results to singular values of general matrices. We subsequently extend the study to Schur-convex functionals that act on several partial traces simultaneously and present sufficient conditions for sharpness. In cases where closed-form maximizers cannot be identified, we present quadratic programs that yield new computable upper bounds for any Schur-convex functional. We additionally present examples demonstrating improvements over previously known bounds. Finally, we conclude with the study of optimal bounds for an $n$-qubit system and its subsystems of dimension $2$. - oai:arXiv.org:2601.14158v1 - quant-ph - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Pablo Costa Rico, Pavel Shteyner - - - Purity of branch and critical locus - https://arxiv.org/abs/1003.5872 - arXiv:1003.5872v5 Announce Type: replace -Abstract: To a dominant morphism $X/S \to Y/S$ of N\oe therian integral $S$-schemes one has the inclusion $C_{X/Y}\subset B_{X/Y}$ of the critical locus in the branch locus of $X/Y$. Starting from the notion of locally complete intersection morphisms, we give conditions on the modules of relative differentials $\Omega_{X/Y}$, $\Omega_{X/S}$, and $\Omega_{Y/S}$ that imply bounds on the codimensions of $ C_{X/Y}$ and $ B_{X/Y}$. These bounds generalise to a wider class of morphisms the classical purity results for finite morphisms by Zariski-Nagata-Auslander, and Faltings and Grothendieck, and van der Waerden's purity for birational morphisms. - oai:arXiv.org:1003.5872v5 - math.AG - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - J. Algebra, vol 379, 2013 - Rolf K\"allstr\"om - - - On the birational geometry of the parameter space for codimension 2 complete intersections - https://arxiv.org/abs/1212.4862 - arXiv:1212.4862v2 Announce Type: replace -Abstract: Codimension 2 complete intersections in P^N have a natural parameter space \bar{H}: a projective bundle over a projective space given by the choice of the lower degree equation and of the higher degree equation up to a multiple of the first. Motivated by the question of existence of complete families of smooth complete intersections, we study the birational geometry of \bar{H}. In a first part, we show that the first contraction of the MMP for \bar{H} always exists and we describe it. Then, we show that it is possible to run the full MMP for \bar{H}, and we describe it, in two degenerate cases. As an application, we prove the existence of complete curves in the punctual Hilbert scheme of complete intersection subschemes of A^2. - oai:arXiv.org:1212.4862v2 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Olivier Benoist - - - Hyperbolicity via Geodesic Stability - https://arxiv.org/abs/1504.06863 - arXiv:1504.06863v3 Announce Type: replace -Abstract: A geodesic $g$ is Morse, for every $L \geq 1, A \geq 0$ there exists a $C=C_g(L,A)$ such that any $(L,A)$-quasi-geodesic connecting two points on $g$ stays $C$-close to $g$. The Morse lemma implies that in a hyperbolic space every geodesic is Morse. Here we prove the converse: If a homogeneous proper geodesic space is such that for every geodesic $g$ and every $L\geq 1, A \geq 0$ there exists a constant $C=C_g(L,A)$ such that any $(L,A)$-quasi-geodesic between any two points on $g$ stays $C$-close, then the space is hyperbolic. This applies in particular to infinite groups in which all geodesics are Morse. - oai:arXiv.org:1504.06863v3 - math.MG - math.GR - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Elisabeth Fink - - - Chernoff bounds for branching random walks - https://arxiv.org/abs/1604.00056 - arXiv:1604.00056v4 Announce Type: replace -Abstract: Concentration inequalities, which have proved very useful in a variety of fields, provide fairly tight bounds on large deviation probabilities while central limit theorem (CLT) describes the asymptotic distribution around the mean (at the $\sqrt{n}$ scale). Harris (1963) conjectured that for a supercritical branching random walk (BRW) of i.i.d offspring and i.i.d displacements, positions of individuals in $nth$ generation approach to Gaussian distribution -- central limit theorem. This conjecture was later proved by Stam (1966) and Kaplan \& Asmussen (1976). Refinements and extensions followed. However, to the best of our knowledge, there is no corresponding existing work on concentration inequalities for BRWs. In this note, we propose a new definition of BRW, providing a more general framework. Owing to this definition, a Chernoff-type (subgaussian) bound for BRWs follows directly from the Chernoff bound for random walk. The relation between RW (random walk) and BRW is discussed. - oai:arXiv.org:1604.00056v4 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Changqing Liu - - - Some properties of the A$_{\infty}$-nerve - https://arxiv.org/abs/1609.00566 - arXiv:1609.00566v2 Announce Type: replace -Abstract: The aim of this paper is to prove that the A$_{\infty}$-nerve of two quasi-equivalent A$_{\infty}$-categories (linear over a commutative ring) are weak-equivalent in the Joyal model structure. As a consequence we prove that the A$_{\infty}$-nerve of a pretriangulated A$_{\infty}$-category is a stable $\infty$-category. - oai:arXiv.org:1609.00566v2 - math.AG - math.CT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mattia Ornaghi - - - On a certain subclass of strongly starlike functions - https://arxiv.org/abs/1811.01271 - arXiv:1811.01271v4 Announce Type: replace -Abstract: Let $\mathcal{S}^*(\alpha_1,\alpha_2)$, where $ \alpha_1, \alpha_2 \in (0,1]$, represent the class of functions $f$ that are analytic in the open unit disk $\mathbb{D}$, normalized by $f(0) = f'(0) - 1=0$, and satisfying the following double-sided inequality: \begin{equation*} - -\frac{\pi\alpha_1}{2}< \arg\left\{\frac{zf'(z)}{f(z)}\right\} <\frac{\pi\alpha_2}{2}, \quad (z\in\mathbb{D}). \end{equation*} In this manuscript, we estimate the coefficients and logarithmic coefficients associated with functions that belong to the class $\mathcal{S}^*(\alpha_1,\alpha_2)$. As a result, we provide a general bound for the coefficients of a strongly starlike function, which has been an open question until now. Finally, we derive upper and lower bounds for the expression ${\rm Re}\{zf'(z)/f(z)\}$, where $f\in \mathcal{S}^*(\alpha_1,\alpha_2)$. - oai:arXiv.org:1811.01271v4 - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - 10.1007/s13324-026-01165-y - Analysis and Mathematical Physics, 2026 - R. Kargar, J. Sok\'o{\l}, H. Mahzoon - - - Linear resolution of products of monomial ideals related to maximal minors - https://arxiv.org/abs/1902.09748 - arXiv:1902.09748v3 Announce Type: replace -Abstract: Let $ X $ be an $ m \times n $ matrix of distinct indeterminates over a field $ K $, where $ m \le n $. Set the polynomial ring $ K[X] := K[X_{ij} : 1 \le i \le m, 1 \le j \le n] $. Let $ 1 \le k < l \le n $ be such that $ l - k + 1 \ge m $. Consider the submatrix $ Y_{kl} $ of consecutive columns of $ X $ from $ k $th column to $ l $th column. Let $ J_{kl} $ be the ideal generated by `diagonal monomials' of all $ m \times m $ submatrices of $ Y_{kl} $, where the diagonal monomial of a square matrix means product of its main diagonal entries. We show that $ J_{k_1 l_1} J_{k_2 l_2} \cdots J_{k_s l_s} $ has a linear free resolution, where $ k_1 \le k_2 \le \cdots \le k_s $ and $ l_1 \le l_2 \le \cdots \le l_s $. This result is a variation of a theorem due to Bruns and Conca. Moreover, our proof is self-contained, elementary and combinatorial. - oai:arXiv.org:1902.09748v3 - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Arindam Banerjee, Dipankar Ghosh, S. Selvaraja - - - Branching rules for winding subalgebras of the affine Kac--Moody algebras $A^{(1)}_1$ and $A^{(2)}_2$ - https://arxiv.org/abs/1911.03316 - arXiv:1911.03316v5 Announce Type: replace -Abstract: We study branching problems for affine Kac--Moody algebras. Unlike the finite-dimensional case, an affine Kac--Moody algebra may contain proper subalgebras isomorphic to itself, such as winding subalgebras obtained by rescaling the loop parameter. We investigate the restriction of integrable highest-weight representations to such subalgebras. The restriction remains integrable and decomposes into irreducible components with finite multiplicities, encoded by pairs of highest weights. We show that this set is closed under addition, extending a result of Brion and Knop to the affine setting. We also give a partial description of this set and provide explicit results for types $A^{(1)}_1$ and $A^{(2)}_2$. - oai:arXiv.org:1911.03316v5 - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Khanh Nguyen Duc - - - Boosted optimal weighted least-squares - https://arxiv.org/abs/1912.07075 - arXiv:1912.07075v3 Announce Type: replace -Abstract: This paper is concerned with the approximation of a function $u$ in a given approximation space $V_m$ of dimension $m$ from evaluations of the function at $n$ suitably chosen points. The aim is to construct an approximation of $u$ in $V_m$ which yields an error close to the best approximation error in $V_m$ and using as few evaluations as possible. Classical least-squares regression, which defines a projection in $V_m$ from $n$ random points, usually requires a large $n$ to guarantee a stable approximation and an error close to the best approximation error. This is a major drawback for applications where $u$ is expensive to evaluate. One remedy is to use a weighted least squares projection using $n$ samples drawn from a properly selected distribution. In this paper, we introduce a boosted weighted least-squares method which allows to ensure almost surely the stability of the weighted least squares projection with a sample size close to the interpolation regime $n=m$. It consists in sampling according to a measure associated with the optimization of a stability criterion over a collection of independent $n$-samples, and resampling according to this measure until a stability condition is satisfied. A greedy method is then proposed to remove points from the obtained sample. Quasi-optimality properties are obtained for the weighted least-squares projection, with or without the greedy procedure. The proposed method is validated on numerical examples and compared to state-of-the-art interpolation and weighted least squares methods. - oai:arXiv.org:1912.07075v3 - math.NA - cs.NA - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1090/mcom/3710 - Math. Comp. (2022) - C\'ecile Haberstich, Anthony Nouy, Guillaume Perrin - - - A Control-Theoretic Perspective on Optimal High-Order Optimization - https://arxiv.org/abs/1912.07168 - arXiv:1912.07168v5 Announce Type: replace -Abstract: We provide a control-theoretic perspective on optimal tensor algorithms for minimizing a convex function in a finite-dimensional Euclidean space. Given a function $\Phi: \mathbb{R}^d \rightarrow \mathbb{R}$ that is convex and twice continuously differentiable, we study a closed-loop control system that is governed by the operators $\nabla \Phi$ and $\nabla^2 \Phi$ together with a feedback control law $\lambda(\cdot)$ satisfying the algebraic equation $(\lambda(t))^p\|\nabla\Phi(x(t))\|^{p-1} = \theta$ for some $\theta \in (0, 1)$. Our first contribution is to prove the existence and uniqueness of a local solution to this system via the Banach fixed-point theorem. We present a simple yet nontrivial Lyapunov function that allows us to establish the existence and uniqueness of a global solution under certain regularity conditions and analyze the convergence properties of trajectories. The rate of convergence is $O(1/t^{(3p+1)/2})$ in terms of objective function gap and $O(1/t^{3p})$ in terms of squared gradient norm. Our second contribution is to provide two algorithmic frameworks obtained from discretization of our continuous-time system, one of which generalizes the large-step A-HPE framework and the other of which leads to a new optimal $p$-th order tensor algorithm. While our discrete-time analysis can be seen as a simplification and generalization of~\citet{Monteiro-2013-Accelerated}, it is largely motivated by the aforementioned continuous-time analysis, demonstrating the fundamental role that the feedback control plays in optimal acceleration and the clear advantage that the continuous-time perspective brings to algorithmic design. A highlight of our analysis is that we show that all of the $p$-th order optimal tensor algorithms that we discuss minimize the squared gradient norm at a rate of $O(k^{-3p})$, which complements the recent analysis. - oai:arXiv.org:1912.07168v5 - math.OC - cs.CC - cs.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tianyi Lin, Michael. I. Jordan - - - Hochschild homology and the derived de Rham complex revisited - https://arxiv.org/abs/2007.02576 - arXiv:2007.02576v3 Announce Type: replace -Abstract: We characterize two objects by universal property: the derived de Rham complex and Hochschild homology together with its Hochschild-Kostant-Rosenberg (HKR) filtration. This involves endowing these objects with extra structure, built on notions of "homotopy-coherent cochain complex" and "filtered circle action" that we study here. We use these universal properties to give a conceptual proof that the associated graded of the HKR filtration identifies with the derived de Rham complex, as well as to give a new construction of the filtrations on cyclic, negative cyclic, and periodic cyclic homology that relate these invariants to derived de Rham cohomology. - oai:arXiv.org:2007.02576v3 - math.AG - math.KT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Arpon Raksit - - - A new characterization of the Hardy space and of other spaces of analytic functions - https://arxiv.org/abs/2009.10937 - arXiv:2009.10937v2 Announce Type: replace -Abstract: The Fock space can be characterized (up to a positive multiplicative factor) as the only Hilbert space of entire functions in which the adjoint of derivation is multiplication by the complex variable. Similarly (and still up to a positive multiplicative factor) the Hardy space is the only space of functions analytic in the open unit disk for which the adjoint of the backward shift operator is the multiplication operator. In the present paper we characterize the Hardy space and some related reproducing kernel Hilbert spaces in terms of the adjoint of the differentiation operator. We use reproducing kernel methods, which seem to also give a new characterization of the Fock space. - oai:arXiv.org:2009.10937v2 - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.26650/ijmath.2023.00005 - Natanael Alpay - - - Invertibility Conditions for the Admittance Matrices of Balanced Power Systems - https://arxiv.org/abs/2012.04087 - arXiv:2012.04087v5 Announce Type: replace -Abstract: The admittance matrix encodes the network topology and electrical parameters of a power system in order to relate the current injection and voltage phasors. Since admittance matrices are central to many power engineering analyses, their characteristics are important subjects of theoretical studies. This paper focuses on the key characteristic of \emph{invertibility}. Previous literature has presented an invertibility condition for admittance matrices. This paper first identifies and fixes a technical issue in the proof of this previously presented invertibility condition. This paper then extends this previous work by deriving new conditions that are applicable to a broader class of systems with lossless branches and transformers with off-nominal tap ratios. - oai:arXiv.org:2012.04087v5 - math.OC - cs.SY - eess.SY - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1109/TPWRS.2022.3206285 - Daniel Turizo, Daniel K. Molzahn - - - Moduli space of homologically trivial parabolic (Higgs) bundles on the projective line and applications - https://arxiv.org/abs/2101.01913 - arXiv:2101.01913v3 Announce Type: replace -Abstract: We establish an isomorphism between the moduli space of homologically trivial parabolic (Higgs) bundles on $\mathbb{P}^1$ and the quiver variety associated to a star-shaped quiver. As applications, we deduce a closed formula for the Littlewood-Richardson coefficients from the Verlinde formula, and solve the nilpotent case of the Deligne-Simpson problem via geometric methods. - oai:arXiv.org:2101.01913v3 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xueqing Wen - - - Coarse Ricci curvature of hypergraphs and its generalization - https://arxiv.org/abs/2102.00698 - arXiv:2102.00698v5 Announce Type: replace -Abstract: In the present paper, we introduce a concept of Ricci curvature on hypergraphs for a nonlinear Laplacian. We prove that our definition of the Ricci curvature is a generalization of Lin-Lu-Yau coarse Ricci curvature for graphs to hypergraphs. We also show a lower bound of nonzero eigenvalues of Laplacian, gradient estimate of heat flow, and diameter bound of Bonnet-Myers type for our curvature notion. This research leads to understanding how nonlinearity of Laplacian causes complexity of curvatures. - oai:arXiv.org:2102.00698v5 - math.MG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - MasaHiro Ikeda, Yu Kitabeppu, Yuuki Takai, Takato Uehara - - - Value existence for zero-sum ergodic stochastic differential games - https://arxiv.org/abs/2106.15894 - arXiv:2106.15894v4 Announce Type: replace -Abstract: In this paper we investigate two-player zero-sum stochastic differential games with an ergodic payoff, in which the diffusion coefficient does not need to be non-degenerate. We first establish the existence of a viscosity solution to the associated ergodic Hamilton-Jacobi-Bellman-Isaacs equation under a dissipativity condition. With the help of this viscosity solution, we then derive estimates for the upper and the lower ergodic value functions by constructing a series of non-degenerate approximating processes combined with the sup- and inf-convolution techniques. Finally, we prove the existence of a value for the game under the Isaacs condition and provide its representation formulae. As an application, we study the pollution accumulation problem with a long-run average social welfare to illustrate our theoretical results. - oai:arXiv.org:2106.15894v4 - math.OC - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Juan Li, Wenqiang Li, Yanwei Li, Huaizhong Zhao - - - Three-chromatic geometric hypergraphs - https://arxiv.org/abs/2112.01820 - arXiv:2112.01820v2 Announce Type: replace -Abstract: We prove that for any planar convex body C there is a positive integer m with the property that any finite point set P in the plane can be three-colored such that there is no translate of C containing at least m points of P, all of the same color. As a part of the proof, we show a strengthening of the Erd\H{o}s-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture. - oai:arXiv.org:2112.01820v2 - math.CO - cs.DM - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - G\'abor Dam\'asdi, D\"om\"ot\"or P\'alv\"olgyi - - - p-adic adelic metrics and Quadratic Chabauty I - https://arxiv.org/abs/2112.03873 - arXiv:2112.03873v4 Announce Type: replace -Abstract: We give a new construction of $p$-adic heights on varieties over number fields using $p$-adic Arakelov theory. In analogy with Zhang's construction of real-valued heights in terms of adelic metrics, these heights are given in terms of $p$-adic adelic metrics on line bundles. In particular, we describe a construction of canonical $p$-adic heights on abelian varieties and we show that we recover the canonical Mazur--Tate height and, for Jacobians, the height constructed by Coleman and Gross. Our main application is a new and simplified approach to the Quadratic Chabauty method for the computation of rational points on certain curves over the rationals, by pulling back the canonical height on the Jacobian with respect to a carefully chosen line bundle. We show that our construction allows us to reprove, without using $p$-adic Hodge theory or arithmetic fundamental groups, several results due to Balakrishnan and Dogra. Our method also extends to primes $p$ of bad reduction. One consequence of our work is that for any canonical height ($p$-adic or $\mathbb{R}$-valued) on an abelian variety (and hence on pull-backs to other varieties), the local contribution at a finite prime $q$ can be constructed using $q$-analytic methods. - oai:arXiv.org:2112.03873v4 - math.NT - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1515/crelle-2025-0063 - Journal f\"ur die reine und angewandte Mathematik (Crelles Journal), vol. 2025, no. 828, 2025, pp. 223-305 - Amnon Besser, J. Steffen M\"uller, Padmavathi Srinivasan - - - Reconstruction for the time-dependent coefficients of a quasilinear dynamical Schr{\"o}dinger equation - https://arxiv.org/abs/2201.09809 - arXiv:2201.09809v2 Announce Type: replace -Abstract: We study an inverse problem related to the dynamical Schr{\"o}dinger equation in a bounded domain of $\Rb^n,n\geq 2$. Since the concerned non-linear Schr\"odinger equation possesses a trivial solution, we linearize the equation around the trivial solution. Demonstrating the well-posedness of the direct problem under appropriate conditions on initial and boundary data, it is observed that the solution admits $\eps$-expansion. By taking into account the fact that the terms $\Oh(|\nabla u(t,x)|^3)$ are negligible in this context, we shall reconstruct the time-dependent coefficients such as electric potential and vector-valued function associated with quadratic nonlinearity from the knowledge of input-output map using the geometric optics solution and Fourier inversion. - oai:arXiv.org:2201.09809v2 - math.AP - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Gen Nakamura, Tanmay sarkar, Manmohan Vashisth - - - Improvements in $L^2$ Restriction bounds for Neumann Data along closed curves - https://arxiv.org/abs/2203.01208 - arXiv:2203.01208v2 Announce Type: replace -Abstract: We seek to improve the restriction bounds of Neumann data of Laplace eigenfunctions $u_h$ by studying the $L^2$ restriction bounds of Neumann data and their $L^2$ concentration as measured by defect measures. Let $\gamma$ be a closed smooth curve with unit exterior normal $\nu$. We can show that $\| h \partial_\nu u_{h} \|_{L^2(\Gamma)}=o(1)$ if $\{u_h\}$ is tangentially concentrated with respect to $\gamma$. As a key ingredient of the proof, we give a detailed analysis of the $L^2$ norms over $\gamma$ of the Neumann data $h\partial_\nu u_h$ when mircolocalized away the cotangential direction. - oai:arXiv.org:2203.01208v2 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Wu Xianchao - - - Holomorphic foliations of degree two and arbitrary dimension - https://arxiv.org/abs/2207.12880 - arXiv:2207.12880v5 Announce Type: replace -Abstract: We prove a complete classification of degree-$2$ foliations on $\mathbb{P}^n$ in any dimension, assuming they are not algebraically integrable. If $\mathcal{F}$ is such a foliation, then either $\mathcal{F}$ is the linear pull-back of a degree-$2$ foliation by curves on $\mathbb{P}^{n-k+1}$, or a logarithmic foliation of type $(1^{n-k+1},2)$, or a logarithmic foliation of type $(1^{n-k+3})$, or the linear pull-back of a degree-$2$ foliation of dimension $2$ on $\mathbb{P}^{n-k+2}$ tangent to an action of the Lie algebra $\mathfrak{aff}(\mathbb{C})$. Meanwhile, we prove that any $2$-dimensional foliation tangent to a global vector field must satisfy that its tangent sheaf is either not locally free or has a direct summand isomorphic to $\mathcal{O}_{\mathbb{P}^{n}}(a)$, with $a\in\{0,1\}$. As a byproduct of our classification, we describe the geometry of Poisson structures on $\mathbb{P}^{4}$ with generic rank two. - oai:arXiv.org:2207.12880v5 - math.AG - math.CV - math.DS - math.SG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Maur\'icio Corr\^ea, Alan Muniz - - - Tame class field theory over local fields - https://arxiv.org/abs/2209.02953 - arXiv:2209.02953v2 Announce Type: replace -Abstract: For a quasi-projective scheme $X$ admitting a smooth compactification over a local field of residue characteristic $p > 0$, we construct a continuous reciprocity homomorphism from a tame class group to the abelian tame etale fundamental group of $X$. We describe the prime-to-$p$ parts of its kernel and cokernel. This generalizes the higher dimensional unramified class field theory over local fields by Jannsen-Saito and Forre. We also prove a finiteness theorem for the geometric part of the abelian tame etale fundamental group, generalizing the results of Grothendieck and Yoshida for the unramified fundamental group. - oai:arXiv.org:2209.02953v2 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Rahul Gupta, Amalendu Krishna, Jitendra Rathore - - - Functional dimension of feedforward ReLU neural networks - https://arxiv.org/abs/2209.04036 - arXiv:2209.04036v2 Announce Type: replace -Abstract: It is well-known that the parameterized family of functions representable by fully-connected feedforward neural networks with ReLU activation function is precisely the class of piecewise linear functions with finitely many pieces. It is less well-known that for every fixed architecture of ReLU neural network, the parameter space admits positive-dimensional spaces of symmetries, and hence the local functional dimension near any given parameter is lower than the parametric dimension. In this work we carefully define the notion of functional dimension, show that it is inhomogeneous across the parameter space of ReLU neural network functions, and continue an investigation - initiated in [14] and [5] - into when the functional dimension achieves its theoretical maximum. We also study the quotient space and fibers of the realization map from parameter space to function space, supplying examples of fibers that are disconnected, fibers upon which functional dimension is non-constant, and fibers upon which the symmetry group acts non-transitively. - oai:arXiv.org:2209.04036v2 - math.MG - cs.LG - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.aim.2025.110636 - Advances in Mathematics, Volume 482, Part C, December 2025, 110636 - J. Elisenda Grigsby, Kathryn Lindsey, Robert Meyerhoff, Chenxi Wu - - - Nondivergence of Reductive group action on Homogeneous Spaces - https://arxiv.org/abs/2209.06463 - arXiv:2209.06463v2 Announce Type: replace -Abstract: Let $X=G/\Gamma$ be the quotient of a semisimple Lie group $G$ by its non-cocompact arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $X$. We give several equivalent algebraic conditions on $H$ for the existence of a fixed compact set in $X$ intersecting \textit{every} $H$-orbit. This generalizes previous results concerning certain special reductive group action on $X$ in this setting. - When $G$ is of real rank one, $\Gamma$ is a non-cocompact lattice of $G$ and $H<G$ is an algebraic group, we also obtain an algebraic condition on $H$ which is equivalent to the return of \textit{every} $H$-orbit to a single compact set in $X$. This complements our results in the case of arithmetic lattice. - oai:arXiv.org:2209.06463v2 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Han Zhang, Runlin Zhang - - - Prime Solutions of Diagonal Diophantine Systems - https://arxiv.org/abs/2209.06934 - arXiv:2209.06934v2 Announce Type: replace -Abstract: An asymptotic formula for the number of prime solutions of a general diagonal system of Diophantine equations is established, contingent on the existence of an appropriate mean value bound and on local solvability. In conjunction with the Vinogradov Mean Value Theorem this yields an asymptotic formula for solutions of Vinogradov systems and in conjunction with Hooley's work on seven cubes this yields a conditional result for the Waring-Goldbach problem on seven cubes of primes, contingent on Hooley's form of the Riemann hypothesis. - oai:arXiv.org:2209.06934v2 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alan Talmage - - - Expander graphs are globally synchronizing - https://arxiv.org/abs/2210.12788 - arXiv:2210.12788v4 Announce Type: replace -Abstract: The Kuramoto model is fundamental to the study of synchronization. It consists of a collection of oscillators with interactions given by a network, which we identify respectively with vertices and edges of a graph. In this paper, we show that a graph with sufficient expansion must be globally synchronizing, meaning that a homogeneous Kuramoto model of identical oscillators on such a graph will converge to the fully synchronized state with all the oscillators having the same phase, for every initial state up to a set of measure zero. In particular, we show that for any $\varepsilon > 0$ and $p \geq (1 + \varepsilon) (\log n) / n$, the homogeneous Kuramoto model on the Erd\H{o}s-R\'enyi random graph $G(n, p)$ is globally synchronizing with probability tending to one as $n$ goes to infinity. This improves on a previous result of Kassabov, Strogatz, and Townsend and solves a conjecture of Ling, Xu, and Bandeira. We also show that the model is globally synchronizing on any $d$-regular Ramanujan graph, and on typical $d$-regular graphs, for large enough degree $d$. - oai:arXiv.org:2210.12788v4 - math.CO - math.DS - math.OC - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.aim.2025.110773 - Advances in Mathematics, vol 488, 2026, p. 110773 - Pedro Abdalla, Afonso S. Bandeira, Martin Kassabov, Victor Souza, Steven H. Strogatz, Alex Townsend - - - Turing meets Moore-Penrose: Computing the Pseudoinverse on Turing Machines - https://arxiv.org/abs/2212.02940 - arXiv:2212.02940v2 Announce Type: replace -Abstract: The pseudoinverse of a matrix, a generalized notion of the inverse, is of fundamental importance in linear algebra and, thereby, in many different fields. Despite its proven existence, an algorithmic approach is typically necessary to obtain the pseudoinverse in practical applications. Therefore, we analyze if and to what degree the pseudoinverse can be computed on perfect digital hardware platforms modeled as Turing machines. For this, we utilize the notion of an effective algorithm that describes a provably correct computation: upon an input of any error parameter, the algorithm provides an approximation within the given error bound with respect to the unknown solution. We prove that a universal effective algorithm for computing the pseudoinverse of any matrix with a finite error bound does not exist on Turing machines. However, for specific classes of matrices, we show that provably correct algorithms exist and obtain a characterization of the properties of the input set, leading to the effective computability breakdown. - oai:arXiv.org:2212.02940v2 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Holger Boche, Adalbert Fono, Gitta Kutyniok - - - Diophantine Criterion for Non-trivial Shafarevich-Tate Groups - https://arxiv.org/abs/2301.03486 - arXiv:2301.03486v3 Announce Type: replace -Abstract: The solvability of Diophantine quartic equations is a contemporary area of interest due to its connection with \textit{generalized Fermat's equation}. In this work, we are interested in the integer solutions of a similar quartic equation $pu^{2} = v^{2}+w^{2}$. For a particular form of $u,v$, and $w$, we prove that the elliptic curves $E_p: y^2 = x(x-1)(x+p^2)$, for primes $p \equiv 1 \pmod{8}$ where $q = (p^2+1)/2$ is also prime, exhibit a sharp dichotomy based on the solution of the aforementioned Diophantine equation: either $\mathrm{rank}(E_p(\mathbb{Q})) = 2$ with trivial Shafarevich-Tate group or $\mathrm{rank} = 0$ with $\Sha(E_p/\mathbb{Q})[2] \cong (\mathbb{Z}/2\mathbb{Z})^2$. - oai:arXiv.org:2301.03486v3 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Vinodkumar Ghale, Gayatri Panicker, Debopam Chakraborty - - - Relations between e, $\pi$, golden ratios and $\sqrt{2}$ - https://arxiv.org/abs/2301.09643 - arXiv:2301.09643v4 Announce Type: replace -Abstract: We write out relations between the base of natural logarithms ($e$), the ratio of the circumference of a circle to its diameter ($\pi$), the golden ratios ($\Phi_p$) of the additive $p$-sequences, and the ratio of the diagonal of a square to its side ($\sqrt{2}$). An additive $p$-sequence is a natural extension of the Fibonacci sequence in which every term is the sum of $p$-previous terms given $p \ge 1$ initial values called seeds. - oai:arXiv.org:2301.09643v4 - math.GM - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Asutosh Kumar - - - Multiperiodic Processes: Ergodic Sources with a Sublinear Entropy - https://arxiv.org/abs/2302.09049 - arXiv:2302.09049v5 Announce Type: replace -Abstract: We construct multiperiodic processes -- a simple example of stationary ergodic (but not mixing) processes over natural numbers that enjoy the vanishing entropy rate under a mild condition. Multiperiodic processes are supported on randomly shifted deterministic sequences called multiperiodic sequences, which can be efficiently generated using an algorithm called the Infinite Clock. Under a suitable parameterization, multiperiodic sequences exhibit relative frequencies of particular numbers given by Zipf's law. Exactly in the same setting, the respective multiperiodic processes satisfy an asymptotic power-law growth of block entropy, called Hilberg's law. Hilberg's law is deemed to hold for statistical language models, in particular. - oai:arXiv.org:2302.09049v5 - cs.IT - cs.LG - math.IT - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - {\L}ukasz D\k{e}bowski - - - A relative Nadel-type vanishing theorem - https://arxiv.org/abs/2302.11080 - arXiv:2302.11080v3 Announce Type: replace -Abstract: Let $f:X\rightarrow Y$ be a K\"{a}hler fibration from a complex manifold $X$ to an analytic space $Y$. We show several relative Nadel-type vanishing theorems. - oai:arXiv.org:2302.11080v3 - math.AG - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jingcao Wu - - - Finite Presentability of Brin-Higman-Thompson Monoids via Free J\'onsson-Tarski Algebras - https://arxiv.org/abs/2303.16044 - arXiv:2303.16044v3 Announce Type: replace -Abstract: We show that the monoids totM_{k,1} introduced by Birget and their generalizations tot nM_{k,r} which extend the Brin-Higman-Thompson groups, can be realized as the endomorphism monoids of higher-dimensional J\'onsson-Tarski algebras. We also show how elements of these monoids can be thought of as "rewrite rules". Using these representations, we show that the monoids are finitely presented. - oai:arXiv.org:2303.16044v3 - math.RA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bill de Witt, Luna Elliott - - - Controlled Interacting Branching Diffusion Processes: Relaxed Formulation in the Mean-Field Regime - https://arxiv.org/abs/2304.07064 - arXiv:2304.07064v3 Announce Type: replace -Abstract: The focus of this article is studying an optimal control problem for branching diffusion processes. Initially, we introduce the problem in its strong formulation and expand it to include linearly growing drifts. Then, we present a relaxed formulation that provides a suitable characterization based on martingale measures. Considering weak controls, we prove they are equivalent to strong controls in the relaxed setting, and establish the equivalence between the strong and relaxed problem, under a Filippov--type convexity condition. Furthermore, by defining control rules, we can restate the problem as the minimization of a lower semi-continuous function over a compact set, leading to the existence of optimal controls both for the relaxed problem and the strong one. Finally, with a useful embedding technique, we show that the value function solves a system of HJB equations, establishing a verification theorem. We then apply it to a linear-quadratic example and a kinetic one. - oai:arXiv.org:2304.07064v3 - math.PR - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Antonio Ocello - - - Continuous-time extensions of discrete-time cocycles - https://arxiv.org/abs/2305.07338 - arXiv:2305.07338v2 Announce Type: replace -Abstract: We consider linear cocycles taking values in $\textup{SL}_d(\mathbb{R})$ driven by homeomorphic transformations of a smooth manifold, in discrete and continuous time. We show that any discrete-time cocycle can be extended to a continuous-time cocycle, while preserving its characteristic properties. We provide a necessary and sufficient condition under which this extension is canonical in the sense that the base is extended to an associated suspension flow and that the discrete-time cocycle is recovered as the time-1 map of the continuous-time cocycle. Further, we refine our general result for the case of (quasi-)periodic driving. We use our findings to construct a non-uniformly hyperbolic continuous-time cocycle in $\SL{2}$ over a uniquely ergodic driving. - oai:arXiv.org:2305.07338v2 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Robin Chemnitz, Maximilian Engel, P\'eter Koltai - - - Quasimaps to quivers with potentials - https://arxiv.org/abs/2306.01302 - arXiv:2306.01302v2 Announce Type: replace -Abstract: This paper is concerned with a non-compact GIT quotient of a vector space, in the presence of an abelian group action and an equivariant regular function (potential) on the quotient. We define virtual counts of quasimaps from prestable curves to the critical locus of the potential, and prove a gluing formula in the formalism of cohomological field theories. - The main examples studied in this paper is when the above setting arises from quivers with potentials, where the above construction gives quantum correction to the equivariant Chow homology of the critical locus. Following similar ideas as in quasimaps to Nakajima quiver varieties studied by the Okounkov school, we analyse vertex functions in several examples, including Hilbert schemes of points on $\mathbb{C}^3$, moduli spaces of perverse coherent systems on the resolved conifold, and a quiver which defines higher $\mathfrak{sl}_2$-spin chains. Bethe equations are calculated in these cases. - The construction in the present paper is based on the theory of gauged linear sigma models as well as shifted symplectic geometry of Pantev, To\"en, Vaquie and Vezzosi, and uses the virtual pullback formalism of symmetric obstruction theory of Park, which arises from the recent development of Donaldson-Thomas theory of Calabi-Yau 4-folds. - oai:arXiv.org:2306.01302v2 - math.AG - hep-th - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yalong Cao, Gufang Zhao - - - The distribution of Ridgeless least squares interpolators - https://arxiv.org/abs/2307.02044 - arXiv:2307.02044v2 Announce Type: replace -Abstract: The Ridgeless minimum $\ell_2$-norm interpolator in overparametrized linear regression has attracted considerable attention in recent years in both machine learning and statistics communities. While it seems to defy conventional wisdom that overfitting leads to poor prediction, recent theoretical research on its $\ell_2$-type risks reveals that its norm minimizing property induces an `implicit regularization' that helps prediction in spite of interpolation. - This paper takes a further step that aims at understanding its precise stochastic behavior as a statistical estimator. Specifically, we characterize the distribution of the Ridgeless interpolator in high dimensions, in terms of a Ridge estimator in an associated Gaussian sequence model with positive regularization, which provides a precise quantification of the prescribed implicit regularization in the most general distributional sense. Our distributional characterizations hold for general non-Gaussian random designs and extend uniformly to positively regularized Ridge estimators. - As a direct application, we obtain a complete characterization for a general class of weighted $\ell_q$ risks of the Ridge(less) estimators that are previously only known for $q=2$ by random matrix methods. These weighted $\ell_q$ risks not only include the standard prediction and estimation errors, but also include the non-standard covariate shift settings. Our uniform characterizations further reveal a surprising feature of the commonly used generalized and $k$-fold cross-validation schemes: tuning the estimated $\ell_2$ prediction risk by these methods alone lead to simultaneous optimal $\ell_2$ in-sample, prediction and estimation risks, as well as the optimal length of debiased confidence intervals. - oai:arXiv.org:2307.02044v2 - math.ST - cs.IT - math.IT - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Qiyang Han, Xiaocong Xu - - - A First-Order Algorithm for Decentralised Min-Max Problems - https://arxiv.org/abs/2308.11876 - arXiv:2308.11876v2 Announce Type: replace -Abstract: In this work, we consider a connected network of finitely many agents working cooperatively to solve a min-max problem with convex-concave structure. We propose a decentralised first-order algorithm which can be viewed as a non-trivial combination of two algorithms: PG-EXTRA for decentralised minimisation problems and the forward reflected backward method for (non-distributed) min-max problems. In each iteration of our algorithm, each agent computes the gradient of the smooth component of its local objective function as well as the proximal operator of its nonsmooth component, following by a round of communication with its neighbours. Our analysis shows that the sequence generated by the method converges under standard assumptions with non-decaying stepsize. - oai:arXiv.org:2308.11876v2 - math.OC - cs.DC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yura Malitsky, Matthew K. Tam - - - Fast and Inverse-Free Algorithms for Deflating Subspaces - https://arxiv.org/abs/2310.00193 - arXiv:2310.00193v4 Announce Type: replace -Abstract: This paper explores a key question in numerical linear algebra: how can we compute projectors onto the deflating subspaces of a regular matrix pencil $(A,B)$, in particular without using matrix inversion or defaulting to an expensive Schur decomposition? We focus specifically on spectral projectors, whose associated deflating subspaces correspond to sets of eigenvalues/eigenvectors. In this work, we present a high-level approach to computing these projectors, which combines rational function approximation with an inverse-free arithmetic of Benner and Byers [Numerische Mathematik 2006]. The result is a numerical framework that captures existing inverse-free methods, generates an array of new options, and provides straightforward tools for pursuing efficiency on structured problems (e.g., definite pencils). To exhibit the efficacy of this framework, we consider a handful of methods in detail, including Implicit Repeated Squaring and iterations based on the matrix sign function. In an appendix, we demonstrate that recent, randomized divide-and-conquer eigensolvers -- which are built on fast methods for individual projectors -- can be adapted to produce the generalized Schur form of any matrix pencil in nearly matrix multiplication time. - oai:arXiv.org:2310.00193v4 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - 10.1016/j.laa.2026.01.014 - Linear Algebra and its Applications (2026) - James Demmel, Ioana Dumitriu, Ryan Schneider - - - Global well-posedness and large-time behavior of the compressible Navier-Stokes equations with hyperbolic heat conduction - https://arxiv.org/abs/2310.13461 - arXiv:2310.13461v2 Announce Type: replace -Abstract: The classical Fourier's law, which states that the heat flux is proportional to the temperature gradient, induces the paradox of infinite propagation speed for heat conduction. To accurately simulate the real physical process, the hyperbolic model of heat conduction named Cattaneo's law was proposed, which leads to the finite speed of heat propagation. A natural question is that whether the large-time behavior of the heat flux for compressible flow would be different for these two laws. In this paper, we aim to address this question by studying the global well-posedness and optimal time-decay rates of classical solutions to the compressible Navier-Stokes system with Cattaneo's law. By designing a new method, we obtain the optimal time-decay rates for the highest derivatives of the heat flux, which cannot be derived for the system with Fourier's law by Matsumura and Nishida [Proc. Japan Acad. Ser. A Math. Sci., 55(9):337-342, 1979]. In this sense, our results first reveal the essential differences between the two laws. - oai:arXiv.org:2310.13461v2 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - 10.1016/j.jde.2026.114111 - Journal of Differential Equations (2026) - Fucai Li, Houzhi Tang, Shuxing Zhang - - - The asymptotic behavior of rarely visited edges of the simple random walk - https://arxiv.org/abs/2310.16657 - arXiv:2310.16657v2 Announce Type: replace -Abstract: In this paper, we study the asymptotic behavior of the number of rarely visited edges (i.e., edges that visited only once) of a simple symmetric random walk on $\mathbb{Z}$. Let $\alpha(n)$ be the number of rarely visited edges up to time $n$. First, we evaluate $\mathbb{E}(\alpha(n))$, show that $n\to \mathbb{E}(\alpha(n))$ is non-decreasing in $n$ and that $\lim\limits_{n\to+\infty}\mathbb{E}(\alpha(n))=2$. Then we study the asymptotic behavior of $\mathbb{P} (\alpha(n)>a(\log n)^2)$ for any $a>0$ and use it to show that there exists a constant $C\in(1/32,1/2]$ such that $\limsup\limits_{n\to+\infty}\frac{\alpha(n)}{(\log n)^2}=C$ almost surely. - oai:arXiv.org:2310.16657v2 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ze-Chun Hu, Xue Peng, Renming Song, Yuan Tan - - - Factorization structures, cones, and polytopes - https://arxiv.org/abs/2311.07328 - arXiv:2311.07328v4 Announce Type: replace -Abstract: Factorization structures occur in toric differential and discrete geometry, and can be viewed in multiple ways, e.g., as objects determining substantial classes of explicit toric Sasaki and K\"ahler geometries, as special coordinates on such, or as an apex generalisation of cyclic polytopes featuring a generalised Gale's evenness condition. This article presents a comprehensive study of factorization structures. It establishes their structure theory and introduces their use in the geometry of cones and polytopes. The article explains the construction of polytopes and cones compatible with a given factorization structure, and exemplifies it for product Segre-Veronese and Veronese factorization structures, where the latter case includes cyclic polytopes. Further, it derives the generalised Gale's evenness condition for compatible cones, polytopes and their duals, and explicitly describes faces of these. Factorization structures naturally provide generalised Vandermonde identities, which relate normals of any compatible polytope, and which are used for Veronese factorization structure to find examples of Delzant and rational Delzant compatible polytopes. The article offers a myriad of factorization structure examples, which are later characterised to be precisely factorization structures with decomposable curves, and raises the question if these encompass all factorization structures, i.e., the existence of an indecomposable factorization curve. - oai:arXiv.org:2311.07328v4 - math.CO - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1017/prm.2025.10046 - Roland P\'u\v{c}ek - - - Abelian gauge-like groups of $L_\infty$-algebras - https://arxiv.org/abs/2311.08512 - arXiv:2311.08512v3 Announce Type: replace -Abstract: Given a finite type degree-wise nilpotent $L_\infty$-algebra, we construct an abelian group that acts on the set of Maurer-Cartan elements of the given $L_\infty$-algebra so that the quotient by this action becomes the moduli space of equivalence classes of Maurer-Cartan elements. Specializing this to degree-wise nilpotent dg Lie algebras, we find that the associated ordinary gauge group of the dg Lie algebra with the Baker-Campbell-Hausdorff multiplication might be substituted by the underlying additive group. This additive group acts on the Maurer-Cartan elements, and the quotient by this action yields the moduli space of gauge-equivalence classes of Maurer-Cartan elements. - oai:arXiv.org:2311.08512v3 - math.RA - math.AT - math.QA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Bashar Saleh - - - Quaternion-Valued Wavelets on the Plane: A Construction via the Douglas-Rachford Approach - https://arxiv.org/abs/2311.12614 - arXiv:2311.12614v2 Announce Type: replace -Abstract: This paper presents a reformulation of the construction of nonseparable multiresolution quaternion-valued wavelets on the plane as a feasibility problem. The constraint sets in the feasibility problem are derived from the standard conditions of smoothness, compact support, and orthonormality. To solve the resulting feasibility problems, we employ a product space formulation of the Douglas-Rachford algorithm. This approach yields novel examples of nonseparable, multiresolution, compactly supported, smooth, and orthonormal quaternion-valued wavelets on the plane. Additionally, by introducing a symmetry-promoting constraint, we construct symmetric quaternion-valued scaling functions on the plane. - oai:arXiv.org:2311.12614v2 - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Neil D. Dizon, Jeffrey A. Hogan - - - On the equivalence of static and dynamic weak optimal transport - https://arxiv.org/abs/2311.13872 - arXiv:2311.13872v3 Announce Type: replace -Abstract: We show that there is a PDE formulation in terms of Fokker-Planck equations for weak optimal transport problems. The main novelty is that we introduce a minimization problem involving Fokker-Planck equations in the extended sense of measure-valued solutions and prove that it is equal to the associated weak transport problem. - oai:arXiv.org:2311.13872v3 - math.OC - math.AP - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bohdan Bulanyi - - - Ancient mean curvature flows from minimal hypersurfaces - https://arxiv.org/abs/2311.15278 - arXiv:2311.15278v4 Announce Type: replace -Abstract: For $n\geq 2$, we construct $I$-dimensional family of embedded ancient solutions to mean curvature flow arise from an unstable minimal hypersurface $\Sigma$ with finite total curvature in $\mathbb{R}^{n+1}$, where $I$ is the Morse index of the Jacobi operator on $\Sigma$. - oai:arXiv.org:2311.15278v4 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yongheng Han - - - Propagating front solutions in a time-fractional Fisher-KPP equation - https://arxiv.org/abs/2311.15651 - arXiv:2311.15651v5 Announce Type: replace -Abstract: In this paper, we treat the Fisher-KPP equation with a Caputo-type time fractional derivative and discuss the propagation speed of the solution. The equation is a mathematical model that describes the processes of sub-diffusion, proliferation, and saturation. We first consider a traveling wave solution to study the propagation of the solution, but we cannot define it in the usual sense due to the time fractional derivative in the equation. We therefore assume that the solution asymptotically approaches a traveling wave solution, and the asymptotic traveling wave solution is formally introduced as a potential asymptotic form of the solution. The existence and the properties of the asymptotic traveling wave solution are discussed using a monotone iteration method. Finally, the behavior of the solution is analyzed by numerical simulations based on the result for asymptotic traveling wave solutions. - oai:arXiv.org:2311.15651v5 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.3934/dcdsb.2024173 - Discrete and Continuous Dynamical Systems - B, 2025, 30(7): 2460-2482 - Hiroshi Ishii - - - Non-vanishing of Kolyvagin systems and Iwasawa theory - https://arxiv.org/abs/2312.09301 - arXiv:2312.09301v2 Announce Type: replace -Abstract: Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime. In 1991 Kolyvagin conjectured that the system of cohomology classes for torsion quotients of the $p$-adic Tate module of $E$ derived from Heegner points over ring class fields of a suitable imaginary quadratic field $K$ (i.e., the Heegner point Kolyvagin system of $E/K$) is non-trivial. In this paper we prove Kolyvagin's conjecture when $p$ is a prime of good ordinary reduction for $E$ that splits in $K$. In particular, our results cover many cases where $p$ is an Eisenstein prime for $E$, complementing Wei Zhang's earlier results on the conjecture by a different approach. - Our methods also yield a proof of a refinement of Kolyvagin's conjecture expressing the divisibility index of the Heegner point Kolyvagin system in terms of the Tamagawa numbers of $E$, as conjectured by Wei Zhang in 2014, as well as proofs of analogous results for the Kolyvagin system obtained from Kato's Euler system. - oai:arXiv.org:2312.09301v2 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ashay Burungale, Francesc Castella, Giada Grossi, Christopher Skinner - - - On the mixed monotonicity of polynomial functions - https://arxiv.org/abs/2312.15517 - arXiv:2312.15517v2 Announce Type: replace -Abstract: In this paper, it is shown that every polynomial function is mixed monotone globally with a polynomial decomposition function. For univariate polynomials, the decomposition functions can be constructed from the Gram matrix representation of polynomial functions. The tightness of polynomial decomposition functions is discussed. Several examples are provided. An example is provided to show that polynomial decomposition functions, in addition to being global decomposition functions, can be much tighter than local decomposition functions constructed using local Jacobian bounds. Furthermore, an example is provided to demonstrate the application to reachable set over-approximation. - oai:arXiv.org:2312.15517v2 - math.OC - cs.SY - eess.SY - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Adam M Tahir - - - Virtual Holonomic and Nonholonomic Constraints on Lie groups - https://arxiv.org/abs/2312.17531 - arXiv:2312.17531v2 Announce Type: replace -Abstract: This paper develops a geometric framework for virtual constraints on Lie groups, with emphasis on mechanical systems modeled as affine connection systems. Virtual holonomic and virtual nonholonomic constraints, including linear and affine nonholonomic constraints, are formulated directly at the level of the Lie algebra and characterized as feedback--invariant manifolds. For each class of constraint, we establish existence and uniqueness conditions for enforcing feedback laws and show that the resulting closed--loop trajectories evolve as the dynamics of mechanical systems endowed with induced constrained connections, generalizing classical holonomic and nonholonomic reductions. Beyond stabilization, the framework enables the systematic generation of low--dimensional motion primitives on Lie groups by enforcing invariant, possibly affine, manifolds and shaping nontrivial dynamical regimes. The approach is illustrated through representative examples, including quadrotor UAVs and a rigid body with an internal rotor, where classical control laws are recovered as special cases and affine constraint--induced motion primitives are obtained. - oai:arXiv.org:2312.17531v2 - math.OC - cs.SY - eess.SY - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - A. Anahory Simoes, A. Bloch, L. Colombo, E. Stratoglou - - - Separable homology of graphs and the separability complex - https://arxiv.org/abs/2401.01320 - arXiv:2401.01320v2 Announce Type: replace -Abstract: We introduce the separability complex, a one-complex associated to a finite regular cover of the rose and show that it is connected if and only if the fundamental group of the associated cover is generated by its intersection with the set of elements in proper free factors of $\mathbf{F}_n$. The separability complex admits an action of $\mathrm{Out}(\mathbf{F}_n)$ by isometries if the associated cover corresponds to a characteristic subgroup of $\mathbf{F}_n$. We prove that the separability complex of the rose has infinite diameter and is nonhyperbolic, implying it is not quasi-isometric to the free splitting complex or the free factor complex. As a consequence, we obtain that the Cayley graph of $\mathbf{F}_n$ with generating set consisting of all primitive elements of $\mathbf{F}_n$ is nonhyperbolic. - oai:arXiv.org:2401.01320v2 - math.GT - math.GR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Becky Eastham - - - A classification of neighborhoods around leaves of a singular foliation - https://arxiv.org/abs/2401.05966 - arXiv:2401.05966v3 Announce Type: replace -Abstract: We classify singular foliations admitting a given leaf and a given transverse singular foliation. - oai:arXiv.org:2401.05966v3 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Simon-Raphael Fischer, Camille Laurent-Gengoux - - - Local dimension spectrum for dominated planar self-affine sets - https://arxiv.org/abs/2401.13626 - arXiv:2401.13626v2 Announce Type: replace -Abstract: The local dimension spectrum provides a framework for quantifying the fractal properties of a measure, and it is well understood for non-overlapping self-similar measures. In this article, we study the local dimension spectrum for dominated self-affine measures. By analyzing exact dimensionality, we obtain deterministic results that extend the scope of the local dimension spectrum beyond the almost-sure setting. - oai:arXiv.org:2401.13626v2 - math.DS - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alex Batsis, Antti K\"aenm\"aki, Tom Kempton - - - On generalized Beauville decompositions - https://arxiv.org/abs/2402.08861 - arXiv:2402.08861v2 Announce Type: replace -Abstract: Motivated by the Beauville decomposition of an abelian scheme and the "Perverse = Chern" phenomenon for a compactified Jacobian fibration, we study in this paper splittings of the perverse filtration for compactified Jacobian fibrations. - On the one hand, we prove for the Beauville-Mukai system associated with an irreducible curve class on a $K3$ surface the existence of a Fourier-stable multiplicative splitting of the perverse filtration, which extends the Beauville decomposition for the nonsingular fibers. Our approach is to construct a Lefschetz decomposition associated with a Fourier-conjugate $\mathfrak{sl}_2$-triple, which relies heavily on recent work concerning the interaction between derived equivalences and LLV algebras for hyper-K\"ahler varieties. Motivic lifting and connections to the Beauville-Voisin conjectures are also discussed. - On the other hand, we construct for any $g\geq 2$ a compactified Jacobian fibration of genus $g$ curves such that each curve is integral with at worst simple nodes and the (multiplicative) perverse filtration does not admit a multiplicative splitting. Our argument relies on the recently established universal double ramification cycle relations. This shows that in general an extension of the Beauville decomposition cannot exist for compactified Jacobian fibrations even when the simplest singular point appears. - oai:arXiv.org:2402.08861v2 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Younghan Bae, Davesh Maulik, Junliang Shen, Qizheng Yin - - - Uniform bounds for bilinear symbols with linear K-quasiconformally embedded singularity - https://arxiv.org/abs/2402.11661 - arXiv:2402.11661v2 Announce Type: replace -Abstract: We prove bounds in the strict local $L^{2}(\mathbb{R}^{d})$ range for trilinear Fourier multiplier forms with a $d$-dimensional singular subspace. Given a fixed parameter $K \ge 1$, we treat multipliers with non-degenerate singularity that are push-forwards by $K$-quasiconformal matrices of suitable symbols. As particular applications, our result recovers the uniform bounds for the one-dimensional bilinear Hilbert transforms in the strict local $L^{2}$ range, and it implies the uniform bounds for two-dimensional bilinear Beurling transforms, which are new, in the same range. - oai:arXiv.org:2402.11661v2 - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.2140/apde.2025.18.2293 - Analysis & PDE 18 (2025) 2293-2323 - Marco Fraccaroli, Olli Saari, Christoph Thiele - - - On transverse-universality of twist knots - https://arxiv.org/abs/2402.12585 - arXiv:2402.12585v3 Announce Type: replace -Abstract: In the search for transverse-universal knots in the standard contact structure on $\mathbb{S}^3$, we present a classification of the transverse twist knots with maximal self-linking number, that admit only overtwisted contact branched covers. As a direct consequence, we obtain an infinite family of transverse knots in $(\mathbb{S}^3,\xi_{std})$ that are not transverse-universal, although they are universal in the topological sense. - oai:arXiv.org:2402.12585v3 - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sebastian Zapata - - - Hyperuniformity and optimal transport of point processes - https://arxiv.org/abs/2402.13705 - arXiv:2402.13705v4 Announce Type: replace -Abstract: We examine optimal matchings or transport between two stationary random measures. It covers allocation from the Lebesgue measure to a point process and matching a point process to a regular (shifted) lattice. The main focus of the article is the impact of hyperuniformity(reduced variance fluctuations in point processes) to optimal transport: in dimension 2, we show that the typical matching cost has finite second moment under a mild logarithmic integrability condition on the reduced pair correlation measure, showing that most planar hyperuniform point processes are L2-perturbed lattices. Our method also retrieves known sharp bounds in finite windows for neutral integrable systems such as Poisson processes, and also applies to hyperfluctuating systems. Further, in three dimensions onwards, all point processes with an integrable pair correlation measure are L2-perturbed lattices without requiring hyperuniformity. - oai:arXiv.org:2402.13705v4 - math.PR - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Rapha\"el Lachi\`eze-Rey, D. Yogeshwaran - - - Bilinear Rough Singular Integrals near the Critical Integrability via Sharp Fourier Multiplier Criteria - https://arxiv.org/abs/2402.15785 - arXiv:2402.15785v2 Announce Type: replace -Abstract: We establish boundedness results for bilinear singular integral operators with rough homogeneous kernels whose restriction to the unit sphere belongs to the Orlicz space $L(\log L)^\alpha$. This improves the previously best known condition for boundedness of such bilinear operators obtained in the paper of the first and third authors, and provides estimates close to the conjectured endpoint of integrability suggested by the linear theory. The proof is based on a new sharp boundedness criterion for bilinear Fourier multiplier operators associated with sums of dyadic dilations of a fixed symbol $m_0$, compactly supported away from the origin. This criterion admits the best possible behavior with respect to a modulation of $m_0$ and is intimately connected with sharp shifted square function estimates. - oai:arXiv.org:2402.15785v2 - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Georgios Dosidis, Bae Jun Park, Lenka Slavikova - - - Counting rationals and diophantine approximation in missing-digit Cantor sets - https://arxiv.org/abs/2402.18395 - arXiv:2402.18395v2 Announce Type: replace -Abstract: We establish a new upper bound for the number of rationals up to a given height in a missing-digit set, making progress towards a conjecture of Broderick, Fishman, and Reich. This enables us to make novel progress towards another conjecture of those authors about the corresponding intrinsic diophantine approximation problem. Moreover, we make further progress towards conjectures of Bugeaud--Durand and Levesley--Salp--Velani on the distribution of diophantine exponents in missing-digit sets. - A key tool in our study is Fourier $\ell^1$ dimension introduced by the last named author in [H. Yu, Rational points near self-similar sets, arXiv:2101.05910]. An important technical contribution of the paper is a method to compute this quantity. - oai:arXiv.org:2402.18395v2 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sam Chow, P\'eter P. Varj\'u, Han Yu - - - A restricted additive smoother for finite cell flow problems - https://arxiv.org/abs/2403.11636 - arXiv:2403.11636v2 Announce Type: replace -Abstract: In this work, we propose an adaptive geometric multigrid method for the solution of large-scale finite cell flow problems. The finite cell method seeks to circumvent the need for a boundary-conforming mesh through the embedding of the physical domain in a regular background mesh. As a result of the intersection between the physical domain and the background computational mesh, the resultant systems of equations are typically numerically ill-conditioned, rendering the appropriate treatment of cutcells a crucial aspect of the solver. To this end, we propose a smoother operator with favorable parallel properties and discuss its memory footprint and parallelization aspects. We propose three cache policies that offer a balance between cached and on-the-fly computation and discuss the optimization opportunities offered by the smoother operator. It is shown that the smoother operator, on account of its additive nature, can be replicated in parallel exactly with little communication overhead, which offers a major advantage in parallel settings as the geometric multigrid solver is consequently independent of the number of processes. The convergence and scalability of the geometric multigrid method is studied using numerical examples. It is shown that the iteration count of the solver remains bounded independent of the problem size and depth of the grid hierarchy. The solver is shown to obtain excellent weak and strong scaling using numerical benchmarks with more than 665 million degrees of freedom. The presented geometric multigrid solver is, therefore, an attractive option for the solution of large-scale finite cell problems in massively parallel high-performance computing environments. - oai:arXiv.org:2403.11636v2 - math.NA - cs.NA - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - M. Saberi, A. Vogel - - - What Is a Good Imputation Under MAR Missingness? - https://arxiv.org/abs/2403.19196 - arXiv:2403.19196v5 Announce Type: replace -Abstract: Missing values pose a persistent challenge in modern data science. Consequently, there is an ever-growing number of publications introducing new imputation methods in various fields. The present paper attempts to take a step back and provide a more systematic analysis. Starting from an in-depth discussion of the Missing at Random (MAR) condition for nonparametric imputation, we first investigate whether the widely used fully conditional specification (FCS) approach indeed identifies the correct conditional distributions. Based on this analysis, we propose three essential properties an ideal imputation method should meet, thus enabling a more principled evaluation of existing methods and more targeted development of new methods. In particular, we introduce a new imputation method, denoted mice-DRF, that meets two out of the three criteria. We also discuss ways to compare imputation methods, based on distributional distances. Finally, numerical experiments illustrate the points made in this discussion. - oai:arXiv.org:2403.19196v5 - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jeffrey N\"af, Erwan Scornet, Julie Josse - - - Analytic holonomicity of real C$^{{\mathrm{exp}}}$-class distributions - https://arxiv.org/abs/2403.20167 - arXiv:2403.20167v2 Announce Type: replace -Abstract: We introduce a notion of distributions on $\mathbb{R}^n$, called distributions of C$^{{\mathrm{exp}}}$-class, based on wavelet transforms of distributions and the theory from Cluckers, Comte, Miller, Rolin, Servi (2018) about C$^{{\mathrm{exp}}}$-class functions. We prove that the framework of C$^{{\mathrm{exp}}}$-class distributions is closed under natural operations, like push-forward, pull-back, derivation and anti-derivation, and, in the tempered case, Fourier transforms. Our main result is the (real analytic) holonomicity of all distributions of C$^{{\mathrm{exp}}}$-class. - oai:arXiv.org:2403.20167v2 - math.AG - math.LO - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Avraham Aizenbud, Raf Cluckers, Michel Raibaut, Tamara Servi - - - Constructive proofs for some semilinear PDEs on $H^2(e^{|x|^2/4},\mathbb{R}^d)$ - https://arxiv.org/abs/2404.04054 - arXiv:2404.04054v2 Announce Type: replace -Abstract: We develop computer-assisted tools to study semilinear equations of the form \begin{equation*} -\Delta u -\frac{x}{2}\cdot \nabla{u}= f(x,u,\nabla u) ,\quad x\in\mathbb{R}^d. \end{equation*} Such equations appear naturally in several contexts, and in particular when looking for self-similar solutions of parabolic PDEs. We develop a general methodology, allowing us not only to prove the existence of solutions, but also to describe them very precisely. We introduce a spectral approach based on an eigenbasis of $\mathcal{L}:= -\Delta -\frac{x}{2}\cdot \nabla$ in spherical coordinates, together with a quadrature rule allowing to deal with nonlinearities, in order to get accurate approximate solutions. We then use a Newton-Kantorovich argument, in an appropriate weighted Sobolev space, to prove the existence of a nearby exact solution. We apply our approach to nonlinear heat equations, to nonlinear Schr\"odinger equations and to a generalised viscous Burgers equation, and obtain both radial and non-radial self-similar profiles. - oai:arXiv.org:2404.04054v2 - math.AP - cs.NA - math.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - 10.1007/s00211-025-01504-4 - Maxime Breden, Hugo Chu - - - Monochromatic polynomial sumset structures on $\mathbb{N}$ - https://arxiv.org/abs/2404.05226 - arXiv:2404.05226v4 Announce Type: replace -Abstract: In the paper, we search for monochromatic infinite additive structures involving polynomials over $\mathbb{N}$. It is proved that for any $r\in \mathbb{N}$, any two distinct natural numbers $a,b$, and any $2$-coloring of $\mathbb{N}$, there exist two sets $B,C\subset \mathbb{N}$ with $|B|=r$ and $|C|=\infty$ such that there exists some color containing $B+aC$ and $B+bC$. - oai:arXiv.org:2404.05226v4 - math.CO - math.DS - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Zhengxing Lian, Rongzhong Xiao - - - Cohen-Macaulay representations of Artin-Schelter Gorenstein algebras of dimension one - https://arxiv.org/abs/2404.05925 - arXiv:2404.05925v4 Announce Type: replace -Abstract: Tilting theory is one of the central tools in modern representation theory, in particular in the study of Cohen-Macaulay representations. We study Cohen-Macaulay representations of $\mathbb N$-graded Artin-Schelter Gorenstein algebras $A$ of dimension one, without assuming the connectedness condition. This framework covers a broad class of noncommutative Gorenstein rings, including classical $\mathbb N$-graded Gorenstein orders. We prove that the stable category $\underline{\mathsf{CM}}_0^{\mathbb Z}A$ admits a silting object if and only if $A_0$ has finite global dimension. In this case we give such a silting object explicitly. Assuming that $A$ is ring-indecomposable, we further show that $\underline{\mathsf{CM}}_0^{\mathbb Z}A$ admits a tilting object if and only if either $A$ is Artin-Schelter regular or the average Gorenstein parameter of $A$ is non-positive. These results generalize those of Buchweitz, Iyama, and Yamaura. We give two proofs of the second result: one via Orlov-type semiorthogonal decompositions, and the other via a direct calculation. As an application, we show that for a Gorenstein tiled order $A$, the category $\underline{\mathsf{CM}}^{\mathbb Z}A$ is equivalent to the derived category of the incidence algebra of an explicitly constructed poset. - We also apply our results and Koszul duality to study smooth noncommutative projective quadric hypersurfaces $\mathsf{qgr}\,B$ of arbitrary dimension. We prove that $\mathsf{D}^{\mathrm b}(\mathsf{qgr}\,B)$ admits an explicitly constructed tilting object, which contains the tilting object of $\underline{\mathsf{CM}}^{\mathbb Z}B$ due to Smith and Van den Bergh as a direct summand via Orlov's semiorthogonal decomposition. - oai:arXiv.org:2404.05925v4 - math.RT - math.AC - math.AG - math.RA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Osamu Iyama, Yuta Kimura, Kenta Ueyama - - - Lower bound for the first eigenvalue of $p-$Laplacian and applications in asymptotically hyperbolic Einstein manifolds - https://arxiv.org/abs/2405.02669 - arXiv:2405.02669v2 Announce Type: replace -Abstract: This paper investigates the first Dirichlet eigenvalue for the $p$-Laplacian in Riemannian manifolds. Firstly, we establish a lower bound for this eigenvalue under the condition that the domain includes a specific function which fulfills certain criteria related to divergence and gradient conditions. In the subsequent section, we introduce an enhanced lower bound for the eigenvalue, which is linked to the distance function defined in the domain. As a practical application, we provide an estimation for the first Dirichlet eigenvalue of geodesic balls with large radius in asymptotically hyperbolic Einstein manifolds. - oai:arXiv.org:2405.02669v2 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xiaoshang Jin - - - Fast Two-Time-Scale Stochastic Gradient Method with Applications in Reinforcement Learning - https://arxiv.org/abs/2405.09660 - arXiv:2405.09660v4 Announce Type: replace -Abstract: Two-time-scale optimization is a framework introduced in Zeng et al. (2024) that abstracts a range of policy evaluation and policy optimization problems in reinforcement learning (RL). Akin to bi-level optimization under a particular type of stochastic oracle, the two-time-scale optimization framework has an upper level objective whose gradient evaluation depends on the solution of a lower level problem, which is to find the root of a strongly monotone operator. In this work, we propose a new method for solving two-time-scale optimization that achieves significantly faster convergence than the prior arts. The key idea of our approach is to leverage an averaging step to improve the estimates of the operators in both lower and upper levels before using them to update the decision variables. These additional averaging steps eliminate the direct coupling between the main variables, enabling the accelerated performance of our algorithm. We characterize the finite-time convergence rates of the proposed algorithm under various conditions of the underlying objective function, including strong convexity, Polyak-Lojasiewicz condition, and general non-convexity. These rates significantly improve over the best-known complexity of the standard two-time-scale stochastic approximation algorithm. When applied to RL, we show how the proposed algorithm specializes to novel online sample-based methods that surpass or match the performance of the existing state of the art. Finally, we support our theoretical results with numerical simulations in RL. - oai:arXiv.org:2405.09660v4 - math.OC - cs.LG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sihan Zeng, Thinh T. Doan - - - On a Conjecture by Hayashi on Finite Connected Quandles - https://arxiv.org/abs/2405.11660 - arXiv:2405.11660v2 Announce Type: replace -Abstract: A quandle is an algebraic structure whose binary operation is idempotent, right-invertible and right self-distributive. Right-invertibility ensures right translations are permutations and right self-distributivity ensures further they are automorphisms. For finite connected quandles, all right translations have the same cycle structure, called the profile of the connected quandle. Hayashi conjectured that the longest length in the profile of a finite connected quandle is a multiple of the remaining lengths. We prove that this conjecture is true for profiles with at most five lengths. - oai:arXiv.org:2405.11660v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - 10.26493/1855-3974.3529.4e5 - Antonio Lages, Pedro Lopes - - - Rate Optimality and Phase Transition for User-Level Local Differential Privacy - https://arxiv.org/abs/2405.11923 - arXiv:2405.11923v3 Announce Type: replace -Abstract: Most of the literature on differential privacy considers the item-level case where each user has a single observation, but a growing field of interest is that of user-level privacy where each of the $n$ users holds $T$ observations and wishes to maintain the privacy of their entire collection. - In this paper, we derive a general minimax lower bound, which shows that, for locally private user-level estimation problems, the risk cannot, in general, be made to vanish for a fixed number of users even when each user holds an arbitrarily large number of observations. We then derive matching, up to logarithmic factors, lower and upper bounds for univariate and multidimensional mean estimation, sparse mean estimation and non-parametric density estimation. In particular, with other model parameters held fixed, we observe phase transition phenomena in the minimax rates as $T$ the number of observations each user holds varies. - In the case of (non-sparse) mean estimation and density estimation, we see that, for $T$ below a phase transition boundary, the rate is the same as having $nT$ users in the item-level setting. Different behaviour is however observed in the case of $s$-sparse $d$-dimensional mean estimation, wherein consistent estimation is impossible when $d$ exceeds the number of observations in the item-level setting, but is possible in the user-level setting when $T \gtrsim s \log (d)$, up to logarithmic factors. This may be of independent interest for applications as an example of a high-dimensional problem that is feasible under local privacy constraints. - oai:arXiv.org:2405.11923v3 - math.ST - stat.ME - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexander Kent, Thomas B. Berrett, Yi Yu - - - Distribution Steering for Discrete-Time Uncertain Ensemble Systems - https://arxiv.org/abs/2405.12415 - arXiv:2405.12415v2 Announce Type: replace -Abstract: Ensemble systems appear frequently in many engineering applications and, as a result, they have become an important research topic in control theory. These systems are best characterized by the evolution of their underlying state distribution. Despite the work to date, few results exist dealing with the problem of directly modifying (i.e., ``steering'') the distribution of an ensemble system. In addition, in most existing results, the distribution of the states of an ensemble of discrete-time systems is assumed to be Gaussian. However, in case the system parameters are uncertain, it is not always realistic to assume that the distribution of the system follows a Gaussian distribution, thus complicating the solution of the overall problem. In this paper, we address the general distribution steering problem for first-order discrete-time ensemble systems, where the distributions of the system parameters and the states are arbitrary with finite first few moments. Linear system dynamics are considered using the method of power moments to transform the original infinite-dimensional problem into a finite-dimensional one. We also propose a control law for the ensuing moment system, which allows us to obtain the power moments of the desired control inputs. Finally, we solve the inverse problem to obtain the feasible control inputs from their corresponding power moments. We provide a numerical example to validate our theoretical developments. - oai:arXiv.org:2405.12415v2 - math.OC - cs.SY - eess.SY - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Guangyu Wu, Panagiotis Tsiotras, Anders Lindquist - - - On non-topologizable semigroups - https://arxiv.org/abs/2405.16992 - arXiv:2405.16992v3 Announce Type: replace -Abstract: We find anti-isomorphic submonoids $\mathscr{C}_{+}(a,b)$ and $\mathscr{C}_{-}(a,b)$ of the bicyclic monoid $\mathscr{C}(a,b)$ with the following properties: every Hausdorff left-continuous (right-continuous) topology on $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) is discrete and there exists a compact Hausdorff topological monoid $S$ which contains $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) as a submonoid. Also, we construct a non-discrete right-continuous (left-continuous) topology $\tau_p^+$ ($\tau_p^-$) on the semigroup $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) which is not left-continuous (right-continuous). - oai:arXiv.org:2405.16992v3 - math.GR - math.GN - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.30970/vmm.2024.96.025-036 - Visnyk of the Lviv University. Series Mechanics and Mathematics 96 (2024), 25-36 - Oleg Gutik - - - CHANI: Correlation-based Hawkes Aggregation of Neurons with bio-Inspiration - https://arxiv.org/abs/2405.18828 - arXiv:2405.18828v2 Announce Type: replace -Abstract: The present work aims at proving mathematically that a neural network inspired by biology can learn a classification task thanks to local transformations only. In this purpose, we propose a spiking neural network named CHANI (Correlation-based Hawkes Aggregation of Neurons with bio-Inspiration), whose neurons activity is modeled by Hawkes processes. Synaptic weights are updated thanks to an expert aggregation algorithm, providing a local and simple learning rule. We were able to prove that our network can learn on average and asymptotically. Moreover, we demonstrated that it automatically produces neuronal assemblies in the sense that the network can encode several classes and that a same neuron in the intermediate layers might be activated by more than one class, and we provided numerical simulations on synthetic dataset. This theoretical approach contrasts with the traditional empirical validation of biologically inspired networks and paves the way for understanding how local learning rules enable neurons to form assemblies able to represent complex concepts. - oai:arXiv.org:2405.18828v2 - math.ST - stat.ML - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sophie Jaffard (CSBD, MPI-CBG), Samuel Vaiter (CNRS, LJAD), Patricia Reynaud-Bouret (CNRS, LJAD) - - - Uniform Resolvent Estimates for Subwavelength Resonators: The Minnaert Bubble Case - https://arxiv.org/abs/2406.02192 - arXiv:2406.02192v4 Announce Type: replace -Abstract: Subwavelength resonators are small scaled objects that exhibit contrasting medium properties (eigher in intensity or sign) while compared to the ones of a uniform background. Such contrasts allow them to resonate at specific frequencies. There are two ways to mathematically define these resonances. First, as the frequencies for which the related system of integral equations is not injective. Second, as the frequencies for which the related resolvent operator of the natural Hamiltonian, given by the wave-operator, has a pole. - In this work, we consider, as the subwavelength resonator, the Minneart bubble. We show that these two mentioned definitions are equivalent. Most importantly, - 1. we derive the related resolvent estimates which are uniform in terms of the size/contrast of the resonators. As a by product, we show that the resolvent operators have no scattering resonances in the upper half complex plane while they exhibit two scattering resonances in the lower half plane which converge to the real axis, as the size of the bubble tends to zero. As these resonances are poles of the natural Hamiltonian, given by the wave-operator, and have the Minnaert frequency as their dominating real part, this justifies calling them Minnaert resonances. - 2. we derive the asymptotic estimates of the generated scattered fields which are uniform in terms of the incident frequency and which are valid everywhere in space (i.e. inside or outside the bubble). - The dominating parts, for both the resolvent operator and the scattered fields, are given by the ones of the point-scatterer supported at the location of the bubble. In particular, these dominant parts are non trivial (not the same as those of the background medium) if and only if the used incident frequency identifies with the Minnaert one. - oai:arXiv.org:2406.02192v4 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Long Li, Mourad Sini - - - Solutions to the exercises from the book "Albert algebras over commutative rings" - https://arxiv.org/abs/2406.02933 - arXiv:2406.02933v3 Announce Type: replace -Abstract: This document presents the solutions to the exercises in the book "Albert algebras over commutative rings" published by Cambridge University Press, 2024, as well as errata and addenda. - oai:arXiv.org:2406.02933v3 - math.RA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Skip Garibaldi, Holger P. Petersson, Michel L. Racine - - - Asymptotics for $t$-Core Partitions and Stanton's Conjecture - https://arxiv.org/abs/2406.02982 - arXiv:2406.02982v3 Announce Type: replace -Abstract: A partition is a $t$-core partition if $t$ is not one of its hook lengths. Let $c_t(N)$ be the number of $t$-core partitions of $N$. In 1999, Stanton conjectured $c_t(N) \le c_{t+1}(N)$ if $4 \le t \ne N-1$. This was proved for $t$ fixed and $N$ sufficiently large by Anderson, and for small values of $t$ by Kim and Rouse. In this paper, we prove Stanton's conjecture in general. Our approach is to find a saddle point asymptotic formula for $c_t(N)$, valid in all ranges of $t$ and $N$. This includes the known asymptotic formulas for $c_t(N)$ as special cases, and shows that the behavior of $c_t(N)$ depends on how $t^2$ compares in size to $N$. For example, our formula implies that if $t^2 = \kappa N + o(t)$, then $c_t(N) = \frac{\exp\left(2\pi\sqrt{A N}\right)}{B N} (1 + o(1))$ for suitable constants $A$ and $B$ defined in terms of $\kappa$. - oai:arXiv.org:2406.02982v3 - math.NT - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Matthew Tyler - - - Exact formulae for ranks of partitions - https://arxiv.org/abs/2406.06294 - arXiv:2406.06294v2 Announce Type: replace -Abstract: In 2009, Bringmann arXiv:0708.0691 [math.NT] used the circle method to prove an asymptotic formula for the Fourier coefficients of rank generating functions. In this paper, we prove that Bringmann's formula, when summing up to infinity and in the case of prime modulus, gives a Rademacher-type exact formula involving sums of vector-valued Kloosterman sums. As a corollary, in another paper arXiv:2406.07469 [math.NT], we will provide a new proof of Dyson's conjectures by showing that the certain Kloosterman sums vanish. - oai:arXiv.org:2406.06294v2 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1090/tran/9539 - Transactions of the American Mathematical Society, vol. 379, no. 3, 2026, pp. 1649-1707 - Qihang Sun - - - Multivariate extreme values for dynamical systems - https://arxiv.org/abs/2406.14807 - arXiv:2406.14807v3 Announce Type: replace -Abstract: We establish a theory for multivariate extreme value analysis of dynamical systems. Namely, we provide conditions adapted to the dynamical setting which enable the study of dependence between extreme values of the components of $\R^d$-valued observables evaluated along the orbits of the systems. We study this cross-sectional dependence, which results from the combination of a spatial and a temporal dependence structures. We give several illustrative applications, where concrete systems and dependence sources are introduced and analysed. - oai:arXiv.org:2406.14807v3 - math.DS - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Romain Aimino, Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, Mike Todd - - - Geometric structures for maximal representations and pencils - https://arxiv.org/abs/2407.01254 - arXiv:2407.01254v2 Announce Type: replace -Abstract: We study fibrations of the projective model for the symmetric space associated with $\text{SL}(2n,\mathbb{R})$ by codimension $2$ projective subspaces, or pencils of quadrics. In particular we show that if such a smooth fibration is equivariant with respect to a representation of a closed surface group, the representation is quasi-isometrically embedded, and even Anosov if the pencils in the image contain only non-degenerate quadrics. We use this to characterize maximal representations among representations of a closed surface group into $\text{Sp}(2n,\mathbb{R})$ by the existence of an equivariant continuous fibration of the associated symmetric space, satisfying an additional technical property. These fibrations extend to fibrations of the projective structures associated to maximal representations by bases of pencils of quadrics. - oai:arXiv.org:2407.01254v2 - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Colin Davalo - - - Heights of Ceresa and Gross-Schoen cycles - https://arxiv.org/abs/2407.01304 - arXiv:2407.01304v4 Announce Type: replace -Abstract: We study the Beilinson-Bloch heights of Ceresa and Gross-Schoen cycles in families. We construct that for any $g\ge 3$, a Zariski open dense subset $\mathcal{M}_g^{\mathrm{amp}}$ of $\mathcal{M}_g$, the coarse moduli of curves of genus $g$ over $\mathbb{Q}$, such that the heights of Ceresa cycles and Gross-Schoen cycles over $\mathcal{M}_g^{\mathrm{amp}}$ have a lower bound and satisfy the Northcott property. - oai:arXiv.org:2407.01304v4 - math.AG - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ziyang Gao, Shou-Wu Zhang - - - A uniform-in-time nonlocal approximation of the standard Fokker-Planck equation - https://arxiv.org/abs/2407.03870 - arXiv:2407.03870v4 Announce Type: replace -Abstract: We study a nonlocal approximation of the Fokker-Planck equation in which we can estimate the speed of convergence to equilibrium in a way which does not degenerate as we approach the local limit of the equation. This uniform estimate cannot be easily obtained with standard inequalities or entropy methods, but can be obtained through the use of Harris's theorem, finding interesting links to quantitative versions of the central limit theorem in probability. As a consequence one can prove that solutions of this nonlocal approximation converge to solutions of the usual Fokker-Planck equation uniformly in time-hence we show the approximation is asymptotic-preserving in this sense. The associated equilibrium has some interesting tail and regularity properties, which we also study. - oai:arXiv.org:2407.03870v4 - math.AP - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.3934/dcds.2025103 - DCDS 2026, Volume 46: 348-386 - Jos\'e A. Ca\~nizo, Niccol\`o Tassi - - - Lipschitz regularity for solutions to an orthotropic $q$-Laplacian-type equation in the Heisenberg group - https://arxiv.org/abs/2407.07548 - arXiv:2407.07548v3 Announce Type: replace -Abstract: We establish the local Lipschitz regularity for solutions to an orthotropic q-Laplacian-type equation within the Heisenberg group. Our approach is largely inspired by the works of X. Zhong, who investigated the q-Laplacian in the same setting and proved the H\"older regularity for the gradient of solutions. Due to the degeneracy of the current equation, such regularity for the gradient of solutions is not even known in the Euclidean setting for dimensions greater than 2, where only boundedness is expected. - oai:arXiv.org:2407.07548v3 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Advances in Differential Equations, 2026 - Michele Circelli, Giovanna Citti, Albert Clop - - - An Adaptive Proximal ADMM for Nonconvex Linearly Constrained Composite Programs - https://arxiv.org/abs/2407.09927 - arXiv:2407.09927v3 Announce Type: replace -Abstract: This paper develops an adaptive proximal alternating direction method of multipliers (ADMM) for solving linearly constrained, composite optimization problems under the assumption that the smooth component of the objective is weakly convex, while the non-smooth component is convex and block-separable. The proposed method is adaptive to all problem parameters, including smoothness and weak convexity constants, and allows each of its block proximal subproblems to be inexactly solved. Each iteration of our adaptive proximal ADMM consists of two steps: the sequential solution of each block proximal subproblem; and adaptive tests to decide whether to perform a full Lagrange multiplier and/or penalty parameter update(s). Without any rank assumptions on the constraint matrices, it is shown that the adaptive proximal ADMM obtains an approximate first-order stationary point of the constrained problem in a number of iterations that matches the state-of-the-art complexity for the class of proximal ADMM's. The three proof-of-concept numerical experiments that conclude the paper suggest our adaptive proximal ADMM enjoys significant computational benefits. - oai:arXiv.org:2407.09927v3 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Leandro Farias Maia, David H. Gutman, Renato D. C. Monteiro, Gilson N. Silva - - - A short nonstandard proof of the Spectral Theorem for unbounded self-adjoint operators - https://arxiv.org/abs/2407.16136 - arXiv:2407.16136v3 Announce Type: replace -Abstract: By nonstandard analysis, a very short and elementary proof of the Spectral Theorem for unbounded self-adjoint operators is given. - oai:arXiv.org:2407.16136v3 - math.SP - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Takashi Matsunaga - - - Winners with Confidence: Discrete Argmin Inference with an Application to Model Selection - https://arxiv.org/abs/2408.02060 - arXiv:2408.02060v4 Announce Type: replace -Abstract: We study the problem of finding the index of the minimum value of a vector from noisy observations. This problem is relevant in population/policy comparison, discrete maximum likelihood, and model selection. We develop an asymptotically normal test statistic, even in high-dimensional settings and with potentially many ties in the population mean vector, by integrating concepts and tools from cross-validation and differential privacy. The key technical ingredient is a central limit theorem for globally dependent data. We also propose practical ways to select the tuning parameter that adapts to the signal landscape. Numerical experiments and data examples demonstrate the ability of the proposed method to achieve a favorable bias-variance trade-off in practical scenarios. - oai:arXiv.org:2408.02060v4 - math.ST - stat.ME - stat.ML - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Tianyu Zhang, Hao Lee, Jing Lei - - - Long-time behaviour of a multidimensional age-dependent branching process with a singular jump kernel modelling telomere shortening - https://arxiv.org/abs/2408.02476 - arXiv:2408.02476v2 Announce Type: replace -Abstract: In this article, we investigate the ergodic behaviour of a multidimensional age-dependent branching process with a singular jump kernel, motivated by studying the phenomenon of telomere shortening in cell populations. Our model tracks individuals evolving within a continuous-time framework indexed by a binary tree, characterised by age and a multidimensional trait. Branching events occur with rates dependent on age, where offspring inherit traits from their parent with random increase or decrease in some coordinates, while the most of them are left unchanged. Exponential ergodicity is obtained at the cost of an exponential normalisation, despite the fact that we have an unbounded age-dependent birth rate that may depend on the multidimensional trait, and a non-compact transition kernel. These two difficulties are respectively treated by stochastically comparing our model to Bellman-Harris processes, and by using a weak form of a Harnack inequality. We conclude this study by giving examples where the assumptions of our main result are verified. - oai:arXiv.org:2408.02476v2 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Electronic Journal of Probability, 2026, 31 - Jules Olay\'e (IMT), Milica Tomasevic (CMAP, MERGE) - - - Poisson Approximation of prime divisors of shifted primes - https://arxiv.org/abs/2408.03803 - arXiv:2408.03803v4 Announce Type: replace -Abstract: We develop an analog for shifted primes of the Kubilius model of prime factors of integers. We prove a total variation distance estimate for the difference between the model and actual prime factors of shifted primes, and apply it to show that the prime factors of shifted primes in disjoint sets behave like independent Poisson variables. As a consequence, we establish a transference principle between the anatomy of random integers up to x and of random shifted primes p+a with p < x. - oai:arXiv.org:2408.03803v4 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kevin Ford - - - On a family of arithmetic series related to the M\"obius function - https://arxiv.org/abs/2409.02754 - arXiv:2409.02754v5 Announce Type: replace -Abstract: Let $P^-(n)$ denote the smallest prime factor of a natural integer $n>1$. Furthermore let $\mu$ and $\omega$ denote respectively the M\"obius function and the number of distinct prime factors function. We show that, given any set ${{\scr P}}$ of prime numbers with a natural density, we have $\sum_{P^-(n)\in \scr P}\mu(n)\omega(n)/n=0$ and provide a effective estimate for the rate of convergence. This extends a recent result of Alladi and Johnson, who considered the case when ${\scr P}$ is an arithmetic progression. - oai:arXiv.org:2409.02754v5 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - G\'erald Tenenbaum - - - Counting points on generic character varieties - https://arxiv.org/abs/2409.04735 - arXiv:2409.04735v3 Announce Type: replace -Abstract: We count points on character varieties associated with punctured surfaces and regular semisimple generic conjugacy classes in reductive groups. We find that the number of points are palindromic polynomials. This suggests a $P=W$ conjecture for these varieties. We also count points on the corresponding additive character varieties and find that the number of points are also polynomials, which we conjecture have non-negative coefficients. These polynomials can be considered as the reductive analogues of the Kac polynomials of comet-shaped quivers. - oai:arXiv.org:2409.04735v3 - math.AG - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Masoud Kamgarpour, GyeongHyeon Nam, Bailey Whitbread, Stefano Giannini - - - A Complexity Dichotomy for Temporal Valued Constraint Satisfaction Problems - https://arxiv.org/abs/2409.07285 - arXiv:2409.07285v2 Announce Type: replace -Abstract: We study the complexity of the valued constraint satisfaction problem (VCSP) for every valued structure with the domain ${\mathbb Q}$ that is preserved by all order-preserving bijections. Such VCSPs will be called temporal, in analogy to the (classical) constraint satisfaction problem: a relational structure is preserved by all order-preserving bijections if and only if all its relations have a first-order definition in $({\mathbb Q};<)$, and the CSPs for such structures are called temporal CSPs. Many optimization problems that have been studied intensively in the literature can be phrased as a temporal VCSP. We prove that a temporal VCSP is in P, or NP-complete. Our analysis uses the concept of fractional polymorphisms. This is the first dichotomy result for VCSPs over infinite domains which is complete in the sense that it treats all valued structures with a given automorphism group. - oai:arXiv.org:2409.07285v2 - math.LO - cs.CC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Manuel Bodirsky, \'Edouard Bonnet, \v{Z}aneta Semani\v{s}inov\'a - - - Matrix perturbation analysis of methods for extracting singular values from approximate singular subspaces - https://arxiv.org/abs/2409.09187 - arXiv:2409.09187v2 Announce Type: replace -Abstract: Given (orthonormal) approximations $\tilde{U}$ and $\tilde{V}$ to the left and right subspaces spanned by the leading singular vectors of a matrix $A$, we discuss methods to approximate the leading singular values of $A$ and study their accuracy. In particular, we focus our analysis on the generalized Nystr\"om approximation, as surprisingly, it is able to obtain significantly better accuracy than classical methods, namely Rayleigh-Ritz and (one-sided) projected SVD. - A key idea of the analysis is to view the methods as finding the exact singular values of a perturbation of $A$. In this context, we derive a matrix perturbation result that exploits the structure of such $2\times2$ block matrix perturbation. Furthermore, we extend it to block tridiagonal matrices. We then obtain bounds on the accuracy of the extracted singular values. This leads to sharp bounds that predict well the approximation error trends and explain the difference in the behavior of these methods. Finally, we present an approach to derive an a-posteriori version of those bounds, which are more amenable to computation in practice. - oai:arXiv.org:2409.09187v2 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Lorenzo Lazzarino, Hussam Al Daas, Yuji Nakatsukasa - - - Averaging theory and catastrophes - https://arxiv.org/abs/2409.11054 - arXiv:2409.11054v2 Announce Type: replace -Abstract: When a dynamical system is subject to a periodic perturbation, the averaging method can be applied to obtain an autonomous leading order "guiding system", placing the time dependence at higher orders. Recent research focused on investigating invariant structures in non-autonomous differential systems arising from hyperbolic structures in the guiding system, such as periodic orbits and invariant tori. Complementarily, the effect that bifurcations in the guiding system have on the original non-autonomous one has also been recently explored, albeit less frequently. This paper extends this study by providing a broader description of the dynamics that can emerge from non-hyperbolic structures of the guiding system. Specifically, we prove here that $\mathcal{K}$-universal bifurcations in the guiding system `persist' in the original non-autonomous one, while non-versal bifurcations, such as the transcritical and pitchfork, do not. We illustrate the results on examples of a fold, a transcritical, a pitchfork, and a saddle-focus. - oai:arXiv.org:2409.11054v2 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.jde.2026.114126 - Pedro C. C. R. Pereira, Mike R. Jeffrey, Douglas D. Novaes - - - QMC integration based on arbitrary (t,m,s)-nets yields optimal convergence rates on several scales of function spaces - https://arxiv.org/abs/2409.12879 - arXiv:2409.12879v2 Announce Type: replace -Abstract: We study the integration problem over the $s$-dimensional unit cube on four types of Banach spaces of integrands. First we consider Haar wavelet spaces, consisting of functions whose Haar wavelet coefficients exhibit a certain decay behavior measured by a parameter $\alpha >0$. We study the worst case error of integration over the norm unit ball and provide upper error bounds for quasi-Monte Carlo (QMC) cubature rules based on arbitrary $(t,m,s)$-nets as well as matching lower error bounds for arbitrary cubature rules. These results show that using arbitrary $(t,m,s)$-nets as sample points yields the best possible rate of convergence. Afterwards we study spaces of integrands of fractional smoothness $\alpha \in (0,1)$ and state a sharp Koksma-Hlawka-type inequality. More precisely, we show that on those spaces the worst case error of integration is equal to the corresponding fractional discrepancy. Those spaces can be continuously embedded into tensor product Bessel potential spaces, also known as Sobolev spaces of dominated mixed smoothness, with the same set of parameters. The latter spaces can be embedded into suitable Besov spaces of dominating mixed smoothness $\alpha$, which in turn can be embedded into the Haar wavelet spaces with the same set of parameters. Therefore our upper error bounds on Haar wavelet spaces for QMC cubatures based on $(t,m,s)$-nets transfer (with possibly different constants) to the corresponding spaces of integrands of fractional smoothness and to Sobolev and Besov spaces of dominating mixed smoothness. Moreover, known lower error bounds for periodic Sobolev and Besov spaces of dominating mixed smoothness show that QMC integration based on arbitrary $(t,m,s)$-nets yields the best possible convergence rate on periodic as well as on non-periodic Sobolev and Besov spaces of dominating smoothness. - oai:arXiv.org:2409.12879v2 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Michael Gnewuch, Josef Dick, Lev Markhasin, Winfried Sickel - - - D2D Coded Caching from Two Classes of Optimal DPDAs using Cross Resolvable Designs - https://arxiv.org/abs/2409.14350 - arXiv:2409.14350v2 Announce Type: replace -Abstract: Device to device (D2D) communication is one of the most promising techniques for fifth-generation and beyond wireless communication systems. This paper considers coded caching in a wireless D2D network, in which a central server initially places the data in the user cache memories, and all user demands are served through inter-user coded multicast transmissions. D2D placement delivery array (DPDA) was proposed as a tool for designing coded caching schemes with reduced subpacketization levels in a D2D network. In this paper, we first constructed three classes of DPDAs using a cross resolvable design, a group divisible design, and a newly developed block design. The resulting D2D schemes achieve low subpacketization levels while meeting the known lower bound on the transmission load of a DPDA. These classes of constructed DPDAs either simplify or generalize all existing DPDA constructions that achieve the known lower bound and have low subpacketization levels. Furthermore, a new lower bound on the transmission load of a DPDA is proposed. Two new classes of DPDAs are then constructed using a cross resolvable design and a newly developed block design, respectively. These constructions yield low-subpacketization D2D schemes and achieve the proposed lower bound on the transmission load. Compared to existing schemes with the same system parameters as those obtained from the proposed DPDAs, the proposed schemes have an advantage in either transmission load or subpacketization level or both. - oai:arXiv.org:2409.14350v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rashid Ummer N. T., B. Sundar Rajan - - - Crystallinity for syntomic cohomology, \'etale cohomology, and algebraic $K$-theory - https://arxiv.org/abs/2409.20543 - arXiv:2409.20543v2 Announce Type: replace -Abstract: We prove for $n\geq c-1$ that the functor taking an animated ring $R$ to its mod $(p^c,v_1^{p^n})$ syntomic cohomology factors through the functor $R \mapsto R/p^{c(n+2)}$, a phenomenon we term crystallinity for mod $(p^c,v_1^{p^n})$ syntomic cohomology. As an application, we completely and explicitly compute the mod $(p,v_1 ^{p^{n}-1})$ algebraic $K$-theory of $\mathbb Z/p^{k}$ whenever $k \geq n+2$ and $p>2$. As a second application, we deduce crystallinity for the mod $p^c$ syntomic complexes associated to smooth $p$-adic formal schemes, and in particular for the Galois equivariant mod $p^c$ \'etale cohomologies of their adic generic fibers. Finally, we strengthen known $p$-adic convergence theorems for the topological Hochschild homology of ring spectra, and as a result relate crystallinity for algebraic $K$-theory to Lichtenbaum--Quillen theorems. - oai:arXiv.org:2409.20543v2 - math.KT - math.AG - math.AT - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jeremy Hahn, Ishan Levy, Andrew Senger - - - Entropy contraction of the Gibbs sampler under log-concavity - https://arxiv.org/abs/2410.00858 - arXiv:2410.00858v2 Announce Type: replace -Abstract: The Gibbs sampler (a.k.a. Glauber dynamics and heat-bath algorithm) is a popular Markov Chain Monte Carlo algorithm which iteratively samples from the conditional distributions of a probability measure $\pi$ of interest. Under the assumption that $\pi$ is strongly log-concave, we show that the random scan Gibbs sampler contracts in relative entropy and provide a sharp characterization of the associated contraction rate. Assuming that evaluating conditionals is cheap compared to evaluating the joint density, our results imply that the number of full evaluations of $\pi$ needed for the Gibbs sampler to mix grows linearly with the condition number and is independent of the dimension. If $\pi$ is non-strongly log-concave, the convergence rate in entropy degrades from exponential to polynomial. Our techniques are versatile and extend to Metropolis-within-Gibbs schemes and the Hit-and-Run algorithm. A comparison with gradient-based schemes and the connection with the optimization literature are also discussed. - oai:arXiv.org:2410.00858v2 - math.PR - math.ST - stat.CO - stat.ML - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Filippo Ascolani, Hugo Lavenant, Giacomo Zanella - - - The Conjecture of Dixmier for the first Weyl algebra is true - https://arxiv.org/abs/2410.06959 - arXiv:2410.06959v5 Announce Type: replace -Abstract: Let $K$ be a field of characteristic zero, let $A_1=K[x][\partial ]$ be the first Weyl algebra. In this paper we prove that the Dixmier conjecture for the first Weyl algebra is true, i.e. each algebra endomorphism of the algebra $A_1$ is an automorphism. - oai:arXiv.org:2410.06959v5 - math.RA - math-ph - math.AG - math.MP - math.QA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Alexander Zheglov - - - Uniform Space and Time Behavior for Acoustic Resonators - https://arxiv.org/abs/2410.09630 - arXiv:2410.09630v2 Announce Type: replace -Abstract: We deal with the time-domain acoustic wave propagation in the presence of a subwavelength resonator given by a Minneart bubble. This bubble is small scaled and enjoys high contrasting mass density and bulk modulus. It is well known that, under certain regimes between these scales, such a bubble generates a single low-frequency (or subwavelength) resonance called Minnaert resonance. In this paper, we study the wave propagation governed by Minnaert resonance effects in time domain. We derive the point-approximation expansion of the wave field. The dominant part is a sum of two terms. - 1. The first one, which we call the primary wave, is the wave field generated in the absence of the bubble. - 2. The second one, which we call the resonant wave, is generated by the interaction between the bubble and the background. It is related to a Dirac-source, in space, that is modulated, in time, with a coefficient which is a solution of a $1$D Cauchy problem, for a second order differential equation, having as propagation and attenuation parameters the real and the imaginary parts, respectively, of the Minnaert resonance. - We show that the evolution of the resonant wave remains valid for a large time of the order $\epsilon^{-1}$, where $\epsilon$ is the radius of the bubble, after which it collapses by exponentially decaying. Precisely, we confirm that such resonant wave have life-time inversely proportional to the imaginary part of the related subwavelength resonances, which is in our case given by the Minnaert one. In addition, the real part of this resonance fixes the period of the wave. - oai:arXiv.org:2410.09630v2 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Long Li, Mourad Sini - - - Safety on the Fly: Constructing Robust Safety Filters via Policy Control Barrier Functions at Runtime - https://arxiv.org/abs/2410.11157 - arXiv:2410.11157v3 Announce Type: replace -Abstract: Control Barrier Functions (CBFs) have proven to be an effective tool for performing safe control synthesis for nonlinear systems. However, guaranteeing safety in the presence of disturbances and input constraints for high relative degree systems is a difficult problem. In this work, we propose the Robust Policy CBF (RPCBF), a practical approach for constructing robust CBF approximations online via the estimation of a value function. We establish conditions under which the approximation qualifies as a valid CBF and demonstrate the effectiveness of the RPCBF-safety filter in simulation on a variety of high relative degree input-constrained systems. Finally, we demonstrate the benefits of our method in compensating for model errors on a hardware quadcopter platform by treating the model errors as disturbances. Website including code: www.oswinso.xyz/rpcbf/ - oai:arXiv.org:2410.11157v3 - math.OC - cs.RO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1109/LRA.2025.3597847 - IEEE Robotics and Automation Letters, Volume 10, Issue 10, 2025, pages 10058-10065 - Luzia Knoedler, Oswin So, Ji Yin, Mitchell Black, Zachary Serlin, Panagiotis Tsiotras, Javier Alonso-Mora, Chuchu Fan - - - Subspace method based on neural networks for eigenvalue problems - https://arxiv.org/abs/2410.13358 - arXiv:2410.13358v2 Announce Type: replace -Abstract: In this paper, we propose a subspace method based on neural networks for eigenvalue problems with high accuracy and low cost. We first construct a neural network-based orthogonal basis by some deep learning method and dimensionality reduction technique, and then calculate the Galerkin projection of the eigenvalue problem onto the subspace spanned by the orthogonal basis and obtain an approximate solution. Numerical experiments show that we can obtain approximate eigenvalues and eigenfunctions with very high accuracy but low cost. - oai:arXiv.org:2410.13358v2 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xiaoying Dai, Yunying Fan, Zhiqiang Sheng - - - Two-stage Online Reusable Resource Allocation: Reservation, Overbooking and Confirmation Call - https://arxiv.org/abs/2410.15245 - arXiv:2410.15245v2 Announce Type: replace -Abstract: We study a two-stage online reusable resource allocation problem over T days involving advance reservations and walk-ins. Each day begins with a reservation stage (Stage I), where reservation requests arrive sequentially. When service starts (Stage II), both reserved and walk-in customers arrive to check in and occupy resources for several days. Reserved customers can cancel without penalty before or during a confirmation call initiated by the decision maker (DM) before day's end. The DM must immediately accept or reject each booking or check-in request, potentially overbooking by accepting more reservations than capacity. An overbooking loss occurs if a reserved customer's check-in is rejected in Stage II; a reward is obtained for each occupied resource unit daily. Our goal is to develop an online policy that controls bookings and check-ins to maximize total revenue over the T-day horizon. We show that due to cancellation uncertainties and complex correlations between occupancy durations, any online policy incurs a regret of \Omega(T) compared to the offline optimal policy when the \textit{busy season} assumption does not hold. To address this, we introduce decoupled adaptive safety stocks, which use only single-day information to hedge against overbooking risks and reduce resource idling. Under the busy season condition, our policy decouples the overall offline optimal into single-day offline optimal policies. Consequently, the regret between our policy and the offline optimal decays exponentially with the time between the confirmation call and day's end, suggesting the DM can delay confirmation calls while maintaining near-optimal performance. We validate our algorithm through sythetic experiments and empirical data from an Algarve resort hotel. - oai:arXiv.org:2410.15245v2 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ruicheng Ao, Hengyu Fu, David Simchi-levi - - - Entrance boundary for standard processes with no negative jumps and its application to exponential convergence to the Yaglom limit - https://arxiv.org/abs/2410.15447 - arXiv:2410.15447v3 Announce Type: replace -Abstract: We study standard processes with no negative jumps under the entrance boundary condition. Similarly to one-dimensional diffusions, we show that the process can be made into a Feller process by attaching the boundary point to the state space. We investigate the spectrum of the infinitesimal generator in detail via the scale function, characterizing it as the zeros of an entire function. As an application, we prove that under the strong Feller property, the convergence to the Yaglom limit of the process killed on hitting the boundary is exponentially fast. - oai:arXiv.org:2410.15447v3 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kosuke Yamato - - - Information-Based Martingale Optimal Transport - https://arxiv.org/abs/2410.16339 - arXiv:2410.16339v3 Announce Type: replace -Abstract: Randomised arcade processes are a class of continuous stochastic processes that interpolate in a strong sense, i.e., omega by omega, between any given ordered set of random variables, at fixed pre-specified times. Utilising these processes as generators of partial information, a class of continuous-time martingale -- the filtered arcade martingales (FAMs) -- are constructed. FAMs interpolate through a sequence of target random variables, which form a discrete-time martingale. The research presented in this paper relaxes the FAM setting to the interpolation between probability measures instead and treats the problem of selecting the worst martingale coupling for given, convexly ordered, probability measures contingent on the paths of FAMs that are constructed using the martingale coupling. This optimisation problem, that we term the information-based martingale optimal transport problem (IB-MOT), can be viewed from different perspectives. It can be understood as a model-free construction of FAMs, in the case where the coupling is not determined a priori. It can also be considered from the vantage point of optimal transport (OT), where the problem is concerned with introducing a noise factor in martingale optimal transport, similarly to how the entropic regularisation of optimal transport introduces noise in OT. The IB-MOT problem is static in its nature, since its aim is to find a coupling. However, a corresponding dynamical solution can be found by considering the FAM constructed with the identified optimal coupling. The existence and uniqueness of its solution are shown and an algorithm for empirical measures is proposed. - oai:arXiv.org:2410.16339v3 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Georges Kassis, Andrea Macrina - - - Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Approximations and regularity - https://arxiv.org/abs/2410.20370 - arXiv:2410.20370v3 Announce Type: replace -Abstract: We study various regularization operators on plurisubharmonic functions that preserve Lelong classes with growth given by certain compact convex sets. The purpose is to show that the weighted Siciak-Zakharyuta functions associated with these Lelong classes are lower semicontinuous. These operators are given by integral, infimal, and supremal convolutions. Continuity properties of the logarithmic supporting function are studied and a precise description is given of when it is uniformly continuous. This gives a contradiction to published results about the H\"older continuity of these Siciak-Zakharyuta functions. - oai:arXiv.org:2410.20370v3 - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bergur Snorrason - - - Erd\H{o}s-P\'osa property of $A$-paths in unoriented group-labelled graphs - https://arxiv.org/abs/2411.05372 - arXiv:2411.05372v2 Announce Type: replace -Abstract: We characterize the obstructions to the Erd\H{o}s-P\'osa property of $A$-paths in unoriented group-labelled graphs. As a result, we prove that for every finite abelian group $\Gamma$ and for every subset $\Lambda$ of $\Gamma$, the family of $\Gamma$-labelled $A$-paths whose lengths are in $\Lambda$ satisfies the half-integral Erd\H{o}s-P\'osa property. Moreover, we give a characterization of such $\Gamma$ and $\Lambda\subseteq\Gamma$ for which the same family of $A$-paths satisfies the full Erd\H{o}s-P\'osa property. - oai:arXiv.org:2411.05372v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - O-joung Kwon, Youngho Yoo - - - Torsion and semi-degeneracy of second-order maximally superintegrable systems - https://arxiv.org/abs/2411.06994 - arXiv:2411.06994v3 Announce Type: replace -Abstract: The isotropic harmonic oscillator and the Kepler-Coulomb system are pivotal models in the Sciences. They are two examples of second-order (maximally) superintegrable (Hamiltonian) systems. These systems are classified in dimension two. A partial classification exists in dimension three. - In this paper, our focus is on second-order superintegrable systems with a $(n+1)$-parameter potential with $n\geq3$. We find that these systems are underpinned by an information-geometric structure, namely the structure of a statistical manifold with torsion. - We obtain a necessary and sufficient condition for such systems to extend to non-degenerate systems, i.e. to admit a maximal family of compatible potentials. The condition is geometric: we show that a $(n+1)$-parameter potential is the restriction of a non-degenerate potential if and only if a certain trace-free tensor field vanishes. We interpret this condition as the requirement that a certain affine connection has vectorial torsion. - We also show that the condition for a system to be extendable is conformally invariant, allowing us to extend our results to second-order conformally superintegrable systems with a $(n+1)$-parameter potential. - oai:arXiv.org:2411.06994v3 - math.DG - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jeremy Nugent, Andreas Vollmer - - - Accelerating Benders decomposition for solving a sequence of sample average approximation replications - https://arxiv.org/abs/2411.09091 - arXiv:2411.09091v2 Announce Type: replace -Abstract: Sample average approximation (SAA) is a technique for obtaining approximate solutions to stochastic programs that uses the average from a random sample to approximate the expected value that is being optimized. Since the outcome from solving an SAA is random, statistical estimates on the optimal value of the true problem can be obtained by solving multiple SAA replications with independent samples. We study techniques to accelerate the solution of this set of SAA replications, when solving them sequentially via Benders decomposition. We investigate how to exploit similarities in the problem structure, as the replications just differ in the realizations of the random samples. Our extensive computational experiments provide empirical evidence that our techniques for using information from solving previous replications can significantly reduce the solution time of later replications. - oai:arXiv.org:2411.09091v2 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Harshit Kothari, James R. Luedtke - - - Harmonic forms on ALE Ricci-flat 4-manifolds - https://arxiv.org/abs/2411.09561 - arXiv:2411.09561v4 Announce Type: replace -Abstract: In this paper, we compute the expansion of some harmonic functions and 1-forms on ALE Ricci-flat 4-manifolds. - oai:arXiv.org:2411.09561v4 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gao Chen, Hao Yan - - - Equivalent spectral theory for fundamental graph cut problems - https://arxiv.org/abs/2411.11077 - arXiv:2411.11077v2 Announce Type: replace -Abstract: We introduce and develop equivalent spectral graph theory for several fundamental graph cut problems including maxcut, mincut, Cheeger cut, anti-Cheeger cut, dual Cheeger problem and their useful variants. A specified strategy for achieving an equivalent eigenproblem is proposed for a general graph cut problem via the set-pair Lov\'asz extension and the Dinkelbach scheme. For a class of 2-cut and 3-cut problems, we reveal the intrinsic difference-of-submodularity for the fractional formulations and show that their set-pair Lov\'asz extensions yield equivalent difference-of-convex structures. Building on the Dinkelbach scheme, we finally establish a unified research roadmap for nonlinear spectral theory that provides a one-to-one correspondence between certain eigenpairs and the optimal graph cut problems. The finer structure of the eigenvectors, the Courant nodal domain theorem and the graphic feature of eigenvalues are studied systematically in the setting of these new nonlinear eigenproblems. - oai:arXiv.org:2411.11077v2 - math.CO - math.SP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sihong Shao, Chuan Yang, Dong Zhang, Weixi Zhang - - - A star is born: Explosive Crump-Mode-Jagers branching processes - https://arxiv.org/abs/2411.18749 - arXiv:2411.18749v2 Announce Type: replace -Abstract: We study a family of Crump--Mode--Jagers branching processes in random environment that explode, i.e. that grow infinitely large in finite time with positive probability. Building on recent work of the author and Iyer (``On the structure of genealogical trees associated with explosive Crump--Mode--Jagers branching processes", arXiv:2311.14664, 2023), we weaken certain assumptions required to prove that the branching process, at the time of explosion, contains a (unique) individual with infinite offspring. We then apply these results to super-linear preferential attachment models. In particular, we fill gaps in some of the cases analysed in Appendix A of the work of the author and Iyer and study a large range of previously unattainable cases. - oai:arXiv.org:2411.18749v2 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Bas Lodewijks - - - Characterization of Trees with Maximum Security - https://arxiv.org/abs/2411.19188 - arXiv:2411.19188v3 Announce Type: replace -Abstract: The rank (also known as protection number or leaf-height) of a vertex in a rooted tree is the minimum distance between the vertex and any of its leaf descendants. We consider the sum of ranks over all vertices (known as the security) in binary trees, and produce a classification of families of binary trees for which the security is maximized. In addition, extremal results relating to maximum rank among all vertices in families of trees is discussed. - oai:arXiv.org:2411.19188v3 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Alex S. A. Alochukwu, Audace A. V. Dossou-Olory, Fadekemi J. Osaye, Valisoa R. M. Rakotonarivo, Shashank Ravichandran, Sarah J. Selkirk, Hua Wang, Hays Whitlatch - - - Arithmetic level raising theorem for some unitary Shimura varieties mod $p$ - https://arxiv.org/abs/2412.03519 - arXiv:2412.03519v4 Announce Type: replace -Abstract: Let $F$ be a real quadratic field in which a fixed prime $p$ is inert, and $E_0$ be an imaginary quadratic field in which $p$ splits; put $E=E_0 F$. Let ${{\rm Sh}}_{1,n-1}$ be the special fiber over $\mathbb{F}_{p^2}$ of the Shimura variety for $G(U(1,n-1)\times U(n-1,1))$ with hyperspecial level structure at $p$ for some integer $n\geq 2$. Let ${{\rm Sh}}_{1,n-1}(K_{\mathfrak{p}}^{1})$ be the special fiber over $\mathbb{F}_{p^2}$ of a Shimura variety for $G(U(1,n-1)\times U(n-1,1))$ with parahoric level structure at $p$ for some integer $n\geq 2$. We exhibit elements in the higher Chow group of the supersingular locus of ${{\rm Sh}}_{1,n-1}$ and study the stratification of ${{\rm Sh}}_{1,n-1}.$ Moreover, we study the geometry of ${{\rm Sh}}_{1,n-1}(K_{\mathfrak{p}}^{1})$ and prove a form of Ihara lemma. With Ihara lemma, we prove the the arithmetic level raising map is surjective for $n=2,3.$ - oai:arXiv.org:2412.03519v4 - math.NT - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zijie Tao - - - Rational First Integrals and Relative Killing Tensors - https://arxiv.org/abs/2412.04151 - arXiv:2412.04151v2 Announce Type: replace -Abstract: We relate rational integrals of the geodesic flow of a (pseudo-)Riemannian metric to relative Killig tensors, describe the spaces they span and discuss upper bounds on their dimensions. - oai:arXiv.org:2412.04151v2 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Boris Kruglikov - - - MPAX: Mathematical Programming in JAX - https://arxiv.org/abs/2412.09734 - arXiv:2412.09734v3 Announce Type: replace -Abstract: We present MPAX (Mathematical Programming in JAX), an open-source first-order solver for large-scale linear programming (LP) and convex quadratic programming (QP) built natively in JAX. The primary goal of MPAX is to exploit modern machine learning infrastructure for large-scale mathematical programming, while also providing advanced mathematical programming algorithms that are easy to integrate into machine learning workflows. MPAX implements two PDHG variants, r2HPDHG for LP and rAPDHG for QP, together with diagonal preconditioning, adaptive restarts, adaptive step sizes, primal-weight updates, infeasibility detection, and feasibility polishing. Leveraging JAX's compilation and parallelization ecosystem, MPAX provides across-hardware portability, batched solving, distributed optimization, and automatic differentiation. We evaluate MPAX on CPUs, NVIDIA GPUs, and Google TPUs, observing substantial GPU speedups over CPU baselines and competitive performance relative to GPU-based codebases on standard LP/QP benchmarks. Our numerical experiments further demonstrate MPAX's capabilities in high-throughput batched solving, near-linear multi-GPU scaling for dense LPs, and efficient end-to-end differentiable training. The solver is publicly available at https://github.com/MIT-Lu-Lab/MPAX. - oai:arXiv.org:2412.09734v3 - math.OC - cs.LG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Haihao Lu, Zedong Peng, Jinwen Yang - - - Finite type as fundamental objects even non-single-valued and non-continuous - https://arxiv.org/abs/2412.11675 - arXiv:2412.11675v5 Announce Type: replace -Abstract: In this paper, inspired by the elegant work of Good and Meddaugh \cite{GM} and the graph models for zero-dimensional systems developed by several authors, like Gambaudo and Martens \cite{GM06}, Shimomura \cite{Sh14}. We try to discover a connection among some objects, such as finite directed graph, shift of finite type and shadowing property by employing the Closed Graph Theorem for multivalued maps. From the perspective of structure theorems, we demonstrate that every closed relation (multivalued map) on a compact, totally disconnected space is represented as an inverse limit of finite directed graph homomorphisms satisfying the Mittag-Leffler condition. Moreover, from dichotomy-theorem point of view, we prove that an inverse limit of finite directed graph homomorphisms possesses the shadowing property if and only if its induced space of infinite graph walks (as a shift of finite type) satisfies the Mittag-Leffler condition. As an application, a question raised by Boro\'nski, Bruin and Kucharski \cite{BBK} is also concerned. Furthermore, we show that under a multivalued dynamical system, the resulting dynamical behaviors exhibit greater diversity and counterintuitively compared to those observed in single-valued continuous systems. - oai:arXiv.org:2412.11675v5 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Zhengyu Yin - - - Rare events statistics for $\mathbb Z^d$ map lattices coupled by collision - https://arxiv.org/abs/2412.12803 - arXiv:2412.12803v3 Announce Type: replace -Abstract: Understanding the statistics of collisions among locally confined gas particles poses a major challenge. In this work we investigate $\mathbb Z^d$-map lattices coupled by collision with simplified local dynamics that offer significant insights for the above challenging problem. We obtain a first order approximation for the first collision rate at a site $\textbf{p}^*\in \mathbb Z^d$ and we prove a distributional convergence for the first collision time to an exponential, with sharp error term. Moreover, we prove that the number of collisions at site $\textbf{p}^*$ converge in distribution to a compound Poisson distributed random variable. Key to our analysis in this infinite dimensional setting is the use of transfer operators associated with the decoupled map lattice at site $\textbf{p}^*$. - oai:arXiv.org:2412.12803v3 - math.DS - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Wael Bahsoun, Maxence Phalempin - - - On some Sobolev and P\'olya-Szeg\"o type inequalities with weights and applications - https://arxiv.org/abs/2412.15490 - arXiv:2412.15490v3 Announce Type: replace -Abstract: We are motivated by studying a boundary-value problem for a class of semilinear degenerate elliptic equations - \begin{align}\tag{P}\label{P} - \begin{cases} - - \Delta_x u - |x|^{2\alpha} \dfrac{\partial^2 u}{\partial y^2} = f(x,y,u), & \textrm{in } \Omega, - u = 0, & \textrm{on } \partial \Omega, - \end{cases} - \end{align} - where $x = (x_1, x_2) \in \mathbb{R}^2$, $\Omega$ is a bounded smooth domain in $\mathbb{R}^3$, $(0,0,0) \in \Omega $, and $\alpha > 0$. - In this paper, we will study this problem by establishing embedding theorems for weighted Sobolev spaces. To this end, we need a new P\'olya-Szeg\"o type inequality, which can be obtained by studying an isoperimetric problem for the corresponding weighted area. Our results then extend the existing ones in \cite{nga, Luyen2} to the three-dimensional context. - oai:arXiv.org:2412.15490v3 - math.AP - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Trung Hieu Giang, Nguyen Minh Tri, Dang Anh Tuan - - - Dynamics, data and reconstruction - https://arxiv.org/abs/2412.19734 - arXiv:2412.19734v5 Announce Type: replace -Abstract: The goal of data-driven learning of dynamical systems is to interpret time series as a continuous observation of an underlying dynamical system. This task is not well-posed for a variety of reasons - such as multiple co-existing sub-systems, topologically inter-weaving of these sub-systems; and more importantly, the non-injectivity of the correspondence between dynamical systems and time series. We show how these ambiguities are circumvented if one considers dynamical systems and measurement maps collectively. Dynamical systems, observed dynamical systems, and time series data - each of these three collections have an extensive network of relations within them, which gives them the mathematical structure of a category. One of the new concepts proposed is a rigorous definition of time series data as a chain of measurement sequences with decreasing information content. This definition subsumes the familiar notions of sequences, time series and even subshifts. Using these notions it is shown that the entire process of converting an observed dynamical systems into a time series object is functorial, and passes through a number of phases each bearing its own categorical structure. This discovery sheds new light on the nature of reconstruction algorithms. Under mild conditions of consistency, reconstruction itself is shown to be functorial operation. This provides a new category theoretic perspective on the nature and limits of reconstruction. - oai:arXiv.org:2412.19734v5 - math.DS - math.CT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Suddhasattwa Das, Tomoharu Suda - - - On the ergodicity of anti-symmetric skew products with singularities and its applications - https://arxiv.org/abs/2412.21067 - arXiv:2412.21067v2 Announce Type: replace -Abstract: We introduce a novel method for proving ergodicity for skew products of interval exchange transformations (IETs) with piecewise smooth cocycles having singularities at the ends of exchanged intervals. This approach is inspired by Borel-Cantelli-type arguments from Fayad and Lema\'nczyk (2006). The key innovation of our method lies in its applicability to singularities beyond the logarithmic type, whereas previous techniques were restricted to logarithmic singularities. Our approach is particularly effective for proving the ergodicity of skew products for symmetric IETs and antisymmetric cocycles. Moreover, its most significant advantage is its ability to study the equidistribution of error terms in the spectral decomposition of Birkhoff integrals for locally Hamiltonian flows on compact surfaces, applicable not only when all saddles are perfect (harmonic) but also in the case of some non-perfect saddles. - oai:arXiv.org:2412.21067v2 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Przemys{\l}aw Berk, Krzysztof Fr\k{a}czek, Frank Trujillo - - - Stratification in equivariant Kasparov theory - https://arxiv.org/abs/2412.21109 - arXiv:2412.21109v3 Announce Type: replace -Abstract: We study stratification, that is the classification of localizing tensor ideal subcategories by geometric means, in the context of Kasparov's equivariant KK-theory of C*-algebras. We introduce a straightforward countable analog of the notion of stratification by Balmer-Favi supports and conjecture that it holds for the equivariant bootstrap subcategory of every finite group G. We prove this conjecture for groups whose nontrivial elements all have prime order, and we verify it rationally for arbitrary finite groups. In all these cases we also compute the Balmer spectrum of compact objects. In our proofs we use larger versions of the equivariant Kasparov categories which admit not only countable coproducts but all small ones; they are constructed in an Appendix using infinity-categorical enhancements and adapting ideas of Bunke-Engel-Land. - oai:arXiv.org:2412.21109v3 - math.KT - math.OA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ivo Dell'Ambrogio, Rub\'en Martos - - - Subspaces of $L^2(\mathbb{R}^n)$ Invariant Under Crystallographic Shifts - https://arxiv.org/abs/2501.02130 - arXiv:2501.02130v2 Announce Type: replace -Abstract: In this thesis we consider crystal groups in dimension $n$ and their natural unitary representation on $L^2(\mathbb{R}^n)$. We show that this representation is unitarily equivalent to a direct integral of factor representations, and use this to characterize the subspaces of $L^2(\mathbb{R}^n)$ invariant under crystal symmetry shifts. Finally, by giving an explicit unitary equivalence of the natural crystal group representation, we find the \textit{central decomposition} guaranteed by direct integral theory. - oai:arXiv.org:2501.02130v2 - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Tom Potter - - - Towards a constructive framework for control theory - https://arxiv.org/abs/2501.02267 - arXiv:2501.02267v2 Announce Type: replace -Abstract: This work presents a framework for control theory based on constructive analysis to account for discrepancy between mathematical results and their implementation in a computer, also referred to as computational uncertainty. In control engineering, the latter is usually either neglected or considered submerged into some other type of uncertainty, such as system noise, and addressed within robust control. However, even robust control methods may be compromised when the mathematical objects involved in the respective algorithms fail to exist in exact form and subsequently fail to satisfy the required properties. For instance, in general stabilization using a control Lyapunov function, computational uncertainty may distort stability certificates or even destabilize the system despite robustness of the stabilization routine with regards to system, actuator and measurement noise. In fact, battling numerical problems in practical implementation of controllers is common among control engineers. Such observations indicate that computational uncertainty should indeed be addressed explicitly in controller synthesis and system analysis. The major contribution here is a fairly general framework for proof techniques in analysis and synthesis of control systems based on constructive analysis which explicitly states that every computation be doable only up to a finite precision thus accounting for computational uncertainty. A series of previous works is overviewed, including constructive system stability and stabilization, approximate optimal controls, eigenvalue problems, Caratheodory trajectories, measurable selectors. Additionally, a new constructive version of the Danskin's theorem, which is crucial in adversarial defense, is presented. - oai:arXiv.org:2501.02267v2 - math.OC - cs.AI - cs.SY - eess.SY - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1109/LCSYS.2021.3076972 - in IEEE Control Systems Letters, vol. 6, pp. 379-384, 2022 - Pavel Osinenko - - - The{N/D}-Conjecture for Nonresonant Hyperplane Arrangements - https://arxiv.org/abs/2501.05189 - arXiv:2501.05189v3 Announce Type: replace -Abstract: This paper studies Bernstein--Sato polynomials $b_{f,0}$ for homogeneous polynomials $f$ of degree $d$ with $n$ variables. It is open to know when $-{n\over d}$ is a root of $b_{f,0}$. For essential indecomposable hyperplane arrangements, this is a conjecture by Budur, Musta\c{t}\u{a} and Teitler and implies the strong topological monodromy conjecture for arrangements. Walther gave a sufficient condition that a certain differential form does not vanish in the top cohomology group of Milnor fiber. We use Walther's result to verify the $n\over d$-conjecture for weighted hyperplane arrangements satisfying the nonresonant condition. - oai:arXiv.org:2501.05189v3 - math.AG - math.AC - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Baiting Xie, Chenglong Yu - - - A Quadratically-Constrained Convex Approximation for the AC Optimal Power Flow - https://arxiv.org/abs/2501.05623 - arXiv:2501.05623v2 Announce Type: replace -Abstract: We introduce a quadratically-constrained approximation (QCAC) of the AC optimal power flow (AC-OPF) problem. Unlike existing approximations like the DC-OPF, our model does not rely on typical assumptions such as high reactance-to-resistance ratio, near-nominal voltage magnitudes, or small angle differences, and preserves the structural sparsity of the original AC power flow equations, making it suitable for decentralized power systems optimization problems. To achieve this, we reformulate the AC-OPF problem as a quadratically constrained quadratic program. The nonconvex terms are expressed as differences of convex functions, which are then convexified around a base point derived from a warm start of the nodal voltages. If this linearization results in a non-empty constraint set, the convexified constraints form an inner convex approximation. Our experimental results, based on Power Grid Library instances of up to 30,000 buses, demonstrate the effectiveness of the QCAC approximation with respect to other well-documented conic relaxations and a linear approximation. We further showcase its potential advantages over the well-documented second-order conic relaxation of the power flow equations in two proof-of-concept case studies: optimal reactive power dispatch in transmission networks and PV hosting capacity in distribution grids. - oai:arXiv.org:2501.05623v2 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Gonzalo E. Constante-Flores, Can Li - - - Positive solutions for fractional-order boundary value problems with or without dependence of integer-order ones - https://arxiv.org/abs/2501.05810 - arXiv:2501.05810v2 Announce Type: replace -Abstract: We investigate the existence, non-existence, uniqueness, and multiplicity of positive solutions to the following problem: \begin{align}\label{P} - \left\{ - \begin{array}{l} - D_{0+}^\alpha u + h(t)f(u) = 0, \quad 0<t<1, \\[1ex] - u(0)=u(1)=0, - \end{array} - \right. \end{align} where $D_{0+}^\alpha$ is the Riemann-Liouville fractional derivative of order $\alpha\in(1,2]$. Firstly, by considering the first eigenvalue $\lambda_1(\alpha)$ of the corresponding eigenvalue problem, we establish the existence of positive solutions for both sublinear and superlinear cases involving $\lambda_1(\alpha)$, thereby extending existing results in the literature. In addition, we address the issue of non-existence, which reinforces the sharpness of both hypotheses. Secondly, we demonstrate the uniqueness of positive solutions. For the sublinear case, we impose certain monotonicity conditions on $f$. For the superlinear case, we assume that $h$ satisfies a specific condition to ensure the uniqueness of positive solutions when $\alpha =2.$ Near $\alpha =2,$ we prove uniqueness by leveraging the non-degeneracy of the unique solution, which represents a novel approach to studying fractional-order differential equations. Finally, we apply this methodology to establish the multiple existence of at least three positive solutions for H\'{e}non-type problems, which is also a new contribution. - oai:arXiv.org:2501.05810v2 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Inbo Sim, Satoshi Tanaka - - - The Born approximation for the fixed energy Calder\'on problem - https://arxiv.org/abs/2501.05889 - arXiv:2501.05889v2 Announce Type: replace -Abstract: The Born approximation of a potential in the context of the Calder\'on inverse problem is an object that can be formally defined in terms of spectral data of the Dirichlet-to-Neumann map of the corresponding Schr\"odinger operator. In this article, we prove, in the case of radial potentials in the Euclidean ball and any fixed energy, that the Born approximation is well-defined as a compactly supported radial distribution, and that the Calder\'on problem can be reformulated as recovering a potential from its Born approximation. In addition, we show that the Born approximation depends locally on the potential and captures exactly its singularities, and that the functional that maps the Born approximation to the potential is H\"older continuous. We also prove that the Born approximation converges to the potential in the high-energy limit. Moreover, we give an explicit formula for the Fourier transform of the Born approximation at any fixed energy, and illustrate how it can be used as the basis of an accurate procedure to approximate a potential from its Dirichlet-to-Neumann map. - oai:arXiv.org:2501.05889v2 - math.AP - math.SP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1017/prm.2025.10092 - Proc. Roy. Soc. Edinburgh Sect. A., 2025 - Fabricio Maci\`a, Crist\'obal Mero\~no, Daniel S\'anchez-Mendoza - - - Tensorization of neural networks for improved privacy and interpretability - https://arxiv.org/abs/2501.06300 - arXiv:2501.06300v3 Announce Type: replace -Abstract: We present a tensorization algorithm for constructing tensor train/matrix product state (MPS) representations of functions, drawing on sketching and cross interpolation ideas. The method only requires black-box access to the target function and a small set of sample points defining the domain of interest. Thus, it is particularly well-suited for machine learning models, where the domain of interest is naturally defined by the training dataset. We show that this approach can be used to enhance the privacy and interpretability of neural network models. Specifically, we apply our decomposition to (i) obfuscate neural networks whose parameters encode patterns tied to the training data distribution, and (ii) estimate topological phases of matter that are easily accessible from the MPS representation. Additionally, we show that this tensorization can serve as an efficient initialization method for optimizing MPS in general settings, and that, for model compression, our algorithm achieves a superior trade-off between memory and time complexity compared to conventional tensorization methods of neural networks. - oai:arXiv.org:2501.06300v3 - math.NA - cs.LG - cs.NA - physics.comp-ph - quant-ph - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - 10.21468/SciPostPhysCore.8.4.095 - SciPost Phys. Core 8, 095 (2025) - Jos\'e Ram\'on Pareja Monturiol, Alejandro Pozas-Kerstjens, David P\'erez-Garc\'ia - - - Eco-evolutionary dynamics of a trait-structured predator-prey model - https://arxiv.org/abs/2501.07379 - arXiv:2501.07379v3 Announce Type: replace -Abstract: The coupling between evolutionary and ecological changes (eco-evolutionary dynamics) has been shown to be relevant among diverse species, and is also of interest outside of ecology, i.e. in cancer evolution. These dynamics play an important role in determining survival in response to climate change, motivating the need for mathematical models to capture this often complex interplay. Models incorporating eco-evolutionary dynamics often sacrifice analytical tractability to capture the complexity of real systems, do not explicitly consider the effect of population heterogeneity, or focus on long-term behaviour. In order to capture population heterogeneity, both transient, and long-term dynamics, while retaining tractability, we generalise a moment-based method applicable in the regime of small segregational variance to the case of time-dependent mortality and birth. These results are applied to a predator-prey model, where ecological parameters such as the contact rate between species are trait-structured. The trait-distribution of the prey species is shown to be approximately Gaussian with constant variance centered on the mean trait, which is asymptotically governed by an autonomous ODE. In this way, we make explicit the impact of eco-evolutionary dynamics on the transient behaviour and long-term fate of the prey species. - oai:arXiv.org:2501.07379v3 - math.AP - q-bio.PE - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Manh Hong Duong, Fabian Spill, Blaine van Rensburg - - - Asymptotic-Preserving Neural Networks based on Even-odd Decomposition for Multiscale Gray Radiative Transfer Equations - https://arxiv.org/abs/2501.08166 - arXiv:2501.08166v2 Announce Type: replace -Abstract: We present a novel Asymptotic-Preserving Neural Network (APNN) approach utilizing even-odd decomposition to tackle the nonlinear gray radiative transfer equations (GRTEs). Our AP loss demonstrates consistent stability concerning the small Knudsen number, ensuring the neural network solution uniformly converges to the diffusion limit solution. This APNN method alleviates the rigorous conservation requirements while simultaneously incorporating an auxiliary deep neural network, distinguishing it from the APNN method based on micro-macro decomposition for GRTE. Several numerical problems are examined to demonstrate the effectiveness of our proposed APNN technique. - oai:arXiv.org:2501.08166v2 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Keke Wu, Xizhe Xie, Wengu Chen, Han Wang, Zheng Ma - - - Stable determination of the potential for the Helmholtz equation in the high frequency limit from boundary measurements - https://arxiv.org/abs/2501.10751 - arXiv:2501.10751v2 Announce Type: replace -Abstract: We establish a triple logarithmic stability estimate of determining the potential in a Helmholtz equation from a partial Dirichlet-to-Neumann map in the high frequency limit. This estimate is proved under the assumption that the potential is known near the boundary of a domain when the dimension is greater than or equal to $3$. In addition, we show a triple logarithmic stability for an interior impedance problem. - oai:arXiv.org:2501.10751v2 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mourad Choulli, Hiroshi Takase - - - D2D Coded Caching Schemes for Multiaccess Networks with Combinatorial Access Topology - https://arxiv.org/abs/2501.10756 - arXiv:2501.10756v2 Announce Type: replace -Abstract: This paper considers wireless device-to-device (D2D) coded caching in a multiaccess network, where the users communicate with each other and each user can access multiple cache nodes. Access topologies derived from two combinatorial designs known as the $t$-design and $t$-group divisible design ($t$-GDD), referred to as the $t$-design and $t$-GDD topologies respectively, which subsume a few other known topologies, have been studied for the multiaccess coded caching (MACC) network by Cheng \textit{et al.} in \cite{MACC_des}. These access topologies are extended to a multiaccess D2D coded caching (MADCC) network and novel MADCC schemes are proposed. MADCC network has been studied so far only for the cyclic wrap-around topology. Apart from the proposed novel MADCC schemes, MADCC schemes are also derived from the existing MACC schemes in \cite{MACC_des}. To compare the performance of different MADCC schemes, the metrics of load per user and subpacketization level are used while keeping the number of caches and cache memory size same. The proposed MADCC scheme with $t$-design topology performs better in terms of subpacketization level while achieving the same load per user compared to the MADCC scheme derived from the MACC scheme with $t$-design topology in \cite{MACC_des}. The proposed MADCC scheme with $t$-GDD topology performs better in terms of load per user while achieving the same subpacketization level compared to the MADCC scheme derived from the MACC scheme with $t$-GDD topology in \cite{MACC_des} in some cases. Compared to the existing MADCC scheme with cyclic wrap-around topology, the proposed MADCC scheme with $t$-design topology performs better in terms of load per user, and the proposed MADCC scheme with $t$-GDD topology performs better in terms of subpacketization level at the expense of an increase in load per user. - oai:arXiv.org:2501.10756v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rashid Ummer N. T., B. Sundar Rajan - - - Surfaces with flat normal connection in 4-dimensional space forms - https://arxiv.org/abs/2501.15780 - arXiv:2501.15780v3 Announce Type: replace -Abstract: Let $N$ be a Riemannian, Lorentzian or neutral $4$-dimensional space form with constant sectional curvature $L_0$. In this paper, noticing the linearly dependent condition, we obtain characterizations of space-like surfaces in $N$ with flat normal connection and parallel normal vector fields. In addition, we obtain a generic characterization of space-like surfaces in $N$ with flat normal connection and $K\equiv L_0$ which do not admit any parallel normal vector fields. For time-like surfaces in $N$ with flat normal connection, we obtain analogous results. - oai:arXiv.org:2501.15780v3 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Naoya Ando, Ryusei Hatanaka - - - A Lyapunov analysis of Korpelevich's extragradient method with fast and flexible extensions - https://arxiv.org/abs/2502.00119 - arXiv:2502.00119v2 Announce Type: replace -Abstract: We develop a Lyapunov-based analysis of Korpelevich's extragradient method and show that it achieves an $o(1/k)$ last-iterate convergence rate of the constructed Lyapunov function. This Lyapunov function simultaneously upper bounds several standard measures of optimality, which allows our analysis to sharpen existing last-iterate convergence guarantees for these measures. Moreover, the same analysis enables the design of a class of flexible extensions of the extragradient method in which extragradient steps are adaptively blended with user-specified directions via a Lyapunov-guided line-search procedure. These extensions retain global convergence under practical assumptions and can attain superlinear rates when the directions are chosen appropriately. Numerical experiments confirm the simplicity and efficiency of the proposed framework. - oai:arXiv.org:2502.00119v2 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - 10.1007/s10107-025-02322-0 - Manu Upadhyaya, Puya Latafat, Pontus Giselsson - - - Extreme-Scale EV Charging Infrastructure Planning for Last-Mile Delivery Using High-Performance Parallel Computing - https://arxiv.org/abs/2502.05152 - arXiv:2502.05152v2 Announce Type: replace -Abstract: This paper addresses stochastic charger location and allocation problems under queue congestion for last-mile delivery using electric vehicles (EVs). The objective is to decide where to open charging stations and how many chargers of each type to install, subject to budgetary and waiting-time constraints. We formulate the problem as a mixed-integer non-linear program, where each station-charger pair is modeled as a multiserver queue with stochastic arrivals and service times to capture the notion of waiting in fleet operations. The model is extremely large, with billions of variables and constraints for a typical metropolitan area; even loading the model in solver memory is difficult, let alone solving it. To address this challenge, we develop a Lagrangian-based dual decomposition framework that decomposes the problem by station and leverages parallelization on high-performance computing systems, where the subproblems are solved by using a cutting plane method and their solutions are collected at the master level. We also develop a three-step rounding heuristic to transform the fractional subproblem solutions into feasible integral solutions. Computational experiments on data from the Chicago metropolitan area with hundreds of thousands of households and thousands of candidate stations show that our approach produces high-quality solutions in cases where existing exact methods cannot even load the model in memory. We also analyze various policy scenarios, demonstrating that combining existing depots with newly built stations under multiagency collaboration substantially reduces costs and congestion. These findings offer a scalable and efficient framework for developing sustainable large-scale EV charging networks. - oai:arXiv.org:2502.05152v2 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Waquar Kaleem, Taner Cokyasar, Jeffrey Larson, Omer Verbas, Tanveer Hossain Bhuiyan, Anirudh Subramanyam - - - Hankel continued fractions and Hankel determinants for $q$-deformed metallic numbers - https://arxiv.org/abs/2502.05993 - arXiv:2502.05993v2 Announce Type: replace -Abstract: Fix $n$ a positive integer. Take the $n$-th metallic number $\phi_n=\frac{n+\sqrt{n^2+4}}{2}$ (e.g. $\phi_1$ is the golden number) and let $\Phi_n(q)$ be its $q$-deformation in the sense of S. Morier-Genoud and V. Ovsienko. This is an algebraic continued fraction which admits an expansion into a Taylor series around $q=0$, with integral coefficients. By using the notion of Hankel continued fraction introduced by the first author in 2016 we determine explicitly the first $n+2$ sequences of shifted Hankel determinants of $\Phi_n$ and show that they satisfy the following properties: - 1) They are periodic and consist of $-1,0,1$ only. - 2) They satisfy a three-term Gale-Robinson recurrence, i.e. they form discrete integrable dynamical systems. - 3) They are all completely determined by the first sequence. - This article thus validates a conjecture formulated by V. Ovsienko and the second author in a recent paper and establishes new connections between $q$-deformations of real numbers and sequences of Catalan or Motzkin numbers. - oai:arXiv.org:2502.05993v2 - math.NT - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Guo-Niu Han, Emmanuel Pedon - - - Homeomorphism groups of basilica, rabbit and airplane Julia sets - https://arxiv.org/abs/2502.07762 - arXiv:2502.07762v2 Announce Type: replace -Abstract: The airplane, the basilica and the Douady rabbit (and, more generally, rabbits with more than two ears) are well-known Julia sets of complex quadratic polynomials. - In this paper we study the groups of all homeomorphisms of such fractals and of all automorphisms of their laminations. - In particular, we identify them with some kaleidoscopic group or universal groups and thus realize them as Polish permutation groups. - From these identifications, we deduce algebraic, topological and geometric properties of these groups. - oai:arXiv.org:2502.07762v2 - math.DS - math.GN - math.GR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Bruno Duchesne, Matteo Tarocchi - - - Newton-Mandelbrot set and Murase-Mandelbrot set - https://arxiv.org/abs/2502.14872 - arXiv:2502.14872v2 Announce Type: replace -Abstract: We obtain four extended Newton's methods and three extended Mandelbrot's recurrence formulas from the Wasan (Japanese mathematics in the Edo period (1603-1868)). Furthermore, two extended Newton's methods relate to one of the extended Mandelbrot's recurrence formulas. We lead four types of extended Mandelbrot recurrence formulas. Next, we show that these become the same extended Mandelbrot set, and connected, closed set. These show the originality of Wasan. - oai:arXiv.org:2502.14872v2 - math.GM - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Shunji Horiguchi - - - A Jacobian-free Newton-Krylov method for cell-centred finite volume solid mechanics - https://arxiv.org/abs/2502.17217 - arXiv:2502.17217v2 Announce Type: replace -Abstract: This study investigates the efficacy of Jacobian-free Newton-Krylov methods in finite-volume solid mechanics. Traditional Newton-based approaches require explicit Jacobian matrix formation and storage, which can be computationally expensive and memory-intensive. In contrast, Jacobian-free Newton-Krylov methods approximate the Jacobian's action using finite differences, combined with Krylov subspace solvers such as the generalised minimal residual method (GMRES), enabling seamless integration into existing segregated finite-volume frameworks without major code refactoring. This work proposes and benchmarks the performance of a compact-stencil Jacobian-free Newton-Krylov method against a conventional segregated approach on a suite of test cases, encompassing varying geometric dimensions, nonlinearities, dynamic responses, and material behaviours. Key metrics, including computational cost, memory efficiency, and robustness, are evaluated, along with the influence of preconditioning strategies and stabilisation scaling. Results show that the proposed Jacobian-free Newton-Krylov method outperforms the segregated approach in all linear and nonlinear elastic cases, achieving order-of-magnitude speedups in many instances; however, divergence is observed in elastoplastic cases, highlighting areas for further development. It is found that preconditioning choice impacts performance: a LU direct solver is fastest in small to moderately-sized cases, while a multigrid method is more effective for larger problems. The findings demonstrate that Jacobian-free Newton-Krylov methods are promising for advancing finite-volume solid mechanics simulations, particularly for existing segregated frameworks where minimal modifications enable their adoption. The described implementations are available in the solids4foam toolbox for OpenFOAM, inviting the community to explore, extend, and compare these procedures. - oai:arXiv.org:2502.17217v2 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Philip Cardiff, Dylan Armfield, \v{Z}eljko Tukovi\'c, Ivan Batisti\'c - - - Multiplier modules of Hilbert C*-modules revisited - https://arxiv.org/abs/2502.17959 - arXiv:2502.17959v3 Announce Type: replace -Abstract: The theory of multiplier modules of Hilbert C*-modules is reconsidered to obtain more properties of these special Hilbert C*-modules. The property of a Hilbert C*-module to be a multiplier C*-module is shown to be an invariant with respect to the consideration as a left or right Hilbert C*-module in the sense of a imprimitivity bimodule in strong Morita equivalence theory. The interrelation of the C*-algebras of ''compact'' operators, the Banach algebras of bounded module operators and the Banach spaces of bounded module operators of a Hilbert C*-module to its C*-dual Banach C*-module, are characterized for pairs of Hilbert C*-modules and their respective multiplier modules. The structures on the latter are always isometrically embedded into the respective structures on the former. Examples are given for which continuation of these kinds of bounded module operators from the initial Hilbert C*-module to its multiplier module fails. However, existing continuations turn out to be always unique. Similarly, bounded modular functionals from both kinds of Hilbert C*-modules to their respective C*-algebras of coefficients are compared, and eventually existing continuations are shown to be unique. - oai:arXiv.org:2502.17959v3 - math.OA - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Michael Frank - - - Rigidity of the escaping set of certain H\'enon maps - https://arxiv.org/abs/2502.19358 - arXiv:2502.19358v3 Announce Type: replace -Abstract: Let $H$ be a H\'enon map of the form $H(x,y)=(y,p(y)-ax)$. We prove that the escaping set $U^+$ (or equivalently, the non-escaping set $K^+$), of $H$ is rigid under the actions of automorphisms of $\mathbb{C}^2$ if the degree of $H=d\le |a|$. Specifically, every automorphism of $\mathbb{C}^2$ that preserves $U^+$, essentially takes the form $C \circ H^s$ where $s \in \mathbb{Z}$, and $C(x,y)=(\eta x, \eta^d y)$ with $\eta$ some $(d^2-1)$-root of unity. - Consequently, we show that the automorphisms of the short $\mathbb{C}^2$'s, obtained as the sub-level sets of the (positive) Green's function corresponding to the H\'enon map $H$ for strictly positive values, are essentially linear maps of $\mathbb{C}^2$ preserving the escaping set $U^+$. Hence, the automorphism groups of these short $\mathbb{C}^2$'s are the same, finite, and form a subgroup of $\mathbb{Z}_{d^2-1}$. - oai:arXiv.org:2502.19358v3 - math.CV - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Sayani Bera - - - Abelian congruences and similarity in varieties with a weak difference term - https://arxiv.org/abs/2502.20517 - arXiv:2502.20517v3 Announce Type: replace -Abstract: This is the first of three papers motivated by the author's desire to understand and explain "algebraically" one aspect of Dmitriy Zhuk's proof of the CSP Dichotomy Theorem. In this paper we study abelian congruences in varieties having a weak difference term. Each class of the congruence supports an abelian group structure; if the congruence is minimal, each class supports the structure of a vector space over a division ring determined by the congruence. A construction due to J. Hagemann, C. Herrmann and R. Freese in the congruence modular setting extends to varieties with a weak difference term, and provides a "universal domain" for the abelian groups or vector spaces that arise from the classes of the congruence within a single class of the annihilator of the congruence. The construction also supports an extension of Freese's similarity relation (between subdirectly irreducible algebras) from the congruence modular setting to varieties with a weak difference term. - oai:arXiv.org:2502.20517v3 - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ross Willard - - - Zhuk's bridges, centralizers, and similarity - https://arxiv.org/abs/2503.03551 - arXiv:2503.03551v2 Announce Type: replace -Abstract: This is the second of three papers motivated by the author's desire to understand and explain "algebraically" one aspect of Dmitriy Zhuk's proof of the CSP Dichotomy Theorem. In this paper we extend Zhuk's "bridge" construction to arbitrary meet-irreducible congruences of finite algebras in locally finite varieties with a Taylor term. We then connect bridges to centrality and similarity. In particular, we prove that Zhuk's bridges and our "similarity bridges" (defined in our first paper) convey the same information in locally finite Taylor varieties. - oai:arXiv.org:2503.03551v2 - math.LO - cs.LO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ross Willard - - - A positive product formula of integral kernels of $k$-Hankel transforms - https://arxiv.org/abs/2503.03554 - arXiv:2503.03554v4 Announce Type: replace -Abstract: The $k$-Hankel transform $F_{k,1}$ (or the $(k,1)$-generalized Fourier transform) is the Dunkl analogue of the unitary inversion operator in the minimal representation of a conformal group initiated by T. Kobayashi and G. Mano. It is one of the two most significant cases in $(k,a)$-generalized Fourier transforms. We will establish a positive radial product formula for the integral kernels of $F_{k,1}$. Such a product formula is equivalent to a representation of the generalized spherical mean operator in terms of the probability measure $\sigma_{x,t}^{k,1}(\xi)$. We will then study the representing measure $\sigma_{x,t}^{k,1}(\xi)$ and analyze the support of the measure, and derive a weak Huygens's principle for the deformed wave equation in $(k,1)$-generalized Fourier analysis. - oai:arXiv.org:2503.03554v4 - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Wentao Teng - - - Consecutive Patterns, Kostant's Problem and Type $A_6$ - https://arxiv.org/abs/2503.07809 - arXiv:2503.07809v2 Announce Type: replace -Abstract: For a permutation $w$ in the symmetric group $\mathfrak{S}_{n}$, let $L(w)$ denote the simple highest weight module in the principal block of the BGG category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_{n}(\mathbb{C})$. We first prove that $L(w)$ is Kostant negative whenever $w$ consecutively contains certain patterns. We then provide a complete answer to Kostant's problem in type $A_{6}$ and show that the indecomposability conjecture also holds in type $A_{6}$, that is, applying an indecomposable projective functor to a simple module outputs either an indecomposable module or zero. - oai:arXiv.org:2503.07809v2 - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Samuel Creedon, Volodymyr Mazorchuk - - - Exotic spherical flexible octahedra and counterexamples to the Modified Bellows Conjecture - https://arxiv.org/abs/2503.09582 - arXiv:2503.09582v2 Announce Type: replace -Abstract: In 2014 the author showed that in the three-dimensional spherical space, alongside with three classical types of flexible octahedra constructed by Bricard, there exists a new type of flexible octahedra, which was called exotic. In the present paper we give a geometric construction for exotic flexible octahedra, describe their configuration spaces, and calculate their volumes. We show that the volume of an exotic flexible octahedron is nonconstant during the flexion, and moreover the volume remains nonconstant if we replace any set of vertices of the octahedron with their antipodes. So exotic flexible octahedra are counterexamples to the Modified Bellows Conjecture proposed by the author in 2015. - oai:arXiv.org:2503.09582v2 - math.MG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexander A. Gaifullin - - - Rigorous analysis of shape transitions in frustrated elastic ribbons - https://arxiv.org/abs/2503.11779 - arXiv:2503.11779v2 Announce Type: replace -Abstract: Ribbons are elastic bodies of thickness $t$ and width $w$ with $t\ll w\ll 1$ (after appropriate nondimensionalization). Many ribbons in nature have a non-trivial internal geometry, making them incompatible with Euclidean space. This incompatibility -- expressed mathematically as a failure of the Gauss-Codazzi equations for surfaces -- can trigger shape transitions between narrow and wide ribbons. These transitions depend on the internal geometry: ribbons whose incompatibility arises from failure of the Gauss equation always exhibit a transition, whereas those whose incompatibility arises from failure of the Codazzi equations, may or may not. We give the first rigorous analysis of this phenomenon, mainly for ribbons whose first fundamental form is flat. For Gauss-incompatible ribbons we identify the natural energy scaling of the problem and prove the existence of a shape transition. For Codazzi-incompatible ribbons we give a necessary condition for a transition to occur. Furthermore, our study reveals a fundamental distinction: the transition is "microscopic" for Gauss-incompatible ribbons, persisting as the width tends to $0$, whereas it is "mesoscopic" for Codazzi-incompatible ribbons, observable only at small but finite width. The results are obtained by calculating the $\Gamma$-limits, as $t,w\to 0$, for narrow ribbons ($w^2 \ll t$), and wide ribbons (taking $t$ to zero and then $w$), in the natural energy scalings dictated by the internal geometry. - oai:arXiv.org:2503.11779v2 - math.AP - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Cy Maor, Maria Giovanna Mora - - - Spanning trees in directed square cycles - https://arxiv.org/abs/2503.12561 - arXiv:2503.12561v2 Announce Type: replace -Abstract: We classify weakly connected spanning closed (WCSC) subgraphs of $\overrightarrow{C_n^2}$, the square of a directed $n$-vertex cycle. Then we show that every spanning tree of $\overrightarrow{C_n^2}$ is contained in a unique nontrivial WCSC subgraph of $\overrightarrow{C_n^2}$. As a result, we obtain a purely combinatorial derivation of the formula for the number of directed spanning trees of $\overrightarrow{C_n^2}$. Moreover, we obtain the formula for the number of directed spanning trees of $\overrightarrow{C_n^2}$, which is a Jacobsthal number. - oai:arXiv.org:2503.12561v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yuuho Tanaka - - - Characterising 1-rectifiable metric spaces via connected tangent spaces - https://arxiv.org/abs/2503.15159 - arXiv:2503.15159v2 Announce Type: replace -Abstract: We prove that in a complete metric space $X$, $1$-rectifiability of a set $E\subset X$ with $\mathcal{H}^1(E)<\infty$ and positive lower density $\mathcal{H}^1$-a.e. is implied by the property that all tangent spaces are connected metric spaces. - oai:arXiv.org:2503.15159v2 - math.MG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - David Bate, Phoebe Valentine - - - Canonical torus action on symplectic singularities - https://arxiv.org/abs/2503.15791 - arXiv:2503.15791v3 Announce Type: replace -Abstract: We show that any symplectic singularity lying on a smoothable projective symplectic variety locally admits a good action of an algebraic torus of dimension $r \geq 1$, which is canonical. Under mild assumptions, $r=1$ is also confirmed. In particular, it admits (canonical) good $\mathbb{C}^*$-action. This proves Kaledin's conjecture conditionally but in a substantially stronger form. Our key idea is to use Donaldson-Sun theory on local Kahler metrics in complex differential geometry to connect with the theory of Poisson deformations of symplectic varieties. - For general symplectic singularities, we prove the same assertion -- namely, the existence of a canonical (local) torus action -- assuming that the Donaldson-Sun theory extends to such singularities along with suitable singular (hyper)Kahler metrics. Conversely, our results can be also used to study local behaviour of such metrics around the germ. For instance, we show that such singular hyperKahler metric around isolated singularity is close to a metric cone in a polynomial order, and satisfies $r=1$ i.e., has a good canonical (local) $\mathbb{C}^*$-action, as the complexification of the cone metric rescaling. Our theory also fits well to singularities on many hyperKahler reductions. - oai:arXiv.org:2503.15791v3 - math.AG - hep-th - math.DG - math.RT - math.SG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Yoshinori Namikawa, Yuji Odaka - - - High-dimensional sparse recovery from function samples Decoders, guarantees and instance optimality - https://arxiv.org/abs/2503.16209 - arXiv:2503.16209v2 Announce Type: replace -Abstract: We investigate the reconstruction of multivariate functions from samples using sparse recovery techniques. For Square Root Lasso, Orthogonal Matching Pursuit, and Compressive Sampling Matching Pursuit, we demonstrate both theoretically and empirically that they allow us to recover functions from a small number of random samples. In contrast to Basis Pursuit Denoising, the deployed decoders only require a search space $V_J$ spanned by dictionary elements indexed by $J$ and a sparsity parameter $n$ to guarantee an $L_2$-approximation error decaying no worse than a best $n$-term approximation error and the truncation error with respect to the search space $V_J$ and the uniform norm. We show that this happens simultaneously for all admissible functions if the number of samples scales as $n\log^2 n\log |J|$, coming from known bounds for the RIP for matrices built upon bounded orthonormal systems. As a consequence, we obtain bounds for sampling widths in function classes. In addition, we establish lower bounds on the required sample complexity, which show that the log-factor in $\vert J \vert$ is indeed necessary to obtain such {\em instance-optimal} error guarantees. Finally, we conduct several numerical experiments to show that our theoretical bounds are reasonable and compare the discussed decoders in practice. - oai:arXiv.org:2503.16209v2 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Moritz Moeller, Sebastian Neumayer, Kateryna Pozharska, Tizian Sommerfeld, Tino Ullrich - - - Two billiard domains whose billiard maps are Lazutkin conjugates are the same - https://arxiv.org/abs/2503.19550 - arXiv:2503.19550v3 Announce Type: replace -Abstract: This paper aimed to show that two billiards whose billiard maps share the same expression in Lazutkin coordinates are isometric. However, the results are incorrect as they rely on erroneous computations in Lazutkin's book concerning the expansion of the billiard map in Lazutkin coordinates. - oai:arXiv.org:2503.19550v3 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Corentin Fierobe - - - The prime number theorem over integers of power-free polynomial values - https://arxiv.org/abs/2504.00804 - arXiv:2504.00804v3 Announce Type: replace -Abstract: Let $f(x)\in \mathbb{Z}[x]$ be an irreducible polynomial of degree $d\ge 1$. Let $k\ge2$ be an integer. The number of integers $n$ such that $f(n)$ is $k$-free is widely studied in the literature. In principle, one expects that $f(n)$ is $k$-free infinitely often, if $f$ has no fixed $k$-th power divisor. In 2022, Bergelson and Richter established a new dynamical generalization of the prime number theorem (PNT). Inspired by their work, one may expect that this generalization of the PNT also holds over integers of power-free polynomial values. In this note, we establish such variants of Bergelson and Richter's theorem for several polynomials studied by Estermann, Hooley, Heath-Brown, Booker and Browning. - oai:arXiv.org:2504.00804v3 - math.NT - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Biao Wang, Shaoyun Yi - - - A Tail-Respecting Explicit Numerical Scheme for L\'evy-Driven SDEs With Superlinear Drifts - https://arxiv.org/abs/2504.07255 - arXiv:2504.07255v2 Announce Type: replace -Abstract: We present an explicit numerical approximation scheme, denoted by $\{X^n\}$, for the effective simulation of solutions $X$ to a multivariate stochastic differential equation (SDE) with a superlinearly growing $\kappa$-dissipative drift, where $\kappa>1$, driven by a multiplicative heavy-tailed L\'evy process that has a finite $p$-th moment, with $p>0$. We show that the strong $L^q$-convergence $\sup_{t\in[0,T]}\mathbf E \|X^n_t-X_t\|^q=\mathcal O (h_n^{\gamma})$ holds true for any $q\in (0,p+\kappa-1)$, which is exactly the range where the $q$-moment of the solution is known to be finite. Additionally, for any $q\in (0,p)$ we establish strong uniform convergence: $\mathbf E\sup_{t\in[0,T]} \|X^n_t-X_t\|^q=\mathcal{O} ( h_n^{\delta} )$. In both cases we determine the convergence rates $\gamma$ and $\delta$. In the special case of SDEs driven solely by a Brownian motion, our numerical scheme preserves super-exponential moments of the solution. The scheme $\{X^n\}$ is realized as a combination of a well-known Euler method with a Lie-Trotter type splitting technique. - oai:arXiv.org:2504.07255v2 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Olga Aryasova, Oleksii Kulyk, Ilya Pavlyukevich - - - Linear complementary dual quasi-cyclic codes of index 2 - https://arxiv.org/abs/2504.09126 - arXiv:2504.09126v3 Announce Type: replace -Abstract: We provide a polynomial approach to investigate linear complementary dual (LCD) quasi-cyclic codes over finite fields. We establish necessary and sufficient conditions for LCD quasi-cyclic codes of index 2 with respect to the Euclidean, Hermitian, and symplectic inner products. As a consequence of these characterizations, we derive necessary and sufficient conditions for LCD one-generator quasi-cyclic codes. Furthermore, using these characterizations, we construct some new quasi-cyclic LCD codes over small fields. - oai:arXiv.org:2504.09126v3 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Kanat Abdukhalikov, Duy Ho, San Ling, Gyanendra K. Verma - - - Estimate for the first Dirichlet eigenvalue of $p-$Laplacian on non-compact manifolds - https://arxiv.org/abs/2504.09856 - arXiv:2504.09856v2 Announce Type: replace -Abstract: In this paper, we establish a sharp lower bound for the first Dirichlet eigenvalue of the $p$-Laplacian on bounded domains of a complete, non-compact Riemannian manifold with non-negative Ricci curvature. - oai:arXiv.org:2504.09856v2 - math.DG - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xiaoshang Jin, Zhiwei L\"u - - - A structure-preserving numerical method for quasi-incompressible Navier-Stokes-Maxwell-Stefan systems - https://arxiv.org/abs/2504.11892 - arXiv:2504.11892v2 Announce Type: replace -Abstract: A conforming finite element scheme with mixed explicit-implicit time discretization for quasi-incompressible Navier-Stokes-Maxwell-Stefan systems in a bounded domain with periodic boundary conditions is presented. The system consists of the Navier-Stokes equations, together with a quasi-incompressibility constraint, coupled with the cross-diffusion Maxwell-Stefan equations. The numerical scheme preserves the partial masses and the quasi-incompressibility constraint and dissipates the discrete energy. Numerical experiments in two space dimensions illustrate the convergence of the scheme and the structure-preserving properties. - oai:arXiv.org:2504.11892v2 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Aaron Brunk, Ansgar J\"ungel, Maria Luk\'a\v{c}ov\'a-Medvid'ov\'a - - - A functional limit theorem for a dynamical system with an observable maximised on a Cantor set - https://arxiv.org/abs/2504.12534 - arXiv:2504.12534v2 Announce Type: replace -Abstract: We consider heavy-tailed observables maximised on a dynamically defined Cantor set and prove convergence of the associated point processes as well as functional limit theorems. The Cantor structure, and its connection to the dynamics, causes clustering of large observations: this is captured in the `decorations' on our point processes and functional limits, an application of the theory developed in a paper by the latter three authors. - oai:arXiv.org:2504.12534v2 - math.DS - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.physd.2025.134989 - Phys. D, 483, Paper No. 134989, 2025 - Raquel Couto, Ana Cristina Moreira Freitas, Jorge Milhazes Freitas, Mike Todd - - - Linear Complementary Pairs of Quasi-Cyclic and Quasi-Twisted Codes - https://arxiv.org/abs/2504.15231 - arXiv:2504.15231v2 Announce Type: replace -Abstract: In this paper, we provide a polynomial characterization of linear complementary pairs of quasi-cyclic and quasi-twisted codes of index 2. We also give several examples of linear complementary pairs of quasi-cyclic and quasi-twisted codes with optimal security parameters. - oai:arXiv.org:2504.15231v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Kanat Abdukhalikov, Duy Ho, San Ling, Gyanendra K. Verma - - - $k$-Inductive and Interpolation-Inspired Barrier Certificates for Stochastic Dynamical Systems - https://arxiv.org/abs/2504.15412 - arXiv:2504.15412v2 Announce Type: replace -Abstract: In this paper, we introduce two new types of barrier certificates that are based on multiple functions rather than a single one. A conventional barrier certificate for a stochastic dynamical system is a nonnegative real-valued function whose expected value does not increase as the system evolves. This requirement guarantees that the barrier certificate forms a nonnegative supermartingale and can be used to derive a lower bound on the probability that the system remains safe. A key advantage of such certificates is that they can be automatically searched for using tools such as optimization programs instantiated with a fixed template. When this search is unsuccessful, the common practice is to modify the template and attempt the synthesis again. Drawing inspiration from logical interpolation, we first propose an alternative framework that uses a collection of functions to jointly serve as a barrier certificate. We refer to this construct as an interpolation-inspired barrier certificate. Nonetheless, we observe that these certificates still require one function in the collection to satisfy a supermartingale condition. Motivated by recent work in the literature, we next combine k-induction with interpolation-inspired certificates to relax this supermartingale constraint. We develop a general and more flexible notion of barrier certificates, which we call k-inductive interpolation-inspired barrier certificates. This formulation encompasses multiple ways of integrating interpolation-inspired barrier certificates with k-induction. We highlight two specific instantiations among these possible combinations. For polynomial systems, we employ sum-of-squares (SOS) programming to synthesize the corresponding set of functions. Finally, through our case studies, we show that the proposed methods enable the use of simpler templates and yield tighter lower bounds on the safety probability. - oai:arXiv.org:2504.15412v2 - math.OC - cs.SY - eess.SY - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mohammed Adib Oumer, Vishnu Murali, Majid Zamani - - - $\eta$-Einstein Sasakian Lie algebras - https://arxiv.org/abs/2504.16033 - arXiv:2504.16033v2 Announce Type: replace -Abstract: We study $\eta$-Einstein Sasakian structures on Lie algebras, that is, Sasakian structures whose associated Ricci tensor satisfies an Einstein-like condition. We divide into the cases in which the Lie algebra's centre is non-trivial (and necessarily one-dimensional) and those where it is zero. In the former case we show that any Sasakian structure on a unimodular Lie algebra is $\eta$-Einstein. As for centreless Sasakian Lie algebras, we devise a complete characterisation under certain dimensional assumptions regarding the action of the Reeb vector. Using this result, together with the theory of normal $j$-algebras and modifications of Hermitian Lie algebras, we construct new examples of $\eta$-Einstein Sasakian Lie algebras and solvmanifolds, and provide effective restrictions for their existence. - oai:arXiv.org:2504.16033v2 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Adri\'an M. Andrada, Simon G. Chiossi, Alberth J. Nu\~nez - - - Modeling and Simulation of Open Membranes in Stokes Flow with Mixed-Dimensional Coupling - https://arxiv.org/abs/2504.16823 - arXiv:2504.16823v2 Announce Type: replace -Abstract: In this work, we present a mathematical and computational framework to model the dynamics of open lipid bilayer membranes interacting with ambient Stokes flow. The model explicitly couples the three-dimensional viscous fluid, the two-dimensional membrane surface, and its one-dimensional free edge. We develop an axisymmetric hybrid BEM-FEM method that solves the problem with an effective one-dimensional formulation. A key component is a local mesh refinement strategy designed to accurately resolve singularities and boundary layers originating at the membrane edge. Several numerical examples are provided to showcase its ability to capture intricate edge dynamics and multiscale fluid-membrane coupling. - oai:arXiv.org:2504.16823v2 - math.NA - cs.NA - physics.comp-ph - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Han Zhou, Yuan-Nan Young, Yoichiro Mori - - - A new approach to the classification of almost contact metric manifolds via intrinsic endomorphisms - https://arxiv.org/abs/2504.16900 - arXiv:2504.16900v2 Announce Type: replace -Abstract: In 1990, D. Chinea and C. Gonzalez gave a classification of almost contact metric manifolds into $2^{12}$ classes, based on the behaviour of the covariant derivative $\nabla^g\Phi$ of the fundamental $2$-form $\Phi$. This large number makes it difficult to deal with this class of manifolds. We propose a new approach to almost contact metric manifolds by introducing two intrinsic endomorphisms $S$ and $h$, which bear their name from the fact that they are, basically, the entities appearing in the intrinsic torsion. We present a new classification scheme for them by providing a simple flowchart based on algebraic conditions involving $S$ and $h$, which then naturally leads to a regrouping of the Chinea-Gonzalez classes, and, in each step, to a further refinement, eventually ending in the single classes. This method allows a more natural exposition and derivation of both known and new results, like a new characterization of almost contact metric manifolds admitting a characteristic connection in terms of intrinsic endomorphisms. We also describe in detail the remarkable (and still very large) subclass of $\mathcal{H}$-parallel almost contact manifolds, defined by the condition $(\nabla^g_X\Phi)(Y,Z)=0$ for all horizontal vector fields, $X,Y,Z\in\mathcal{H}$. - oai:arXiv.org:2504.16900v2 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Ilka Agricola, Dario Di Pinto, Giulia Dileo, Marius Kuhrt - - - A finite volume Simo-Reissner beam method for moored floating body dynamics - https://arxiv.org/abs/2504.18248 - arXiv:2504.18248v2 Announce Type: replace -Abstract: This paper presents a novel finite volume mooring line model based on the geometrically exact Simo-Reissner beam model for analysing the interaction between a floating rigid body and its mooring lines. The coupled numerical model is implemented entirely within a finite volume-based discretisation framework using a popular computational fluid dynamics C++ toolbox, OpenFOAM. Unlike existing methods for modelling mooring lines, which rely on lumped mass models or finite element-based approaches, this work simulates the mooring cables using non-linear beam models implemented in a finite volume framework to account for bending, tensile, and torsional loading. This advancement makes the current work particularly valuable for simulating extreme sea conditions. The coupled model developed in this study has been validated and verified using experimental and numerical data for a floating box moored with four catenary mooring lines under regular wave conditions featuring different wave heights and periods. The results demonstrate strong agreement with both experimental and numerical data, highlighting the model's accuracy in capturing mooring dynamics and floating body motion. - oai:arXiv.org:2504.18248v2 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - 10.1016/j.apor.2025.104845 - Amirhossein Taran, Seevani Bali, Zeljko Tukovic, Vikram Pakrashi, Philip Cardiff - - - Homogeneity in Coxeter groups and split crystallographic groups - https://arxiv.org/abs/2504.18354 - arXiv:2504.18354v2 Announce Type: replace -Abstract: We prove that affine Coxeter groups, even hyperbolic Coxeter groups and one-ended hyperbolic Coxeter groups are homogeneous in the sense of model theory. More generally, we prove that many (Gromov) hyperbolic groups generated by torsion elements are homogeneous. In contrast, we construct split crystallographic groups that are not homogeneous, and hyperbolic (in fact, virtually free) Coxeter groups that are not homogeneous (or, to be more precise, not $\mathrm{EAE}$-homogeneous). We also prove that, on the other hand, irreducible split crystallographic groups and torsion-generated hyperbolic groups are almost homogeneous. We also prove that finitely generated abelian-by-finite groups are homogeneous if and only if they are profinitely homogeneous, i.e., any tuple of words from the group is profinitely rigid. We use this to deduce that affine Coxeter groups are profinitely homogeneous, a result of independent interest in the profinite context. - oai:arXiv.org:2504.18354v2 - math.GR - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Simon Andr\'e, Gianluca Paolini - - - On the Relation Between Treewidth, Tree-Independence Number, and Tree-Chromatic Number of Graphs - https://arxiv.org/abs/2504.19751 - arXiv:2504.19751v2 Announce Type: replace -Abstract: We investigate two recently introduced graph parameters, both of which measure the complexity of the tree decompositions of a given graph. Recall that the treewidth ${\rm tw}(G)$ of a graph $G$ measures the largest number of vertices required in a bag of every tree decomposition of $G$. Similarly, the tree-independence number ${\rm tree\textnormal{-}}\alpha(G)$ and the tree-chromatic number ${\rm tree\textnormal{-}}\chi(G)$ measure the largest independence number, respectively the largest chromatic number, required in a bag of every tree decomposition of $G$. - Recently, Dallard, Milani\v{c}, and \v{S}torgel asked (JCTB, 2024) whether for all graphs $G$ it holds that ${\rm tw}(G)+1 \leq {\rm tree\textnormal{-}}\alpha(G) \cdot {\rm tree\textnormal{-}}\chi(G)$. We provide a negative answer for this question in a strong form: for every function $f\colon {\mathbb N} \rightarrow {\mathbb N}$, there exists a graph $G$ such that ${\rm tw}(G) > {\rm tree\textnormal{-}}\alpha(G) \cdot f({\rm tree\textnormal{-}}\chi(G))$. On the other hand, we complement this result with an upper bound, by showing that ${\rm tw}(G)+1 \leq {\rm tree\textnormal{-}}\alpha(G)^2 \cdot {\rm tree\textnormal{-}}\chi(G)$ for every graph $G$. - oai:arXiv.org:2504.19751v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Alex Koutsoutis, Kilian Krause, Chun-Hung Liu, Mirza Redzic, Torsten Ueckerdt - - - Global Activity Scores - https://arxiv.org/abs/2505.00711 - arXiv:2505.00711v4 Announce Type: replace -Abstract: We introduce a new global sensitivity measure, the global activity scores. We establish its theoretical connection with Sobol' sensitivity indices and demonstrate its performance through numerical examples. In these examples, we compare global activity scores with Sobol' sensitivity indices, derivative-based sensitivity measures, and activity scores. The results show that in the presence of noise or high variability, global activity scores outperform derivative-based measures and activity scores, while in noiseless settings the three approaches yield similar results. - oai:arXiv.org:2505.00711v4 - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ruilong Yue, Giray \"Okten - - - p-adic Heisenberg-Robertson-Schrodinger and p-adic Maccone-Pati Uncertainty Principles - https://arxiv.org/abs/2505.02838 - arXiv:2505.02838v2 Announce Type: replace -Abstract: Let $\mathcal{X}$ be a p-adic Hilbert space. Let $A:\mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X}$ and $B: \mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X}$ be possibly unbounded self-adjoint linear operators. For $x \in \mathcal{D}(A)$ with $\langle x, x \rangle =1$, define $ \Delta_x(A):= \|Ax- \langle Ax, x \rangle x \|.$ Then for all $x \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\langle x, x \rangle =1$, we show that \begin{align*} (1) \quad \quad \quad \max\{\Delta_x(A), \Delta_x(B)\}\geq \frac{\sqrt{\bigg|\big\langle [A,B]x, x \big\rangle ^2+\big(\langle \{A,B\}x, x \rangle -2\langle Ax, x \rangle\langle Bx, x \rangle\big)^2\bigg|}}{\sqrt{|2|}} \end{align*} and \begin{align*} (2) \quad \quad \quad \max\{\Delta_x(A), \Delta_x(B)\} \geq |\langle (A+B)x, y \rangle |, \quad \forall y \in \mathcal{X} \text{ satisfying } \|y\|\leq 1, \langle x, y \rangle =0. \end{align*} We call Inequality (1) as p-adic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as p-adic Maccone-Pati uncertainty principle. - oai:arXiv.org:2505.02838v2 - math.FA - math.OA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - K. Mahesh Krishna - - - Rationality patterns - https://arxiv.org/abs/2505.07151 - arXiv:2505.07151v2 Announce Type: replace -Abstract: In this paper, we establish general categorical frameworks that extend Loewy's classification scheme for finite-dimensional real irreducible representations of groups and Borel--Tits' criterion for the existence of rational forms of representations of $\bar{F}\otimes_F G$ for a connected reductive algebraic group $G$ over a field $F$ of characteristic zero and its algebraic closure $\bar{F}$. We also discuss applications of these general formalisms to the theory of Harish-Chandra modules, specifically to classify irreducible Harish-Chandra modules over fields $F$ of characteristic zero and to identify smaller fields of definition of irreducible Harish-Chandra modules over $\bar{F}$, particularly in the case of cohomological irreducible essentially unitarizable modules. - oai:arXiv.org:2505.07151v2 - math.RT - math.CT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Takuma Hayashi - - - High-Order Hermite Optimization: Fast and Exact Gradient Computation in Open-Loop Quantum Optimal Control using a Discrete Adjoint Approach - https://arxiv.org/abs/2505.09857 - arXiv:2505.09857v3 Announce Type: replace -Abstract: This work introduces the High-Order Hermite Optimization (HOHO) method, an open-loop discrete adjoint method for quantum optimal control. Our method is the first of its kind to efficiently compute exact (discrete) gradients when using continuous, parameterized control pulses while solving the forward equations (e.g. Schrodinger's equation or the Linblad master equation) with an arbitrarily high-order Hermite Runge-Kutta method. The HOHO method is implemented in QuantumGateDesign$.$jl (https://github.com/leespen1/QuantumGateDesign.jl), an open-source software package for the Julia programming language, which we use to perform numerical experiments comparing the method to Juqbox$.$jl (https://github.com/LLNL/Juqbox.jl). For realistic model problems we observe speedups up to 775x. - oai:arXiv.org:2505.09857v3 - math.NA - cs.NA - quant-ph - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Spencer Lee, Daniel Appelo - - - Long-Term Average Impulse Control with Mean Field Interactions - https://arxiv.org/abs/2505.11345 - arXiv:2505.11345v2 Announce Type: replace -Abstract: This paper analyzes and explicitly solves a class of long-term average impulse control problems with a specific mean-field interaction. The underlying process is a general one-dimensional diffusion with appropriate boundary behavior. The model is motivated by applications such as the optimal long-term management of renewable resources and financial portfolio management. Each individual agent seeks to maximize her long-term average reward, which consists of a running reward and income from discrete impulses, where the unit intervention price depends on the market through a stationary supply rate, the specific mean field variable to be considered. In a competitive market setting, we establish the existence of and explicitly characterize an equilibrium strategy within a large class of policies under mild conditions. Additionally, we formulate and solve the mean field control problem, in which agents cooperate with each other, aiming to realize a common maximal long-term average profit. To illustrate the theoretical results, we examine a stochastic logistic growth model and a population growth model in a stochastic environment with impulse control. - oai:arXiv.org:2505.11345v2 - math.OC - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-sa/4.0/ - K. L. Helmes, R. H. Stockbridge, C. Zhu - - - Existence of Friedrich-Wintgen Bound States in the Continuum: Cavity with a Thin Waveguide Opening - https://arxiv.org/abs/2505.12297 - arXiv:2505.12297v2 Announce Type: replace -Abstract: Bound states in the continuum (BICs) are localized states embedded within a continuum of propagating waves. Perturbations that disrupt BICs typically induce ultra-strong resonances, a phenomenon enabling diverse applications in photonics. This work investigates the existence of BICs in two-dimensional electromagnetic cavities coupled to thin waveguides for H-polarized waves. Our focus is on Friedrich-Wintgen BICs (FW-BICs), which arise from destructive interference between two resonant modes and were identified numerically in rectangular cavities with waveguide openings by Lyapina et al. [J. Fluid Mech., 780 (2015), pp. 370--387]. Here, we rigorously establish the existence of FW-BICs in a broader class of cavity geometries by introducing perturbations to the refractive index under regularity constraints. We show that BICs correspond to intersections of two curves derived implicitly from the governing equations constructed via the mode-matching method. Crucially, we prove that such intersections are guaranteed for sufficiently small waveguide widths, provided that two eigenvalues of the cavity cross and the associated eigenfunctions exhibit non-vanishing coupling to the radiation channel at the cavity-waveguide interface. Furthermore, our approach remains applicable for studying the emergence of FW-BICs under parameter-dependent boundary perturbations to the cavity. - oai:arXiv.org:2505.12297v2 - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jiaxin Zhou, Wangtao Lu, Ya Yan Lu - - - Semiparametric Off-Policy Inference for Optimal Policy Values under Possible Non-Uniqueness - https://arxiv.org/abs/2505.13809 - arXiv:2505.13809v4 Announce Type: replace -Abstract: Off-policy evaluation (OPE) constructs confidence intervals for the value of a target policy using data generated under a different behavior policy. Most existing inference methods focus on fixed target policies and may fail when the target policy is estimated as optimal, particularly when the optimal policy is non-unique or nearly deterministic. - We study inference for the value of optimal policies in Markov decision processes. We characterize the existence of the efficient influence function and show that non-regularity arises under policy non-uniqueness. Motivated by this analysis, we propose a novel \textit{N}onparametric \textit{S}equenti\textit{A}l \textit{V}alue \textit{E}valuation (NSAVE) method, which achieves semiparametric efficiency and retains the double robustness property when the optimal policy is unique, and remains stable in degenerate regimes beyond the scope of existing asymptotic theory. We further develop a smoothing-based approach for valid inference under non-unique optimal policies, and a post-selection procedure with uniform coverage for data-selected optimal policies. - Simulation studies support the theoretical results. An application to the OhioT1DM mobile health dataset provides patient-specific confidence intervals for optimal policy values and their improvement over observed treatment policies. - oai:arXiv.org:2505.13809v4 - math.ST - econ.EM - stat.ML - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Haoyu Wei - - - Periodic operators over a component domain and homogenization of some class of quasi-linear elliptic problems in two-component domain with interfacial resistance - https://arxiv.org/abs/2505.14944 - arXiv:2505.14944v3 Announce Type: replace -Abstract: This paper addresses the periodic homogenization of quasilinear elliptic PDEs in a two-component domain with an interfacial thermal barrier. It introduces a periodic extension operator that ensures strong convergence of function sequences in the Sobolev space. Moreover, two families of quasilinear elliptic problems in two-component domains with interfacial resistance will be considered here. One family with \(L^2\) data and another family with \(L^1\) data. - oai:arXiv.org:2505.14944v3 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Rodolfo E. Maza - - - An adaptive proximal safeguarded augmented Lagrangian method for nonsmooth DC problems with convex constraints - https://arxiv.org/abs/2505.15369 - arXiv:2505.15369v2 Announce Type: replace -Abstract: A proximal safeguarded augmented Lagrangian method for minimizing the difference of convex (DC) functions over a nonempty, closed and convex set with additional linear equality as well as convex inequality constraints is presented. Thereby, all functions involved may be nonsmooth. Iterates (of the primal variable) are obtained by solving convex optimization problems as the concave part of the objective function gets approximated by an affine linearization. Under the assumption of a modified Slater constraint qualification, both convergence of the primal and dual variables to a generalized Karush-Kuhn-Tucker (KKT) point is proven, at least on a subsequence. Numerical experiments and comparison with existing solution methods are presented using some classes of constrained and nonsmooth DC problems. - oai:arXiv.org:2505.15369v2 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Christian Kanzow, Tanja Neder - - - Free monoids and Riguet congruences - https://arxiv.org/abs/2505.15767 - arXiv:2505.15767v3 Announce Type: replace -Abstract: We begin by associating to $\mathbf{A}^{\star}$, the free monoid on a set $A$, a category $\mathsf{C}(\mathbf{A}^{\star})$ -- an instance of the free coproduct completion of a discrete category -- which is in general non-skeletal, and by proving that it is equivalent to $\mathsf{Set}^{A}_{\mathrm{f}}$, the category of finite $A$-sorted sets. Next, as a step toward constructing a skeletal quotient category of $\mathsf{C}(\mathbf{A}^{\star})$ via the notion of a Riguet congruence on a category, we recall this notion, correct and complete it, and examine its relationship with generalized congruences from both lattice-theoretic and category-theoretic perspectives. In particular, after introducing the notion of strong generalized congruence on a category, we prove that, for any category $\mathsf{C}$, there exists an isotone and Scott continuous morphism from $(\mathrm{RCgr}(\mathsf{C}),\subseteq)$, the bounded directed-complete ordered set of Riguet congruences on $\mathsf{C}$ to $(\mathrm{GCgr}(\mathsf{C}),\subseteq)$, the algebraic lattice of generalized congruences on $\mathsf{C}$, that sends a Riguet congruence $\Phi$ on $\mathsf{C}$ to the strong generalized congruence $\Phi^{\natural}$ on $\mathsf{C}$. Finally, for a suitable Riguet congruence on $\mathsf{C}(\mathbf{A}^{\star})$, denoted by $\equiv^{A}$, we construct a skeletal quotient category $\mathsf{Q}(\mathbf{A}^{\star})$ of $\mathsf{C}(\mathbf{A}^{\star})$ and prove that it is equivalent to $\mathsf{Set}^{A}_{\mathrm{f}}$ and also to $\mathsf{C}(\mathbf{A}^{\star})/{\equiv^{A\natural}}$, where $\equiv^{A\natural}$ is the strong generalized congruence on $\mathsf{C}(\mathbf{A}^{\star})$ canonically associated to $\equiv^{A}$. - oai:arXiv.org:2505.15767v3 - math.CT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Juan Climent Vidal, Enric Cosme Ll\'opez, Ra\'ul Ruiz Mora - - - On groups with EDT0L word problem - https://arxiv.org/abs/2505.20057 - arXiv:2505.20057v3 Announce Type: replace -Abstract: We prove that the word problem for the infinite cyclic group is not EDT0L, and obtain as a corollary that a finitely generated group with EDT0L word problem must be torsion. In addition, we show that the property of having an EDT0L word problem is invariant under change of generating set and passing to finitely generated subgroups. This represents significant progress towards the conjecture that all groups with EDT0L word problem are finite (i.e. precisely the groups with regular word problem). - oai:arXiv.org:2505.20057v3 - math.GR - cs.FL - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alex Bishop, Murray Elder, Alex Evetts, Paul Gallot, Alex Levine - - - Locating Extremal Periodic Orbits for the Planar Circular Restricted Three Body Problem using Polynomial Sum-of-Squares Optimization - https://arxiv.org/abs/2505.23430 - arXiv:2505.23430v2 Announce Type: replace -Abstract: With an increasing interest in the design of long and complex space missions, the search for orbits that require the least amount of fuel is of fundamental interest. This paper develops existing computational models for locating Unstable Periodic Orbits (UPOs) in polynomial dynamical systems using Sum-of-Squares (SOS) optimization technique and proposes a numerical framework to converge UPOs for the Planar Circular Restricted Three-Body Problem (PCR3BP) in astrodynamics. This is done by developing the polynomial SOS optimization technique with extension to systems with non-polynomial and Hamiltonian dynamics. First, we demonstrate and exploit the dependency of convergence of tight bounds on an observable of interest with varying scaling factors for large polynomial degrees. SOS optimization is then used to compute nonnegative polynomials, the minimization sublevel sets of which, approximately localise parts of the corresponding UPO. Improvements in current non-linear optimization techniques are suggested to compute a large number of points inside the relevant sublevel sets. Such points provide good initial conditions for UPO computations with existing algorithms. The distinguishing feature of such UPOs is that they optimize the long-time average of an input observable of interest which is a function of state variables. For the PCR3BP this means that such orbits in space can be traversed indefinitely in time without continuous fuel expenditure. As practical applications to space mission designs, we converge UPOs that minimise transmitted power required by satellites for the Earth-Moon system in a communication relay problem by minimizing the infinite-time average of sum of squares of distances of a satellite from Earth and the Moon. - oai:arXiv.org:2505.23430v2 - math.DS - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Vinay Sharma, Sergei I Chernyshenko - - - Rydberg Atomic Receivers for Multi-Band Communications and Sensing - https://arxiv.org/abs/2505.24168 - arXiv:2505.24168v3 Announce Type: replace -Abstract: Harnessing multi-level electron transitions, Rydberg Atomic REceivers (RAREs) can detect wireless signals across a wide range of frequency bands, from Megahertz to Terahertz. This capability enables multi-band wireless communications and sensing (CommunSense). Existing research on multi-band RAREs primarily focuses on experimental demonstrations, lacking a tractable model to mathematically characterize their mechanisms. This issue leaves the multi-band RARE as a black box and poses challenges in its practical applications. To fill in this gap, this paper investigates the underlying mechanism of multiband RAREs and explores their optimal performance. For the first time, an analytical transfer function with a closed-form expression for multi-band RAREs is derived by solving the quantum response of Rydberg atoms. It shows that a multiband RARE simultaneously serves as a multi-band atomic mixer for down-converting multi-band signals and a multi-band atomic amplifier that reflects its sensitivity to each band. Further analysis of the atomic amplifier unveils that the intrinsic gain at each frequency band can be decoupled into a global gain term and a Rabi attention term. The former determines the overall sensitivity of a RARE to all frequency bands of wireless signals. The latter influences the allocation of the overall sensitivity to each frequency band, representing a unique attention mechanism of multi-band RAREs. The optimal design of the global gain is provided to maximize the overall sensitivity of multi-band RAREs. Subsequently, the optimal Rabi attentions are also derived to maximize the practical multi-band CommunSense performance. An experiment platform is built to validate the effectiveness of the derived transfer function, and numerical results confirm the superiority of multi-band RAREs. - oai:arXiv.org:2505.24168v3 - cs.IT - eess.SP - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/publicdomain/zero/1.0/ - Mingyao Cui, Qunsong Zeng, Minze Chen, Zhanwei Wang, Tianqi Mao, Dezhi Zheng, Kaibin Huang - - - Convergence of spectra of digraph limits - https://arxiv.org/abs/2506.04426 - arXiv:2506.04426v2 Announce Type: replace -Abstract: The relation between densities of cycles and the spectrum of a graphon, which implies that the spectra of convergent graphons converge, fundamentally relies on the self-adjointness of the linear operator associated with a graphon. In this short paper, we consider the setting of digraphons, which are limits of directed graphs, and prove that the spectra of convergent digraphons converge. Using this result, we establish the relation between densities of directed cycles and the spectrum of a digraphon. - oai:arXiv.org:2506.04426v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jan Greb\'ik, Daniel Kr\'al', Xizhi Liu, Oleg Pikhurko, Julia Slipantschuk - - - Rejection-Sampled Linear Codes for Lossy Compression and Channel Simulation - https://arxiv.org/abs/2506.09239 - arXiv:2506.09239v2 Announce Type: replace -Abstract: We show that linear codes combined with rejection sampling can yield a capacity-achieving scheme for simulating additive exchangeable noise channels. Specifically, our scheme achieves an amount of communication within $\log e + 1$ bits from the excess functional information lower bound. Hence, it can be used in lossy source coding to achieve the rate-distortion function. We discuss practical implementations based on BCH codes and polar codes. For the simulation of binary symmetric channels, the BCH-based construction with a blocklength of $n = 63$ attains a rate comparable to the PolarSim with $n = 4096$, while significantly reducing the latency. The polar-based construction asymptotically achieves the channel capacity with polynomial average complexity. Furthermore, using the idea from greedy rejection sampling, we propose an algorithm to construct capacity-achieving schemes based on any linear codes. Experiments reveal that our construction can outperform conventional covering codes for lossy source coding with Hamming distortion for a certain range of distortion levels, and performs well even when the blocklength is small (e.g., $n = 24$). - oai:arXiv.org:2506.09239v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jianguo Zhao, Cheuk Ting Li - - - Sobolev regularity for the $\overline\partial$-Neumann operator and transverse vector fields - https://arxiv.org/abs/2506.10372 - arXiv:2506.10372v2 Announce Type: replace -Abstract: On a bounded smooth pseudoconvex domain in $\mathbb{C}^n$ with $n >2$, inspired by the compactness condition introduced by Yue Zhang, we present the new sufficient condition for the exact regularity of the $\overline\partial$-Neumann operator via the transverse vector fields. - oai:arXiv.org:2506.10372v2 - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Qianyun Wang, Yuan Yuan, Xu Zhang - - - Two axes in non-commutative algebras with a Frobenius form - https://arxiv.org/abs/2506.11303 - arXiv:2506.11303v5 Announce Type: replace -Abstract: Throughout this paper $A$ is a commutative non-associative algebra over a field $\mathbb{F}$ of characteristic not $2.$ In addition $A$ posses a Frobenius form. We obtain detailed information about the multiplication in $A$ given two axes of type half in $A.$ - oai:arXiv.org:2506.11303v5 - math.RA - math.GR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/publicdomain/zero/1.0/ - Yoav Segev - - - Serre's question on thin sets in projective space - https://arxiv.org/abs/2506.13471 - arXiv:2506.13471v2 Announce Type: replace -Abstract: We answer a question of Serre from the 1980s on rational points of bounded height on projective thin sets, in degree at least $4$. For degrees $2$ and $3$ we improve the known bounds in general. The focus is on thin sets of type II, namely corresponding to the images of ramified dominant quasi-finite covers of projective space, as thin sets of type I are already well understood via dimension growth results by the third author in 2002 (published in 2023) by a global variant of Heath-Brown's $p$-adic determinant method. For type II, we obtain a uniform affine variant of Serre's question which implies the projective case and for which the implicit constant is furthermore polynomial in the degree. We are able to avoid logarithmic factors when the degree is at least $5$ and we prove our results over any global field, of any characteristic. A key ingredient for obtaining the affine variant comes from Binyamini-Cluckers-Novikov (2024) and Binyamini-Cluckers-Kato (2025) where a question of the third author was answered by providing bounds, for rational points on irreducible curves, which are quadratic in the degree. A second key ingredient is an adaptation of Salberger's global determinant method to the case of weighted polynomials. The third key ingredient is the design of our affine variant of Serre's question, for weighted polynomials which are not necessarily weighted homogeneous. - oai:arXiv.org:2506.13471v2 - math.NT - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tijs Buggenhout, Raf Cluckers, Per Salberger, Tim Santens, Floris Vermeulen - - - Instantaneous blowup for interacting SDEs with superlinear drift - https://arxiv.org/abs/2506.21164 - arXiv:2506.21164v2 Announce Type: replace -Abstract: We consider a system of interacting SDEs on the integer lattice with multiplicative noise and a drift satisfying the finite Osgood's condition. We show instantaneous everywhere blowup for initial profiles decaying slower than $\exp \left( -\sqrt{\big|\log |x|\big|}\right)$. We employ the splitting-up method to compare the interacting system to a one-dimensional SDE which blows up. - oai:arXiv.org:2506.21164v2 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Mathew Joseph, Shubham Ovhal - - - On a result by Meshulam - https://arxiv.org/abs/2506.22553 - arXiv:2506.22553v2 Announce Type: replace -Abstract: In 1996, Meshulam proved that every sequence generated by applying projections onto affine subspaces, drawn from a finite collection in Euclidean space, must be bounded. - In this paper, we extend his result not only from affine subspaces to convex polyhedral subsets, but also from Euclidean to general Hilbert space. Various examples are provided to illustrate the sharpness of the results. - oai:arXiv.org:2506.22553v2 - math.OC - cs.NA - math.FA - math.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Heinz H. Bauschke, Tran Thanh Tung - - - Degrees of non-Gorenstein canonical Fano threefolds with Picard number one - https://arxiv.org/abs/2507.04615 - arXiv:2507.04615v3 Announce Type: replace -Abstract: We show that the optimal upper bound for the anticanonical degrees of non-Gorenstein $\mathbb{Q}$-factorial canonical Fano threefolds with Picard number one is 200/3. - oai:arXiv.org:2507.04615v3 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Minyou Li - - - Weinstein neighbourhood theorems for stratified subspaces - https://arxiv.org/abs/2507.04897 - arXiv:2507.04897v2 Announce Type: replace -Abstract: By analogy with Weinstein's neighbourhood theorem, we prove a uniqueness result for symplectic neighbourhoods of a large family of stratified subspaces. This result generalizes existing constructions, e.g., in the search for exotic Lagrangians. Along the way, we prove a strong version of Moser's trick and a (non-symplectic) tubular neighbourhood theorem for these stratified subspaces. - oai:arXiv.org:2507.04897v2 - math.SG - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yael Karshon, Sara B. Tukachinsky, Yoav Zimhony - - - Elementary equivalence and diffeomorphism groups of smooth manifolds - https://arxiv.org/abs/2507.07427 - arXiv:2507.07427v3 Announce Type: replace -Abstract: Let $M$ and $N$ be smooth manifolds, with $M$ closed and connected. If the $C^r$--diffeomorphism group of $M$ is elementarily equivalent to the $C^s$--diffeomorphism group of $N$ for some $r,s\in[1,\infty)\cup\{0,\infty\}$, then $r=s$ and $M$ and $N$ are $C^r$--diffeomorphic. This strengthens a previously known result by Takens and Filipkiewicz, which asserts that for integer regularities, a group isomorphism between diffeomorphism groups of closed manifolds necessarily arises from a diffeomorphism of the underlying manifolds. We prove an analogous result for groups of diffeomorphisms preserving smooth volume forms, in dimension at least two. - oai:arXiv.org:2507.07427v3 - math.GR - math.GT - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sang-hyun Kim, Thomas Koberda, J. de la Nuez Gonz\'alez - - - $f$-algebra products on AL and AM-spaces - https://arxiv.org/abs/2507.08435 - arXiv:2507.08435v2 Announce Type: replace -Abstract: We characterize all $f$-algebra products on AM-spaces by constructing a canonical AM-space $W_X$ associated to each AM-space $X$, such that the $f$-algebra products on $X$ correspond bijectively to the positive cone $(W_X)_+$. This generalizes the classical description of $f$-algebra products on $C(K)$ spaces. We also identify the unique product (when it exists) that embeds $X$ as a closed subalgebra of $C(K)$, and study AM-spaces for which this product exists -- the so-called AM-algebras. Finally, we investigate AM-spaces that admit only the zero product, providing a characterization in the AL-space case and examples showing that no simple characterization exists in general. - oai:arXiv.org:2507.08435v2 - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - David Mu\~noz-Lahoz - - - Classification of curl forces for all space dimensions - https://arxiv.org/abs/2507.09817 - arXiv:2507.09817v2 Announce Type: replace -Abstract: We present a decomposition of classical potentials into a conservative (gradient) component and a non-conservative component. The latter generalizes the curl component of the force in the three-dimensional case. The force is transformed into a differential $1$-form, known as the work form. This work form is decomposed into an exact (gradient) component and an antiexact component, which in turn generalizes the curl part of the force. The antiexact component is subsequently decomposed using the Frobenius theorem. This local decomposition is a useful tool for identifying the specific components of classical potentials. - oai:arXiv.org:2507.09817v2 - math-ph - math.DG - math.MP - physics.class-ph - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rados{\l}aw Antoni Kycia - - - Finite approximation of free groups II: the Theorems of Ash, Herwig-Lascar and Ribes-Zalesskii -- revisited and strengthened - https://arxiv.org/abs/2507.11685 - arXiv:2507.11685v3 Announce Type: replace -Abstract: Relations and interactions between the theorems of Ash, Herwig-Lascar and Ribes-Zalesskii are discussed and it is shown that these three theorems are equivalent in the sense that each of them can be derived from each other one. Some strengthening of these theorems that can be obtained by use of the groups provided by the third author's construction are also considered. - oai:arXiv.org:2507.11685v3 - math.GR - math.CO - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - K. Auinger, J. Bitterlich, M. Otto - - - Refinement of the theory and convergence of the Sinc convolution -- beyond Stenger's conjecture - https://arxiv.org/abs/2507.12406 - arXiv:2507.12406v3 Announce Type: replace -Abstract: The Sinc convolution is an approximate formula for indefinite convolutions proposed by Stenger. The formula was derived based on the Sinc indefinite integration formula combined with the single-exponential transformation. Although its efficiency has been confirmed in various fields, several theoretical issues remain unresolved. The first contribution of this study is to resolve those issues by refining the underlying theory of the Sinc convolution. This contribution includes an essential resolution of Stenger's conjecture. The second contribution of this study is to improve the convergence rate by replacing the single-exponential transformation with the double-exponential transformation. Theoretical analysis and numerical experiments confirm that the modified formula achieves superior convergence compared to Stenger's original formula. - oai:arXiv.org:2507.12406v3 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Tomoaki Okayama - - - Degenerations of families of bands and strings for gentle algebras - https://arxiv.org/abs/2507.13945 - arXiv:2507.13945v2 Announce Type: replace -Abstract: Let $A$ be a gentle algebra. For every collection of string and band diagrammes, we consider the constructible subset of the variety of representations containing all modules with this underlying diagramme. We study degenerations of such sets. We show that these sets are defined by vectors of integers which we call $h$-vectors and which are related to a restricted version of the hom-order. We provide combinatorial criteria for the existence of a degeneration, involving the removal of an arrow or the resolving of a type of configuration called "reaching". - oai:arXiv.org:2507.13945v2 - math.RT - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Judith Marquardt - - - The inference of Fokker-Planck equations via transport maps - https://arxiv.org/abs/2507.15091 - arXiv:2507.15091v3 Announce Type: replace -Abstract: We present a framework, which, from the trajectories detailing the spatiotemporal dynamics of a population, simultaneously reconstructs a transport map as well as the Fokker-Planck equation governing the coarse-grained probability distribution. Leveraging the Knothe-Rosenblatt rearrangement, we model the transport map from a fixed reference distribution to the target distribution, and derive the velocity fields of the flows from the trajectory of transport maps. Exploiting the velocity fields, we circumvent spatial gradients to infer the Fokker-Planck equation's potential and diffusivity. The sparsity of trajectories injects uncertainty, which we treat in a Bayesian setting using variational inference. The approach is applied to inferring the Fokker-Planck dynamics in spaces of up to five dimensions, demonstrating both accurate identification of the system and efficiency with respect to data size. - oai:arXiv.org:2507.15091v3 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Saem Han, Krishna Garikipati - - - Structured linear factor models for tail dependence - https://arxiv.org/abs/2507.16340 - arXiv:2507.16340v2 Announce Type: replace -Abstract: A common object to describe the extremal dependence of a $d$-variate random vector $X$ is the stable tail dependence function $L$. Various parametric models have emerged, with a popular subclass consisting of those stable tail dependence functions that arise for linear and max-linear factor models with heavy tailed factors. The stable tail dependence function is then parameterized by a $d \times K$ matrix $A$, where $K$ is the number of factors and where $A$ can be interpreted as a factor loading matrix. We study estimation of $L$ under an additional assumption on $A$ called the `pure variable assumption'. Both $K \in \{1, \dots, d\}$ and $A \in [0, \infty)^{d \times K}$ are treated as unknown, which constitutes an unconventional parameter space that does not fit into common estimation frameworks. We suggest two algorithms that allow to estimate $K$ and $A$, and provide finite sample guarantees for both algorithms. Remarkably, the guarantees allow for the case where the dimension $d$ is larger than the sample size $n$. The results are illustrated with numerical experiments and two case studies. - oai:arXiv.org:2507.16340v2 - math.ST - stat.ME - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexis Boulin, Axel B\"ucher - - - Analytic Theory on the Space of Blaschke Products: Simultaneous Uniformization and Pressure Metric - https://arxiv.org/abs/2507.17077 - arXiv:2507.17077v2 Announce Type: replace -Abstract: In this paper, we study complex analytic aspects of the moduli space $\Bcal_d^{fm}$ of degree $d\ge2$ fixed-point-marked Blaschke products. We define a complex structure on $\Bcal_d^{fm}$ and prove the simultaneous uniformization theorem for fixed-point-marked quasi-Blaschke products. As an application, we show that the pressure semi-norms on the space of Blaschke products are non-degenerate outside of the super-attracting locus $\mathcal{SA}^{fm}_d$, which is a codimension-1 subspace of $\Bcal^{fm}_d$. - oai:arXiv.org:2507.17077v2 - math.DS - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yan Mary He, Homin Lee, Insung Park - - - The norms for symmetric and antisymmetric tensor products of the weighted shift operators - https://arxiv.org/abs/2507.17181 - arXiv:2507.17181v2 Announce Type: replace -Abstract: In the present paper, we study the norms for symmetric and antisymmetric tensor products of weighted shift operators. By proving that for $n\geq 2$, - $$\|S_{\alpha}^{l_1}\odot\cdots \odot S_{\alpha}^{l_k}\odot S_{\alpha}^{*l_{k+1}}\odot\cdots \odot S_{\alpha}^{*l_{n}}\| =\mathop{\prod}_{i=1}^n\left \| S_{\alpha}^{{l_{i}}}\right\|, \text{ for any} \ (l_1,l_2\cdots l_n)\in\mathbb N^n$$ if and only if the weight satisfies the regularity condition, we partially solve \cite[Problem 6 and Problem 7]{GA}. It will be seen that most weighted shift operators on function spaces, including weighted Bergman shift, Hardy shift, Dirichlet shift, etc, satisfy the regularity condition. Moreover, at the end of the paper, we solve \cite[Problem 1 and Problem 2]{GA}. - oai:arXiv.org:2507.17181v2 - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Xiance Tian, Penghui Wang, Zeyou Zhu - - - Finite generation of abelianizations of the genus 3 Johnson kernel and the commutator subgroup of the Torelli group for $\mathrm{Out}(F_3)$ - https://arxiv.org/abs/2507.20710 - arXiv:2507.20710v2 Announce Type: replace -Abstract: Let $\Sigma_g^b$ be a compact oriented surface of genus $g$ with $b$ boundary components, where $b\in\{0,1\}$. The Johnson kernel $\mathcal{K}_g^b$ is the subgroup of the mapping class group $\mathrm{Mod}(\Sigma_g^b)$ generated by Dehn twists about separating simple closed curves. Let $F_n$ be a free group with $n$ generators. The Torelli group for $\mathrm{Out}(F_n)$ is the subgroup $\mathrm{IO}_n\subset\mathrm{Out}(F_n)$ consisting of all outer automorphisms that act trivially on the abelianization of $F_n$. Long standing questions are whether the groups $\mathcal{K}_g^b$ and $[\mathrm{IO}_n,\mathrm{IO}_n]$ or their abelianizations $(\mathcal{K}_g^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_n,\mathrm{IO}_n]^{\mathrm{ab}}$ are finitely generated for $g\ge3$ (respectively, $n\ge3$). During the last 15 years, these questions were answered positively for $g\ge4$ and $n\ge4$, respectively. Nevertheless, the cases of $g=3$ and $n=3$ remained completely unsettled. In this paper, we prove that the abelianizations $(\mathcal{K}_3^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_3,\mathrm{IO}_3]^{\mathrm{ab}}$ are finitely generated. Our approach is based on a new general sufficient condition for a module over a Laurent polynomial ring to be finitely generated as an abelian group. - oai:arXiv.org:2507.20710v2 - math.GR - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexander A. Gaifullin - - - On admissibility in post-hoc hypothesis testing - https://arxiv.org/abs/2508.00770 - arXiv:2508.00770v3 Announce Type: replace -Abstract: The validity of classical hypothesis testing requires the significance level $\alpha$ be fixed before any statistical analysis takes place. This is a stringent requirement. For instance, it prohibits updating $\alpha$ during (or after) an experiment due to changing concern about the cost of false positives, or to reflect unexpectedly strong evidence against the null. Perhaps most disturbingly, witnessing a p-value $p\ll\alpha$ vs $p= \alpha- \epsilon$ for tiny $\epsilon > 0$ has no (statistical) relevance for any downstream decision-making. Following recent work of Gr\"unwald (2024), we develop a theory of post-hoc hypothesis testing, enabling $\alpha$ to be chosen after seeing and analyzing the data. To study "good" post-hoc tests we introduce $\Gamma$-admissibility, where $\Gamma$ is a set of adversaries which map the data to a significance level. We classify the set of $\Gamma$-admissible rules for various sets $\Gamma$, showing they must be based on e-values, and recover the Neyman-Pearson lemma when $\Gamma$ is the constant map. - oai:arXiv.org:2508.00770v3 - math.ST - stat.ME - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ben Chugg, Tyron Lardy, Aaditya Ramdas, Peter Gr\"unwald - - - Exceptional dual pair correspondences; case of real groups of split rank one - https://arxiv.org/abs/2508.01551 - arXiv:2508.01551v2 Announce Type: replace -Abstract: Exceptional real groups have quaternionic forms of split rank 4 that contain dual pairs $G\times G'$, where $G'$ is the split Lie group of the type $G_2$, and $G$ a Lie group of split rank one. In this paper we restrict the minimal representation of the quaternionic group to the dual pair and prove some significant results for the resulting correspondence of representations. - oai:arXiv.org:2508.01551v2 - math.RT - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Petar Bakic, Hung Yean Loke, Gordan Savin - - - Large AI Models for Wireless Physical Layer - https://arxiv.org/abs/2508.02314 - arXiv:2508.02314v2 Announce Type: replace -Abstract: Large artificial intelligence models (LAMs) are transforming wireless physical layer technologies through their robust generalization, multitask processing, and multimodal capabilities. This article reviews recent advancements in applying LAMs to physical layer communications, addressing obstacles of conventional AI-based approaches. LAM-based solutions are classified into two strategies: leveraging pre-trained LAMs and developing native LAMs designed specifically for physical layer tasks. The motivations and key frameworks of these approaches are comprehensively examined through multiple use cases. Both strategies significantly improve performance and adaptability across diverse wireless scenarios. Future research directions, including efficient architectures, interpretability, standardized datasets, and collaboration between large and small models, are proposed to advance LAM-based physical layer solutions for next-generation communication systems. - oai:arXiv.org:2508.02314v2 - cs.IT - cs.AI - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jiajia Guo, Yiming Cui, Shi Jin, Jun Zhang - - - Coordinate-independent model reductions of chemical reaction networks based on geometric singular perturbation theory - https://arxiv.org/abs/2508.03304 - arXiv:2508.03304v2 Announce Type: replace -Abstract: The quasi-steady-state approximation (QSSA) is a standard technique for reducing the complexity of chemical reaction networks (CRNs). The validity of any QSSA-based model is restricted to specific parameter regimes. Selecting the appropriate reduction is not always straightforward. At times, QSSAs are misused outside of their validity regions and, even when a particular QSSA is considered valid in a given parameter regime, other QSSAs may be simultaneously valid, creating ambiguity. - Here, we employ a more powerful alternative: a constructive model reduction framework based on coordinate-independent geometric singular perturbation theory (ci-GSPT) and the parametrization method. A key advantage of this approach is its ability to derive reduced models independent of a clear timescale separation in the variables for a specific parameter configuration. - We demonstrate our approach on two benchmark systems. For the Michaelis-Menten (MM) reaction, we show that the framework provides a systematic approach by exploring parameter configurations across three orders of magnitude: asymptotically large, small, and `order one'. A consequence of this systematic analysis is a geometric classification that categorizes the resulting model reductions and provides a point of comparison between our approach and common QSSA variants in the literature. For the more complex Kim-Forger model, we show that this approach successfully produces a reduction without the need for a coordinate transformation, showcasing its applicability to larger systems. - oai:arXiv.org:2508.03304v2 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Timothy Earl Figueroa Lapuz, Martin Wechselberger - - - A dimensional mass transference principle from balls to open sets and applications to dynamical Diophantine approximation - https://arxiv.org/abs/2508.03359 - arXiv:2508.03359v2 Announce Type: replace -Abstract: The mass transference principle of Beresnevich and Velani is a powerful mechanism for determining the Hausdorff dimension/measure of $\limsup$ sets that arise naturally in Diophantine approximation. However, in the setting of dynamical Diophantine approximation, this principle often fails to apply effectively, as the radii of the balls defining the dynamical $\limsup$ sets generally depend on the orbit of the point $x$ itself. - In this paper, we develop a dimensional mass transference principle that enables us to recover and extend classical results on shrinking target problems, particularly for the $\beta$-transformation and the Gauss map. Moreover, our result shows that the corresponding $\limsup$ sets have large intersection properties. A potentially interesting feature of our method is that, in many cases, shrinking target problems are closely related to finding an appropriate Gibbs measure, which may reveal new aspects of the link between thermodynamic formalism and dynamical Diophantine approximation. - oai:arXiv.org:2508.03359v2 - math.NT - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yubin He - - - Computable Bounds for Strong Approximations with Applications - https://arxiv.org/abs/2508.03833 - arXiv:2508.03833v2 Announce Type: replace -Abstract: The Koml\'os$\unicode{x2013}$Major$\unicode{x2013}$Tusn\'ady (KMT) inequality for partial sums is one of the most celebrated results in probability theory. Yet its practical application has been hindered by a lack of practical constants. This paper addresses this limitation for bounded i.i.d. random variables. At the cost of an additional logarithmic factor, we propose a computable version of the KMT inequality that depends only on the variables' range and standard deviation. We also derive an empirical version of the inequality that achieves nominal coverage even when the standard deviation is unknown. We then demonstrate the practicality of our bounds through applications to online change point detection and first hitting time probabilities. As a byproduct of our analysis, we obtain a Cram\'er-type moderate deviation bound for normalized centered partial sums. - oai:arXiv.org:2508.03833v2 - math.ST - math.PR - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Haoyu Ye, Morgane Austern - - - Nonparametric Estimation from Correlated Copies of a Drifted Process - https://arxiv.org/abs/2508.05259 - arXiv:2508.05259v2 Announce Type: replace -Abstract: This paper presents several situations leading to the observation of multiple correlated copies of a drifted process, and then non-asymptotic risk bounds are established on nonparametric estimators of the drift function $b_0$ and its derivative. For drifted Gaussian processes with a regular enough covariance function, a sharper risk bound is established on the estimator of $b_0'$, and a model selection procedure is provided with theoretical guarantees. - oai:arXiv.org:2508.05259v2 - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Nicolas Marie - - - Entropy and polynomial entropy of derived autoequivalences of derived discrete algebras - https://arxiv.org/abs/2508.05794 - arXiv:2508.05794v2 Announce Type: replace -Abstract: The aim of this paper is to calculate entropy in the sense of Dimitrov-Haiden-Katzarkov-Kontsevich and polynomial entropy as defined by Fan-Fu-Ouchi of derived autoequivalences of derived discrete algebras over an algebraically closed field. - oai:arXiv.org:2508.05794v2 - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Tomasz Ciborski - - - Optimum 1-Step Majority-Logic Decoding of Binary Reed-Muller Codes - https://arxiv.org/abs/2508.08736 - arXiv:2508.08736v3 Announce Type: replace -Abstract: The classical majority-logic decoder proposed by Reed for Reed-Muller codes RM(r, m) of order r and length 2^m, unfolds in r+1 sequential steps, decoding message symbols from highest to lowest degree. Several follow-up decoding algorithms reduced the number of steps, but for a limited set of parameters, or at the expense of reduced performance, or relying on the existence of some combinatorial structures. We show that any one-step majority-logic decoder-that is, a decoder performing all majority votes in one step simultaneously without sequential processing-can correct at most d_min/4 errors for all values of r and m, where d_min denotes the code's minimum distance. We then introduce a new hard-decision decoder that completes the decoding in a single step and attains this error-correction limit. It applies to all r and m, and can be viewed as a parallel realization of Reed's original algorithm, decoding all message symbols simultaneously. Remarkably, we also prove that the decoder is optimum in the erasure setting: it recovers the message from any erasure pattern of up to d_min-1 symbols-the theoretical limit. To our knowledge, this is the first 1-step decoder for RM codes that achieves both optimal erasure correction and the maximum one-step error correction capability. - oai:arXiv.org:2508.08736v3 - cs.IT - math.CO - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Hoang Ly, Emina Soljanin - - - Tatuzawa's theorem for Rankin-Selberg $L$-functions - https://arxiv.org/abs/2508.10844 - arXiv:2508.10844v3 Announce Type: replace -Abstract: Let $\pi$ and $\pi'$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ over a number field $F$. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,\pi\times\pi')$, generalizing Tatuzawa's refinement of Siegel's work on Dirichlet $L$-functions. As a corollary, we show that for all $\varepsilon>0$, there exists an effectively computable constant $c>0$ depending only on $(n,n',[F:\mathbb{Q}],\varepsilon)$ such that $L(s,\pi\times\pi')$ has at most one zero (necessarily simple) in the region \[ \mathrm{Re}(s)\geq 1-c/(C(\pi)C(\pi')(|\mathrm{Im}(s)|+1))^{\varepsilon}, \] where $C(\pi)$ and $C(\pi')$ are the analytic conductors. A crucial component of our proof is a new standard zero-free region for any twist of $L(s,\pi\times\widetilde{\pi})$ by an idele class character $\chi$ apart from a possible single exceptional zero (necessarily real and simple) that can occur only when $\pi\otimes\chi^2=\pi$. This extends earlier work of Humphries and Thorner. - oai:arXiv.org:2508.10844v3 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Gergely Harcos, Jesse Thorner - - - High-Capacity and Low-PAPR BICM-OFDM Systems Using Non-Equiprobable and Non-Uniform Constellation Shaping With Clipping and Filtering - https://arxiv.org/abs/2508.15639 - arXiv:2508.15639v2 Announce Type: replace -Abstract: We address a design of high-capacity and low-peak-to-average power ratio (PAPR) orthogonal frequency-division multiplexing (OFDM) systems based on bit-interleaved coded modulation (BICM) utilizing non-equiprobable and non-uniform (NENU) constellations as well as clipping and filtering (CAF). The proposed constellations are generated using a truncated Gaussian distribution and the merging of constellation points, where the former creates a non-uniform constellation (NUC), and the latter adjusts the number of signal points for further improving the total bit-wise mutual information (BMI). Unlike other exhaustive search-based approaches, the proposed constellations are uniquely determined by only two parameters associated with NUC and cardinality. Due to this property of limited degrees of freedom, the complexity required for the numerical optimization process can be significantly low. We focus on the constellation design based on one dimension, i.e., pulse amplitude modulation (PAM), which facilitates the reduction of demapping complexity for the BICM receiver. The use of CAF at the transmitter can efficiently reduce the PAPR of OFDM signals; however, it introduces clipping noise that may degrade error rate performance, making the application of clipping noise cancellation (CNC) at the receiver essential. Therefore, we optimize the NENU constellations in the presence of CAF and CNC. Simulation results demonstrate that the combination of constellation shaping with CAF and CNC enables BICM-OFDM systems to simultaneously achieve low PAPR and high spectral efficiency over additive white Gaussian noise (AWGN) as well as frequency-selective fading channels. Furthermore, comparative studies confirm that the proposed system significantly outperforms the single-carrier counterpart (i.e., DFT-precoded BICM-OFDM) in terms of PAPR and bit error rate (BER) performance over fading channels. - oai:arXiv.org:2508.15639v2 - cs.IT - eess.SP - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Eito Kurihara, Hideki Ochiai - - - Physics-Informed Kolmogorov-Arnold Networks for multi-material elasticity problems in electronic packaging - https://arxiv.org/abs/2508.16999 - arXiv:2508.16999v2 Announce Type: replace -Abstract: This paper proposes a Physics-Informed Kolmogorov-Arnold Network for analyzing elasticity problems in multi-material electronic packaging structures. The method replaces traditional Multi-Layer Perceptrons with Kolmogorov-Arnold Networks within an energy-based Physics-Informed Neural Network framework. By constructing admissible displacement fields satisfying essential boundary conditions and optimizing network parameters through numerical integration, the proposed method effectively handles material property discontinuities. Unlike traditional methods that require domain decomposition and interface constraints for multi-material problems, Kolmogorov-Arnold Networks' trainable B-spline activation functions provide inherent piecewise characteristics. This capability stems from B-splines' local support, which enables effective approximation of discontinuities despite their individual smoothness. Consequently, this approach enables accurate approximation across the entire domain using a single network and simplifying the computational framework. Numerical experiments demonstrate that the proposed method achieves excellent accuracy and robustness in multi-material elasticity problems, validating its practical potential for electronic packaging analysis. Source codes are available at https://github.com/yanpeng-gong/PIKAN-MultiMaterial. - oai:arXiv.org:2508.16999v2 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Yanpeng Gong, Yida He, Yue Mei, Xiaoying Zhuang, Fei Qin, Timon Rabczuk - - - Local Well-Posedness of the Cahn-Hilliard-Biot System - https://arxiv.org/abs/2508.17893 - arXiv:2508.17893v2 Announce Type: replace -Abstract: We show short-time well-posedness of a diffuse interface model describing the flow of a fluid through a deformable porous medium consisting of two phases. The system non-linearly couples Biot's equations for poroelasticity, including phase-field dependent material properties, with the Cahn-Hilliard equation to model the evolution of the solid, where we further distinguish between the absence and presence of a visco-elastic term of Kelvin-Voigt type. While both problems will be reduced to a fixed-point equation that can be solved using maximal regularity theory along with a contraction argument, the first case relies on a semigroup approach over suitable Hilbert spaces, whereas treating the second case under minimal assumptions with respect to spatial regularity necessitates the application of Banach scales. - oai:arXiv.org:2508.17893v2 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Helmut Abels, Jonas Haselb\"ock - - - Calogero-Sutherland hyperbolic system and Heckman-Opdam $\mathfrak{gl}_n$ hypergeometric function - https://arxiv.org/abs/2508.18864 - arXiv:2508.18864v3 Announce Type: replace -Abstract: We prove equivalence of two integral representations for the wave functions of hyperbolic Calogero-Sutherland system. For this we study two families of Baxter operators related to hyperbolic Calogero-Sutherland and rational Ruijsenaars models; the first one as a limit from hyperbolic Ruijsenaars system, while the second one independently. Besides, computing asymptotics of integral representations and also the value at zero point, we identify them with renormalized Heckman-Opdam $\mathfrak{gl}_n$ hypergeometric function. - oai:arXiv.org:2508.18864v3 - math-ph - math.CA - math.MP - math.RT - nlin.SI - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - N. Belousov, L. Cherepanov, S. Derkachov, S. Khoroshkin - - - Asymptotic behavior of the Bergman kernel and associated invariants in weakly pseudoconvex domains - https://arxiv.org/abs/2509.00301 - arXiv:2509.00301v3 Announce Type: replace -Abstract: In this paper, we present an explicit description for the boundary behavior of the Bergman kernel function, the Bergman metric, and the associated curvatures along certain sequences converging to an $h$-extendible boundary point. - oai:arXiv.org:2509.00301v3 - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ninh Van Thu - - - Hochschild-Kostant-Rosenberg isomorphism for derived Deligne-Mumford stacks - https://arxiv.org/abs/2509.00501 - arXiv:2509.00501v3 Announce Type: replace -Abstract: We prove a Hochschild--Konstant--Rosenberg (HKR) theorem for arbitrary derived Deligne--Mumford (DM) stacks, extending the results of Arinkin-C\u{a}ld\u{a}raru-Hablicsek in the smooth, global quotient case, although with different methods. To formulate our result, we introduce the notion of orbifold inertia stack of a derived DM stack; this supplies a finely tuned derived enhancement of the classical inertia stack, which does not always coincide with the classical truncation of the free loop space. We show that, in characteristic 0, given a derived DM stack, the shifted tangent bundle of its orbifold inertia stack is equivalent to its free loop space. This yields a canonical HKR isomorphism of algebras between the Hochschild homology of a derived DM stack and the cohomology of differential forms on its orbifold inertia stack. Moreover, this isomorphism intertwines the natural circle action and the de Rham differential. Similarly, HKR theorems for derived DM stacks are established for Hochschild cohomology, cyclic homology, negative cyclic homology, and periodic cyclic homology. As applications, we provide a rich supply of computations of Hochschild homology and Hochschild cohomology for interesting derived DM stacks, such as weighted projective lines, root stacks, quotients by algebraic groups, and mapping stacks, among others. - oai:arXiv.org:2509.00501v3 - math.AG - math.AT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lie Fu, Mauro Porta, Sarah Scherotzke, Nicol\`o Sibilla - - - Uniformly S-essential submodules and uniformly S-injective uniformly S-envelopes - https://arxiv.org/abs/2509.01638 - arXiv:2509.01638v2 Announce Type: replace -Abstract: In this paper, we introduce the notion of uniformly S-essential (u-S-essential) submodules. Let R be a commutative ring and S a multiplicative subset of R. A submodule K of an R-module M is said to be u-S-essential in M if for any submodule L of M, s1(K \cap L) = 0 for some s1 \in S implies s2L = 0 for some s2 \in S. Several properties of this notion are studied. The notions of a u-S-uniform module and a u-S-injective u-S-envelope are also introduced, and we show that these notions are characterized by u-S-essential submodules. - oai:arXiv.org:2509.01638v2 - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/publicdomain/zero/1.0/ - Mohammad Adarbeh, Mohammad Saleh - - - Uniformly S-projective relative to a module and its dual - https://arxiv.org/abs/2509.01646 - arXiv:2509.01646v2 Announce Type: replace -Abstract: In this article, we introduce the notion of u-S-projective relative to a module. Let S be a multiplicative subset of a ring R and M an R-module. An R-module P is said to be u-S-projective relative to M if for any u-S-epimorphism f : M \to N, the induced map HomR(P, f) : HomR(P,M) \to HomR(P,N) is a u-S-epimorphism. Dually, we also introduce u-S-injective relative to a module. Some properties of these notions are discussed. Several characterizations of u-S-semisimple modules in terms of these notions are given. The notions of u-S-quasi-projective and u-S-quasi-injective modules are also introduced and some of their properties are discussed. - oai:arXiv.org:2509.01646v2 - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/publicdomain/zero/1.0/ - Mohammad Adarbeh, Mohammad Saleh - - - On base point freeness for rank one foliations - https://arxiv.org/abs/2509.03109 - arXiv:2509.03109v2 Announce Type: replace -Abstract: We prove the base point free theorem for log canonical foliated pairs of rank one on a Q-factorial projective klt threefold. Moreover, we show abundance in the case of numerically trivial log canonical foliated pairs of rank one in any dimension. - oai:arXiv.org:2509.03109v2 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Paolo Cascini, Calum Spicer - - - Recovery of Sturm-Liouville operators from partial boundary spectral data and applications - https://arxiv.org/abs/2509.04289 - arXiv:2509.04289v2 Announce Type: replace -Abstract: We study the inverse Sturm-Liouville problem on a finite interval from partial knowledge of spectral data. Specifically, we show that the potential can be uniquely reconstructed from the knowledge of a fraction of Dirichlet eigenvalues together with the normal derivatives of the corresponding eigenfunctions at both endpoints. We present two novel applications of our spectral result in inverse coefficient determination problems for evolutionary PDEs that include passive wave-based imaging of a medium and active imaging for the time-dependent Schr\"odinger equation with unknown internal sources. Our results yield finite time measurement bounds for such inverse coefficient determination problems. A central innovation is the use of Kahane's interpolation theorem to analyze endpoint time traces of solutions, enabling the recovery without requiring analyticity assumptions or infinite-time data, as in previous approaches. Finally, in the appendix, we present a spectral interpolation theorem for one-dimensional Schr\"odinger operators, which may be of independent interest. - oai:arXiv.org:2509.04289v2 - math.AP - math.SP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ali Feizmohammadi, Yavar Kian - - - $\mathbb R^{\omega_1}$-Factorizable Spaces and Groups - https://arxiv.org/abs/2509.05105 - arXiv:2509.05105v2 Announce Type: replace -Abstract: A topological space $X$ is $\mathbb R^{\omega_1}$-factorizable if any continuous function $f\colon X\to \mathbb R^{\omega_1}$ factors through a continuous function from $X$ to a second-countable space. It is shown that a Tychonoff space $X$ is $\mathbb R^{\omega_1}$-factorizable if and only if $X\times D(\omega_1)$, where $D(\omega_1)$ is a discrete space of cardinality $\omega_1$, is $z$-embedded in the product $\beta X\times \beta D(\omega_1)$ of the Stone--Cech compactifications. It is also proved that $\mathbb R^{\omega_1}$-factorizability is hereditary and countably multiplicative, that any $\mathbb R^{\omega_1}$-factorizable space is hereditarily Lindel\"of and hereditarily separable, and that the existence of nonmetrizable $\mathbb R^{\omega_1}$-factorizable topological spaces and groups is independent of ZFC: under CH, all $\mathbb R^{\omega_1}$-factorizable spaces are second-countable, while under MA + $\lnot$CH, the countable Fr\'echet--Urysohn fan is $\mathbb R^{\omega_1}$-factorizable. - oai:arXiv.org:2509.05105v2 - math.GN - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Anton Lipin, Evgenii Reznichenko, Ol'ga Sipacheva - - - Serrin's overdetermined theorem within Lipschitz domains - https://arxiv.org/abs/2509.05155 - arXiv:2509.05155v3 Announce Type: replace -Abstract: Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain. We prove that, $\Omega$ satisfies the following Serrin-type overdetermined system - $$u \in W^{1,2}(\mathbb R^n), \quad u=0\ \text{ a.e. in }\mathbb R^n\setminus \Omega,\quad \Delta u=\mathbf{c}\mathscr{H}^{n-1}|_{\partial^*\Omega} - \mathbf{1}_{\Omega}\,dx,$$ - in the weak sense if and only if $\Omega$ is a ball. Here $\mathscr H^{n-1}$ denotes the $(n-1)$-dimensional Hausdorff measure. Moreover, a generalization of our method in the anisotropic setting is discussed. Our approach offers an alternative proof to [15] in the case of Lipschitz domains, introducing a novel viewpoint to settle [18, Question 7.1]. - oai:arXiv.org:2509.05155v3 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Hongjie Dong, Yi Ru-Ya Zhang - - - Uniformly S-pseudo-injective modules - https://arxiv.org/abs/2509.05843 - arXiv:2509.05843v2 Announce Type: replace -Abstract: This paper introduces the notion of uniformly-S-pseudo-injective (u-S-pseudo-injective) modules as a generalization of u-S-injective modules. Let R be a ring and S a multiplicative subset of R. An R-module E is said to be u-S-pseudo-injective if for any submodule K of E, there is s in S such that for any u-S-monomorphism f : K \to E, sf can be extended to an endomorphism g : E \to E. Several properties of this notion are studied. For example, we show that an R-module M is u-S-quasi-injective if and only if M \oplus M is u-S-pseudo-injective. Two classes of rings related to the class of QI-rings are introduced and characterized. - oai:arXiv.org:2509.05843v2 - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/publicdomain/zero/1.0/ - Mohammad Adarbeh, Mohammad Saleh - - - Fujita Phenomenon for Mixed Local--Nonlocal Diffusion Equations - https://arxiv.org/abs/2509.07405 - arXiv:2509.07405v3 Announce Type: replace -Abstract: We investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local-nonlocal diffusion operator of the form \[ \partial_t u - (\Delta - (-\Delta)^{\mathsf{s}})u = \mathsf{h}(t)|x|^{-b}|u|^p + t^\varrho \mathbf{w}(x), \qquad (x,t)\in \mathbb{R}^N\times (0,\infty), \] where $\mathsf{s}\in (0,1)$, $p>1$, $b\geq 0$, and $\varrho>-1$. The function $\mathsf{h}(t)$ is assumed to belong to the generalized class of regularly varying functions, while $\mathbf{w}$ is a prescribed spatial source. We first revisit the unforced case and establish sharp blow-up and global existence criteria in terms of the critical Fujita exponent, thereby extending earlier results to the wider class of time-dependent coefficients. For the forced problem, we derive nonexistence of global weak solutions under suitable growth conditions on $\mathsf{h}$ and integrability assumptions on $\mathbf{w}$. Furthermore, we provide sufficient smallness conditions on the initial data and the forcing term ensuring global-in-time mild solutions. Our analysis combines semigroup estimates for the mixed operator, test function methods, and asymptotic properties of regularly varying functions. To our knowledge, this is the first study addressing blow-up phenomena for nonlinear diffusion equations with such a class of time-dependent coefficients. - oai:arXiv.org:2509.07405v3 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Rihab Ben Belgacem, Mohamed Majdoub - - - Quasi-optimal time-space discretizations for a class of nonlinear parabolic PDEs - https://arxiv.org/abs/2509.08645 - arXiv:2509.08645v2 Announce Type: replace -Abstract: We consider parabolic evolution equations with Lipschitz continuous and strongly monotone spatial operators. By introducing an additional variable, we construct an equivalent system where the operator is a Lipschitz continuous mapping from a Hilbert space $Y \times X$ to its dual, with a Lipschitz continuous inverse. Resulting Galerkin discretizations can be solved with an inexact Uzawa type algorithm. Quasi-optimality of the Galerkin approximations is guaranteed under an inf-sup condition on the selected `test' and `trial' subspaces of $Y$ and $X$. To circumvent the restriction imposed by this inf-sup condition, an a posteriori condition for quasi-optimality is developed that is shown to be satisfied whenever the test space is sufficiently large. - oai:arXiv.org:2509.08645v2 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Nina Beranek, Robin Smeets, Rob Stevenson - - - 2-Distance Coloring of Planar Graphs with Specific Maximum Degree - https://arxiv.org/abs/2509.10861 - arXiv:2509.10861v2 Announce Type: replace -Abstract: A k-distance r-coloring of a graph is a coloring of the vertices of the graph such that if the distance between 2 vertices x and y is less or equal to k, then x and y must have distinct colors. A planar graph is a graph that can be drawn with no edge crossing. We will study the 2-distance coloring of planar graphs with maximum degree at least 6. - oai:arXiv.org:2509.10861v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Sara Al Hajjar - - - Failure of Lichnerowicz-type result in parabolic geometries of real rank at least $3$ - https://arxiv.org/abs/2509.12542 - arXiv:2509.12542v4 Announce Type: replace -Abstract: Given a Yamaguchi nonrigid parabolic model geometry $(G,P)$ with $G$ simple of real rank at least $3$, we use techniques developed by Erickson to establish the existence of closed, nonflat, essential, regular, normal Cartan geometries modeled on $(G,P)$. Yamaguchi nonrigidity is a necessary condition for admitting nonflat, regular, normal examples. This rules out Lichnerowicz-type conjectures for these model geometries. - oai:arXiv.org:2509.12542v4 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/publicdomain/zero/1.0/ - Toby Aldape - - - Learning the Influence Graph of a Markov Process that Randomly Resets to the Past - https://arxiv.org/abs/2509.16129 - arXiv:2509.16129v2 Announce Type: replace -Abstract: Learning the influence graph G of a high-dimensional Markov process is central to many application domains, including social networks, neuroscience, and financial risk analysis. However, in many of these applications, future states of the process are occasionally and unpredictably influenced by a distant past state, thus destroying the Markovianity. To study this practical issue, we propose the past influence model (PIM), which captures the occasional "random resets to past" by modifying the Markovian dynamics in [1], which, in turn, is a non-linear generalization of the dynamics studied in [2], [3]. The recursive greedy algorithm proposed in this paper recovers any bounded degree $G$ when the number of ``jumps back in time" is order-wise smaller than the total number of samples, and the algorithm does not require memory. - oai:arXiv.org:2509.16129v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Sudharsan Senthil, Avhishek Chatterjee - - - Perfect Divisibility and Coloring of Some Bull-Free Graphs - https://arxiv.org/abs/2509.18856 - arXiv:2509.18856v3 Announce Type: replace -Abstract: A graph $G$ is {\em perfectly divisible} if, for each induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\omega(H[B])<\omega(H)$. A {\em bull} is a graph consisting of a triangle with two disjoint pendant edges, a {\em fork } is a graph obtained from $K_{1,3}$ by subdividing an edge once, and an {\em odd torch} is a graph obtained from an odd hole by adding an edge $xy$ such that $x$ is non-adjacent to any vertex on the odd hole and the set of neighbors of $y$ on the odd hole is a stable set. - Chudnovsky and Sivaraman [J. Graph Theory 90 (2019) 54-60] proved that every (odd hole, bull)-free graph and every ($P_5$, bull)-free graph are perfectly divisible. Karthick {\em et al.} [The Electron. J. of Combin. 29 (2022) P3.19.] proved that every (fork, bull)-free graph is perfectly divisible. Chen and Xu [Discrete Appl. Math. 372 (2025) 298-307.] proved that every ($P_7,C_5$, bull)-free graph is perfectly divisible. Let $H\in$\{\{odd~torch\}, $\{P_8,C_5\}\}$. In this paper, we prove that every ($H$, bull)-free graph is perfectly divisible. We also prove that a ($P_6$, bull)-free graph is perfectly divisible if and only if it contains no Mycielski-Gr\"{o}tzsch graph as an induced subgraph. As corollaries, these graphs are $\binom{\omega+1}{2}$-colorable. Notice that every odd torch contains an odd hole, an induced $P_5$, and an induced fork. Therefore, our results generalize their findings. Moreover, we prove that every ($P_6$, bull)-free graph $G$ satisfies $\chi(G)\leq\omega(G)^7$. - oai:arXiv.org:2509.18856v3 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ran Chen, Di Wu, Junran Yu, Xiaowen Zhang - - - Souriau-Fisher metric and Completely integrable system on Lie groups SO(2) and SO(3) - https://arxiv.org/abs/2509.20910 - arXiv:2509.20910v3 Announce Type: replace -Abstract: We study the generalize Fisher metric on SO(2) and SO(3) via the thermodynamics Lie group theories of Souriau. Then we give the effect of 2-cocycle on the integrability of gradient systems due to the Fisher metric and Souriau-Fisher metric. In addition, we show how the cocycle can locally modify the Fisher metric on a coadjoint orbit, in explicit terms of brackets and central extensions on the Lie groups SO(2) and SO(3). - oai:arXiv.org:2509.20910v3 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/publicdomain/zero/1.0/ - Prosper Rosaire Mama Assandje, Michel Bertrand Djiadeu Ngaha, Romain Nimpa Pefoukeu, Salomon Joseph Mbatakou - - - A nonlocal Aw-Rascle-Zhang system with linear pressure term - https://arxiv.org/abs/2509.20973 - arXiv:2509.20973v2 Announce Type: replace -Abstract: In this paper, we study a nonlocal extension of the Aw-Rascle-Zhang traffic model, where the pressure-like term is modeled as a convolution between vehicle density and a kernel function. This formulation captures nonlocal driver interactions and aligns structurally with the Euler-alignment system studied in [23]. Using a sticky particle approximation, we construct entropy solutions to the equation for the cumulative density and prove convergence of approximate solutions to weak solutions of the nonlocal system. The analysis includes well-posedness, stability estimates, and an entropic selection principle. - oai:arXiv.org:2509.20973v2 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Debora Amadori, Felisia Angela Chiarello, Gianmarco Cipollone - - - Motivic Classes of Isotropic Degeneracy Loci and Symmetric Orbit Closures - https://arxiv.org/abs/2509.24348 - arXiv:2509.24348v2 Announce Type: replace -Abstract: We provide explicit formulas for computing the motivic Chern and Hirzebruch classes of degeneracy loci, especially those coming from the symplectic and odd orthogonal Grassmannians. The Chern--Schwartz--MacPherson classes, K-theory classes, and Cappell--Shaneson L-classes arise as specializations of the motivic Chern and Hirzebruch classes. Our result is the analogue of the result of Anderson--Chen--Tarasca for the degeneracy loci from the ordinary Grassmannians. As applications, we obtain the motivic Chern and Hirzebruch classes of orthogonal and symplectic orbit closures in flag varieties. - oai:arXiv.org:2509.24348v2 - math.AG - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Minyoung Jeon - - - Squared Bessel processes under nonlinear expectation - https://arxiv.org/abs/2509.24481 - arXiv:2509.24481v2 Announce Type: replace -Abstract: In this paper, we define the squared G-Bessel process as the square of the modulus of a class of G-Brownian motions and establish that it is the unique solution to a stochastic differential equation. We then derive several path properties of the squared G-Bessel process, which are more profound in the capacity sense. Furthermore, we provide upper and lower bounds for the Laplace transform of the squared G-Bessel process. Finally, we prove that the time-space transformed squared G-Bessel process is a G'-CIR process. - oai:arXiv.org:2509.24481v2 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mingshang Hu, Renxing Li, Xue Zhang - - - Bounds on the propagation radius in power domination - https://arxiv.org/abs/2510.02211 - arXiv:2510.02211v2 Announce Type: replace -Abstract: Let $G$ be a graph and let $S \subseteq V(G)$. It is said that $S$ \textit{dominates} $N[S]$. We say that $S$ \textit{monitors} vertices of $G$ as follows. Initially, all dominated vertices are monitored. This step is called the \textit{domination} step. Thereafter, the set of unmonitored vertices of which each is the only unmonitored neighbour of a monitored vertex, is monitored. This step is called a \textit{propagation} step and is repeated until the process terminates. The process terminates when the there are no monitored vertices which have exactly one unmonitored neighbour. This combined process of initial domination and subsequent propagation is called \textit{power domination}. If all vertices of $G$ are monitored at termination, then $S$ is said to be a \textit{power dominating set (PDS) of $G$}. The \textit{power domination number of $G$}, denoted as $\gamma_p(G)$, is the minimum cardinality of a PDS of $G$. The \textit{propagation radius of $G$} is the minimum number of steps it takes a minimum PDS to monitor $V(G)$. In this paper we determine an upper bound on the propagation radius of $G$ with regards to power domination, in terms of $\delta$ and $n$. We show that this bound is only attained when $\gamma_p(G)=1$ and then improve this bound for $\gamma_p(G)\geq 2$. Sharpness examples for these bounds are provided. We also present sharp upper bounds on the propagation radius of split graphs. We present sharpness results for a known lower bound of the propagation radius for all $\Delta\geq 3$. - oai:arXiv.org:2510.02211v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Imran Allie, Brandon du Preez, Dean Reagon, Adriana Roux - - - Certain Inequalities for the generalized polar derivative of a polynomial - https://arxiv.org/abs/2510.02775 - arXiv:2510.02775v2 Announce Type: replace -Abstract: Recently Rather et al. \cite{NT} considered the generalized derivative and the generalized polar derivative and studied the relative position of zeros of generalized derivative and generalized polar derivative with respect to the zeros of polynomial.\\ \indent In this paper, we establish some inequalities that estimate the maximum modulus of generalized derivative and the generalized polar derivative of the polynomial $P(z)$, which is also the extension of recently known results. - oai:arXiv.org:2510.02775v2 - math.CA - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - N. A. Rathe, D. R. Bhat, I. Dar - - - Perspectives on Stochastic Localization - https://arxiv.org/abs/2510.04460 - arXiv:2510.04460v2 Announce Type: replace -Abstract: We survey different perspectives on the stochastic localization process of Eldan, a powerful construction that has had many exciting recent applications in high-dimensional probability and algorithm design. Unlike prior surveys on this topic, our focus is on giving a self-contained presentation of all known alternative constructions of Eldan's stochastic localization, with an emphasis on connections between different constructions. Our hope is that by collecting these perspectives, some of which had primarily arisen within a particular community (e.g., probability theory, theoretical computer science, information theory, or machine learning), we can broaden the accessibility of stochastic localization, and ease its future use. - oai:arXiv.org:2510.04460v2 - math.PR - cs.DS - cs.LG - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Bobby Shi, Kevin Tian, Matthew S. Zhang - - - Relaxation of quasi-convex functionals with variable exponent growth - https://arxiv.org/abs/2510.04672 - arXiv:2510.04672v3 Announce Type: replace -Abstract: We prove a relaxation result for a quasi-convex bulk integral functional with variable exponent growth in a suitable space of bounded variation type. A key tool is a decomposition under mild assumptions of the energy into absolutely continuous and singular parts weighted via a recession function. - oai:arXiv.org:2510.04672v3 - math.AP - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Giacomo Bertazzoni, Petteri Harjulehto, Peter H\"ast\"o, Elvira Zappale - - - Log-majorizations between quasi-geometric type means for matrices - https://arxiv.org/abs/2510.04691 - arXiv:2510.04691v3 Announce Type: replace -Abstract: In this paper, for $\alpha\in(0,\infty)\setminus\{1\}$, $p>0$ and positive semidefinite matrices $A$ and $B$, we consider the quasi-extension $\mathcal{M}_{\alpha,p}(A,B):=\mathcal{M}_\alpha(A^p,B^p)^{1/p}$ of several $\alpha$-weighted geometric type matrix means $\mathcal{M}_\alpha(A,B)$ such as the $\alpha$-weighted geometric mean in Kubo--Ando's sense, the R\'enyi mean, etc. The log-majorization $\mathcal{M}_{\alpha,p}(A,B)\prec_{\log}\mathcal{N}_{\alpha,q}(A,B)$ is examined for pairs $(\mathcal{M},\mathcal{N})$ of those $\alpha$-weighted geometric type means. The joint concavity/convexity of the trace functions $\mathrm{Tr}\,\mathcal{M}_{\alpha,p}$ is also discussed based on theory of quantum divergences. - oai:arXiv.org:2510.04691v3 - math.FA - quant-ph - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.laa.2026.01.003 - Linear Algebra and its Applications 735 (2026), 123--174 - Fumio Hiai - - - Inverse scattering for $N$-body time-decaying harmonic oscillators - https://arxiv.org/abs/2510.04702 - arXiv:2510.04702v4 Announce Type: replace -Abstract: In the previous study (Ishida, 2025), the author proved the uniqueness of short-range potential functions using the Enss-Weder time-dependent method (Enss and Weder, 1995) for a two-body quantum system described by time-decaying harmonic oscillators. In this study, we extend the result of Ishida (2025) to the $N$-body case. We use the approaches developed in Enss and Weder (1995), Weder (1996), and Valencia and Weder (2012) to prove that the high-velocity limit of the scattering operator uniquely determines all the pairwise interaction potentials among the $N$ particles, focusing respectively on each fixed pair of particles. - oai:arXiv.org:2510.04702v4 - math-ph - math.AP - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Atsuhide Ishida - - - Locally similar distances and equality of the induced intrinsic distances - https://arxiv.org/abs/2510.05574 - arXiv:2510.05574v2 Announce Type: replace -Abstract: Let $X$ be a set and $d_1,d_2$ be two distances on $X$. We say that $d_1$ and $d_2$ are locally similar and write $d_1\cong d_2$ if $d_1$ and $d_2$ are topologically equivalent and, for every $a$ in $X$, \[ \lim_{x\to a} \frac{d_2(x,a)}{d_1(x,a)}=1. \] We prove that if $d_1\cong d_2$, then the intrinsic distances induced by $d_1$ and $d_2$ coincide. We also provide sufficient conditions for $d_1\cong d_2$ and consider several examples related to reproducing kernel Hilbert spaces. - oai:arXiv.org:2510.05574v2 - math.MG - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Erick Lee-Guzm\'an, Egor A. Maximenko, Enrique Abdeel Mu\~noz-de-la-Colina, Marco Iv\'an Ruiz-Carmona - - - Rearrangements of distributions on integers that minimize variance - https://arxiv.org/abs/2510.07899 - arXiv:2510.07899v2 Announce Type: replace -Abstract: Which permutations of a probability distribution on integers minimize variance? - Let $X$ be a random variable on a set of integers $\{x_1, \dots, x_N\}$ such that $\mathbb{P}(X_i = x_i) = p_i$, $i \in \{1,\dots,N\}$. Let $(p^{(1)}, \dots, p^{(N)})$ be the sequence $(p_1, \dots, p_N)$ ordered non-increasingly. Let $X^+$ be the random variable defined by $\mathbb{P}(X^+=0)=p^{(1)}$, $\mathbb{P}(X^+=1) = p^{(2)}$, $\mathbb{P}(X^+=-1)=p^{(3)}, \dots, \mathbb{P}(X^+=(-1)^N \lfloor \frac {N} 2 \rfloor)=p^{(N)}$. In this short note we generalize and prove the inequality $\mathrm{Var}\, X^+ \le \mathrm{Var}\, X$. - oai:arXiv.org:2510.07899v2 - math.CO - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Aistis Atminas, Valentas Kurauskas - - - On covering properties of end and ray spaces - https://arxiv.org/abs/2510.10825 - arXiv:2510.10825v3 Announce Type: replace -Abstract: We provide new results on combinatorial characterizations of covering properties in end spaces and ray spaces. In particular, we characterize the Lindel\"of degree, the extent, the Rothberger property, $\sigma$-compactness and the Menger property for ray, end and edge-end spaces. We show that $\sigma$-compactness and the Menger property are equivalent for these spaces, and that they are all $D$-spaces. As an application of some of these characterizations, we are able to provide combinatorial characterizations of graphs with countably many ends and edge-ends. - oai:arXiv.org:2510.10825v3 - math.CO - math.GN - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Rodrigo Rey Carvalho, Matheus Duzi, Vinicius de Oliveira Rodrigues - - - Denominators of R-matrices, higher Dorey's rules and a generalization of T-systems for quantum affine algebras - https://arxiv.org/abs/2510.10874 - arXiv:2510.10874v2 Announce Type: replace -Abstract: We construct a higher level analogue of Dorey's rule, which describe certain surjective morphisms between Kirillov--Reshetikhin (KR) modules over quantum affine algebras. Building on this, we establish a generalized T-system of short exact sequences and prove the denominator formula between KR modules in all nonexceptional types, except with only mild ambiguities persisting in type $C_n^{(1)}$. As a consequence, we can completely classify when a tensor product of KR modules is simple. These results have further applications to Schur positivity statements, quiver Hecke algebras, and the recently introduced $\mathfrak{d}$-invariants in monoidal categories over quantum affine algebras and quiver Hecke algebras. - oai:arXiv.org:2510.10874v2 - math.QA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/publicdomain/zero/1.0/ - Se-jin Oh, Travis Scrimshaw - - - A preorder on the set of links with applications to symmetric unions - https://arxiv.org/abs/2510.12372 - arXiv:2510.12372v2 Announce Type: replace -Abstract: For a link $L$ in the $3$-sphere, the $\pi$-orbifold group $G^\mathrm{orb}(L)$ is defined as a quotient of the link group of $L$. When there exists an epimorphism $G^\mathrm{orb}(L)\to G^\mathrm{orb}(L')$, we denote this by $L\succeq L'$ and explore the relationships between the two links. Specifically, we prove that if $L\succeq L'$ and $L$ is a Montesinos link with $r$ rational tangles $(r\geq 3)$, then $L'$ is either a Montesinos link with at most $r+1$ rational tangles or a certain connected sum. We further show that if $L$ is a small link, then there are only finitely many links $L'$ satisfying $L\succeq L'$. In contrast, if $L$ has determinant zero, then $L\succeq L'$ for every $2$-bridge link $L'$. Our main applications concern symmetric unions of knots. In particular, we provide a criterion showing that a given knot does not admit a symmetric union presentation. - oai:arXiv.org:2510.12372v2 - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Michel Boileau, Teruaki Kitano, Yuta Nozaki - - - Complete gradient Einstein-type Sasakian manifolds with $\alpha=0$ - https://arxiv.org/abs/2510.12441 - arXiv:2510.12441v3 Announce Type: replace -Abstract: Catino, Mastrolia, Monticelli, and Rigoli have launched an ambitious program to study known geometric solitons from a unified perspective, which they term Einstein-type manifolds. This framework allows one to treat Ricci solitons, Yamabe solitons, and all of their generalizations simultaneously. Einstein-type manifolds are characterized by four constants $\alpha, \beta, \mu$ and $\rho$. In this paper, we show that when $\alpha = 0$, complete gradient Einstein-type Sasakian manifolds are trivial or isometric to the unit sphere. As a consequence, many geometric solitons on Sasakian manifolds turn out to be trivial or isometric to the unit sphere. - oai:arXiv.org:2510.12441v3 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/publicdomain/zero/1.0/ - Shun Maeta - - - Existence of 3 anti-cocircular truncated M\"obius planes and constructions of strength-4 covering arrays - https://arxiv.org/abs/2510.13122 - arXiv:2510.13122v2 Announce Type: replace -Abstract: Two projective (affine) planes with the same point sets are orthogoval if the common intersection of any two lines, one from each, has size at most two. The existence of a pair of orthogoval projective planes has been proven and published independently many times. A strength-$t$ covering array, denoted by CA$(N; t, k, v)$, is an $N \times k$ array over a $v$-set such that in any $t$-set of columns, each $t$-tuple occurs at least once in a row. A pair of orthogoval projective planes can be used to construct a strength-$3$ covering array CA$(2q^3-1; 3, q^2 + q + 1, q)$. Our work extends this result to construct arrays of strength $4$. A $k$-cap in a projective geometry is a set of $k$ points no three of which are collinear. In $PG(3,q)$, an ovoid is a maximum-sized $k$-cap with $k =q^2+1$. Its plane sections (circles) are the blocks of a $3-(q^2 + 1, q + 1, 1)$ design, called a M\"obius plane of order $q$. For $q$ an odd prime power, we prove the existence of three truncated M\"obius planes, such that for any choice of these circles, one from each plane, their intersection size is at most three. From this, we construct a strength-$4$ covering array CA$(3q^4-2; 4, \frac{q^2+1}{2}, q)$. For $q \geq 11$, these covering arrays improve the size of the best-known covering arrays with the same parameters by almost 25 percent. The CA$(3q^4 -3; 4, \frac{q^2 +1}{2}, q)$ is used as the main ingredient in a recursive construction to obtain a CA$(5q^4 - 4q^3 - q^2 + 2q; 4, q^2 +1, q)$. Some improvements are obtained in the size of the best-known arrays using these covering arrays. - oai:arXiv.org:2510.13122v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kianoosh Shokri, Lucia Moura, Brett Stevens - - - A central limit theorem for partitions involving generalised divisor functions - https://arxiv.org/abs/2510.19740 - arXiv:2510.19740v2 Announce Type: replace -Abstract: We define an $f$-restricted partition $p_f(n,k)$ of fixed length $k$ given by the bivariate generating series \begin{align*} Q_f(z,u) \coloneqq 1+\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} p_f(n,k) u^kz^n =\prod_{k=1}^{\infty}(1+uz^k)^{\Delta_f(k)}, \end{align*} where $\Delta_f(n)=f(n+1)-f(n)$. In this article, we establish a central limit theorem for the number of summands in such partitions when $f(n)=\sigma_r(n)$ denotes the generalised divisor function, defined as $\sigma_r(n)=\sum_{d|n}d^r$ for integer $r\geq 2$. This can be considered as a generalisation of the work of Lipnik, Madritsch, and Tichy, who previously studied this problem for $f(n)=\lfloor{n}^{\alpha}\rfloor$ with $0<\alpha<1$. A key element of our proof relies on the analytic behaviour of the Dirichlet series \begin{align*} \sum_{n=1}^{\infty}\frac{\sigma_r(n+1)}{n^s}, \end{align*} for $\mathrm{Re}(s)>1$. We study this problem employing the identity involving the Ramanujan sum. Furthermore, we analyse the Euler product arising from the above Dirichlet series by adopting the argument of Alkan, Ledoan and Zaharescu. - oai:arXiv.org:2510.19740v2 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Madhuparna Das, Nicolas Robles - - - Qualitative Behavior of Solutions to a Forced Nonlocal Thin-Film Equation - https://arxiv.org/abs/2510.20289 - arXiv:2510.20289v2 Announce Type: replace -Abstract: We study a one-dimensional nonlocal degenerate fourth-order parabolic equation with inhomogeneous forces relevant to hydraulic fracture modeling. Employing a regularization scheme, modified energy/entropy methods, and novel differential inequality techniques, we establish global existence and long-time behavior results for weak solutions under both time-and space-dependent and time-and space-independent inhomogeneous forces. Specifically, for the time-and space-dependent force $S(t, x)$, we prove that the solution converges to $\bar{u}_0+\frac{1}{|\Omega|}\int_0^\infty \int_\Omega S(r, x)\, dxdr $, where $\bar{u}_0=\frac{1}{|\Omega|}\int_{\Omega}u_{0}(x)\,dx$ is the spatial average of the initial data, and we provide bilateral estimates for the convergence rate. For the time-and space-independent force $S_0$, we show that the solution approaches the linear function $\bar{u}_0 + tS_0$ at an exponential rate. - oai:arXiv.org:2510.20289v2 - math.AP - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jinhong Zhao, Bin Guo - - - Binno: A 1st-order method for Bi-level Nonconvex Nonsmooth Optimization for Matrix Factorizations - https://arxiv.org/abs/2510.21390 - arXiv:2510.21390v2 Announce Type: replace -Abstract: In this work, we develop a method for nonconvex, nonsmooth bi-level optimization and we introduce Binno, a first order method that leverages proximal constructions together with carefully designed descent conditions and variational analysis. Within this framework, Binno provably enforces a descent property for the overall objective surrogate associated with the bi-level problem. Each iteration performs blockwise proximal-gradient updates for the upper and the lower problems separately and then forms a calibrated, block-diagonal convex combination of the two tentative iterates. A linesearch selects the combination weights to enforce simultaneous descent of both level-wise objectives, and we establish conditions guaranteeing the existence of such weights together with descent directions induced by the associated proximal-gradient maps. We also apply Binno in the context of sparse low-rank factorization, where the upper level uses elementwise $\ell_1$ penalties and the lower level uses nuclear norms, coupled via a Frobenius data term. We test Binno on synthetic matrix and a real traffic-video dataset, attaining lower relative reconstruction error and higher peak signal-to-noise ratio than some standard methods. - oai:arXiv.org:2510.21390v2 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Laura Selicato, Flavia Esposito, Andersen Ang - - - Universal Maximum Likelihood (List) Decoding via Fast Vector-Matrix Multiplication - https://arxiv.org/abs/2510.21414 - arXiv:2510.21414v3 Announce Type: replace -Abstract: Maximum-likelihood (ML) decoding for arbitrary block codes remains fundamentally hard, with worst-case time complexity-measured by the total number of multiplications-being no better than straightforward exhaustive search, which requires $q^{k} n$ operations for an $[n,k]_q$ code. This paper introduces a simple, code-agnostic framework that reduces the worst-case complexity by a factor of $n$, down to $q^{k}$ operations, a highly desirable reduction in practice. The result holds for both linear and nonlinear block codes over general memoryless channels and under both hard-decision and soft-decision decoding. It naturally extends to intersymbol-interference (ISI) channels and ML list decoding with only a negligible increase in complexity. Our core insight is that, upon receipt of each sequence at the receiver, the conditional probability of that sequence for each codeword in the codebook (i.e., the \emph{likelihood}) can be expressed as the inner product of two carefully constructed vectors -- the first depending on the received sequence, and the second on that codeword itself. As a result, evaluating the likelihoods for all codewords in the codebook reduces to a single vector-matrix multiplication, and ML decoding (MLD) becomes the simple task of picking the maximum entry in the resulting vector. The only non-trivial cost lies in the vector-matrix product. However, our matrix construction allows the use of the Mailman algorithm to reduce this cost. This time reduction is achieved at the cost of high space complexity, requiring $\mathcal{O}(q^{k+1} n)$ space to store the pre-computed codebook matrix. - oai:arXiv.org:2510.21414v3 - cs.IT - cs.DS - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Hoang Ly, Emina Soljanin, Michael Schleppy - - - Cellular flow control design for mixing based on the least action principle - https://arxiv.org/abs/2510.22703 - arXiv:2510.22703v2 Announce Type: replace -Abstract: We consider a novel approach for the enhancement of fluid mixing via pure stirring strategies building upon the Least Action Principle (LAP) for incompressible flows. The LAP is formally analogous to the Benamou--Brenier formulation of optimal transport, but imposes an incompressibility constraint. Our objective is to find a velocity field, generated by Hamiltonian flows, that minimizes the kinetic energy while ensuring that the initial scalar distribution reaches a prescribed degree of mixedness by a finite time. This formulation leads to a ``point to set" type of optimization problem which relaxes the requirement on controllability of the system compared to the classic LAP framework. In particular, we assume that the velocity field is induced by a finite set of cellular flows that can be controlled in time. We justify the feasibility of this constraint set and leverage Benamou--Brenier's results to establish the existence of a global optimal solution. Finally, we derive the corresponding optimality conditions for solving the optimal time control and conduct numerical experiments demonstrating the effectiveness of our control design. - oai:arXiv.org:2510.22703v2 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Weiwei Hu, Ming-Jun Lai, Hao-Ning Wu - - - Numerical Spectrum Linking: Identification of Governing PDE via Koopman-Chebyshev Approximation - https://arxiv.org/abs/2510.23078 - arXiv:2510.23078v3 Announce Type: replace -Abstract: A numerical framework is proposed for identifying partial differential equations (PDEs) governing dynamical systems directly from their observation data using Chebyshev polynomial approximation. In contrast to data-driven approaches such as dynamic mode decomposition (DMD), which approximate the Koopman operator without a clear connection to differential operators, the proposed method constructs finite-dimensional Koopman matrices by projecting the dynamics onto a Chebyshev basis, thereby capturing both differential and nonlinear terms. This establishes a numerical link between the Koopman and differential operators. Numerical experiments on benchmark dynamical systems confirm the accuracy and efficiency of the approach, underscoring its potential for interpretable operator learning. The framework also lays a foundation for future integration with symbolic regression, enabling the construction of explicit mathematical models directly from data. - oai:arXiv.org:2510.23078v3 - math.NA - cs.NA - eess.SP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Phonepaserth Sisaykeo, Shogo Muramatsu - - - Tree-Cotree-Based IETI-DP for Eddy Current Problems in Time-Domain - https://arxiv.org/abs/2510.23446 - arXiv:2510.23446v2 Announce Type: replace -Abstract: For low-frequency electromagnetic problems, where wave-propagation effects can be neglected, eddy current formulations are commonly used as a simplification of the full Maxwell's equations. In this setup, time-domain simulations, needed to capture transient startup responses or nonlinear behavior, are often computationally expensive. We propose a novel tearing and interconnecting approach for eddy currents in time-domain and investigate its scalability. - oai:arXiv.org:2510.23446v2 - math.NA - cs.CE - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mario Mally, Rafael V\'azquez, Sebastian Sch\"ops - - - Point Convergence of Nesterov's Accelerated Gradient Method: An AI-Assisted Proof - https://arxiv.org/abs/2510.23513 - arXiv:2510.23513v2 Announce Type: replace -Abstract: The Nesterov accelerated gradient method, introduced in 1983, has been a cornerstone of optimization theory and practice. Yet the question of its point convergence had remained open. In this work, we resolve this longstanding open problem in the affirmative. The discovery of the proof was heavily assisted by ChatGPT, a proprietary large language model, and we describe the process through which its assistance was elicited. - oai:arXiv.org:2510.23513v2 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Uijeong Jang, Ernest K. Ryu - - - A relationship between the Kauffman bracket skein algebras and Roger-Yang skein algebras of some small surfaces - https://arxiv.org/abs/2510.23865 - arXiv:2510.23865v2 Announce Type: replace -Abstract: We calculate the Roger-Yang skein algebra of the annulus with two interior punctures, $ \mathcal S^{RY}(\Sigma_{0, 2, 2})$, and show there is a surjective homomorphism from this algebra to the Kauffman bracket skein algebra of the closed torus. Using this homomorphism, we characterize the irreducible, finite-dimensional representations of $ \mathcal S^{RY}(\Sigma_{0, 2, 2})$, showing that they can be described by certain complex data and that the correspondence is unique if certain polynomial conditions are satisfied. We also use the relationship with the skein algebra of the torus to compute structural constants for a bracelets basis for $ \mathcal S^{RY}(\Sigma_{0, 2, 2})$, giving evidence for positivity. - oai:arXiv.org:2510.23865v2 - math.GT - math.QA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Chloe Marple, Helen Wong - - - The multivariate Hermite method for counting real and complex solutions to polynomial systems - https://arxiv.org/abs/2510.23897 - arXiv:2510.23897v2 Announce Type: replace -Abstract: This note presents the multivariate Hermite criterion: a practical and powerful algorithm for determining the number of distinct real and complex roots of a zero-dimensional system of polynomials in any finite number of variables. The final section includes an implementation in Macaulay2, a free and open-source computer algebra system. - oai:arXiv.org:2510.23897v2 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Volodymyr Oleksiyuk - - - Additional Congruences for generalized Color Partitions of Hirschhorn and Sellers - https://arxiv.org/abs/2510.25250 - arXiv:2510.25250v2 Announce Type: replace -Abstract: Let $a_k(n)$ denote the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may be ``colored" with one of $k$ colors, for fixed $k$. In this note, we find some congruences for $a_k(n)$ in the spirit of Ramanujan's congruences. We prove a number of results for $a_k(n)$ modulo powers of $2$ for infinitely many values of $k$. Our approach is truly elementary, relying on generating function manipulations, theta functions and $q$-dissection techniques. We then close by demonstrating an infinite family of congruences modulo 11 which is proven using a result of Ahlgren. - oai:arXiv.org:2510.25250v2 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Anjelin Mariya Johnson, James A. Sellers, S. N. Fathima - - - Cubic Polynomials and Sums of Two Squares - https://arxiv.org/abs/2510.25492 - arXiv:2510.25492v2 Announce Type: replace -Abstract: We establish a lower bound for the frequency with which an irreducible monic cubic polynomial can be expressed as a sum of two squares ($\square_{2}$). This provides a quantitative answer to a question posed by Grechuk (2021) concerning the infinitude of such values. Our proof relies on a two-dimensional unit argument and the arithmetic of degree six number fields. For example, we show that if $h \equiv 2 \pmod{4}$, then \begin{align*} \# \{n : n^3+h \in \square_{2}, \ 1 \leq n \leq x \} \gg x^{1/3-o(1)}. \end{align*} These arguments may be generalised to study the representation of irreducible monic cubic polynomials by the quadratic form $x^2+ny^2$, where $n \in \mathbb{N}$. - oai:arXiv.org:2510.25492v2 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Siddharth Iyer - - - Fine-grained deterministic hardness of the shortest vector problem - https://arxiv.org/abs/2511.01626 - arXiv:2511.01626v2 Announce Type: replace -Abstract: Let $\gamma$-$\mathsf{GapSVP}_p$ be the decision version of the shortest vector problem in the $\ell_p$-norm with approximation factor $\gamma$, let $n$ be the lattice dimension and $0<\varepsilon\leq 1$. We prove that the following statements hold for infinitely many values of $p$. - $(2-\varepsilon)$-$\mathsf{GapSVP}_p$ is not in $O\left(2^{O(p)}\cdot n^{O(1)}\right)$-time, unless $\text{P}=\text{NP}$. - $(2-\varepsilon)$-$\mathsf{GapSVP}_p$ is not in $O\left(2^{2^{o(p)}}\cdot 2^{o(n)}\right)$-time, unless the Strong Exponential Time Hypothesis is false. - The proofs are based on a Karp reduction from a variant of the subset-sum problem that imposes restrictions on vectors orthogonal to the vector of its weights. While more extensive hardness results for the shortest vector problem in all $\ell_p$-norms have already been established under randomized reductions, the results in this paper are fully deterministic. - oai:arXiv.org:2511.01626v2 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Markus Hittmeir - - - Ergodic Rate Analysis of Two-State Pinching-Antenna Systems - https://arxiv.org/abs/2511.01798 - arXiv:2511.01798v2 Announce Type: replace -Abstract: Flexible Antenna Systems (FAS) are a key enabler of next-generation wireless networks, allowing the antenna aperture to be dynamically reconfigured to adapt to channel conditions and service requirements. In this context, pinching-antenna systems (PASs) implemented on software-controllable dielectric waveguides provide the ability to reconfigure both channel characteristics and path loss by selectively exciting discrete radiation points. Existing works, however, typically assume continuously adjustable pinching positions, neglecting the spatial discreteness imposed by practical implementations. This paper develops a closed-form analytical framework for the ergodic rate of two-state PASs, where pinching antennas are fixed and only their activation states are controlled. To quantify the impact of spatial discretization, pinching discretization efficiency is introduced, characterizing the performance gap relative to the ideal continuous case. Finally, numerical results show that near-continuous performance can be achieved with a limited number of pinching points, providing design insights for scalable PASs. - oai:arXiv.org:2511.01798v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Dimitrios Tyrovolas, Sotiris A. Tegos, Yue Xiao, Panagiotis D. Diamantoulakis, Sotiris Ioannidis, Christos Liaskos, George K. Karagiannidis, Stylianos D. Asimonis - - - A decomposition theorem for the Hochschild homology of symmetric powers of a dg category - https://arxiv.org/abs/2511.03269 - arXiv:2511.03269v2 Announce Type: replace -Abstract: We prove a conjecture by Belmans, Fu and Krug concerning the Hochschild homology of the symmetric powers of a small dg category $\mathscr{C}$. More precisely, we show that these groups decompose into pieces that only depend on the Hochschild homology of the dg category $\mathscr{C}$. - oai:arXiv.org:2511.03269v2 - math.CT - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ville Nordstrom - - - The associative-poset point of view on right regular bands - https://arxiv.org/abs/2511.05721 - arXiv:2511.05721v2 Announce Type: replace -Abstract: We present two results on the relation between the class of right regular bands (RRBs) and their underlying *associative posets*. The first one is a construction of a left adjoint to the forgetful functor that takes an RRB $(P,\cdot)$ to the corresponding $(P,\leq)$. The construction of such a left adjoint is actually done in general for any class of relational structures $(X,R)$ obtained from a variety, where $R$ is defined by a finite conjunction of identities. The second result generalizes the "inner" representations of direct product decompositions of semilattices studied by the second author to RRBs having at least one commuting element. - oai:arXiv.org:2511.05721v2 - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Joel Kuperman, Pedro S\'anchez Terraf - - - On the Asymptotic Palindrome Density of Fibonacci Infinite Words - https://arxiv.org/abs/2511.06485 - arXiv:2511.06485v2 Announce Type: replace -Abstract: In this paper, we investigate the combinatorial and density properties of infinite words generated by Fibonacci-type morphisms, focusing on their subword structure, palindrome density, and extremal statistical behaviors. Using the morphism $0 \to 01$, $1 \to 0$, we define a derived ternary word $\mathbb{Y}$ and establish new results relating its density components $\mathrm{dens}(\lambda,n)$, $\mathrm{dens}(\alpha,n)$, and $\mathrm{dens}(\beta,n)$, deriving explicit formulae and bounds on their behavior. We further prove a general density theorem for infinite words with paired subwords, showing that the associated palindromic prefix density is bounded above by $\frac{1}{\varphi_1}$, where $\varphi_1 = (1 + \sqrt{5})/2$ is the golden ratio. Our approach connects the structure of Fibonacci and Thue--Morse sequences with precise asymptotic and combinatorial interpretations for the observed densities. - oai:arXiv.org:2511.06485v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Duaa Abdullah, Jasem Hamoud - - - Analytical estimations of edge states and extended states in large finite-size lattices - https://arxiv.org/abs/2511.07875 - arXiv:2511.07875v2 Announce Type: replace -Abstract: The bulk boundary correspondence, one of the most significant features of topological matter, theoretically connects the existence of edge modes at the boundary with topological invariants of the bulk spectral bands. However, it remains unspecified in realistic examples how large the size of a lattice should be for the correspondence to take effect. In this work, we employ the diatomic chain model to introduce an analytical framework to characterize the dependence of edge states on the lattice size and boundary conditions. In particular, we apply asymptotic estimates to examine the bulk boundary correspondence in long diatomic chains as well as precisely quantify the deviations from the bulk boundary correspondence in finite lattices due to symmetry breaking and finite size effects. Moreover, under our framework the eigenfrequencies near the band edges can be well approximated where two special patterns are detected. These estimates on edge states and eigenfrequencies in linear diatomic chains can be further extended to nonlinear chains to investigate the emergence of new nonlinear edge states and other nonlinear localized states. In addition to one-dimensional diatomic chains, examples of more complicated and higher dimensional lattices are provided to show the universality of our analytical framework. - oai:arXiv.org:2511.07875v2 - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Huajie Song, Haitao Xu - - - Minimal simplicial spherical mappings with a given degree - https://arxiv.org/abs/2511.10870 - arXiv:2511.10870v4 Announce Type: replace -Abstract: This paper studies the minimal number of vertices $\lambda(n,d)$ required in a triangulation of the $n$-sphere to admit a simplicial map to the boundary of a $(n+1)$-simplex with a given degree $d$. We establish upper bounds for $\lambda(n,d)$ in dimensions $n \geq 3$. Furthermore, we provide exact formulas for small values of $d$, showing that $\lambda(n,d)=n+d+3$ for $n \geq 3$ and $d=2,3,4$. A key technical result is the identity $\lambda(n,d) = \lambda(d-1,d) + n - d + 1$ for $n \geq d$, which allows us to reduce higher-dimensional cases to lower-dimensional ones. The proofs involve constructive methods based on local modifications of triangulations and combinatorial arguments. - oai:arXiv.org:2511.10870v4 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ksenia Apolonskaya, Oleg R. Musin - - - SCL Decoding of Non-Binary Linear Block Codes - https://arxiv.org/abs/2511.11256 - arXiv:2511.11256v2 Announce Type: replace -Abstract: Non-binary linear block codes (NB-LBCs) are an important class of error-correcting codes that are especially competent in correcting burst errors. They have broad applications in modern communications and storage systems. However, efficient soft-decision decoding of these codes remains to be further developed. This paper proposes successive cancellation list (SCL) decoding for NB-LBCs that are defined over a finite field of characteristic two, i.e., F_{2^r}, where r is the extension degree. By establishing a one-to-r mapping between the binary composition of each non-binary codeword and $r$ binary polar codewords, SCL decoding of the r polar codes can be performed with a complexity that is sub-quadratic in the codeword length. A simplified path sorting is further proposed to facilitate the decoding. Simulation results on short-length extended Reed-Solomon (eRS) and non-binary extended BCH (NB-eBCH) codes show that SCL decoding can outperform their state-of-the-art soft-decision decoding with fewer finite field arithmetic operations. For length-16 eRS codes, their maximum-likelihood (ML) decoding performances can be approached with a moderate list size. - oai:arXiv.org:2511.11256v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jingyu Lin, Li Chen, Xiaoqian Ye - - - Threshold graphs are globally synchronizing - https://arxiv.org/abs/2511.12646 - arXiv:2511.12646v4 Announce Type: replace -Abstract: The Kuramoto model describes phase oscillators on the unit circle whose interactions are encoded by a graph. Each edge acts like a spring that pulls adjacent oscillators toward each other whenever their phases differ. A central question is to determine which graphs are globally synchronizing, meaning that trajectories of the Kuramoto dynamics converge to the fully synchronized state from almost all initial conditions. This property is tightly linked to the benign nonconvexity of the model's energy landscape. Existing guarantees for global synchronization typically require every node to connect to sufficiently many neighbors, which in turn enforces rather homogeneous degree sequences and highly dense graphs. In this work, we identify threshold graphs as a class of graphs that lie outside this regime yet nevertheless exhibit global synchronization. More precisely, connected threshold graphs realize all admissible edge densities ranging from \(2/n\) to \(1\), and their degree sequences are maximally left-skewed among all graphs with the same edge density, when ordered in nonincreasing order. Our proof relies on a phasor-geometric analysis of the stationary points of the associated energy landscape. - oai:arXiv.org:2511.12646v4 - math.DS - math.CA - math.CO - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Hongjin Wu, Ulrik Brandes - - - Cyclotomic integral points for affine dynamics - https://arxiv.org/abs/2511.13443 - arXiv:2511.13443v2 Announce Type: replace -Abstract: Let $f:\mathbb{A}^N\to\mathbb{A}^N$ be a regular endomorphism of algebraic degree $d\geq2$ (i.e., $f$ extends to an endomorphism on $\mathbb{P}^N$ of algebraic degree $d$) defined over a number field. We prove that if the set of cyclotomic $f$-preperiodic points is Zariski-dense in $\mathbb{A}^N$, then some iterate $f^{\circ l}$ ($l\geq1$) is a quotient of a surjective algebraic group endomorphism $g:\mathbb{G}_m^N\to\mathbb{G}_m^N$, over $\overline{\mathbb{Q}}$. This result generalizes a theorem of Dvornicich and Zannier on cyclotomic preperiodic points of one-variable polynomials to higher dimensions. In fact, we prove a much more general rigidity result for dominant endomorphisms $f$ on an affine variety $X$ defined over a number field, concerning "almost $f$-invariant" Zariski-dense subsets of cyclotomic integral points. We apply our results to backward orbits of regular endomorphisms on $\mathbb{A}^N$ of algebraic degree $d\geq2$, and to periodic points of automorphisms of H\'enon type on $\mathbb{A}^N$. - oai:arXiv.org:2511.13443v2 - math.DS - math.AG - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zhuchao Ji, Junyi Xie, Geng-Rui Zhang - - - CLT for the trace functional of the IDS of magnetic random Schr\"{o}dinger operators - https://arxiv.org/abs/2511.14448 - arXiv:2511.14448v3 Announce Type: replace -Abstract: We consider the existence of the integrated density of states (IDS) of the magnetic Schr\"{o}dinger operator with a random potential on the Hilbert space \( L^2(\mathbb{R}^d) \), as an analogue of the law of large numbers (LLN) for trace functionals. In this work, we establish an analogue of the central limit theorem (CLT), which describes the fluctuations of the trace functionals of the IDS, for a class of test functions denoted by \( C^1_{d,0}(\mathbb{R}) \). This class consists of real-valued, continuously differentiable functions on \( \mathbb{R} \) that decay at the rate \( O(|x|^{-m}) \) as \( |x| \to \infty \), where \( m > d + 1 \). - oai:arXiv.org:2511.14448v3 - math.SP - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Dhriti Ranjan Dolai, Naveen Kumar - - - Nonlinear scalar field equations with a critical Hardy potential - https://arxiv.org/abs/2511.15668 - arXiv:2511.15668v2 Announce Type: replace -Abstract: We study the existence of solutions for the nonlinear scalar field equation $$-\Delta u - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \mbox{in } \mathbb{R}^N \setminus \{0\},$$ where the potential $-\frac{(N-2)^2}{4|x|^2}$ is the critical Hardy potential and $N \geq 3$. The nonlinearity $g$ is continuous and satisfies general subcritical growth assumptions of the Berestycki-Lions type. The problem is approached using variational methods within a non-standard functional setting. The natural energy functional associated with the equation is defined on the space $X^1(\mathbb{R}^N)$, which is the completion of $H^1(\mathbb{R}^N)$ with respect to the norm induced by the quadratic part of the functional. We establish the existence of a nontrivial solution $u_0 \in X^1(\mathbb{R}^N)$ that satisfies the Poho\v{z}aev constraint $\mathcal{M}$ and minimizes the energy functional on $\mathcal{M}$. Furthermore, assuming $g$ is odd, we prove the existence of at least one non-radial solution. - oai:arXiv.org:2511.15668v2 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Bartosz Bieganowski, Daniel Strzelecki - - - Normalization of Puiseux Hypersurfaces - https://arxiv.org/abs/2511.15863 - arXiv:2511.15863v2 Announce Type: replace -Abstract: It is known that the normalization of a quasi-ordinary complex singularity is a Hirzebruch-Jung, see [Gon00; Pop04; AS05]. We extend this result to Puiseux hypersurfaces. Moreover, we prove that Hirzebruch-Jung singularities are precisely normalizations of Puiseux hypersurfaces. Our result holds over an algebraically closed field whose characteristic does not divide the degree of the polynomial defining the hypersurface. Finally, in the analytic complex case, we conclude that the normalization of an irreducible Puiseux hypersurface is the normalization of a complex analytic quasi-ordinary singularity. - oai:arXiv.org:2511.15863v2 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Fuensanta Aroca, Annel Ayala, Oscar Casta\~n\'on, Diana Mendez Penagos, Dami\'an Ochoa, Camille Pl\'enat - - - Subtlety of oscillation indices of oscillatory integrals of real analytic functions - https://arxiv.org/abs/2511.16257 - arXiv:2511.16257v2 Announce Type: replace -Abstract: For a locally defined real analytic function, we study the relation between the oscillation index of oscillatory integrals and the real log canonical threshold. The former is always negative, and its absolute value is greater than or equal to the latter. They coincide very often, but there are certain exceptional cases even in the Newton nondegenerate convenient homogeneous case, for instance if the number of variables is even and smaller than the degree. This does not seem compatible with some standard formula in the literature, and there must be some error somewhere, although it does not seem easy to find it inside this paper. - oai:arXiv.org:2511.16257v2 - math.CV - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - In-Kyun Kim, Morihiko Saito - - - Full flexibility of isometric immersions of metrics with low H\"older regularity in Poznyak theorem's dimension - https://arxiv.org/abs/2511.16305 - arXiv:2511.16305v2 Announce Type: replace -Abstract: A classical result by Poznyak asserts that any smooth $2$-dimensional Riemannian metric $g$, posed on the closure of a simply connected domain $\omega\subset\mathbb{R}^2$, has a smooth isometric immersion into $\mathbb{R}^4$. Using techniques of convex integration, we prove that for any $2$-dimensional $g\in\mathcal{C}^{r,\beta}$, an isometric immersion of regularity $\mathcal{C}^{1,\alpha}(\bar\omega,\mathbb{R}^4)$ for any $\alpha<\min\{\frac{r+\beta}{2},1\}$, may be found arbitrarily close to any short immersion. The fact that this result's regularity reaches $\mathcal{C}^{1,1-}$ for $g\in \mathcal{C}^2$, which is referred to as "full flexibility", should be contrasted with: (i) the regularity $\mathcal{C}^{1,1/3-}$ achieved by Cao, Hirsch and Inauen for isometric immersions into $\mathbb{R}^{3}$ and the lack of flexibility (rigidity) of such isometric immersions with regularity $\mathcal{C}^{1, 2/3+}$ proved by Borisov and then by Conti, de Lellis and Szekelyhidi; (ii) the regularity $\mathcal{C}^{1,1-}$ obtained byt K\"allen for isometric immersions into higher codimensional space $\mathbb{R}^{8}$; and (iii) the regularity $\mathcal{C}^{1,\frac{1}{1+d(d+1)/k}-}$ proved by the author in the general case of $d$-dimensional metrics and $(d+k)$-dimensional immersions for the closely related Monge-Amp\`ere system. - oai:arXiv.org:2511.16305v2 - math.DG - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Marta Lewicka - - - Function-Correcting Codes With Data Protection - https://arxiv.org/abs/2511.18420 - arXiv:2511.18420v2 Announce Type: replace -Abstract: Function-correcting codes (FCCs) are designed to provide error protection for the value of a function computed on the data. Existing work typically focuses solely on protecting the function value and not the underlying data. In this work, we propose a general framework that offers protection for both the data and the function values. Since protecting the data inherently contributes to protecting the function value, we focus on scenarios where the function value requires stronger protection than the data itself. We first introduce a more general approach and a framework for function-correcting codes that incorporates data protection along with protection of function values. A two-step construction procedure for such codes is proposed, and bounds on the optimal redundancy of general FCCs with data protection are reported. Using these results, we exhibit examples that show that data protection can be added to existing FCCs without increasing redundancy. Using our two-step construction procedure, we present explicit constructions of FCCs with data protection for specific families of functions, such as locally bounded functions and the Hamming weight function. We associate a graph called minimum-distance graph to a code and use it to show that perfect codes and maximum distance separable (MDS) codes cannot provide additional protection to function values over and above the amount of protection for data for any function. Then we focus on linear FCCs and provide some results for linear functions, leveraging their inherent structural properties. To the best of our knowledge, this is the first instance of FCCs with a linear structure. Finally, we generalize the Plotkin and Hamming bounds well known in classical error-correcting coding theory to FCCs with data protection. - oai:arXiv.org:2511.18420v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Charul Rajput, B. Sundar Rajan, Ragnar Freij-Hollanti, Camilla Hollanti - - - Elastic scattering by locally rough interfaces - https://arxiv.org/abs/2511.18799 - arXiv:2511.18799v2 Announce Type: replace -Abstract: In this paper, we present the first well-posedness result for elastic scattering by locally rough interfaces in both two and three dimensions. Inspired by the Helmholtz decomposition, we discover a fundamental identity for the stress vector, revealing an intrinsic relationship among the generalized stress vector, the Lame constants and certain tangential differential operators. This identity leads to two key limits for surface integrals involving scattered solutions, from which we deduce the first uniqueness result of direct problem for all frequencies. Through a detailed analysis, applying the steepest descent method, subsequently we derive the existence and uniqueness of the corresponding two-layered Green's tensor along with its explicit expression when the transmission coefficient equals 1. Finally, by leveraging properties of the Green's tensor, we establish the existence of solutions via the variational method and the boundary integral equation, thereby achieving the first well-posedness result for elastic scattering by rough interfaces. - oai:arXiv.org:2511.18799v2 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Chengyu Wu, Yushan Xue, Jiaqing Yang - - - Higher property T and below-rank phenomena of lattices - https://arxiv.org/abs/2511.20192 - arXiv:2511.20192v2 Announce Type: replace -Abstract: The purpose of this paper is twofold. We explore higher property T as an abstract group-theoretic property. In particular, we provide new operator-algebraic characterizations of higher property T. Then we turn to lattices in semisimple Lie groups. We relate higher property T to other cohomological, rigidity and geometric phenomena below the real rank. The second part outlines a conjectural framework that unifies these aspects and reviews recent advances. - oai:arXiv.org:2511.20192v2 - math.GR - math.GT - math.OA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Uri Bader, Roman Sauer - - - Uniform inference for kernel instrumental variable regression - https://arxiv.org/abs/2511.21603 - arXiv:2511.21603v2 Announce Type: replace -Abstract: Instrumental variable regression is a foundational tool for causal analysis across the social and biomedical sciences. Recent advances use kernel methods to estimate nonparametric causal relationships, with general data types, while retaining a simple closed-form expression. Empirical researchers ultimately need reliable inference on causal estimates; however, uniform confidence sets for the method remain unavailable. To fill this gap, we develop valid and sharp confidence sets for kernel instrumental variable regression, allowing general nonlinearities and data types. Computationally, our bootstrap procedure requires only a single run of the kernel instrumental variable regression estimator. Theoretically, it relies on the same key assumptions. Overall, we provide a practical procedure for inference that substantially increases the value of kernel methods for causal analysis. - oai:arXiv.org:2511.21603v2 - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Marvin Lob, Rahul Singh, Suhas Vijaykumar - - - Maximal variation of linear systems - https://arxiv.org/abs/2511.22329 - arXiv:2511.22329v4 Announce Type: replace -Abstract: Let X be a smooth projective complex variety, and L a line bundle on X . We say that the linear system |L| has maximal variation if its elements have the maximum number dim|L| of moduli. We discuss some cases where this situation is expected: hypersurfaces, double coverings of the projective space, K3 surfaces, hyperkahler manifolds, and abelian varieties. - oai:arXiv.org:2511.22329v4 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Arnaud Beauville - - - Convergence rates of self-repelling diffusions on Riemannian manifolds - https://arxiv.org/abs/2511.23333 - arXiv:2511.23333v2 Announce Type: replace -Abstract: We study a class of self-repelling diffusions on compact Riemannian manifolds whose drift is the gradient of a potential accumulated along their trajectory. When the interaction potential admits a suitable spectral decomposition, the dynamics and its environment are equivalent to a finite-dimensional degenerate diffusion. We show that this diffusion is a second-order lift of an Ornstein-Uhlenbeck process whose invariant law corresponds to the Gaussian invariant measure of the environment, and immediately obtain a general upper bound on the rate of convergence to stationarity using the framework of second-order lifts. Furthermore, using a flow Poincar\'e inequality, we develop lower bounds on the convergence rate. We show that, in the periodic case, these lower bounds improve upon those of Bena\"im and Gauthier (Probab. Theory Relat. Fields, 2016), and even match the order of the upper bound in some cases. - oai:arXiv.org:2511.23333v2 - math.PR - math.AP - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Francis L\"orler - - - Finiteness of leaps of modules of integrable derivations of algebras of finite type - https://arxiv.org/abs/2512.00690 - arXiv:2512.00690v4 Announce Type: replace -Abstract: We prove the finiteness of leaps of modules of $m$-integrable derivations for algebras essentially of finite type and, more generally, for schemes essentially of finite type over an algebraically closed field of positive characteristic. This provides an affirmative answer to a question posed by L. Narv\'aez Macarro. As an application, we establish the coherence of the module of $\infty$-integrable derivations. - oai:arXiv.org:2512.00690v4 - math.AG - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Takuya Miyamoto - - - The intrinsic subgroup of an elliptic curve and Mazur's torsion theorem - https://arxiv.org/abs/2512.00787 - arXiv:2512.00787v2 Announce Type: replace -Abstract: We define and study a biadditive symmetric (not necessarily perfect) pairing on the torsion part $\mathrm{Pic}(X)_{\mathrm{tors}}$ of the Picard group of a smooth projective curve $X$ over a field $k$ with values in $k^\times \otimes \mathbb{Q}/\mathbb{Z}$. We call its kernel the intrinsic subgroup of $X$. It turns out that some information on the reduction type of $X$ can be read off from the intrinsic subgroup. Mazur's torsion theorem says that there are exactly 15 isomorphism classes of abelian groups that appear as the rational torsion points of an elliptic curve $X$ over $\mathbb{Q}$ (identified with $\mathrm{Pic}(X)_{\mathrm{tors}}$). We refine this result by determining which subgroups of those 15 groups appear as the intrinsic subgroups. - oai:arXiv.org:2512.00787v2 - math.NT - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Takao Yamazaki, Yifan Yang, Hwajong Yoo, Myungjun Yu - - - Spectrally additive maps on the positive cones of the Wiener algebra - https://arxiv.org/abs/2512.01173 - arXiv:2512.01173v2 Announce Type: replace -Abstract: We study surjective maps between the positive cones of the Wiener algebra that preserve the spectrum of the sum of every two elements. We show that such maps can be extended to isometric real-linear isomorphisms of the Wiener algebra. - oai:arXiv.org:2512.01173v2 - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Shiho Oi, Kaito Sato - - - A discrete approach to Dirichlet L-functions, their special values and zeros - https://arxiv.org/abs/2512.01779 - arXiv:2512.01779v2 Announce Type: replace -Abstract: We obtain new infinite families of identities among special values of Dirichlet $L$-functions using finite spectral sums. More precisely, we study Dirichlet $L$-functions via discrete analogues $L_n$ arising from the spectral theory of cyclic graphs as $n\rightarrow \infty$. Applying a refined Euler-Maclaurin asymptotic expansion due to Sidi, together with an independent polynomiality property of these finite spectral sums at integers, we obtain exact special-value formulas, even starting at $n=1$. This yields new expressions for certain trigonometric sums of interest in physics, and recovers, by a different method, the striking formulas of Xie, Zhao, and Zhao. - Concerning zeros, using the same asymptotic expansion, we prove that for odd primitive characters, an asymptotic functional equation relating $L_n(1-s,\overline{\chi })$ to $L_n(s,\chi)$ is equivalent to the Generalized Riemann Hypothesis for the corresponding Dirichlet $L$-function $L(s,\chi)$. We also provide some remarks about the non-existence of possible real zeros. - oai:arXiv.org:2512.01779v2 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Anders Karlsson, Dylan M\"uller - - - The time fractional stochastic partial differential equations with non-local operator on $\mathbb{R}^{d}$ - https://arxiv.org/abs/2512.03754 - arXiv:2512.03754v2 Announce Type: replace -Abstract: This paper establishes a comprehensive well-posedness and regularity theory for time-fractional stochastic partial differential equations on $\mathbb{R}^d$ driven by mixed Wiener--L\'evy noises. The equations feature a Caputo time derivative $\partial_t^\alpha$ ($0<\alpha<1$) and a spatial nonlocal operator $\phi(\Delta)$ generated by a subordinate Brownian motion, leading to a doubly nonlocal structure. - For the case $p \ge 2$, we prove the existence, uniqueness, and sharp Sobolev regularity of weak solutions in the scale of $\phi$-Sobolev spaces $\mathcal{H}_p^{\phi,\gamma+2}(T)$. Our approach combines harmonic analysis techniques (Fefferman--Stein theorem, Littlewood--Paley theory) with stochastic analysis to handle the combined Wiener and L\'evy noise terms. In the special case of cylindrical Wiener noise, a dimensional constraint $d < 2\kappa_0\bigl(2 - (2\sigma_2 - 2/p)_+/\alpha\bigr)$ is obtained.~For the low-regularity case $1 \le p \le 2$, where maximal function estimates fail, we construct unique local mild solutions in $L_p(\mathbb{R}^d)$ for equations driven by pure-jump L\'evy space-time white noise, using stochastic truncation and fixed-point arguments. - The results unify and extend previous theories by simultaneously incorporating time-space nonlocality and jump-type randomness. - oai:arXiv.org:2512.03754v2 - math.AP - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yong Zhen Yang, Yong Zhou - - - A Second Main Theorem for Entire Curves Intersecting Three Conics - https://arxiv.org/abs/2512.03948 - arXiv:2512.03948v2 Announce Type: replace -Abstract: We establish a Second Main Theorem for entire holomorphic curves \( f: \mathbb{C} \to \mathbb{P}^2 \) intersecting a generic configuration of three conics \(\mathcal{C}= \mathcal{C}_1+ \mathcal{C}_2+ \mathcal{C}_3 \) in the complex projective plane $\mathbb{P}^2$. Using invariant logarithmic $2$-jet differentials with negative twists, we prove the estimate \[ T_f(r) \leqslant 5 \sum_{i=1}^3 N_f^{[1]}(r, \mathcal{C}_i) + o\big(T_f(r)\big)\quad\parallel, \] where \( T_f(r) \) is the Nevanlinna characteristic function, and \( N_f^{[1]}(r, \mathcal{C}_i) \) is the $1$-truncated counting function. - The key innovation of our approach is establishing new vanishing lemmas of the form \[ H^0\bigl(\mathbb{P}^2,\, E_{2,m}T_{\mathbb{P}^2}^*(\log \mathcal{C}) \otimes \mathcal{O}_{\mathbb{P}^2}(-t)\bigr) = 0 \] for specific pairs \((m, t)\), achieved by combining algebro-geometric arguments with computer-assisted computations through a mod-\(p\) reduction technique. This yields a systematic method for proving vanishing results for negatively twisted jet differentials -- a key component in complex hyperbolic geometry. - oai:arXiv.org:2512.03948v2 - math.CV - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lei Hou, Dinh Tuan Huynh, Jo\"el Merker, Song-Yan Xie - - - A Tight-binding Approach for Computing Subwavelength Guided Modes in Crystals with Line Defects - https://arxiv.org/abs/2512.05370 - arXiv:2512.05370v2 Announce Type: replace -Abstract: In this paper, we develop an accurate and efficient framework for computing subwavelength guided modes in high-contrast periodic media with line defects, based on a tight-binding approximation. The physical problem is formulated as an eigenvalue problem for the Helmholtz equation with high-contrast parameters. By employing layer potential theory on unbounded domains, we characterize the subwavelength frequencies via the quasi-periodic capacitance matrix. Our main contribution is the proof of exponential decay of the off-diagonal elements of the associated full and quasi-periodic capacitance matrices. These decay properties provide error bounds for the banded approximation of the capacitance matrices, thereby enabling a tight-binding approach for computing the spectral properties of subwavelength resonators with non-compact defects. Various numerical experiments are presented to validate the theoretical results, including applications to topological interface modes. - oai:arXiv.org:2512.05370v2 - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Habib Ammari, Erik Orvehed Hiltunen, Ping Liu, Borui Miao, Yi Zhu - - - Non-continuous valuations on convex bodies and a new characterization of volume - https://arxiv.org/abs/2512.06745 - arXiv:2512.06745v2 Announce Type: replace -Abstract: This paper investigates the use of automatic continuity techniques in the context of valuations on convex bodies. We first provide an automatic continuity theorem for valuations restricted to parallelotopes with respect to a fixed basis. This result in combination with a counting argument provides a strengthened version of a classical characterization of volume due to Hadwiger. As a byproduct of the proof it is shown that $[0,n-1]\cup\{n\}$ are precisely the possible degrees of homogeneity of bounded translation invariant valuations on $n$-dimensional convex bodies. - oai:arXiv.org:2512.06745v2 - math.MG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Jorge S. Ib\'a\~nez Marcos, Pedro Tradacete, Ignacio Villanueva - - - A duality approach to gradient H\"older estimates for linear divergence form elliptic equations - https://arxiv.org/abs/2512.06979 - arXiv:2512.06979v2 Announce Type: replace -Abstract: We prove a sparse bound in the context of Schauder theory for divergence form elliptic partial differential equations. In addition, we show how an iteration argument inspired by sparse domination bounds can be used to deduce gradient reverse H\"older inequalities for equations with non-constant coefficients from the theory for constant coefficient equations. We deal with coefficient matrices whose entries are either H\"older continuous or just uniformly continuous, leading to different results. The purpose of the approach is to highlight the connection between Schauder theory and duality of local Hardy spaces and local H\"older spaces. - oai:arXiv.org:2512.06979v2 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Olli Saari, Yuanlin Sun, Hua-Yang Wang, Yuanhong Wei - - - Stability for Strichartz inequalities: Existence of minimizers - https://arxiv.org/abs/2512.07174 - arXiv:2512.07174v2 Announce Type: replace -Abstract: We study the quantitative stability associated with the adjoint Fourier restriction inequality, focusing on the paraboloid and two-dimensional sphere cases. We show that these Strichartz-stability inequalities admit minimizers attaining their sharp constants, provided that these sharp constants are strictly smaller than the corresponding spectral-gap constants. Furthermore, for the two-dimensional sphere case, we obtain the existence of minimizers. - oai:arXiv.org:2512.07174v2 - math.CA - math.AP - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Boning Di, Dunyan Yan - - - Entropy-Smooth Structures on Topological Manifolds - https://arxiv.org/abs/2512.07660 - arXiv:2512.07660v3 Announce Type: replace -Abstract: We introduce an information-theoretic framework for smooth structures on topological manifolds, replacing coordinate charts with small-scale entropy data of local probability probes. A concise set of axioms identifies admissible coordinate functions and reconstructs a smooth atlas directly from the quadratic entropy response. We prove that this entropy-smooth structure is equivalent to the classical smooth structure, stable under perturbations, and compatible with products, submanifolds, immersions, and diffeomorphisms. This establishes smoothness as an information-theoretic phenomenon and forms the foundational layer of a broader program linking entropy, diffusion, and differential geometry. - oai:arXiv.org:2512.07660v3 - math.DG - math.GN - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Amandip Sangha - - - Small time asymptotics of spectral heat content of isotropic processes - https://arxiv.org/abs/2512.08595 - arXiv:2512.08595v3 Announce Type: replace -Abstract: The spectral heat content of a domain $\Omega\subset\mathbb{R}^d$ corresponding to a $d$-dimensional stochastic process $X=(X_t)_{t\ge 0}$ is defined as \[Q^{X}_\Omega(t)=\int_{\mathbb{R}^d} \mathbb{P}_x(\tau^X_\Omega>t)dx,\] where $\tau^X_\Omega$ is the first exit time of $X$ from $\Omega$. We provide a novel technique for proving small time asymptotic of spectral heat content for any translation invariant isotropic process satisfying negligible tail probability condition. As a consequence, we recover several existing results in the context of L\'evy processes and Gaussian processes, and provide spectral heat content asymptotics for a class of $\alpha$-stable L\'evy processes time-changed by right inverse of positive, increasing, self-similar Markov processes. The latter has connection to some Cauchy problems that are non-local in both time and space. - oai:arXiv.org:2512.08595v3 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/publicdomain/zero/1.0/ - Rohan Sarkar - - - Graph Quantum Magic Squares and Free Spectrahedra - https://arxiv.org/abs/2512.08797 - arXiv:2512.08797v2 Announce Type: replace -Abstract: Recently De les Coves, Drescher and Netzer showed that an analogue of the Birkhoff--von Neumann theorem fails in the quantum setting. Motivated by this and questions arising in the study of quantum automorphisms of graphs, we introduce a graph-based variant of quantum magic squares and show that the analogue already fails for the cycle \(C_4\), via an explicit counterexample. We also show that they admit monic linear matrix inequality descriptions, hence form compact free spectrahedra. - oai:arXiv.org:2512.08797v2 - math-ph - math.MP - math.OA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Francesca La Piana - - - Characterization of Jordan Vectors of Operator-Valued Functions with Applications in Differential Equations - https://arxiv.org/abs/2512.09178 - arXiv:2512.09178v2 Announce Type: replace -Abstract: A well-known characterization of Jordan vectors of a matrix polynomial $L(z)$ is generalized to a characterization of Jordan vectors of the operator-valued function $Q(z)$ at an eigenvalue $\alpha \in \mathbb{C}$. The results are then applied to solve a system of nonlinear ordinary differential equations. - oai:arXiv.org:2512.09178v2 - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Muhamed Borogovac - - - An Elementary Proof of the Near Optimality of LogSumExp Smoothing - https://arxiv.org/abs/2512.10825 - arXiv:2512.10825v2 Announce Type: replace -Abstract: We consider the design of smoothings of the (coordinate-wise) max function in $\mathbb{R}^d$ in the infinity norm. The LogSumExp function $f(x)=\ln(\sum^d_i\exp(x_i))$ provides a classical smoothing, differing from the max function in value by at most $\ln(d)$. We provide an elementary construction of a lower bound, establishing that every overestimating smoothing of the max function must differ by at least $\sim 0.8145\ln(d)$. Hence, LogSumExp is optimal up to small constant factors. However, in small dimensions, we provide stronger, exactly optimal smoothings attaining our lower bound, showing that the entropy-based LogSumExp approach to smoothing is not exactly optimal. - oai:arXiv.org:2512.10825v2 - math.ST - cs.LG - math.OC - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Thabo Samakhoana, Benjamin Grimmer - - - Complements of discriminants of real parabolic function singularities. II - https://arxiv.org/abs/2512.12738 - arXiv:2512.12738v3 Announce Type: replace -Abstract: We list all connected components of sets of non-discriminant functions within versal deformations of all {\em parabolic} function singularities. These singularities are the second most common family of singularity classes of smooth functions after {\em simple singularities}. Thus, we prove (and improve in one particular case) the corresponding conjectures from the previous work \cite{para} with the same title. As an application, we enumerate all local Petrovskii lacunas near arbitrary parabolic singularities of wavefronts of hyperbolic PDEs. We also show that the complements of the discriminant varieties of $X_9^+$ and $P_8^1$ singularities have nontrivial one-dimensional homology groups, in contrast to all simple singularities. - These results are applications of a general method for investigating non-singular perturbations of real function singularities. An important part of this method is a computer program that formalizes Picard--Lefschetz theory and surgeries of Morse functions. - oai:arXiv.org:2512.12738v3 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - V. A. Vassiliev - - - Integrating ethical, societal and environmental issues into algorithm design courses - https://arxiv.org/abs/2512.13216 - arXiv:2512.13216v2 Announce Type: replace -Abstract: This document, intended for computer science teachers, describes a case study that puts into practice a questioning of ethical, societal and environmental issues when designing or implementing a decision support system. This study is based on a very popular application, namely road navigation software that informs users of real-time traffic conditions and suggests routes between a starting point and a destination, taking these conditions into account (such as Waze). The approach proposes to intertwine technical considerations (optimal path algorithms, data needed for location, etc.) with a broader view of the ethical, environmental and societal issues raised by the tools studied. Based on the authors' experience conducting sessions with students over several years, this document discusses the context of such a study, suggests teaching resources for implementing it, describes ways to structure discussions, and shares scenarios in different teaching contexts. - oai:arXiv.org:2512.13216v2 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Odile Bellenguez (IMT Atlantique - DAPI, LS2N - \'equipe MODELIS, IMT Atlantique, LS2N), Nadia Brauner (G-SCOP\_ROSP, G-SCOP), Christine Solnon (EMERAUDE), Alexis Tsoukias (LAMSADE) - - - Sums of four fourth power of primes - https://arxiv.org/abs/2512.14386 - arXiv:2512.14386v4 Announce Type: replace -Abstract: For any sufficiently large positive integer $\ell$, suppose that $\ell$ can be expressed as $ \ell=p_1^4+p_2^4+p_3^4+p_4^4$, where $p_1, p_2,p_3,p_4$ are primes. For such $\ell$, in this paper we will use circle method and sieves to prove that the proportion of $\ell$ in positive integers is at least $\frac{1}{27241.64}$ . - oai:arXiv.org:2512.14386v4 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Yang Qu, Rong Ma - - - Empirical Laws for Iterated Correlation Matrices - https://arxiv.org/abs/2512.15421 - arXiv:2512.15421v3 Announce Type: replace -Abstract: We study the discrete dynamical system obtained by repeatedly applying the Pearson correlation operator to a real matrix. Each step centers every row, normalizes each centered row to unit Euclidean norm, and forms the Gram matrix of the resulting rows. This produces a nonlinear map that underlies the classical CONCOR and GAP procedures. Despite its simple formulation and long history, the global behavior of this iteration has remained analytically unresolved. - We present a geometric formulation that separates directions associated with changes in row means and row norms from directions that preserve them. This formulation clarifies why local analysis does not extend to a global convergence theorem: the iteration is nonlinear, the structure of its fixed-point set is not fully characterized, and standard uniform contractive or Fejer-type techniques do not directly apply. - Empirically, the iteration stabilizes at a block plus or minus one pattern, exhibits finite total variation, and displays rapid decay once trajectories enter a neighborhood of a fixed pattern. We develop a dimension-uniform experimental framework and perform a large-scale numerical study over dimensions from 3 to 2000 with thousands of random initializations. Using the Frobenius step size, the entrywise step size, and the one-step ratio, we identify four universal empirical laws that persist uniformly across all tested dimensions. These observations provide a quantitative, dimension-uniform description of the iteration and formulate a precise target for future global analysis. - oai:arXiv.org:2512.15421v3 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ishrak Alhajj Hassan - - - Transgressions and Chern characters in coarse homotopy theory - https://arxiv.org/abs/2512.16749 - arXiv:2512.16749v2 Announce Type: replace -Abstract: This paper investigates a variety of coarse homology theories and natural transformations between them. We in particular study the commutativity of a square relating analytical and topological transgressions with algebraic and homotopy theoretic Chern characters. Here a transgression is a natural transformation from a coarse homology theory to a functor which factorizes over the Higson corona functor, and a Chern character is a transformation from a $K$-theory like coarse or Borel-Moore type homology theory to an ordinary version. - oai:arXiv.org:2512.16749v2 - math.AT - math.KT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ulrich Bunke - - - False positive control in time series coincidence detection - https://arxiv.org/abs/2512.17372 - arXiv:2512.17372v2 Announce Type: replace -Abstract: We study the problem of coincidence detection in time series data, where we aim to determine whether the appearance of simultaneous or near-simultaneous events in two time series is indicative of some shared underlying signal or synchronicity, or might simply be due to random chance. This problem arises across many applications, such as astrophysics (e.g., detecting astrophysical events such as gravitational waves, with two or more detectors) and neuroscience (e.g., detecting synchronous firing patterns between two or more neurons). In this work, we consider methods based on time-shifting, where the timeline of one data stream is randomly shifted relative to another, to mimic the types of coincidences that could occur by random chance. Our theoretical results establish rigorous finite-sample guarantees controlling the probability of false positives, under weak assumptions that allow for dependence within the time series data, providing reassurance that time-shifting methods are a reliable tool for inference in this setting. Empirical results with simulated and real data validate the strong performance of time-shifting methods in dependent-data settings. - oai:arXiv.org:2512.17372v2 - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ruiting Liang, Samuel Dyson, Rina Foygel Barber, Daniel E. Holz - - - Forbidding just one intersection for short integer sequences - https://arxiv.org/abs/2512.17544 - arXiv:2512.17544v2 Announce Type: replace -Abstract: In this paper, we study the famous Erd\H{o}s--S\'os forbidden intersection problem for words over an alphabet of size $m$: what is the maximal size of a subfamily $\mathcal{F}$ of $[m]^n$ that does not contain two vectors $x, y$ coinciding on exactly $t - 1$ coordinates? We answer this question provided $m \ge \operatorname{poly}(t)$ and $n \ge \operatorname{poly}(t)$ for some polynomial function $\operatorname{poly}(\cdot)$ of $t$, greatly extending the recent result of Keevash, Lifshitz, Long and Minzer. Our proof combines some of the recently developed methods in extremal combinatorics, including the spread approximation technique of Kupavskii and Zakharov and the hypercontractivity approach developed in a series of works by Keevash, Keller, Lifshitz, Long, Marcus and Minzer. - oai:arXiv.org:2512.17544v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Elizaveta Iarovikova, Fedor Noskov, Georgy Sokolov, Nikolai Terekhov - - - Pseudo-Legendrian and Legendrian Simplicity of Links in 3-Manifolds - https://arxiv.org/abs/2512.18185 - arXiv:2512.18185v2 Announce Type: replace -Abstract: We construct infinite families of non-simple isotopy classes of links in overtwisted contact structures on $S^1$-bundles over surfaces. These examples include: (1) a pair of Legendrian links that are not Legendrian isotopic, but which are isotopic as framed links, homotopic as Legendrian immersed multi-curves, and have Legendrian-isotopic components and (2) a pair of Legendrian links that are not Legendrian isotopic, but are isotopic as framed links, homotopic as Legendrian immersed multi-curves, and which are link-homotopic as Legendrian links. Moreover, we construct examples showing that both of these non-simplicity phenomena can occur in the same smooth isotopy class. To construct these examples, we develop the theory of links transverse to a nowhere-zero vector field in a 3-manifold, and construct analogous examples in the category of links transverse to a vector field. - oai:arXiv.org:2512.18185v2 - math.GT - math.SG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Patricia Cahn, Rima Chatterjee, Vladimir Chernov - - - Kuznecov formulae for fractal measures - https://arxiv.org/abs/2512.18379 - arXiv:2512.18379v2 Announce Type: replace -Abstract: Let $(M,g)$ be a compact, connected Riemannian manifold of dimension $n\ge 2$, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions $-\Delta_g e_j=\lambda_j^2 e_j$. Given a finite Borel measure $\mu$ on $M$, consider the Kuznecov sum \[ - N_\mu(\lambda):=\sum_{\lambda_j\le \lambda}\Bigl|\int_M e_j\,d\mu\Bigr|^2. \] Assume that $\mu$ admits an averaged $s$-density constant $A_\mu$ with correlation dimension $s\in(0,n)$. We prove that \[N_\mu(\lambda)= (2\pi)^{-(n-s)}\,{\rm vol}(B^{\,n-s})\,A_\mu\,\lambda^{n-s}+ o(\lambda^{n-s})\qquad (\lambda\to\infty). \] The averaged $s$-density condition is necessary for such a one-term asymptotic, and in general, the remainder $o(\lambda^{n-s})$ is sharp in the sense that it cannot be improved uniformly to a power-saving error term. This extends the classical Kuznecov formula of Zelditch for smooth submanifold measures to a broad class of singular and fractal measures. - oai:arXiv.org:2512.18379v2 - math.AP - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yakun Xi - - - Induced minors and subpolynomial treewidth - https://arxiv.org/abs/2512.18835 - arXiv:2512.18835v2 Announce Type: replace -Abstract: Given a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-induced-minor-free if no induced minor of $G$ is isomorphic to a member of $\mathcal{H}$, We denote by $W_{t\times t}$ the $t$-by-$t$ hexagonal grid, and by $K_{t,t}$ the complete bipartite graph with both sides of the bipartition of size $t$. We show that the class of $\{K_{t,t},W_{t\times t}\}$-induced minor-free graphs with bounded clique number has subpolynomial treewidth. Specifically, we prove that for every integer $t$ there exist $\epsilon \in (0,1]$ and $c \in \mathbb{N}$ such that every $n$-vertex $\{K_{t,t},W_{t\times t}\}$-induced minor-free graph with no clique of size $t$ has treewidth at most $2^{c\log^{1-\epsilon}n}$. - oai:arXiv.org:2512.18835v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Maria Chudnovsky, Julien Codsi, David Fischer, Daniel Lokshtanov - - - Relative Bruhat decomposition of wonderful compactification - https://arxiv.org/abs/2512.19008 - arXiv:2512.19008v2 Announce Type: replace -Abstract: In the seminal paper of Borel and Tits about reductive groups, they show some fundamental results about Bruhat cells with respect to a minimal parabolic subgroup, e.g., relative Bruhat decomposition and its geometrization, relative Bruhat order and the relation of Zariski closure and topological closure. In this paper, we show analogous results for Bruhat cells of wonderful group compactification in the sense of De Concini and Procesi. Our results can be viewed as the version at infinity of those of Borel and Tits. Our main focus is general base field. When the base field is algebraically closed, most of our results are proved by Brion and Springer. - oai:arXiv.org:2512.19008v2 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Fei Chen, Shang Li - - - Quantization Dimension of $1$-variable Random Self-Similar Measures - https://arxiv.org/abs/2512.19628 - arXiv:2512.19628v2 Announce Type: replace -Abstract: The quantization problem for random fractals presents unique challenges due to the lack of uniform geometric scaling inherent in deterministic systems. In this article, we establish the almost sure quantization dimension for a class of $1$-variable (homogeneously) random self-similar measures. Unlike the deterministic setting, where the dimension is derived from a fixed pressure function, we prove that in the random case, the quantization dimension $\kappa_{r}$ is the unique zero of the expectation of the topological pressure. We rigorously justify this by exploiting the ergodicity of the shift map on the symbolic space to control distortion errors across non-uniform scales. Our results highlight the thermodynamic formalism underlying the quantization of random dynamical systems. - oai:arXiv.org:2512.19628v2 - math.DS - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Akash Banerjee, Alamgir Hossain, Md. Nasim Akhtar - - - Asymptotics for the number of domino tilings of L-shaped Aztec domains - https://arxiv.org/abs/2512.20388 - arXiv:2512.20388v2 Announce Type: replace -Abstract: We obtain precise asymptotics for the weighted number of domino tilings of an L-shaped subset of the Aztec diamond, obtained by removing an approximate rectangle in a corner of the Aztec diamond. By tuning the size of the removed corner, we observe different types of asymptotics. For a small removed corner, the number of tilings is close to that of the full Aztec diamond. Enlarging the removed corner to a critical size, a phase transition described in terms of the Tracy-Widom distribution occurs. Further increasing the size of the removed region, we observe a sharp decrease of the number of tilings, until it is finally approximated by the number of tilings of two smaller disjoint Aztec diamonds. We obtain uniform asymptotics for the number of domino tilings which fully describe these transitions. - oai:arXiv.org:2512.20388v2 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Christophe Charlier, Tom Claeys - - - Linear varieties and matroids with applications to the Cullis' determinant - https://arxiv.org/abs/2512.21098 - arXiv:2512.21098v2 Announce Type: replace -Abstract: Let $V$ be a vector space of rectangular $n\times k$ matrices annihilating the Cullis' determinant. We show that $\dim(V) \le (n-1)k$, extending Dieudonn{\'{e}}'s result on the dimension of vector spaces of square matrices annihilating the ordinary determinant. - Furthermore, for certain values of $n$ and $k$, we explicitly describe such vector spaces of maximal dimension. Namely, we establish that if $k$ is odd, $n \ge k + 2$ and $\dim(V) = (n-1)k$, then $V$ is equal to the space of all $n\times k$ matrices $X$ such that alternating row sum of $X$ is equal to zero. - Our proofs rely on the following observations from the matroid theory that have an independent interest. First, we provide a notion of matroid corresponding to a given linear variety. Second, we prove that if the linear variety is transformed by projections and restrictions, then the behaviour of the corresponding matroid is expressed in the terms of matroid contraction and restriction. Third, we establish that if $M$ is a matroid, $I^*$ its coindependent set $M|S$ and its restriction on a set $S$, then the union of $I^*\setminus S$ with every cobase of $M|S$ is coindependent set of $M$. - oai:arXiv.org:2512.21098v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Alexander Guterman, Andrey Yurkov - - - $x(1-t(x+x^{-1}))F(x;t) = x-tF(0;t)$ - https://arxiv.org/abs/2512.21753 - arXiv:2512.21753v2 Announce Type: replace -Abstract: The purpose of these notes is to introduce some of the problems the enumeration of lattice walks is dedicated to and familiarize with some of the arguments they can be addressed with. We discuss the enumeration of lattice walks, their generating functions, and the functional equations they satisfy. We focus on algebraic methods for manipulating and solving these equations. Elementary power series algebra plays a prominent role, computer algebra too, but we repeatedly digress and present ideas and methods of different kind whenever it is appropriate. The exposition is organized around the most simple yet non-trivial problem: the enumeration of simple walks on the half-line. The intention is to illustrate different techniques without getting technical. - oai:arXiv.org:2512.21753v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Manfred Buchacher - - - Exchangeability and randomness for infinite and finite sequences - https://arxiv.org/abs/2512.22162 - arXiv:2512.22162v2 Announce Type: replace -Abstract: Randomness (in the sense of being generated in an IID fashion) and exchangeability are standard assumptions in nonparametric statistics and machine learning, and relations between them have been a popular topic of research. This short paper draws the reader's attention to the fact that, while for infinite sequences of observations the two assumptions are almost indistinguishable, the difference between them becomes very significant for finite sequences of a given length. - oai:arXiv.org:2512.22162v2 - math.ST - stat.ME - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Vladimir Vovk - - - Research projects and Moscow Mathematical Conference for high school students - https://arxiv.org/abs/2512.22191 - arXiv:2512.22191v2 Announce Type: replace -Abstract: This paper shares some experience in advanced mathematical education. We show how a high school student can be naturally and gradually introduced to basic steps of scientific research: developing intuition by finding and correcting mistakes through discussions and writing a paper, (transparent) anonymous peer review, recognition and award. We show that most of this can be done in research projects not aiming at scientific novelty. We share the experience (both principles and examples) of the Moscow Mathematical Conference of High School Students. - oai:arXiv.org:2512.22191v2 - math.HO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - A. Zaslavskiy, A. Skopenkov - - - Covering in Hamming and Grassmann Spaces: New Bounds and Reed--Solomon-Based Constructions - https://arxiv.org/abs/2512.22911 - arXiv:2512.22911v2 Announce Type: replace -Abstract: We study covering problems in Hamming and Grassmann spaces through a unified coding-theoretic and information-theoretic framework. Viewing covering as a form of quantization in general metric spaces, we introduce the notion of the average covering radius as a natural measure of average distortion, complementing the classical worst-case covering radius. By leveraging tools from one-shot rate-distortion theory, we derive explicit non-asymptotic random-coding bounds on the average covering radius in both spaces, which serve as fundamental performance benchmarks. - On the construction side, we develop efficient puncturing-based covering algorithms for generalized Reed--Solomon (GRS) codes in the Hamming space and extend them to a new family of subspace codes, termed character-Reed--Solomon (CRS) codes, for Grassmannian quantization under the chordal distance. Our results reveal that, despite poor worst-case covering guarantees, these structured codes exhibit strong average covering performance. In particular, numerical results in the Hamming space demonstrate that RS-based constructions often outperform random codebooks in terms of average covering radius. In the one-dimensional Grassmann space, we numerically show that CRS codes over prime fields asymptotically achieve average covering radii within a constant factor of the random-coding bound in the high-rate regime. Together, these results provide new insights into the role of algebraic structure in covering problems and high-dimensional quantization. - oai:arXiv.org:2512.22911v2 - cs.IT - eess.SP - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Samin Riasat, Hessam Mahdavifar - - - MSO logic of the real order with the set quantifiers ranging over the Borel sets - https://arxiv.org/abs/2512.23003 - arXiv:2512.23003v5 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. If we only ask for the decidability in the language where each level of the Borel hierarchy is allowed a quantifier to denote sets of that level in the hierarchy, then we obtain a weaker MSO theory, which is not inly decidable but also interpretable in S2S. - oai:arXiv.org:2512.23003v5 - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Mirna D\v{z}amonja - - - Average first-passage times for character sums - https://arxiv.org/abs/2512.24631 - arXiv:2512.24631v2 Announce Type: replace -Abstract: Let $\varepsilon>0$ and, for an odd prime $p$, set $$ S_\ell(p):=\sum_{n\le \ell}\left(\frac{n}{p}\right). $$ Define the first-passage time $$ f_\varepsilon(p):=\min\{\ell\ge 1:\ S_\ell(p)<\varepsilon\ell\}. $$ We prove that there exists a constant $c_\varepsilon>0$ such that, as $x\to\infty$, $$ \sum_{p\le x} f_\varepsilon(p)\sim c_\varepsilon \frac{x}{\log x}. $$ - oai:arXiv.org:2512.24631v2 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Quanyu Tang, Hao Zhang - - - Collective behaviors of an electron gas in the mean-field regime - https://arxiv.org/abs/2512.24666 - arXiv:2512.24666v3 Announce Type: replace -Abstract: In this paper, we study the momentum distribution of an electron gas in a $3$-dimensional torus. The goal is to compute the occupation number of Fourier modes for some trial state obtained through random phase approximation. We obtain the mean-field analogue of momentum distribution formulas for electron gas in [Daniel and Voskov, Phys. Rev. \textbf{120}, (1960)] in high density limit and [Lam, Phys. Rev. \textbf{3}, (1971)] at metallic density. The analysis in the present paper is majorly based on the work [Christiansen, Hainzl, Nam, Comm. Math. Phys. \textbf{401}, (2023)]. Our findings are related to recent results obtained independently by Benedikter, Lill and Naidu, and the analysis applies to a general class of singular potentials rather than just the Coulomb case. - oai:arXiv.org:2512.24666v3 - math-ph - math.MP - quant-ph - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Dong Hao Ou Yang - - - From Complex-Analytic Models to Sparse Domination: A Dyadic Approach of Hypersingular Operators via Bourgain's Interpolation Method - https://arxiv.org/abs/2512.24972 - arXiv:2512.24972v4 Announce Type: replace -Abstract: Motivated by the work of Cheng--Fang--Wang--Yu on the hypersingular Bergman projection, we develop a real-variable and dyadic framework for hypersingular operators in regimes where strong-type estimates fail at the critical line. The main new input is a hypersingular sparse domination principle combined with Bourgain's interpolation method, which provides a flexible mechanism to establish critical-line (and endpoint) estimates. - In the unit disc setting with $1<t<3/2$, we obtain a full characterization of the $(p,q)$ mapping theory for the dyadic hypersingular maximal operator $\mathcal M_t^{\mathcal D}$, in particular including estimates on the critical line $1/q-1/p=2t-2$ and a weighted endpoint criterion in the radial setting. In addition, we establish a novel two-weight estimate for $\mathcal M_t^{\mathcal D}$ in the range $p>q$, valid for any $t>0$. We also prove endpoint estimates for the hypersingular Bergman projection \[ K_{2t}f(z)=\int_{\mathbb D}\frac{f(w)}{(1-z\overline w)^{2t}}\,dA(w), \] including a restricted weak-type bound at $(p,q)=\bigl(\tfrac{1}{3-2t},1\bigr)$. Finally, we introduce a class of hypersingular cousin of sparse operators in $\mathbb R^n$ associated with \emph{graded} sparse families, quantified by the sparseness $\eta$ and a new structural parameter (the \emph{degree}) $K_{\mathcal S}$, and we characterize the corresponding strong/weak/restricted weak-type regimes in terms of $(n,t,\eta,K_{\mathcal S})$. - Our real-variable perspective addresses an inquiry raised by Cheng--Fang--Wang--Yu on developing effective real-analytic tools in the hypersingular regime for $K_{2t}$, and it also provides a new route toward the critical-line analysis of Forelli--Rudin type operators and related hypersingular operators in both real and complex settings. - oai:arXiv.org:2512.24972v4 - math.CA - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bingyang Hu, Xiaojing Zhou - - - Limit Computation Over Posets via Minimal Initial Functors - https://arxiv.org/abs/2601.00209 - arXiv:2601.00209v2 Announce Type: replace -Abstract: It is well known that limits can be computed by restricting along an initial functor, and that this often simplifies limit computation. We systematically study the algorithmic implications of this idea for diagrams indexed by a finite poset. We say an initial functor $F\colon C\to D$ with $C$ small is \emph{minimal} if the sets of objects and morphisms of $C$ each have minimum cardinality, among the sources of all initial functors with target $D$. For $Q$ a finite poset or $Q\subseteq \mathbb N^d$ an interval (i.e., a convex, connected subposet), we describe all minimal initial functors $F\colon P\to Q$ and in particular, show that $F$ is always a subposet inclusion. We give efficient algorithms to compute a choice of minimal initial functor. In the case that $Q\subseteq \mathbb N^d$ is an interval, we give asymptotically optimal bounds on $|P|$, the number of relations in $P$ (including identities), in terms of the number $n$ of minima of $Q$: We show that $|P|=\Theta(n)$ for $d\leq 3$, and $|P|=\Theta(n^2)$ for $d>3$. We apply these results to give new bounds on the cost of computing $\lim G$ for a functor $G \colon Q\to \mathbf{Vec}$ valued in vector spaces. For $Q$ connected, we also give new bounds on the cost of computing the \emph{generalized rank} of $G$ (i.e., the rank of the induced map $\lim G\to \mathop{\mathrm{colim}} G$), which is of interest in topological data analysis. - oai:arXiv.org:2601.00209v2 - math.AT - math.CT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Tamal K. Dey, Michael Lesnick - - - Hadamard-type formulas for real eigenvalues of canonically symplectic operators - https://arxiv.org/abs/2601.00520 - arXiv:2601.00520v3 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:2601.00520v3 - math.SP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mitchell Curran, Selim Sukhtaiev - - - Complexity of deep computations via topology of function spaces - https://arxiv.org/abs/2601.00528 - arXiv:2601.00528v2 Announce Type: replace -Abstract: We use topological methods to study complexity of deep computations and limit computations. We use topology of function spaces, specifically, the classification Rosenthal compacta, to identify new complexity classes. We use the language of model theory, specifically, the concept of \emph{independence} from Shelah's classification theory, to translate between topology and computation. We use the theory of Rosenthal compacta to characterize approximablility of deep computations, both deterministically and probabilistically. - oai:arXiv.org:2601.00528v2 - math.LO - math.GN - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Eduardo Due\~nez, Jos\'e Iovino, Tonatiuh Matos-Wiederhold, Luciano Salvetti, Franklin D. Tall - - - On oscillator death in the Winfree model - https://arxiv.org/abs/2601.01203 - arXiv:2601.01203v2 Announce Type: replace -Abstract: We show that for the standard sinusoidal Winfree model, a coupling strength exceeding twice the maximal magnitude of the intrinsic frequencies guarantees the convergence of the system for Lebesgue almost every initial data. This is proven by first showing, via an order parameter bootstrapping argument, that the pathwise critical coupling strength is upper bounded by a function of the order parameter, and then showing by a volumetric argument that for Lebesgue almost every data the order parameter cannot stay below and be bounded away from 1 for all time; this is a Winfree model counterpart of the analysis of Ha and the author (2020) performed for the Kuramoto model. Using concentration of measure and the aforementioned volumetric argument, we show that, except possibly on a set of very small measure, oscillator death is observed in finite time; this rigorously demonstrates the existence of the oscillator death regime numerically observed by Ariaratnam and Strogatz (2001). These results are robust under many other choices of interaction functions often considered for the Winfree model. We demonstrate that the asymptotic dynamics described in this paper are sharp by analyzing the equilibria of the Winfree model, and we bound the total number of equilibria using a polynomial description. - oai:arXiv.org:2601.01203v2 - math.CA - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Seung-Yeon Ryoo - - - New discretised polynomial expander and incidence estimates - https://arxiv.org/abs/2601.01264 - arXiv:2601.01264v2 Announce Type: replace -Abstract: We present two applications of recent developments in incidence geometry. One is a $\delta$-discretised version of a particular `Elekes--R\'onyai' expander problem. The second application is an incidence estimate addressing the scenario when both tubes, squares and their shadings satisfy non-concentration assumptions. - oai:arXiv.org:2601.01264v2 - math.CO - math.CA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ciprian Demeter, William O'Regan - - - Algorithmic Information Theory for Graph Edge Grouping and Substructure Analysis - https://arxiv.org/abs/2601.01760 - arXiv:2601.01760v3 Announce Type: replace -Abstract: Understanding natural phenomenon through the interactions of different complex systems has become an increasing focus in scientific inquiry. Defining complexity and actually measuring it is an ongoing debate and no standard framework has been established that is both theoretically sound and computationally practical to use. Currently, one of the fields which attempts to formally define complexity is in the realm of Algorithmic Information Theory. The field has shown advances by studying the complexity values of binary strings and 2-dimensional binary matrices using 1-dimensional and 2-dimensional Turing machines, respectively. Using these complexity values, an algorithm called the Block Decomposition Method developed by Zenil, et al. in 2018, has been created to approximate the complexity of adjacency matrices of graphs which have found relative success in grouping graphs based on their complexity values. We use this method along with another method called edge perturbation to exhaustively determine if an edge can be identified to connect two subgraphs within a graph using the entire symmetric group of its vertices permutation and via unique permutations we call automorphic subsets, which are a special subset of the symmetric group. We also analyze if edges will be grouped closer to their respective subgraphs in terms of the average algorithmic information contribution. This analysis ascertains if Algorithmic Information Theory can serve as a viable theory for understanding graph substructures and as a foundation for frameworks measuring and analyzing complexity. The study found that the connecting edge was successfully identified as having the highest average information contribution in 29 out of 30 graphs, and in 16 of these, the distance to the next edge was greater than log_2(2). Furthermore, the symmetric group outperformed automorphic subsets in edge grouping. - oai:arXiv.org:2601.01760v3 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Gabriel Potestades - - - Hilbert Polynomials of Calabi Yau Hypersurfaces in Toric Varieties and Lattice Points in Polytope Boundaries - https://arxiv.org/abs/2601.02176 - arXiv:2601.02176v2 Announce Type: replace -Abstract: We show that the Hilbert polynomial of a Calabi-Yau hypersurface $Z$ in a smooth toric variety $M$ associated to a convex polytope $\Delta$ is given by a lattice point count in the polytope boundary $\partial \Delta,$ just as the Hilbert polynomial of $M$ is known to be given by a lattice point count in the convex polytope $\Delta.$ Our main tool is a computation of the Euler class in $K$-theory of the normal line bundle to the hypersurface $Z,$ in terms of the Euler classes of the divisors corresponding to the facets of the moment polytope. We observe a remarkable parallel between our expression for the Euler class and the inclusion-exclusion principle in combinatorics. To obtain our result we combine these facts with the known relation between lattice point counts in the facets of $\Delta$ and the Hilbert polynomials of the smooth toric varieties corresponding to these facets. - oai:arXiv.org:2601.02176v2 - math.AG - hep-th - math.CO - math.KT - math.SG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jonathan Weitsman - - - A modern perspective on Tutte's homotopy theorem - https://arxiv.org/abs/2601.02582 - arXiv:2601.02582v2 Announce Type: replace -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:2601.02582v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Matthew Baker, Tong Jin, Oliver Lorscheid - - - Local Asymptotic Normality for Mixed Fractional Brownian Motion Under High-Frequency Observation - https://arxiv.org/abs/2601.02622 - arXiv:2601.02622v3 Announce Type: replace -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. Beyond the case of $H>3/4$, we also present that the $\| \cdot \|_{\mathrm{op}}/\|\cdot\|_F\to0$ method is also useful for the case of $H<3/4$ and the proof will be concise compared with the Whittle translation method. We consider that this method can be applied to this type of problem, including the fractional Ornstein-Uhlenbeck model and mixed fractional O-U process. - oai:arXiv.org:2601.02622v3 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chunhao Cai, Yiwu Shang - - - Coupling Brownian loop soups and random walk loop soups at all polynomial scales - https://arxiv.org/abs/2601.02992 - arXiv:2601.02992v2 Announce Type: replace -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 $N^{-2/3}$). This coupling was subsequently extended by Sapozhnikov and Shiraishi to $\mathbb{Z}^d$ with $d\ge 3$, for loops with time length at least $N^{\theta-2}$, for $\theta \in(2d/(d+4),2)$. - In this paper, we find a simple way to remove the restriction $\theta>2d/(d+4)$, so that such a coupling works for all $\theta\in (0,2)$, i.e. for loops at all polynomial scales. We establish couplings for both discrete-time and continuous-time random walk loop soups on $\mathbb{Z}^d$, for $d\ge 1$. - oai:arXiv.org:2601.02992v2 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Wei Qian - - - On difference sets of dense subsets of $\mathbb{Z}^2$ - https://arxiv.org/abs/2601.03797 - arXiv:2601.03797v2 Announce Type: replace -Abstract: In this article, we study the structure of the difference set $E - E$ for subsets $E \subseteq \mathbb{Z}^2$ of positive upper Banach density. Fish asked in [Proc. Amer. Math. Soc. 146 (2018), 3449-3453] whether, for every such set $E$, there exists a nonzero integer $k$ such that $k \cdot \mathbb{Z} \subseteq \{\, xy : (x,y) \in E - E \,\}.$ Although this question remains open, we establish a relatively weaker form of this conjecture. Specifically, we prove that if $\langle a_j\rangle_{j=1}^m$ is any finite sequence in $\mathbb{N},$ then there exist infinitely many integers $k \in \mathbb{Z}$ and a sequence $\langle x_n \rangle_{n \in \mathbb{N}}$ in $\mathbb{Z}$ such that $k \cdot MT\left(\langle a_j \rangle_{j=1}^m, \langle x_n\rangle_{n}\right) \subseteq \{\, xy : (x,y) \in E - E \,\},$ where $MT\left(\langle a_j \rangle_{j=1}^m, \langle x_n\rangle_{n}\right)$ denotes the milliken-Taylor configuration generated by the sequences $\langle a_j\rangle_{j=1}^m$ and $\langle x_n \rangle_{n \in \mathbb{N}}$. - oai:arXiv.org:2601.03797v2 - math.NT - math.CO - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Sayan Goswami - - - Trade-off between spread and width for tree decompositions - https://arxiv.org/abs/2601.04040 - arXiv:2601.04040v2 Announce Type: replace -Abstract: We study the trade-off between (average) spread and width in tree decompositions, answering several questions from Wood [arXiv:2509.01140]. The spread of a vertex $v$ in a tree decomposition is the number of bags that contain $v$. Wood asked for which $c>0$, there exists $c'$ such that each graph $G$ has a tree decomposition of width $c\cdot tw(G)$ in which each vertex $v$ has spread at most $c'(d(v)+1)$. We show that $c\geq 2$ is necessary and that $c>3$ is sufficient. Moreover, we answer a second question fully by showing that near-optimal average spread can be achieved simultaneously with width $O(tw(G))$. - oai:arXiv.org:2601.04040v2 - math.CO - cs.DM - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hans L. Bodlaender, Carla Groenland - - - Ergodic Theorems and Equivalence of Green's Kernel for Random Walks in Random Environments - https://arxiv.org/abs/2601.04161 - arXiv:2601.04161v3 Announce Type: replace -Abstract: We study the Ergodic Properties of Random Walks in stationary ergodic environments without uniform ellipticity under a minimal assumption. There are two main components in our work. The first step is to adopt the arguments of Lawler to first prove a uniqueness principle. We use a more general definition of environments using~\textit{Environment Functions}. As a corollary, we can deduce an invariance principle under these assumptions for balanced environments under some assumptions. We also use the uniqueness principle to show that any balanced, elliptic random walk must have the same transience behaviour as the simple symmetric random walk. The second is to transfer the results we deduce in balanced environments to general ergodic environments(under some assumptions) using a control technique to derive a measure under which the \textit{local process} is stationary and ergodic. As a consequence of our results, we deduce the Law of Large Numbers for the Random Walk and an Invariance Principle under our assumptions. - oai:arXiv.org:2601.04161v3 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Ayan Ghosh - - - Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms - https://arxiv.org/abs/2601.04965 - arXiv:2601.04965v3 Announce Type: replace -Abstract: We study SOS properties of biquadratic forms. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic form is a sum of squares of bilinear forms. This extends the known result for fully symmetric biquadratic forms. We describe an efficient computational procedure for constructing SOS decompositions, exploiting the Kronecker-product structure of the associated matrix representation. We introduce simple biquadratic forms. For $m \ge 2$, we present a $m \times 2$ PSD biquadratic form and show that it can be expressed as the sum of $m+1$ squares, but cannot be expressed as the sum of $m$ squares. This provides a lower bound for sos rank of $m \times 2$ biquadratic forms, and shows that previously proved results that a $2 \times 2$ PSD biquadratic form can be expressed as the sum of three squares, and a $3 \times 2$ PSD biquadratic form can be expressed as the sum of four squares, are tight. We also present an $3 \times 3$ SOS biquadratic form, which can be expressed as the sum of six squares, but not the sum of five squares.We present a $2 \times 2$ PSD biquadratic form, and show that it can be expressed as the sum of three squares, but cannot be expressed as the sum of two squares. Furthermore, we present a $3 \times 2$ PSD biquadratic form, and show that it can be expressed as the sum of four squares, but cannot be expressed as the sum of three squares. These show that previously proved results that a $2 \times 2$ PSD biquadratic form can be expressed as the sum of three squares, and a $3 \times 2$ PSD biquadratic form can be expressed as the sum of four squares, are tight. Moreover, we establish a universal upper bound SOS-rank$(P) \le mn-1$ for any SOS biquadratic form, which improves the trivial bound $mn$ and is tight in small dimensions. - oai:arXiv.org:2601.04965v3 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Liqun Qi, Chunfeng Cui, Yi Xu - - - Analysis of the Density of Words under Morphism $\{a,b\}$ - https://arxiv.org/abs/2601.06150 - arXiv:2601.06150v2 Announce Type: replace -Abstract: In this paper, we analyze the density of the Fibonacci word and its derived forms by examining the morphisms associated with each. It offers a comparative analysis of the density of Fibonacci numbers alongside other words derived from Fibonacci word. Fibonacci words over the alphabet $\{a,b\}$, we define a novel \emph{power} operation that yields a formal linear combination in the free abelian group generated by all finite words. - oai:arXiv.org:2601.06150v2 - math.GM - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Jasem Hamoud, Duaa Abdullah - - - Upper bound for the total mean curvature of spin fill-ins - https://arxiv.org/abs/2601.06713 - arXiv:2601.06713v2 Announce Type: replace -Abstract: Gromov conjectured that the total mean curvature of the boundary of a compact Riemannian manifold can be estimated from above by a constant depending only on the boundary metric and on a lower bound for the scalar curvature of the fill-in. We prove Gromov's conjecture if the manifolds are spin and the mean curvature is non-negative. One can also allow $H$ to take negative values, but then the constant depends on a (negative) lower bound for $H$. - oai:arXiv.org:2601.06713v2 - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Christian Baer - - - The Greedy Algorithm for Dissociated Sets - https://arxiv.org/abs/2601.07068 - arXiv:2601.07068v3 Announce Type: replace -Abstract: A set $\mathcal S\subset \mathbb N$ is said to be a subset-sum-distinct or dissociated if all of its finite subsets have different sums. Alternately, an equivalent classification is if any equality of the form $$\sum_{s\in \mathcal S} \varepsilon_s \cdot s =0$$ where $\varepsilon_s \in \{-1,0,+1\}$ implies that all the $\varepsilon_s$'s are $0$. For a dissociated set $\mathcal S$, we prove that for $c_\ast = \frac 12 \log_2 \left(\frac \pi 2\right)$ and any $c_\ast-1<C<c_\ast$, we have $$\mathcal S(n) \,:=\, \mathcal S\cap [1,n] \,\le\, \log_2 n +\frac 12 \log_2\log_2 n + C$$ for all $n\in \mathcal N_C$ with asymptotic density $\mathbf d\left(\mathcal N_C\right)=2-2^{c_\ast-C}$. Further, we consider the greedy algorithm for generating these sets and prove that this algorithm always eventually doubles. Finally, we also consider some generalizations of dissociated sets and prove similar results about them. - oai:arXiv.org:2601.07068v3 - math.CO - math.NT - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Sayan Dutta - - - The infinitude of square-free palindromes - https://arxiv.org/abs/2601.07097 - arXiv:2601.07097v2 Announce Type: replace -Abstract: We settle an open problem regarding palindromes; that is, positive integers which are the same when written forwards and backwards. In particular, we prove that for any fixed base $b\geq 2$, there exist infinitely many square-free palindromes in base $b$. We also provide an asymptotic expression for the number of such integers $\leq x$. The core of our proof utilises a hybrid $p$-adic/Archimedean van der Corput process, used in conjunction with an equidistribution estimate of Tuxanidy and Panario, as well as an elementary argument of Cilleruelo, Luca and Shparlinski. - oai:arXiv.org:2601.07097v2 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Daniel R. Johnston, Bryce Kerr - - - Local and global $C^{1,\beta}$-regularity for uniformly elliptic quasilinear equations of $p$-Laplace and Orlicz-Laplace type - https://arxiv.org/abs/2601.07140 - arXiv:2601.07140v2 Announce Type: replace -Abstract: We establish gradient H\"older continuity for solutions to quasilinear, uniformly elliptic equations, including $p$-Laplace and Orlicz-Laplace type operators. We revisit and improve upon the results existing in the literature, proving gradient regularity both in the interior and up to the boundary, under Dirichlet or Neumann boundary conditions. - oai:arXiv.org:2601.07140v2 - math.AP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Carlo Alberto Antonini - - - Resolution of Erd\H{o}s Problem #728: a writeup of Aristotle's Lean proof - https://arxiv.org/abs/2601.07421 - arXiv:2601.07421v4 Announce Type: replace -Abstract: We provide a writeup of a resolution of Erd\H{o}s Problem #728; this is the first Erd\H{o}s problem (a problem proposed by Paul Erd\H{o}s which has been collected in the Erd\H{o}s Problems website) regarded as fully resolved autonomously by an AI system. The system in question is a combination of GPT-5.2 Pro by OpenAI and Aristotle by Harmonic, operated by Kevin Barreto. The final result of the system is a formal proof written in Lean, which we translate to informal mathematics in the present writeup for wider accessibility. - The proved result is as follows. We show a logarithmic-gap phenomenon regarding factorial divisibility: For any constants $0<C_1<C_2$ and $0 < \varepsilon < 1/2$ there exist infinitely many triples $(a,b,n)\in\mathbb N^3$ with $\varepsilon n \le a,b \le (1-\varepsilon)n$ such that \[ a!\,b!\mid n!\,(a+b-n)!\qquad\text{and}\qquad C_1\log n < a+b-n < C_2\log n. \] The argument reduces this to a binomial divisibility $\binom{m+k}{k}\mid\binom{2m}{m}$ and studies it prime-by-prime. By Kummer's theorem, $\nu_p\binom{2m}{m}$ translates into a carry count for doubling $m$ in base $p$. We then employ a counting argument to find, in each scale $[M,2M]$, an integer $m$ whose base-$p$ expansions simultaneously force many carries when doubling $m$, for every prime $p\le 2k$, while avoiding the rare event that one of $m+1,\dots,m+k$ is divisible by an unusually high power of $p$. These "carry-rich but spike-free" choices of $m$ force the needed $p$-adic inequalities and the divisibility. The overall strategy is similar to results regarding divisors of $\binom{2n}{n}$ studied earlier by Erd\H{o}s and by Pomerance. - oai:arXiv.org:2601.07421v4 - math.NT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Nat Sothanaphan - - - On eigenvalues of the Landau Hamiltonian with a periodic electric potential - https://arxiv.org/abs/2601.07495 - arXiv:2601.07495v2 Announce Type: replace -Abstract: We consider the Landau Hamiltonian $\widehat H_B+V$ on $L^2({\mathbb R}^2)$ with a periodic electric potential $V$. For every $m\in {\mathbb N}$ we prove that there exist nonconstant periodic electric potentials $V\in C^{\infty }({\mathbb R}^2;{\mathbb R})$ with zero mean values that analytically depend on a small parameter $\varepsilon \in {\mathbb R}$ such that the Landau level $(2m+1)B$ is an eigenvalue of the Hamiltonian (of infinite multiplicity) where $B>0$ is a strength of a homogeneous magnetic field. - oai:arXiv.org:2601.07495v2 - math-ph - math.MP - math.SP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Leonid Danilov - - - On the Sequence Reconstruction Problem for the Single-Deletion Two-Substitution Channel - https://arxiv.org/abs/2601.07547 - arXiv:2601.07547v2 Announce Type: replace -Abstract: The Levenshtein sequence reconstruction problem studies the reconstruction of a transmitted sequence from multiple erroneous copies of it. A fundamental question in this field is to determine the minimum number of erroneous copies required to guarantee correct reconstruction of the original sequence. This problem is equivalent to determining the maximum possible intersection size of two error balls associated with the underlying channel. Existing research on the sequence reconstruction problem has largely focused on channels with a single type of error, such as insertions, deletions, or substitutions alone. However, relatively little is known for channels that involve a mixture of error types, for instance, channels allowing both deletions and substitutions. In this work, we study the sequence reconstruction problem for the single-deletion two-substitution channel, which allows one deletion and at most two substitutions applied to the transmitted sequence. Specifically, we prove that if two $q$-ary length-$n$ sequences have the Hamming distance $d\geq 2$, where $q\geq 2$ is any fixed integer, then the intersection size of their error balls under the single-deletion two-substitution channel is upper bounded by $(q^2-1)n^2-(3q^2+5q-5)n+O_q(1)$, where $O_q(1)$ is a constant independent from $n$ but dependent on $q$. Moreover, we show that this upper bound is tight up to an additive constant. - oai:arXiv.org:2601.07547v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Wentu Song, Kui Cai, Tony Q. S. Quek - - - Pluripotential theory on algebraic curves - https://arxiv.org/abs/2601.07639 - arXiv:2601.07639v2 Announce Type: replace -Abstract: In previous works, the second author defined directional Robin constants associated to a compact, nonpolar subset $K$ of an algebraic curve $A$ in $\mathbb{C}^N$ and related these to a natural class of Chebyshev constants for $K$. We define a second class of Chebyshev constants for $K$; relate these two classes; and utilize each of them to define two families of extremal-like functions which can be used to recover the Siciak-Zaharjuta extremal function for $K$. - oai:arXiv.org:2601.07639v2 - math.CV - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Norm Levenberg, Sione Ma'u - - - On curves of degree 10 with 12 triple points - https://arxiv.org/abs/2601.07809 - arXiv:2601.07809v2 Announce Type: replace -Abstract: We construct an irreducible rational curve of degree 10 in $CP^2$ which has 12 triple points and a union of three rational quartics with 19 triple points. This gives counter-examples to a conjecture by Dimca, Harbourne, and Sticlaru. We also prove that there exists an analytic family $C_u$ of curves of degree 10 with 12 triple points which tends as $u\to 0$ to the union of the dual Hesse arrangement of lines (9 lines with 12 triple points) with an additional line. We hope that our approach to the proof of the latter fact could be of independent interest. - oai:arXiv.org:2601.07809v2 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - S. Yu. Orevkov - - - On Strong Lefschetz Property of 0-dimensional complete intersections - https://arxiv.org/abs/2601.07874 - arXiv:2601.07874v3 Announce Type: replace -Abstract: We prove that a homogeneous 0-dimensional complete intersection satisfies the Strong Lefschetz Property (SLP) in degree 1 if and only if its associated form has nonzero Hessian. The result is essentially known in the literature, but our proof is different compared with the previous ones. - oai:arXiv.org:2601.07874v3 - math.AG - math.AC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zhenjian Wang - - - Determining the Winner in Alternating-Move Games - https://arxiv.org/abs/2601.08359 - arXiv:2601.08359v2 Announce Type: replace -Abstract: We provide a criterion for determining the winner in two-player win-lose alternating-move games on trees, in terms of the Hausdorff dimension of the target set. We focus our study on special cases, including the Gale-Stewart game on the complete binary tree and a family of Schmidt games. Building on the Hausdorff dimension games originally introduced by Das, Fishman, Simmons, and Urba\'nski, which provide a game-theoretic approach for computing Hausdorff dimensions, we employ a generalized family of these games, and show that they are useful for analyzing sets underlying the win-lose games we study. - oai:arXiv.org:2601.08359v2 - math.DS - cs.GT - math.LO - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Itamar Bella\"iche, Auriel Rosenzweig - - - On the Generalization Error of Differentially Private Algorithms Via Typicality - https://arxiv.org/abs/2601.08386 - arXiv:2601.08386v2 Announce Type: replace -Abstract: We study the generalization error of stochastic learning algorithms from an information-theoretic perspective, with a particular emphasis on deriving sharper bounds for differentially private algorithms. It is well known that the generalization error of stochastic learning algorithms can be bounded in terms of mutual information and maximal leakage, yielding in-expectation and high-probability guarantees, respectively. In this work, we further upper bound mutual information and maximal leakage by explicit, easily computable formulas, using typicality-based arguments and exploiting the stability properties of private algorithms. In the first part of the paper, we strictly improve the mutual-information bounds by Rodr\'iguez-G\'alvez et al. (IEEE Trans. Inf. Theory, 2021). In the second part, we derive new upper bounds on the maximal leakage of learning algorithms. In both cases, the resulting bounds on information measures translate directly into generalization error guarantees. - oai:arXiv.org:2601.08386v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yanxiao Liu, Chun Hei Michael Shiu, Lele Wang, Deniz G\"und\"uz - - - Fluctuations of the Ising free energy on Erd\H{o}s-R\'enyi graphs - https://arxiv.org/abs/2601.08590 - arXiv:2601.08590v2 Announce Type: replace -Abstract: We investigate the ferromagnetic Ising model on the Erd\H{o}s-R\'enyi random graph $\mathbb{G}(n,m)$ with bounded average degree $d=2m/n$. Specifically, we determine the limiting distribution of $\log Z_{\mathbb{G}(n,m)}(\beta,B)$, where $Z_{\mathbb{G}(n,m)}(\beta,B)$ is the partition function at inverse temperature $\beta>0$ and external field $B\geq0$. - If either $B>0$, or $B=0$, $d>1$ and $\beta>\operatorname{ath}(1/d)$ the limiting distribution is a Gaussian whose variance is of order $\Theta(n)$ and is described by a family of stochastic fixed point problems that encode the root magnetisation of two correlated Galton-Watson trees. By contrast, if $B=0$ and either $d\leq1$ or $\beta<\operatorname{ath}(1/d)$ the limiting distribution is an infinite sum of independent random variables and has bounded variance. - oai:arXiv.org:2601.08590v2 - math.CO - math-ph - math.MP - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Amin Coja-Oghlan, Dominik Kaaser, Maurice Rolvien, Pavel Zakharov, Kostas Zampetakis - - - Spectral Fusion Deformations for Locally Compact Quantum Groups - https://arxiv.org/abs/2601.08688 - arXiv:2601.08688v2 Announce Type: replace -Abstract: We develop a deformation framework for $C^*$-algebras equipped with a coaction of a locally compact quantum group, formulated intrinsically at the level of spectral subspaces determined by the coaction. The construction is defined algebraically on a finite spectral core and extended by continuity to a natural Fr\'echet $*$-algebra completion under mild analytic regularity assumptions. - Deformations are governed by scalar fusion data assigning phases to fusion channels of irreducible corepresentations. Associativity and $*$-compatibility are characterized by explicit algebraic identities. The framework recovers a range of known deformation procedures, including Rieffel, Kasprzak, and Drinfeld-type constructions, and also yields genuinely new deformations that do not arise from dual $2$--cocycles or crossed-product methods. - At the $C^*$-level, we identify a minimal reduced setting in which the deformed algebra admits a canonical completion, formulated in terms of boundedness of the deformed left regular action on the Haar--GNS space. This separates algebraic coherence from analytic implementability and clarifies the precise role of higher-order fusion data in deformation theory for locally compact quantum groups. In particular, the framework exhibits explicit associator-level deformations governed by fusion $3$--cocycles that cannot arise from any dual $2$--cocycle or crossed-product construction. - oai:arXiv.org:2601.08688v2 - math.OA - math.FA - math.KT - math.QA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Amandip Sangha - - - Upper and Lower Bounds for The Quantum Dynamics of One-Dimensional Divergence-Type Random Jacobi Operators - https://arxiv.org/abs/2601.08796 - arXiv:2601.08796v2 Announce Type: replace -Abstract: We study quantum transport for the discrete one-dimensional random Jacobi operator of divergence-gradient type. For strictly positive and bounded random variables, we analyze the q-moments of the position operator and establish both upper and lower power-law bounds on their growth. Our approach relies on the asymptotic behavior of the integrated density of states and the Lyapunov exponent near the critical energy 0, previously obtained by Pastur and Figotin. A key ingredient in our analysis is the large deviation-type estimates explored via the phase formalism, which play a central role in deriving bounds on the growth of the transfer matrices. - oai:arXiv.org:2601.08796v2 - math-ph - math.MP - math.SP - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Long Li, Wei Wang, Shiwen Zhang - - - Quantum Heegaard diagrams and knot Floer Homology - https://arxiv.org/abs/2601.08805 - arXiv:2601.08805v2 Announce Type: replace -Abstract: Given a knot presented as a braid closure, we construct a unified intersection model for the Alexander and Jones polynomials of the knot via what we call quantum Heegaard diagrams. These diagrams are obtained by stabilising the disc model of the first author, which we show are doubly-pointed Heegaard diagrams of the knot together with an additional set of base points. We identify the Alexander grading in the disc model with the Alexander grading in the Heegaard diagram. As the Lagrangian intersection Floer homology of the Heegaard tori in the symmetric power of the Heegaard surface is knot Floer homology, we can view knot Floer homology as a natural categorification of the Alexander polynomial arising from the disc model. - The additional base points let us define a new grading on the intersection between the Heegaard tori, which we call quantum Alexander grading. Combining this with the classical Alexander grading, we define a two-variable graded intersection between the Heegaard tori that recovers the Jones and Alexander polynomials as two specialisations of coefficients. The resulting intersection formula for the Jones polynomial opens up a potential avenue to obtaining a new geometric categorification of the Jones polynomial. - oai:arXiv.org:2601.08805v2 - math.GT - math.SG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Cristina Ana-Maria Anghel, Andr\'as Juh\'asz - - - Positive Lyapunov Exponents versus Integrability in Random Conservative Dynamics - https://arxiv.org/abs/2601.08814 - arXiv:2601.08814v2 Announce Type: replace -Abstract: We study random dynamical systems generated by volume-preserving piecewise $C^{1}$ maps. For this class of systems, we establish an invariance principle stating that if all Lyapunov exponents vanish, then there exists a measurable family of probability measures on the projective bundle that is invariant under the projective cocycle induced by the derivative. We apply this principle to two classes of random systems. First, we consider random additive perturbations of the billiard map associated with a strictly convex planar table on a surface of constant curvature. In this setting, we show that the Lyapunov exponents vanish almost everywhere if and only if the billiard table is a geodesic disk. Second, we study random additive perturbations of a standard map and prove that the Lyapunov exponents vanish almost everywhere if and only if the map is integrable. - oai:arXiv.org:2601.08814v2 - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Gianluigi Del Magno, Jo\~ao Lopes Dias, Jos\'e Pedro Gaiv\~ao - - - Two-dimensional Entanglement-assisted Quantum Quasi-cyclic Low-density Parity-check Codes - https://arxiv.org/abs/2601.08927 - arXiv:2601.08927v2 Announce Type: replace -Abstract: For any positive integer $g \ge 2$, we derive general condition for the existence of a $2g$-cycle in the Tanner graph of two-dimensional ($2$-D) classical quasi-cyclic (QC) low-density parity-check (LDPC) codes. Depending on whether $p$ is an odd prime or a composite number, we construct two distinct families of $2$-D classical QC-LDPC codes with girth $>4$ by stacking $p \times p \times p$ tensors. Furthermore, using generalized Behrend sequences, we propose an additional family of $2$-D classical QC-LDPC codes with girth $>6$, constructed via a similar tensor-stacking approach. All the proposed $2\text{-D}$ classical QC-LDPC codes exhibit an erasure correction capability of at least $p \times p$. Based on the constructed $2\text{-D}$ classical QC-LDPC codes, we derive two families of $2\text{-D}$ entanglement-assisted (EA) quantum low-density parity-check (QLDPC) codes. The first family of $2\text{-D}$ EA-QLDPC codes is obtained from a pair of $2\text{-D}$ classical QC-LDPC codes and is designed such that the unassisted part of the Tanner graph of the resulting EA-QLDPC code is free of $4$-cycles, while requiring only a single ebit to be shared across the quantum transceiver. The second family is constructed from a single $2\text{-D}$ classical QC-LDPC code whose Tanner graph is free from $4$-cycles. Moreover, the constructed EA-QLDPC codes inherit an erasure correction capability of $p \times p$, as the underlying classical codes possess the same erasure correction property. - oai:arXiv.org:2601.08927v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pavan Kumar, Shayan Srinivasa Garani - - - Solution to a Problem of Erd\H{o}s Concerning Distances and Points - https://arxiv.org/abs/2601.09102 - arXiv:2601.09102v2 Announce Type: replace -Abstract: In 1997, Erd\H{o}s asked whether for arbitrarily large $n$ there exists a set of $n$ points in $\mathbb{R}^2$ that determines $O(\frac{n}{\sqrt{\log n}})$ distinct distances while satisfying the local constraint that every 4-point subset determines at least 3 distinct pairwise distances. We construct $n$-point sets from an $m\times m$ box of the lattice $L = \{(x,\sqrt{2}y):x,y \in \mathbb{Z}\} \subset \mathbb{R}^2.$ The distinct distance bound follows from applying Bernays' theorem to the number of integers represented by the binary quadratic form $u^2 + 2v^2$. The local 4-point constraint is verified through Perucca's similarity classification of the six similarity types determining exactly two distances. - oai:arXiv.org:2601.09102v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Benjamin Grayzel - - - System Availability Optimization: Integrating Quantity Discounts and Delivery Lead Time Considerations - https://arxiv.org/abs/2601.09194 - arXiv:2601.09194v2 Announce Type: replace -Abstract: Purpose: The model allocates the system components orders to the suppliers to minimize the parts price and the system construction delay penalties and maximize the system availability during its use. It considers the quantity-based discount and variation of delivery lead time by ordering similar components. The model also reflects the prerequisite relationships between construction activities and calculates the delay penalty resulting from parts delivery lead time. Design/methodology/approach: This research presents a model for selecting suppliers of components of an industrial series-parallel multi-state system. A nonlinear binary mathematical program uses the Markov process results to select system components. It minimizes the total system construction phase costs, including the components' price, and the system construction delay penalty, and the system exploitation phase costs, including the system shutdown and working at half capacity. Findings: The model allocates the optimal orders for a typical industrial system's components, composing four elements. The proposed approach combines the nonlinear binary program and the Markov process results to optimize the system life cycle parameters, including the system construction cost and operational availability. Originality/value: Using the Markov chain results in binary nonlinear mathematical programming, this study attempts to strike the right balance between the construction phase's objectives and an industrial unit's operation phase. - oai:arXiv.org:2601.09194v2 - math.OC - cs.SY - eess.SY - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Zahra Sobhani, Mahmoud Shahrokhi - - - On strong law of large numbers for weakly stationary $\varphi$-mixing set-valued random variable sequences - https://arxiv.org/abs/2601.09197 - arXiv:2601.09197v2 Announce Type: replace -Abstract: In this paper we extend the notion of $\varphi$-mixing to set-valued random sequences that take values in the family of closed subsets of a Banach space. Several strong laws of large numbers for such $\varphi$-mixing sequences are stated and proved. Illustrative examples show that the hypotheses of the theorems are both natural and sharp. - oai:arXiv.org:2601.09197v2 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Luc Tri Tuyen - - - On Polar Coding with Feedback - https://arxiv.org/abs/2601.09222 - arXiv:2601.09222v2 Announce Type: replace -Abstract: In this work, we investigate the performance of polar codes with the assistance of feedback in communication systems. Although it is well known that feedback does not improve the capacity of memoryless channels, we show that the finite length performance of polar codes can be significantly improved as feedback enables genie-aided decoding and allows more flexible thresholds for the polar coding construction. To analyze the performance under the new construction, we then propose an accurate characterization of the distribution of the error event under the genie-aided successive cancellation (SC) decoding. This characterization can be also used to predict the performance of the standard SC decoding of polar codes with rates close to capacity. - oai:arXiv.org:2601.09222v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Ling Liu, Qi Cao, Liping Li, Baoming Bai - - - On generalized Tur\'{a}n problems for expansions - https://arxiv.org/abs/2601.09244 - arXiv:2601.09244v2 Announce Type: replace -Abstract: Given a graph $F$, the $r$-expansion $F^r$ of $F$ is the $r$-uniform hypergraph obtained from $F$ by inserting $r-2$ new distinct vertices in each edge of $F$. Given $r$-uniform hypergraphs $\mathcal{H}$ and $\mathcal{F}$, the generalized Tur\'{a}n number, denoted by $\textrm{ex}_r(n,\mathcal{H},\mathcal{F})$, is the maximum number of copies of $\mathcal{H}$ in an $n$-vertex $r$-uniform hypergraph that does not contain $\mathcal{F}$ as a subhypergraph. In the case where $r=2$ (i.e., the graph case), the study of generalized Tur\'{a}n problems was initiated by Alon and Shikhelman [\textit{J. Combin. Theory Series B.} 121 (2016) 146--172]. Motivated by their work, we systematically study generalized Tur\'{a}n problems for expansions and obtain several general and exact results. In particular, for the non-degenerate case, we determine the exact generalized Tur\'{a}n number for expansions of complete graphs, and establish the asymptotics of the generalized Tur\'{a}n number for expansions of the vertex-disjoint union of complete graphs. For the degenerate case, we establish the asymptotics of generalized Tur\'{a}n numbers for expansions of several classes of forests, including star forests, linear forests and star-path forests. - oai:arXiv.org:2601.09244v2 - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Junpeng Zhou, Xiamiao Zhao, Xiying Yuan - - - A Finite-Sample Strong Converse for Binary Hypothesis Testing via (Reverse) R\'enyi Divergence - https://arxiv.org/abs/2601.09550 - arXiv:2601.09550v2 Announce Type: replace -Abstract: This work investigates binary hypothesis testing between $H_0\sim P_0$ and $H_1\sim P_1$ in the finite-sample regime under asymmetric error constraints. By employing the ``reverse" R\'enyi divergence, we derive novel non-asymptotic bounds on the Type II error probability which naturally establish a strong converse result. Furthermore, when the Type I error is constrained to decay exponentially with a rate $c$, we show that the Type II error converges to 1 exponentially fast if $c$ exceeds the Kullback-Leibler divergence $D(P_1\|P_0)$, and vanishes exponentially fast if $c$ is smaller. Finally, we present numerical examples demonstrating that the proposed converse bounds strictly improve upon existing finite-sample results in the literature. - oai:arXiv.org:2601.09550v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Roberto Bruno, Adrien Vandenbroucque, Amedeo Roberto Esposito - - - Counting and Entropy Bounds for Structure-Avoiding Spatially-Coupled LDPC Constructions - https://arxiv.org/abs/2601.09674 - arXiv:2601.09674v2 Announce Type: replace -Abstract: Designing large coupling memory quasi-cyclic spatially-coupled LDPC (QC-SC-LDPC) codes with low error floors requires eliminating specific harmful substructures (e.g., short cycles) induced by edge spreading and lifting. Building on our work~\cite{r15} that introduced a Clique Lov\'asz Local Lemma (CLLL)-based design principle and a Moser--Tardos (MT)-type constructive approach, this work quantifies the size and structure of the feasible design space. Using the quantitative CLLL, we derive explicit lower bounds on the number of feasible edge-spreading and lifting assignments satisfying a given family of structure-avoidance constraints, and further obtain bounds on the number of non-equivalent solutions under row/column permutations. Moreover, via R\'enyi entropy bounds for the MT distribution, we provide a computable lower bound on the number of distinct solutions that the MT algorithm can output, giving a concrete diversity guarantee for randomized constructions. Specializations for eliminating 4-cycles yield closed-form bounds as functions of system parameters, offering a principled way to select the memory and lifting degree and to estimate the remaining search space. - oai:arXiv.org:2601.09674v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lei Huang - - - On the Leaky Private Information Retrieval with Side Information - https://arxiv.org/abs/2601.09960 - arXiv:2601.09960v2 Announce Type: replace -Abstract: This paper investigates the problem of Leaky Private Information Retrieval with Side Information (L-PIR-SI), providing a fundamental characterization of the trade-off among leaky privacy, side information, and download cost. We propose a unified probabilistic framework to design L-PIR-SI schemes under $\varepsilon$-differential privacy variants of both $W$-privacy and $(W, S)$-privacy. Explicit upper bounds on the download cost are derived, which strictly generalize existing results: our bounds recover the capacity of perfect PIR-SI as $\varepsilon \to 0$, and reduce to the known $\varepsilon$-leaky PIR rate in the absence of side information. Furthermore, we conduct a refined analysis of the privacy--utility trade-off at the scaling-law level, demonstrating that the leakage ratio exponent scales as $\mathcal{O}(\log \frac{K}{M + 1})$ under leaky $W$-privacy, and as $\mathcal{O}(\log K)$ under leaky $(W, S)$-privacy in the minimal non-trivial setting $M = 1$, where $K$ and $M$ denote the number of messages and the side information size, respectively. - oai:arXiv.org:2601.09960v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yingying Huangfu, Tian Bai - - - Sharp propagation of chaos in R\'enyi divergence - https://arxiv.org/abs/2601.10076 - arXiv:2601.10076v2 Announce Type: replace -Abstract: We establish sharp rates for propagation of chaos in R\'enyi divergences for interacting diffusion systems at stationarity. Building upon the entropic hierarchy established in Lacker (2023), we show that under strong isoperimetry and weak interaction conditions, one can achieve $\mathsf R_q(\mu^1 \,\lVert\, \pi) = \widetilde O(\frac{d q^2}{N^2})$ bounds on the $q$-R\'enyi divergence. - oai:arXiv.org:2601.10076v2 - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Matthew S. Zhang - - - Infinite-horizon controllability scores for linear time-invariant systems - https://arxiv.org/abs/2601.10260 - arXiv:2601.10260v2 Announce Type: replace -Abstract: We introduce a numerically stable reformulation of controllability scoring based on a scaled controllability Gramian, which remains reliably computable even for unstable systems. The resulting optimization problems define dynamics-aware network centrality measures, referred to as the volumetric controllability score (VCS) and the average energy controllability score (AECS). Building on this stable reformulation, we derive the corresponding infinite-horizon problems, develop an algorithm to solve them, and highlight computational advantages over their finite-horizon counterparts. Under suitable assumptions, we prove that the infinite-horizon VCS and AECS are unique and that the finite-horizon scores converge to them. We further show that VCS and AECS can differ markedly in this limit, because VCS enforces controllability of the full system, whereas AECS accounts only for the stable modes. Finally, numerical experiments on Laplacian dynamics illustrate this convergence. - oai:arXiv.org:2601.10260v2 - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kota Umezu, Kazuhiro Sato - - - Polymultiplicative maps associated with the algebra of Iterated Laurent series and the higher-dimensional Contou-Carrere Symbol - https://arxiv.org/abs/2601.10335 - arXiv:2601.10335v2 Announce Type: replace -Abstract: We study functorial polymultiplicative maps from the multiplicative group of the algebra of $n$-times iterated Laurent series over a commutative ring in $n+1$ variables into the multiplicative group of the ring. It is proven that if such a map is invariant under continuous automorphisms of this algebra, then it coincides, up to a sign, with an integer power of the $n$-dimensional Contou-Carr\`ere symbol. - oai:arXiv.org:2601.10335v2 - math.AG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Vladislav Levashev - - - A Hybrid Reliability--Weight Framework for Construction of Polar Codes - https://arxiv.org/abs/2601.10376 - arXiv:2601.10376v2 Announce Type: replace -Abstract: Polar codes are usually constructed by ranking synthetic bit-channels according to reliability, which guarantees capacity-achieving behavior but can yield poor low-weight spectra at short and moderate lengths. Recent algebraic results express the contribution of individual bit-channels to the multiplicities of minimum and near-minimum weight codewords in closed form. In this work we combine these insights into a mixed (reliability--weight) bit-channel ordering. We define a per-bit cost whose distance term is derived from orbit enumeration of minimum-weight codewords and scaled by a Bhattacharyya-type factor, and show that the resulting mixed construction minimises a truncated SC/ML union-bound surrogate within a class of decreasing monomial codes. We relate the mixed metric to error events in SCL decoding via a pruning/ML decomposition, and prove that mixed designs act as local perturbations of reliability-based constructions whose asymptotic impact vanishes as code-length approaches infinity. Numerical results for short and moderate lengths on BPSK-AWGN, implemented via Gaussian approximation and closed-form weight contributions, illustrate the trade-off between pure reliability-based and mixed constructions in terms of minimum distance, multiplicity, and union-bound approximations. All proofs are deferred to the appendices. - oai:arXiv.org:2601.10376v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Mohammad Rowshan, Vlad-Florin Dragoi - - - Positivity of the third Chern form for Griffiths positive vector bundles - https://arxiv.org/abs/2601.10424 - arXiv:2601.10424v2 Announce Type: replace -Abstract: In this paper, we prove the positivity of the double mixed discriminant associated with a positive linear map between spaces of third-order complex matrices, thereby settling the three-dimensional case of Finski's open problem. As an application, we obtain the weak positivity of the third Chern form for Griffiths positive vector bundles. Moreover, we show that all Schur forms are weakly positive for Griffiths positive vector bundles of rank three over complex threefolds. This yields a complete affirmative answer, in the case where both the rank and the dimension are three, to the question posed by Griffiths in 1969. - oai:arXiv.org:2601.10424v2 - math.AG - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xueyuan Wan - - - Energy-Efficient Probabilistic Semantic Communication Over Visible Light Networks With Rate Splitting - https://arxiv.org/abs/2601.10452 - arXiv:2601.10452v2 Announce Type: replace -Abstract: Visible light communication (VLC) is emerging as a key technology for future wireless communication systems due to its unique physical-layer advantages over traditional radio-frequency (RF)-based systems. However, its integration with higher-layer techniques, such as semantic communication, remains underexplored. This paper investigates the energy efficiency maximization problem in a resource-constrained VLC-based probabilistic semantic communication (PSCom) system. In the considered model, light-emitting diode (LED) transmitters perform semantic compression to reduce data size, which incurs additional computation overhead. The compressed semantic information is transmitted to the users for semantic inference using a shared knowledge base that requires periodic updates to ensure synchronization. In the PSCom system, the knowledge base is represented by probabilistic graphs. To enable simultaneous transmission of both knowledge and information data, rate splitting multiple access (RSMA) is employed. The optimization problem focuses on maximizing energy efficiency by jointly optimizing transmit beamforming, direct current (DC) bias, common rate allocation, and semantic compression ratio, while accounting for both communication and computation costs. To solve this problem, an alternating optimization algorithm based on successive convex approximation (SCA) and Dinkelbach method is developed. Simulation results demonstrate the effectiveness of the proposed approach. - oai:arXiv.org:2601.10452v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zhouxiang Zhao, Zhaohui Yang, Chen Zhu, Xin Tong, Zhaoyang Zhang - - - Basis-Spline Assisted Coded Computing: Strategies and Error Bounds - https://arxiv.org/abs/2601.10616 - arXiv:2601.10616v2 Announce Type: replace -Abstract: Coded computing has emerged as a key framework for addressing the impact of stragglers in distributed computation. While polynomial functions often admit exact recovery under existing coded computing schemes, non-polynomial functions require approximate reconstruction from a finite number of evaluations, posing significant challenges. Consequently, interpolation-based methods for non-polynomial coded computing have gained attention, with Berrut approximated coded computing emerging as a state-of-the-art approach. However, due to the global support of Berrut interpolants, the reconstruction accuracy degrades significantly as the number of stragglers increases. To address this challenge, we propose a coded computing framework based on cubic B-spline interpolation. In our approach, server-side function evaluations are reconstructed at the master using B-splines, exploiting their local support and smoothness properties to enhance stability and accuracy. We provide a systematic methodology for integrating B-spline interpolation into coded computing and derive theoretical bounds on approximation error for certain class of smooth functions. Our analysis demonstrates that the error bounds of our approach exhibit a faster decay with respect to the number of workers compared to the Berrut-based method. Experimental results also confirm that our method offers improved accuracy over Berrut-based methods for various smooth non-polynomial functions. - oai:arXiv.org:2601.10616v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rimpi Borah, J. Harshan, V. Lalitha - - - One-Shot Broadcast Joint Source-Channel Coding with Codebook Diversity - https://arxiv.org/abs/2601.10648 - arXiv:2601.10648v2 Announce Type: replace -Abstract: We study a one-shot joint source-channel coding setting where the source is encoded once and broadcast to $K$ decoders through independent channels. Success is predicated on at least one decoder recovering the source within a maximum distortion constraint. We find that in the one-shot regime, utilizing disjoint codebooks at each decoder yields a codebook diversity gain, distinct from the channel diversity gain that may be expected when several decoders observe independent realizations of the channel's output but share the same codebook. Coding schemes are introduced that leverage this phenomenon, where first- and second-order achievability bounds are derived via an adaptation of the Poisson matching lemma (Li and Anantharam, 2021) which allows for multiple decoders using disjoint codebooks. We further propose a hybrid coding scheme that partitions decoders into groups to optimally balance codebook and channel diversity. Numerical results on the binary symmetric channel demonstrate that the hybrid approach outperforms strategies where the decoders' codebooks are either fully shared or disjoint. - oai:arXiv.org:2601.10648v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Joseph Rowan, Buu Phan, Ashish Khisti - - - Perfect Secret Key Generation for a class of Hypergraphical Sources - https://arxiv.org/abs/2601.10697 - arXiv:2601.10697v2 Announce Type: replace -Abstract: Nitinawarat and Narayan proposed a perfect secret key generation scheme for the so-called \emph{pairwise independent network (PIN) model} by exploiting the combinatorial properties of the underlying graph, namely the spanning tree packing rate. This work considers a generalization of the PIN model where the underlying graph is replaced with a hypergraph, and makes progress towards designing similar perfect secret key generation schemes by exploiting the combinatorial properties of the hypergraph. - Our contributions are two-fold. We first provide a capacity achieving scheme for a complete $t$-uniform hypergraph on $m$ vertices by leveraging a packing of the complete $t$-uniform hypergraphs by what we refer to as star hypergraphs, and designing a scheme that gives $\binom{m-2}{t-2}$ bits of perfect secret key per star graph. Our second contribution is a 2-bit perfect secret key generation scheme for 3-uniform star hypergraphs whose projections are cycles. This scheme is then extended to a perfect secret key generation scheme for generic 3-uniform hypergraphs by exploiting star graph packing of 3-uniform hypergraphs and Hamiltonian packings of graphs. The scheme is then shown to be capacity achieving for certain classes of hypergraphs. - oai:arXiv.org:2601.10697v2 - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Manuj Mukherjee, Sagnik Chatterjee, Alhad Sethi - - - Invariant Algebraic $D$-Modules on Semisimple and General Linear Groups: Classification and Tensor Categories - https://arxiv.org/abs/2601.10934 - arXiv:2601.10934v2 Announce Type: replace -Abstract: We study finite-rank left-translation invariant algebraic $D$-modules on complex affine algebraic groups. Using the standard description of these objects as left-invariant flat algebraic connections on the trivial vector bundle, modulo algebraic gauge transformations, we recast the classification problem as an explicit moduli problem for constant connections. - Our main results treat the semisimple case and the general linear case. For a connected semisimple complex algebraic group, invariant $D$-modules are classified by representations of the finite central kernel of the simply connected cover, yielding an equivalence of tensor categories. For a general linear group, every invariant $D$-module is obtained by pullback along the determinant map, reducing the classification to the one-dimensional torus case and inducing a tensor equivalence with the corresponding invariant category on the torus. - oai:arXiv.org:2601.10934v2 - math.RT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Rudrendra Kashyap, Ruoxi Li - - - Exact Constraint Enforcement in Physics-Informed Extreme Learning Machines using Null-Space Projection Framework - https://arxiv.org/abs/2601.10999 - arXiv:2601.10999v2 Announce Type: replace -Abstract: Physics-informed extreme learning machines (PIELMs) typically impose boundary and initial conditions through penalty terms, yielding only approximate satisfaction that is sensitive to user-specified weights and can propagate errors into the interior solution. This work introduces Null-Space Projected PIELM (NP-PIELM), achieving exact constraint enforcement through algebraic projection in coefficient space. The method exploits the geometric structure of the admissible coefficient manifold, recognizing that it admits a decomposition through the null space of the boundary operator. By characterizing this manifold via a translation-invariant representation and projecting onto the kernel component, optimization is restricted to constraint-preserving directions, transforming the constrained problem into unconstrained least-squares where boundary conditions are satisfied exactly at discrete collocation points. This eliminates penalty coefficients, dual variables, and problem-specific constructions while preserving single-shot training efficiency. Numerical experiments on elliptic and parabolic problems including complex geometries and mixed boundary conditions validate the framework. - oai:arXiv.org:2601.10999v2 - math.NA - cs.LG - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Rishi Mishra, Smriti, Balaji Srinivasan, Sundararajan Natarajan, Ganapathy Krishnamurthi - - - Countable-Support Symmetric Iterations - https://arxiv.org/abs/2601.11008 - arXiv:2601.11008v2 Announce Type: replace -Abstract: We present a framework for iterating symmetric extensions with \emph{countable support}. Assuming the successor-step two-stage symmetric-system construction from the standard finite-support theory, we define the countable-support iteration and its induced automorphism groups, taking $\omega_1$-completions of the successor-stage symmetry filters. At limit stages of uncountable cofinality, countable supports are bounded and we use the resulting direct-limit presentation, and at limits of cofinality $\omega$ we use the inverse-limit presentation induced by restriction maps. We define canonical limit symmetry filters generated by head pullbacks from earlier stages: at limits of cofinality $>\omega$ we take the smallest normal filter containing these generators (and prove it is $\omega_1$--complete by stage-bounding), while at limits of cofinality $\omega$ we take the smallest normal $\omega_1$--complete filter extending the same generators. In either case the resulting limit filter is normal and $\omega_1$--complete. - Using this, we define the class of hereditarily symmetric names at each stage and prove closure under the operations required for the $ZF$ axioms. In particular, the resulting symmetric model satisfies $ZF$; for set-length stages over a $ZFC$ ground, the resulting symmetric model satisfies $DC=DC_\omega$. Finally, when the iteration template is first-order definable over a $GBC$ ground with Global Choice and sufficient class recursion (e.g.\ $GBC+\mathsf{ETR}$), the same scheme extends to class-length iterations, and the final symmetric model is obtained by evaluating hereditarily symmetric class-names. - oai:arXiv.org:2601.11008v2 - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Frank Gilson - - - Constructing Orthogonal Rational Function Vectors with an application in Rational Approximation - https://arxiv.org/abs/2601.11317 - arXiv:2601.11317v2 Announce Type: replace -Abstract: We present two algorithms for constructing orthonormal bases of rational function vectors with respect to a discrete inner product, and discuss how to use them for a rational approximation problem. Building on the pencil-based formulation of the inverse generalized eigenvalue problem by Van Buggenhout et al. (2022), we extend it to rational vectors of arbitrary length $k$, where the recurrence relations are represented by a pair of $k$-Hessenberg matrices, i.e., matrices with possibly $k$ nonzero subdiagonals. An updating algorithm based on similarity transformations using rotations and a Krylov-type algorithm related to the rational Arnoldi method are derived. The performance is demonstrated on the rational approximation of $\sqrt{z}$ on $[0,1]$, where the optimal lightning + polynomial convergence rate of Herremans, Huybrechs, and Trefethen (2023) is successfully recovered. This illustrates the robustness of the proposed methods for handling exponentially clustered poles near singularities. - oai:arXiv.org:2601.11317v2 - math.NA - cs.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Robbe Vermeiren - - - Efficient Channel Autoencoders for Wideband Communications leveraging Walsh-Hadamard interleaving - https://arxiv.org/abs/2601.11407 - arXiv:2601.11407v2 Announce Type: replace -Abstract: This paper investigates how end-to-end (E2E) channel autoencoders (AEs) can achieve energy-efficient wideband communications by leveraging Walsh-Hadamard (WH) interleaved converters. WH interleaving enables high sampling rate analog-digital conversion with reduced power consumption using an analog WH transformation. We demonstrate that E2E-trained neural coded modulation can transparently adapt to the WH-transceiver hardware without requiring algorithmic redesign. Focusing on the short block length regime, we train WH-domain AEs and benchmark them against standard neural and conventional baselines, including 5G Polar codes. We quantify the system-level energy tradeoffs among baseband compute, channel signal-to-noise ratio (SNR), and analog converter power. Our analysis shows that the proposed WH-AE system can approach conventional Polar code SNR performance within 0.14dB while consuming comparable or lower system power. Compared to the best neural baseline, WH-AE achieves, on average, 29% higher energy efficiency (in bit/J) for the same reliability. These findings establish WH-domain learning as a viable path to energy-efficient, high-throughput wideband communications by explicitly balancing compute complexity, SNR, and analog power consumption. - oai:arXiv.org:2601.11407v2 - cs.IT - eess.SP - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-sa/4.0/ - Cel Thys, Rodney Martinez Alonso, Sofie Pollin - - - Frame eversion and contextual geometric rigidity - https://arxiv.org/abs/2601.11455 - arXiv:2601.11455v2 Announce Type: replace -Abstract: We prove rigidity results describing contextually-constrained maps defined on Grassmannians and manifolds of ordered independent line tuples in finite-dimensional vector or Hilbert spaces. One statement in the spirit of the Fundamental Theorem of Projective Geometry classifies maps between full Grassmannians of two $n$-dimensional Hilbert spaces, $n\ge 3$, preserving dimension and lattice operations for pairs with commuting orthogonal projections, as precisely those induced by semilinear injections unique up to scaling. - In a different but related direction, denote the manifolds of ordered orthogonal (linearly-independent) $n$-tuples of lines in an $n$-dimensional Hilbert space $V$ by $\mathbb{F}^{\perp}(V)$ (respectively $\mathbb{F}(V)$) and, for partitions $\pi$ of the set $\{1..n\}$, call two tuples $\pi$-linked if the spans along $\pi$-blocks agree. A Wigner-style rigidity theorem proves that the symmetric maps $\mathbb{F}^{\perp}(\mathbb{C}^n)\to \mathbb{F}(\mathbb{C}^n)$, $n\ge 3$ respecting $\pi$-linkage are precisely those induced by semilinear injections, hence by linear or conjugate-linear maps if also assumed measurable. On the other hand, in the $\mathbb{F}(\mathbb{C}^n)$-defined analogue the only other possibility is a qualitatively new type of purely-contextual-global symmetry transforming a tuple $(\ell_i)_i$ of lines into $\left(\left(\bigoplus_{j\ne i}\ell_j\right)^{\perp}\right)_i$. - oai:arXiv.org:2601.11455v2 - math.FA - math-ph - math.CO - math.MP - math.OA - Wed, 21 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexandru Chirvasitu - - - A New Generation of Brain-Computer Interface Based on Riemannian Geometry - https://arxiv.org/abs/1310.8115 - arXiv:1310.8115v2 Announce Type: replace-cross -Abstract: Based on the cumulated experience over the past 25 years in the field of Brain-Computer Interface (BCI) we can now envision a new generation of BCI. Such BCIs will not require training; instead they will be smartly initialized using remote massive databases and will adapt to the user fast and effectively in the first minute of use. They will be reliable, robust and will maintain good performances within and across sessions. A general classification framework based on recent advances in Riemannian geometry and possessing these characteristics is presented. It applies equally well to BCI based on event-related potentials (ERP), sensorimotor (mu) rhythms and steady-state evoked potential (SSEP). The framework is very simple, both algorithmically and computationally. Due to its simplicity, its ability to learn rapidly (with little training data) and its good across-subject and across-session generalization, this strategy a very good candidate for building a new generation of BCIs, thus we hereby propose it as a benchmark method for the field. - oai:arXiv.org:1310.8115v2 - cs.HC - math.DG - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Marco Congedo, Alexandre Barachant, Anton Andreev - - - UVIP: Model-Free Approach to Evaluate Reinforcement Learning Algorithms - https://arxiv.org/abs/2105.02135 - arXiv:2105.02135v5 Announce Type: replace-cross -Abstract: Policy evaluation is an important instrument for the comparison of different algorithms in Reinforcement Learning (RL). However, even a precise knowledge of the value function $V^{\pi}$ corresponding to a policy $\pi$ does not provide reliable information on how far the policy $\pi$ is from the optimal one. We present a novel model-free upper value iteration procedure ({\sf UVIP}) that allows us to estimate the suboptimality gap $V^{\star}(x) - V^{\pi}(x)$ from above and to construct confidence intervals for \(V^\star\). Our approach relies on upper bounds to the solution of the Bellman optimality equation via the martingale approach. We provide theoretical guarantees for {\sf UVIP} under general assumptions and illustrate its performance on a number of benchmark RL problems. - oai:arXiv.org:2105.02135v5 - cs.LG - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Denis Belomestny, Ilya Levin, Alexey Naumov, Sergey Samsonov - - - Explicit Non-Abelian Gerbes with Connections - https://arxiv.org/abs/2203.00092 - arXiv:2203.00092v4 Announce Type: replace-cross -Abstract: We define the notion of adjustment for strict Lie 2-groups and provide the complete cocycle description for non-Abelian gerbes with connections whose structure 2-group is an adjusted 2-group. Most importantly, we depart from the common fake-flat connections and employ adjusted connections. This is an important generalisation that is needed for physical applications especially in the context of supergravity. We give a number of explicit examples; in particular, we lift the spin structure on $S^4$, corresponding to an instanton-anti-instanton pair, to a string structure, a 2-group bundle with connection. We also outline how categorified forms of Bogomolny monopoles known as self-dual strings can be obtained via a Penrose-Ward transform of string bundles over twistor space. - oai:arXiv.org:2203.00092v4 - hep-th - math-ph - math.DG - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1088/1751-8121/ae2e60 - J. Phys. A 59 (2026) 035201 - Dominik Rist, Christian Saemann, Martin Wolf - - - A Note on Comparator-Overdrive-Delay Conditioning for Current-Mode Control - https://arxiv.org/abs/2206.09340 - arXiv:2206.09340v3 Announce Type: replace-cross -Abstract: Comparator-overdrive-delay conditioning is a new control conditioning approach for high-frequency current-mode control. No existing literature rigorously studies the effect of the comparator overdrive delay on the current-mode control. The results in this paper provide insights into the mechanism of comparator-overdrive-delay conditioning. - oai:arXiv.org:2206.09340v3 - eess.SY - cs.SY - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xiaofan Cui, Guanyu Qian, Al-Thaddeus Avestruz - - - Machine Learning Decoder for 5G NR PUCCH Format 0 - https://arxiv.org/abs/2209.07861 - arXiv:2209.07861v2 Announce Type: replace-cross -Abstract: 5G cellular systems depend on the timely exchange of feedback control information between the user equipment and the base station. Proper decoding of this control information is necessary to set up and sustain high throughput radio links. This paper makes the first attempt at using Machine Learning techniques to improve the decoding performance of the Physical Uplink Control Channel Format 0. We use fully connected neural networks to classify the received samples based on the uplink control information content embedded within them. The trained neural network, tested on real-time wireless captures, shows significant improvement in accuracy over conventional DFT-based decoders, even at low SNR. The obtained accuracy results also demonstrate conformance with 3GPP requirements. - oai:arXiv.org:2209.07861v2 - cs.NI - cs.IT - cs.LG - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by-sa/4.0/ - 10.1109/NCC56989.2023.10067950 - Anil Kumar Yerrapragada, Jeeva Keshav S, Ankit Gautam, Radha Krishna Ganti - - - Two convergent NPA-like hierarchies for the quantum bilocal scenario - https://arxiv.org/abs/2210.09065 - arXiv:2210.09065v5 Announce Type: replace-cross -Abstract: Characterising the correlations that arise from locally measuring a single part of a joint quantum system is one of the main problems of quantum information theory. The seminal work [M. Navascu\'es et al., New J. Phys. 10, 073013 (2008)], known as the Navascu\'es-Pironio-Ac\'in (NPA) hierarchy, reformulated this question as a polynomial optimisation problem over noncommutative variables and proposed a convergent hierarchy of necessary conditions, each testable using semidefinite programming. More recently, the problem of characterising the quantum network correlations, which arise when locally measuring several independent quantum systems distributed in a network, has received considerable interest. Several generalisations of the NPA hierarchy, such as the scalar extension [A. Pozas-Kerstjens et al., Phys. Rev. Lett. 123, 140503 (2019)], were introduced while their converging sets remain unknown. In this work, we introduce a new bilocal factorisation NPA hierarchy, prove its equivalence to a modified bilocal scalar extension NPA hierarchy, and characterise its convergence in the case of the simplest network, the bilocal scenario. We further explore its relations with the other known generalisations. - oai:arXiv.org:2210.09065v5 - quant-ph - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1063/5.0211008 - J. Math. Phys. 67, 012203 (2026) - Marc-Olivier Renou, Xiangling Xu, Laurens T. Ligthart - - - On the Global Convergence of Risk-Averse Natural Policy Gradient Methods with Expected Conditional Risk Measures - https://arxiv.org/abs/2301.10932 - arXiv:2301.10932v5 Announce Type: replace-cross -Abstract: Risk-sensitive reinforcement learning (RL) has become a popular tool for controlling the risk of uncertain outcomes and ensuring reliable performance in highly stochastic sequential decision-making problems. While it has been shown that policy gradient methods can find globally optimal policies in the risk-neutral setting, it remains unclear if the risk-averse variants enjoy the same global convergence guarantees. In this paper, we consider a class of dynamic time-consistent risk measures, named Expected Conditional Risk Measures (ECRMs), and derive natural policy gradient (NPG) updates for ECRMs-based RL problems. We provide global optimality and iteration complexity of the proposed risk-averse NPG algorithm with softmax parameterization and entropy regularization under both exact and inexact policy evaluation. Furthermore, we test our risk-averse NPG algorithm on a stochastic Cliffwalk environment to demonstrate the efficacy of our method. - oai:arXiv.org:2301.10932v5 - cs.LG - math.OC - stat.ML - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xian Yu, Lei Ying - - - Optimal Conditional Inference in Adaptive Experiments - https://arxiv.org/abs/2309.12162 - arXiv:2309.12162v2 Announce Type: replace-cross -Abstract: We study batched bandit experiments and consider the problem of inference conditional on the realized stopping time, assignment probabilities, and target parameter, where all of these may be chosen adaptively using information up to the last batch of the experiment. Absent further restrictions on the experiment, we show that inference using only the results of the last batch is optimal. When the adaptive aspects of the experiment are known to be location-invariant, in the sense that they are unchanged when we shift all batch-arm means by a constant, we show that there is additional information in the data, captured by one additional linear function of the batch-arm means. In the more restrictive case where the stopping time, assignment probabilities, and target parameter are known to depend on the data only through a collection of polyhedral events, we derive computationally tractable and optimal conditional inference procedures. - oai:arXiv.org:2309.12162v2 - stat.ME - cs.LG - econ.EM - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Jiafeng Chen, Isaiah Andrews - - - Hidden Minima in Two-Layer ReLU Networks - https://arxiv.org/abs/2312.16819 - arXiv:2312.16819v4 Announce Type: replace-cross -Abstract: We consider the optimization problem associated with training two-layer ReLU networks with \(d\) inputs under the squared loss, where the labels are generated by a target network. Recent work has identified two distinct classes of infinite families of minima: one whose training loss vanishes in the high-dimensional limit, and another whose loss remains bounded away from zero. The latter family is empirically avoided by stochastic gradient descent, hence \emph{hidden}, motivating the search for analytic criteria that distinguish hidden from non-hidden minima. A key challenge is that prior analyses have shown the Hessian spectra at hidden and non-hidden minima to coincide up to terms of order \(O(d^{-1/2})\), seemingly limiting the discriminative power of spectral methods. We therefore take a different route, studying instead certain curves along which the loss is locally minimized. Our main result shows that arcs emanating from hidden minima exhibit distinctive structural and symmetry properties, arising precisely from \(\Omega(d^{-1/2})\) eigenvalue contributions that are absent from earlier analyses. - oai:arXiv.org:2312.16819v4 - cs.LG - math.OC - stat.ML - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yossi Arjevani - - - AlphaMapleSAT: An MCTS-based Cube-and-Conquer SAT Solver for Hard Combinatorial Problems - https://arxiv.org/abs/2401.13770 - arXiv:2401.13770v2 Announce Type: replace-cross -Abstract: This paper introduces AlphaMapleSAT, a Cube-and-Conquer (CnC) parallel SAT solver that integrates Monte Carlo Tree Search (MCTS) with deductive feedback to efficiently solve challenging combinatorial SAT problems. Traditional lookahead cubing methods, used by solvers such as March, limit their search depth to reduce overhead often resulting in suboptimal partitions. By contrast, AlphaMapleSAT performs a deeper MCTS search guided by deductive rewards from SAT solvers. This approach enables informed exploration of the cubing space while keeping cubing costs low. We demonstrate the efficacy of our technique via extensive evaluations against the widely used and established March cubing solver on three well-known challenging combinatorial benchmarks, including the minimum Kochen-Specker (KS) problem from quantum mechanics, the Murty-Simon Conjecture, and the Ramsey problems from extremal graph theory. We compare AlphaMapleSAT against March using different types of conquering solvers such as SAT Modulo Symmetries (SMS) and SAT+CAS, both built on top of the CaDiCaL SAT solver. We show that in all cases, there is a speedup in elapsed real time (wall clock time) ranging from 1.61x to 7.57x on a 128 core machine for the above-mentioned problems. We also perform cube-level and parallel scaling analysis over 32, 64, and 128 cores, which shows that AlphaMapleSAT outperforms March on all these settings. Our results show that deductively-guided MCTS search technique for cubing in CnC solvers can significantly outperform March on hard combinatorial problems. - oai:arXiv.org:2401.13770v2 - cs.AI - math.CO - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Piyush Jha, Zhengyu Li, Zhengyang Lu, Raymond Zeng, Curtis Bright, Vijay Ganesh - - - Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Loss - https://arxiv.org/abs/2402.00152 - arXiv:2402.00152v4 Announce Type: replace-cross -Abstract: Constructing the architecture of a neural network is a challenging pursuit for the machine learning community, and the dilemma of whether to go deeper or wider remains a persistent question. This paper explores a comparison between deeper neural networks (DeNNs) with a flexible number of layers and wider neural networks (WeNNs) with limited hidden layers, focusing on their optimal generalization error in Sobolev losses. Analytical investigations reveal that the architecture of a neural network can be significantly influenced by various factors, including the number of sample points, parameters within the neural networks, and the regularity of the loss function. Specifically, a higher number of parameters tends to favor WeNNs, while an increased number of sample points and greater regularity in the loss function lean towards the adoption of DeNNs. We ultimately apply this theory to address partial differential equations using deep Ritz and physics-informed neural network (PINN) methods, guiding the design of neural networks. - oai:arXiv.org:2402.00152v4 - cs.LG - cs.NA - math.NA - stat.ML - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yahong Yang, Juncai He - - - On Dirac equations on phase spaces - https://arxiv.org/abs/2402.06404 - arXiv:2402.06404v3 Announce Type: replace-cross -Abstract: We consider Dirac equations on relativistic phase spaces $T^*{\mathbb R}^{p-1,1}$, where ${\mathbb R}^{p-1,1}$ is Minkowski space with $p=2,4$. We use the geometric quantization approach in which the wave functions are polarized sections of a complex line bundle $L_{\sf{v}}$ over $T^*{\mathbb R}^{p-1,1}$. The covariant derivatives with connection $A_{\sf{vac}}$ in this bundle define canonical commutation relations. Fermions are charged with respect to the field $A_{\sf{vac}}$, so lifting the Dirac equations from space-time ${\mathbb R}^{p-1,1}$ to phase space $T^*{\mathbb R}^{p-1,1}$ results in their solutions being localized in the space ${\mathbb R}^{p-1}$ or in space-time ${\mathbb R}^{p-1,1}$. We describe the explicit form of these solutions. - oai:arXiv.org:2402.06404v3 - hep-th - math-ph - math.MP - quant-ph - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexander D. Popov - - - Reflexive graph lenses in univalent foundations - https://arxiv.org/abs/2404.07854 - arXiv:2404.07854v2 Announce Type: replace-cross -Abstract: Martin-L\"of's identity types provide a generic (albeit opaque) notion of identification or "equality" between any two elements of the same type, embodied in a canonical reflexive graph structure $(=_A, \mathbf{refl})$ on any type $A$. The miracle of Voevodsky's univalence principle is that it ensures, for essentially any naturally occurring structure in mathematics, that this the resultant notion of identification is equivalent to the type of isomorphisms in the category of such structures. Characterisations of this kind are not automatic and must be established one-by-one; to this end, several authors have employed reflexive graphs and displayed reflexive graphs to organise the characterisation of identity types. We contribute reflexive graph lenses, a new family of intermediate abstractions lying between families of reflexive graphs and displayed reflexive graphs that simplifies the characterisation of identity types for complex structures. Every reflexive graph lens gives rise to a (more complicated) displayed reflexive graph, and our experience suggests that many naturally occurring displayed reflexive graphs arise in this way. Evidence for the utility of reflexive graph lenses is given by means of several case studies, including the theory of reflexive graphs itself as well as that of polynomial type operators. Finally, we exhibit an equivalence between the type of reflexive graph fibrations and the type of univalent reflexive graph lenses. - oai:arXiv.org:2404.07854v2 - cs.LO - math.CT - math.LO - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Jonathan Sterling - - - Asymmetric canonical correlation analysis of Riemannian and high-dimensional data - https://arxiv.org/abs/2404.11781 - arXiv:2404.11781v3 Announce Type: replace-cross -Abstract: In this paper, we introduce a novel statistical model for the integrative analysis of Riemannian-valued functional data and high-dimensional data. We apply this model to explore the dependence structure between each subject's dynamic functional connectivity -- represented by a temporally indexed collection of positive definite covariance matrices -- and high-dimensional data representing lifestyle, demographic, and psychometric measures. Specifically, we employ a reformulation of canonical correlation analysis that enables efficient control of the complexity of the functional canonical directions using tangent space sieve approximations. Additionally, we enforce an interpretable group structure on the high-dimensional canonical directions via a sparsity-promoting penalty. The proposed method shows improved empirical performance over alternative approaches and comes with theoretical guarantees. Its application to data from the Human Connectome Project reveals a dominant mode of covariation between dynamic functional connectivity and lifestyle, demographic, and psychometric measures. This mode aligns with results from static connectivity studies but reveals a unique temporal non-stationary pattern that such studies fail to capture. - oai:arXiv.org:2404.11781v3 - stat.ME - math.ST - stat.AP - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - 10.1214/25-EJS2468 - Electron. J. Statist. 19 (2) 6077 - 6102, 2025 - James Buenfil, Eardi Lila - - - Non-hyperbolic 3-manifolds and 3D field theories for 2D Virasoro minimal models - https://arxiv.org/abs/2405.16377 - arXiv:2405.16377v3 Announce Type: replace-cross -Abstract: Using 3D-3D correspondence, we construct 3D dual bulk field theories for general Virasoro minimal models $M(P,Q)$. These theories correspond to Seifert fiber spaces $S^2 ((P,P-R),(Q,S),(3,1))$ with two integers $(R,S)$ satisfying $PS-QR =1$. In the unitary case, where $|P-Q|=1$, the bulk theory has a mass gap and flows to a unitary topological field theory (TQFT) in the IR, which is expected to support the chiral Virasoro minimal model at the boundary under an appropriate boundary condition. For the non-unitary case, where $|P-Q|>1$, the bulk theory flows to a 3D $\mathcal{N}=4$ rank-0 superconformal field theory, whose topologically twisted theory supports the chiral minimal model at the boundary. We also provide a concrete field theory description of the 3D bulk theory using $T[SU(2)]$ theories. Our proposals are supported by various consistency checks using 3D-3D relations and direct computations of various partition functions. - oai:arXiv.org:2405.16377v3 - hep-th - math.GT - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Dongmin Gang, Heesu Kang, Seongmin Kim - - - Few for Many: Tchebycheff Set Scalarization for Many-Objective Optimization - https://arxiv.org/abs/2405.19650 - arXiv:2405.19650v3 Announce Type: replace-cross -Abstract: Multi-objective optimization can be found in many real-world applications where some conflicting objectives can not be optimized by a single solution. Existing optimization methods often focus on finding a set of Pareto solutions with different optimal trade-offs among the objectives. However, the required number of solutions to well approximate the whole Pareto optimal set could be exponentially large with respect to the number of objectives, which makes these methods unsuitable for handling many optimization objectives. In this work, instead of finding a dense set of Pareto solutions, we propose a novel Tchebycheff set scalarization method to find a few representative solutions (e.g., 5) to cover a large number of objectives (e.g., $>100$) in a collaborative and complementary manner. In this way, each objective can be well addressed by at least one solution in the small solution set. In addition, we further develop a smooth Tchebycheff set scalarization approach for efficient optimization with good theoretical guarantees. Experimental studies on different problems with many optimization objectives demonstrate the effectiveness of our proposed method. - oai:arXiv.org:2405.19650v3 - cs.LG - cs.AI - cs.NE - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xi Lin, Yilu Liu, Xiaoyuan Zhang, Fei Liu, Zhenkun Wang, Qingfu Zhang - - - U-learning for Prediction Inference via Combinatory Multi-Subsampling: With Applications to LASSO and Neural Networks - https://arxiv.org/abs/2407.15301 - arXiv:2407.15301v2 Announce Type: replace-cross -Abstract: Epigenetic aging clocks play a pivotal role in estimating an individual's biological age through the examination of DNA methylation patterns at numerous CpG (Cytosine-phosphate-Guanine) sites within their genome. However, making valid inferences on predicted epigenetic ages, or more broadly, on predictions derived from high-dimensional inputs, presents challenges. We introduce a novel U-learning approach via combinatory multi-subsampling for making ensemble predictions and constructing confidence intervals for predictions of continuous outcomes when traditional asymptotic methods are not applicable. More specifically, our approach conceptualizes the ensemble estimators within the framework of generalized U-statistics and invokes the H\'ajek projection for deriving the variances of predictions and constructing confidence intervals with valid conditional coverage probabilities. We apply our approach to two commonly used predictive algorithms, Lasso and deep neural networks (DNNs), and illustrate the validity of inferences with extensive numerical studies. We have applied these methods to predict the DNA methylation age (DNAmAge) of patients with various health conditions, aiming to accurately characterize the aging process and potentially guide anti-aging interventions. - oai:arXiv.org:2407.15301v2 - stat.ML - cs.LG - math.ST - q-bio.QM - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zhe Fei, Yi Li - - - Gibbs Sampling gives Quantum Advantage at Constant Temperatures with O(1)-Local Hamiltonians - https://arxiv.org/abs/2408.01516 - arXiv:2408.01516v4 Announce Type: replace-cross -Abstract: Sampling from Gibbs states -- states corresponding to system in thermal equilibrium -- has recently been shown to be a task for which quantum computers are expected to achieve super-polynomial speed-up compared to classical computers, provided the locality of the Hamiltonian increases with the system size (Bergamaschi et al., arXiv: 2404.14639). We extend these results to show that this quantum advantage still occurs for Gibbs states of Hamiltonians with O(1)-local interactions at constant temperature by showing classical hardness-of-sampling and demonstrating such Gibbs states can be prepared efficiently using a quantum computer. In particular, we show hardness-of-sampling is maintained even for 5-local Hamiltonians on a 3D lattice. We additionally show that the hardness-of-sampling is robust when we are only able to make imperfect measurements. - oai:arXiv.org:2408.01516v4 - quant-ph - cond-mat.other - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Joel Rajakumar, James D. Watson - - - Fast-forwarding quantum algorithms for linear dissipative differential equations - https://arxiv.org/abs/2410.13189 - arXiv:2410.13189v2 Announce Type: replace-cross -Abstract: We establish improved complexity estimates of quantum algorithms for linear dissipative ordinary differential equations (ODEs) and show that the time dependence can be fast-forwarded to be sub-linear. Specifically, we show that a quantum algorithm based on truncated Dyson series can prepare history states of dissipative ODEs up to time $T$ with cost $\widetilde{\mathcal{O}}(\log(T) (\log(1/\epsilon))^2 )$, which is an exponential speedup over the best previous result. For final state preparation at time $T$, we show that its complexity is $\widetilde{\mathcal{O}}(\sqrt{T} (\log(1/\epsilon))^2 )$, achieving a polynomial speedup in $T$. We also analyze the complexity of simpler lower-order quantum algorithms, such as the forward Euler method and the trapezoidal rule, and find that even lower-order methods can still achieve $\widetilde{\mathcal{O}}(\sqrt{T})$ cost with respect to time $T$ for preparing final states of dissipative ODEs. As applications, we show that quantum algorithms can simulate dissipative non-Hermitian quantum dynamics and heat processes with fast-forwarded complexity sub-linear in time. - oai:arXiv.org:2410.13189v2 - quant-ph - cs.NA - math.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Dong An, Akwum Onwunta, Gengzhi Yang - - - Simultaneous Inference in Multiple Matrix-Variate Graphs for High-Dimensional Neural Recordings - https://arxiv.org/abs/2410.15530 - arXiv:2410.15530v2 Announce Type: replace-cross -Abstract: We study simultaneous inference for multiple matrix-variate Gaussian graphical models in high-dimensional settings. Such models arise when spatiotemporal data are collected across multiple sample groups or experimental sessions, where each group is characterized by its own graphical structure but shares common sparsity patterns. A central challenge is to conduct valid inference on collections of graph edges while efficiently borrowing strength across groups under both high-dimensionality and temporal dependence. We propose a unified framework that combines joint estimation via group penalized regression with a high-dimensional Gaussian approximation bootstrap to enable global testing of edge subsets of arbitrary size. The proposed procedure accommodates temporally dependent observations and avoids naive pooling across heterogeneous groups. We establish theoretical guarantees for the validity of the simultaneous tests under mild conditions on sample size, dimensionality, and non-stationary autoregressive temporal dependence, and show that the resulting tests are nearly optimal in terms of the testable region boundary. The method relies only on convex optimization and parametric bootstrap, making it computationally tractable. Simulation studies and a neural recording example illustrate the efficacy of the proposed approach. - oai:arXiv.org:2410.15530v2 - stat.ME - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Zongge Liu, Heejong Bong, Zhao Ren, Matthew A. Smith, Robert E. Kass - - - An Adaptive Online Smoother with Closed-Form Solutions and Information-Theoretic Lag Selection for Conditional Gaussian Nonlinear Systems - https://arxiv.org/abs/2411.05870 - arXiv:2411.05870v2 Announce Type: replace-cross -Abstract: Data assimilation (DA) combines partial observations with dynamical models to improve state estimation. Filter-based DA uses only past and present data and is the prerequisite for real-time forecasts. Smoother-based DA exploits both past and future observations. It aims to fill in missing data, provide more accurate estimations, and develop high-quality datasets. However, the standard smoothing procedure requires using all historical state estimations, which is storage-demanding, especially for high-dimensional systems. This paper develops an adaptive-lag online smoother for a large class of complex dynamical systems with strong nonlinear and non-Gaussian features, which has important applications to many real-world problems. The adaptive lag allows the utilization of observations only within a nearby window, thus reducing computational complexity and storage needs. Online lag adjustment is essential for tackling turbulent systems, where temporal autocorrelation varies significantly over time due to intermittency, extreme events, and nonlinearity. Based on the uncertainty reduction in the estimated state, an information criterion is developed to systematically determine the adaptive lag. Notably, the mathematical structure of these systems facilitates the use of closed analytic formulae to calculate the online smoother and adaptive lag, avoiding empirical tunings as in ensemble-based DA methods. The adaptive online smoother is applied to studying three important scientific problems. First, it helps detect online causal relationships between state variables. Second, the advantage of reduced computational storage expenditure is illustrated via Lagrangian DA, a high-dimensional nonlinear problem. Finally, the adaptive smoother advances online parameter estimation with partial observations, emphasizing the role of the observed extreme events in accelerating convergence. - oai:arXiv.org:2411.05870v2 - eess.SY - cs.SY - math.DS - math.PR - physics.data-an - stat.ME - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Marios Andreou, Nan Chen, Yingda Li - - - Quantum particle in the wrong box (or: the perils of finite-dimensional approximations) - https://arxiv.org/abs/2412.15889 - arXiv:2412.15889v3 Announce Type: replace-cross -Abstract: When numerically simulating the unitary time evolution of an infinite-dimensional quantum system, one is usually led to treat the Hamiltonian $H$ as an "infinite-dimensional matrix" by expressing it in some orthonormal basis of the Hilbert space, and then truncate it to some finite dimensions. However, the solutions of the Schr\"odinger equations generated by the truncated Hamiltonians need not converge, in general, to the solution of the Schr\"odinger equation corresponding to the actual Hamiltonian. - In this paper we demonstrate that, under mild assumptions, they converge to the solution of the Schr\"odinger equation generated by a specific Hamiltonian which crucially depends on the particular choice of basis: the Friedrichs extension of the restriction of $H$ to the space of finite linear combinations of elements of the basis. Importantly, this is generally different from $H$ itself; in all such cases, numerical simulations will unavoidably reproduce the wrong dynamics in the limit, and yet there is no numerical test that can reveal this failure, unless one has the analytical solution to compare with. - As a practical demonstration of such results, we consider the quantum particle in the box, and we show that, for a wide class of bases (which include associated Legendre polynomials as a concrete example) the dynamics generated by the truncated Hamiltonians will always converge to the one corresponding to the particle with Dirichlet boundary conditions, regardless the initial choice of boundary conditions. Other such examples are discussed. - oai:arXiv.org:2412.15889v3 - quant-ph - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Felix Fischer, Daniel Burgarth, Davide Lonigro - - - Optimization Insights into Deep Diagonal Linear Networks - https://arxiv.org/abs/2412.16765 - arXiv:2412.16765v3 Announce Type: replace-cross -Abstract: Gradient-based methods successfully train highly overparameterized models in practice, even though the associated optimization problems are markedly nonconvex. Understanding the mechanisms that make such methods effective has become a central problem in modern optimization. To investigate this question in a tractable setting, we study Deep Diagonal Linear Networks. These are multilayer architectures with a reparameterization that preserves convexity in the effective parameter, while inducing a nontrivial geometry in the optimization landscape. Under mild initialization conditions, we show that gradient flow on the layer parameters induces a mirror-flow dynamic in the effective parameter space. This structural insight yields explicit convergence guarantees, including exponential decay of the loss under a Polyak-Lojasiewicz condition, and clarifies how the parametrization and initialization scale govern the training speed. Overall, our results demonstrate that deep diagonal over parameterizations, despite their apparent complexity, can endow standard gradient methods with well-behaved and interpretable optimization dynamics. - oai:arXiv.org:2412.16765v3 - cs.LG - math.OC - stat.ML - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Hippolyte Labarri\`ere, Cesare Molinari, Lorenzo Rosasco, Cristian Vega, Silvia Villa - - - A non-semisimple non-invertible symmetry - https://arxiv.org/abs/2412.19635 - arXiv:2412.19635v2 Announce Type: replace-cross -Abstract: We investigate the action of a non-invertible symmetry on spins chains whose topological lines are labelled by representations of the four-dimensional Taft algebra. The main peculiarity of this symmetry is the existence of junctions between distinct indecomposable lines. Sacrificing Hermiticity, we construct several symmetric, frustration-free, gapped Hamiltonians with real spectra and analyse their ground state subspaces. Our study reveals two intriguing phenomena. First, we identify a smooth path of gapped symmetric Hamiltonians whose ground states transform inequivalently under the symmetry. Second, we find a model where a product state and the so-called W state spontaneously break the symmetry, and propose an explanation for the indistinguishability of these two states in the infinite-volume limit in terms of the symmetry category. - oai:arXiv.org:2412.19635v2 - cond-mat.str-el - hep-th - math-ph - math.MP - quant-ph - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1103/nbf9-ywmd - Clement Delcamp, Edmund Heng, Matthew Yu - - - Prior distributions for structured semi-orthogonal matrices - https://arxiv.org/abs/2501.10263 - arXiv:2501.10263v2 Announce Type: replace-cross -Abstract: Statistical models for multivariate data often include a semi-orthogonal matrix parameter. In many applications, there is reason to expect that the semi-orthogonal matrix parameter satisfies a structural assumption such as sparsity or smoothness. From a Bayesian perspective, these structural assumptions should be incorporated into an analysis through the prior distribution. In this work, we introduce a general approach to constructing prior distributions for structured semi-orthogonal matrices that leads to tractable posterior inference via parameter-expanded Markov chain Monte Carlo. We draw on recent results from random matrix theory to establish a theoretical basis for the proposed approach. We then introduce specific prior distributions for incorporating sparsity or smoothness and illustrate their use through applications to biological and oceanographic data. - oai:arXiv.org:2501.10263v2 - stat.ME - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Michael Jauch, Marie-Christine D\"uker, Peter Hoff - - - A Novel Approach to the Initial Value Problem with a complete validated algorithm - https://arxiv.org/abs/2502.00503 - arXiv:2502.00503v4 Announce Type: replace-cross -Abstract: We consider the first order autonomous differential equation (ODE) ${\bf x}'={\bf f}({\bf x})$ where ${\bf f}: {\mathbb R}^n\to{\mathbb R}^n$ is locally Lipschitz. For ${\bf x}_0\in{\mathbb R}^n$ and $h>0$, the initial value problem (IVP) for $({\bf f},{\bf x}_0,h)$ is to determine if there is a unique solution, i.e., a function ${\bf x}:[0,h]\to{\mathbb R}^n$ that satisfies the ODE with ${\bf x}(0)={\bf x}_0$. Write ${\bf x} ={\tt IVP}_{\bf f}({\bf x}_0,h)$ for this unique solution. - We pose a corresponding computational problem, called the End Enclosure Problem: given $({\bf f},B_0,h,\varepsilon_0)$ where $B_0\subseteq{\mathbb R}^n$ is a box and $\varepsilon_0>0$, to compute a pair of non-empty boxes $(\underline{B}_0,B_1)$ such that $\underline{B}_0\subseteq B_0$, width of $B_1$ is $<\varepsilon_0$, and for all ${\bf x}_0\in \underline{B}_0$, ${\bf x}={\tt IVP}_{\bf f}({\bf x}_0,h)$ exists and ${\bf x}(h)\in B_1$. We provide a complete validated algorithm for this problem. Under the assumption (promise) that for all ${\bf x}_0\in B_0$, ${\tt IVP}_{\bf f}({\bf x}_0,h)$ exists, we prove the halting of our algorithm. This is the first halting algorithm for IVP problems in such a general setting. - We also introduce novel techniques for subroutines such as StepA and StepB, and a scaffold datastructure to support our End Enclosure algorithm. Among the techniques are new ways refine full- and end-enclosures based on a {\bf radical transform} combined with logarithm norms. Our preliminary implementation and experiments show considerable promise, and compare well with current validated algorithms. - oai:arXiv.org:2502.00503v4 - cs.SC - cs.NA - math.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bingwei Zhang, Chee Yap - - - On dimensions of (2+1)D abelian bosonic topological systems on non-orientable manifolds - https://arxiv.org/abs/2502.13532 - arXiv:2502.13532v2 Announce Type: replace-cross -Abstract: We give a framework to describe abelian bosonic topological systems with parity symmetry on a torus in terms of the projective representation of $GL(2,\mathbb{Z})$. However, this information alone does not guarantee that we can assign Hilbert spaces to non-orientable surfaces in a way compatible with the gluing axiom of topological quantum field theory. Here, we show that we may assign Hilbert spaces with integer dimensions to non-orientable surfaces in the case of abelian bosonic topological systems with time-reversal symmetry, which can be seen as a necessary condition for the existence of topological quantum field theories. - oai:arXiv.org:2502.13532v2 - hep-th - cond-mat.str-el - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1093/ptep/ptaf056 - Prog. Theor. Exp. Phys. 2025, 053B01 (2025) - Ippo Orii - - - Quasi Zigzag Persistence: A Topological Framework for Analyzing Time-Varying Data - https://arxiv.org/abs/2502.16049 - arXiv:2502.16049v3 Announce Type: replace-cross -Abstract: In this paper, we propose Quasi Zigzag Persistent Homology (QZPH) as a framework for analyzing time-varying data by integrating multiparameter persistence and zigzag persistence. To this end, we introduce a stable topological invariant that captures both static and dynamic features at different scales. We present an algorithm to compute this invariant efficiently. We show that it enhances the machine learning models when applied to tasks such as sleep-stage detection, demonstrating its effectiveness in capturing the evolving patterns in time-varying datasets. - oai:arXiv.org:2502.16049v3 - cs.LG - math.AT - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Tamal K. Dey, Shreyas N. Samaga - - - Covert Entanglement Generation and Secrecy - https://arxiv.org/abs/2503.21002 - arXiv:2503.21002v4 Announce Type: replace-cross -Abstract: We determine the covert capacity for entanglement generation over a noisy quantum channel. While secrecy guarantees that the transmitted information remains inaccessible to an adversary, covert communication ensures that the transmission itself remains undetectable. The entanglement dimension follows a square root law (SRL) in the covert setting, i.e., $O(\sqrt{n})$ EPR pairs can be distributed covertly and reliably over $n$ channel uses. We begin with covert communication of classical information under a secrecy constraint. We then leverage this result to construct a coding scheme for covert entanglement generation. - oai:arXiv.org:2503.21002v4 - quant-ph - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ohad Kimelfeld, Boulat A. Bash, Uzi Pereg - - - Adaptive Entanglement Distillation - https://arxiv.org/abs/2504.11670 - arXiv:2504.11670v2 Announce Type: replace-cross -Abstract: Quantum network applications impose a variety of requirements on entanglement resources in terms of rate, fidelity, latency, and more. The repeaters in the quantum network must combine good methods for entanglement generation, effective entanglement distillation, and smart routing protocols to satisfy these application requirements. In this work, we focus on entanglement distillation in a linear chain of quantum repeaters. While conventional approaches reuse the same distillation scheme over multiple hop lengths after entanglement swaps, we propose a novel adaptive quantum error correction (QEC) scheme that boosts end-to-end metrics. Specifically, depending on the network operating point, we adapt the code used in distillation over successive rounds to monotonically increase the rate while also improving fidelity. We demonstrate the effectiveness of this strategy using three codes, with parameters [[9,1,3]], [[9,2,3]], [[9,3,3]], and a new performance metric, efficiency, that incorporates both overall rate and fidelity. Since the minimum input fidelity for QEC-based distillation is high, we then extend our study to include non-QEC-based purification protocols, specifically DEJMPS since it outperforms others. We compare the performance of end-to-end DEJMPS against adapting from DEJMPS to QEC once DEJMPS improves the initial fidelity to the threshold for QEC. Through a refined efficiency metric, we illuminate the regime where QEC is beneficial. These results provide a detailed outlook for entanglement purification and distillation in first and second generation quantum repeaters. - oai:arXiv.org:2504.11670v2 - quant-ph - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sijie Cheng, Narayanan Rengaswamy - - - The frequency $K_i$s for symmetrical traveling salesman problem - https://arxiv.org/abs/2504.19608 - arXiv:2504.19608v4 Announce Type: replace-cross -Abstract: The frequency $K_i$s ($i\in[4,n]$) are studied for symmetric traveling salesman problem ($TSP$) to illustrate the structure properties of the edges in optimal Hamiltonian cycle ($OHC$). A frequency $K_i$ is computed with the set of ${{i}\choose{2}}$ optimal $i$-vertex paths with given endpoints (optimal $i$-vertex paths) in one corresponding $K_i$ in $K_n$. Given an $OHC$ edge related to $K_i$, it has certain frequency bigger than $\frac{1}{2}{{i}\choose{2}}$ in the frequency $K_i$, and that of an ordinary edge not in $OHC$ is smaller than $2(n-3)$. Moreover, given a frequency $K_i$ containing an $OHC$ edge related to $K_n$, the frequency of the $OHC$ edge is bigger than $\frac{1}{2}{{i}\choose{2}}$ in the average case. It also found that the probability that an $OHC$ edge is contained in the optimal $i$-vertex paths increases according to $i\in [4, n]$ or keeps stable if it decreases from $i$ to $i+1\leq n$. As the frequency $K_i$s are used to compute the frequency of an edge, each $OHC$ edge reaches its own peak frequency at $i=P_0$ where $P_0=\frac{n}{2} + 2$ for even $n$ or $\frac{n+1}{2} + 1$ for odd $n$. For each ordinary edge out of $OHC$, the probability that they are contained in the optimal $i$-vertex paths decreases according to $i$, respectively, in the average case. Moreover, the average frequency of an ordinary edge will be smaller than $\frac{1}{2}{{i}\choose{2}}$ if $i \geq 2i_d$ where $i_d$ is the smallest number meeting the condition $\frac{(n-2)(n-3) - (i_d-2)(i_d-3)}{(n-2)(n-3) - (i_d-1)(i_d-2)} \geq \sqrt{1 + \frac{2}{i_d(i_d+1)}}$ and $i_d = O(n^{\frac{4}{7}})$. Based on these findings, an algorithm is presented to find $OHC$ in $O(n^2i_d^42^{2i_d})$ time using dynamic programming. The experiments are executed to verify these findings with the benchmark $TSP$ instances. - oai:arXiv.org:2504.19608v4 - cs.DM - math.CO - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Yong Wang - - - ADMM-Based Training for Spiking Neural Networks - https://arxiv.org/abs/2505.05527 - arXiv:2505.05527v2 Announce Type: replace-cross -Abstract: In recent years, spiking neural networks (SNNs) have gained momentum due to their high potential in time-series processing combined with minimal energy consumption. However, they still lack a dedicated and efficient training algorithm. The popular backpropagation with surrogate gradients, adapted from stochastic gradient descent (SGD)-derived algorithms, has several drawbacks when used as an optimizer for SNNs. Specifically, the approximation introduced by the use of surrogate gradients leads to numerical imprecision, poor tracking of SNN firing times at training time, and, in turn, poor scalability. In this paper, we propose a novel SNN training method based on the alternating direction method of multipliers (ADMM). Our ADMM-based training aims to solve the problem of the SNN step function's non-differentiability by taking an entirely new approach with respect to gradient backpropagation. For the first time, we formulate the SNN training problem as an ADMM-based iterative optimization, derive closed-form updates, and empirically show the optimizer's convergence, its great potential, and discuss future and promising research directions to improve the method to different layer types and deeper architectures. - oai:arXiv.org:2505.05527v2 - cs.LG - cs.AI - cs.NE - eess.SP - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Giovanni Perin, Cesare Bidini, Riccardo Mazzieri, Michele Rossi - - - Algebraic Topology Principles behind Topological Quantum Error Correction - https://arxiv.org/abs/2505.06082 - arXiv:2505.06082v2 Announce Type: replace-cross -Abstract: Quantum error correction (QEC) is crucial for realizing scalable quantum technologies, and topological quantum error correction (TQEC) has emerged as the most experimentally advanced paradigm of QEC. Existing homological and topological code constructions, however, are largely confined to orientable two-manifolds with simple boundary conditions. In this work, we develop a unified algebraic-topological framework for TQEC based on homology, cohomology, and intersection theory, which characterizes exactly when an arbitrary-dimensional manifold (with or without boundary) can serve as a quantum memory, thereby extending the standard 2D homological-code picture to arbitrary dimension and to manifolds with boundary via Poincar\'e-Lefschetz duality. Building on this classification, we introduce concrete code families that exploit nontrivial topology beyond the planar and toric settings. These include ``3-torus code'' and higher-dimensional ``volume codes'' on compact manifolds with mixed $X$- and $Z$-type boundaries. We further give a topological construction of qudit TQEC codes on general two-dimensional cell complexes using group presentation complexes, which unifies and extends several known quantum LDPC and homological-product-like constructions within a single geometric language. Finally, we combine the theoretical framework with numerical simulations to demonstrate that changing only the global topology can yield improved logical performance at fixed entanglement resources. Taken together, our results provide a systematic set of topological design principles for constructing and analyzing TQEC codes across dimensions and boundaries, and they open new avenues for topology-aware fault-tolerant quantum architectures. - oai:arXiv.org:2505.06082v2 - quant-ph - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xiang Zou, Hoi-Kwong Lo - - - Fermion Doubling in Quantum Cellular Automata - https://arxiv.org/abs/2505.07900 - arXiv:2505.07900v3 Announce Type: replace-cross -Abstract: A Quantum Cellular Automaton (QCA) is essentially an operator driving the evolution of particles on a lattice, through local unitaries. Because $\Delta_t=\Delta_x = \epsilon$, QCAs constitute a privileged framework to cast the digital quantum simulation of relativistic quantum particles and their interactions with gauge fields, e.g., $(3+1)$D Quantum Electrodynamics (QED). But before they can be adopted, simulation schemes for high-energy physics need prove themselves against specific numerical issues, of which the most infamous is Fermion Doubling (FD). FD is well understood in particular in the real-time, discrete-space \emph{but} continuous-time settings of Hamiltonian Lattice Gauge Theories (LGTs), as the appearance of spurious solutions for all $\Delta_x=\epsilon\neq 0$. We rigorously extend this analysis to the real-time, discrete-space \emph{and} discrete-time schemes that QCAs are. We demonstrate the existence of FD issues in QCAs for $\Delta_t =\Delta_x = \epsilon \neq 0$. By applying a covering map on the Brillouin zone, we provide a flavor-staggering-only way of fixing FD that does not break the chiral symmetry of the massless scheme. We explain how this method coexists with the Nielsen-Ninomiya no-go theorem, and give an example of neutrino-like QCA showing that our model allows to put chiral fermions interacting via the weak interaction on a spacetime lattice, without running into any FD problem. - oai:arXiv.org:2505.07900v3 - quant-ph - hep-lat - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Dogukan Bakircioglu, Pablo Arnault, Pablo Arrighi - - - RDA-PSO: A computational method to quantify the diffusive dispersal of insects - https://arxiv.org/abs/2505.08848 - arXiv:2505.08848v2 Announce Type: replace-cross -Abstract: This article introduces a computational method, called "Recapture of Diffusive Agents & Particle Swarm Optimization" (RDA-PSO), designed to estimate the dispersal parameter of diffusive insects in mark-release-recapture (MRR) field experiments. In addition to describing the method, its properties are discussed, with particular focus on robustness in estimating the observed diffusion coefficient in the presence of uncertainty. It is shown that RDA-PSO provides a simple and reliable approach to quantify insect dispersal that can handle low recapture rates and uneven capture site distributions without the need for area corrections. Tests on synthetic data, for which the actual diffusion coefficient is known, show the method outperforms three techniques based on the solution of the diffusion equation, which are also introduced in this work. Examples of application to real field data for the yellow fever mosquito are provided. - oai:arXiv.org:2505.08848v2 - q-bio.QM - math.OC - q-bio.PE - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1007/s11538-025-01577-0 - Bull Math Biol 88, 24 (2026) - Lidia Mrad, Joceline Lega - - - Statistical properties of non-linear observables of fractal Gaussian fields with a focus on spatial-averaging observables and on composite operators - https://arxiv.org/abs/2505.16356 - arXiv:2505.16356v2 Announce Type: replace-cross -Abstract: The statistical properties of non-linear observables of the fractal Gaussian field $\phi(\vec x)$ of negative Hurst exponent $H<0$ in dimension $d$ are revisited with a focus on spatial-averaging observables and on the properties of the finite parts $\phi_n(\vec x)$ of the ill-defined composite operators $\phi^n(\vec x) $. For the special case $n=2$ of quadratic observables, explicit results include the cumulants of arbitrary order, the L\'evy-Khintchine formula for the characteristic function and the anomalous large deviations properties. The case of observables of arbitrary order $n>2$ is analyzed via the Wiener-Ito chaos-expansion for functionals of the white noise: the multiple stochastic Ito integrals are useful to identify the finite parts $\phi_n(\vec x)$ of the ill-defined composite operators $\phi^n(\vec x) $ and to compute their correlations involving the Hurst exponents $H_n=nH$. - oai:arXiv.org:2505.16356v2 - cond-mat.stat-mech - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Cecile Monthus - - - Flagged Extensions and Numerical Simulations for Quantum Channel Capacity: Bridging Theory and Computation - https://arxiv.org/abs/2506.03429 - arXiv:2506.03429v2 Announce Type: replace-cross -Abstract: I will investigate the capacities of noisy quantum channels through a combined analytical and numerical approach. First, I introduce novel flagged extension techniques that embed a channel into a higher-dimensional space, enabling single-letter upper bounds on quantum and private capacities. My results refine previous bounds and clarify noise thresholds beyond which quantum transmission vanishes. Second, I present a simulation framework that uses coherent information to estimate channel capacities in practice, focusing on two canonical examples: the amplitude damping channel (which we confirm is degradable and thus single-letter) and the depolarizing channel (whose capacity requires multi-letter superadditivity). By parameterizing input qubit states on the Bloch sphere, I numerically pinpoint the maximum coherent information for each channel and validate the flagged extension bounds. Notably, I capture the abrupt transition to zero capacity at high noise and observe superadditivity for moderate noise levels. - oai:arXiv.org:2506.03429v2 - quant-ph - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Vahid Nourozi - - - Dynamic Hybrid Modeling: Incremental Identification and Model Predictive Control - https://arxiv.org/abs/2506.18344 - arXiv:2506.18344v2 Announce Type: replace-cross -Abstract: Mathematical models are crucial for optimizing and controlling chemical processes, yet they often face significant limitations in terms of computational time, algorithm complexity, and development costs. Hybrid models, which combine mechanistic models with data-driven models (i.e. models derived via the application of machine learning to experimental data), have emerged as a promising solution to these challenges. However, the identification of dynamic hybrid models remains difficult due to the need to integrate data-driven models within mechanistic model structures. - We present an incremental identification approach for dynamic hybrid models that decouples the mechanistic and data-driven components to overcome computational and conceptual difficulties. Our methodology comprises four key steps: (1) regularized dynamic parameter estimation to determine optimal time profiles for flux variables, (2) correlation analysis to evaluate relationships between variables, (3) data-driven model identification using advanced machine learning techniques, and (4) hybrid model integration to combine the mechanistic and data-driven components. This approach facilitates early evaluation of model structure suitability, accelerates the development of hybrid models, and allows for independent identification of data-driven components. - Three case studies are presented to illustrate the robustness, reliability, and efficiency of our incremental approach in handling complex systems and scenarios with limited data. - oai:arXiv.org:2506.18344v2 - eess.SY - cs.LG - cs.SY - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - 10.1016/j.compchemeng.2025.109413 - Computers & Chemical Engineering, Volume 204, January 2026, 109413 - Adrian Caspari, Thomas Bierweiler, Sarah Fadda, Daniel Labisch, Maarten Nauta, Franzisko Wagner, Merle Warmbold, Constantinos C. Pantelides - - - Modeling Hierarchical Spaces: A Review and Unified Framework for Surrogate-Based Architecture Design - https://arxiv.org/abs/2506.22621 - arXiv:2506.22621v3 Announce Type: replace-cross -Abstract: Simulation-based problems involving mixed-variable inputs frequently feature domains that are hierarchical, conditional, heterogeneous, or tree-structured. These characteristics pose challenges for data representation, modeling, and optimization. This paper reviews extensive literature on these structured input spaces and proposes a unified framework that generalizes existing approaches. - In this framework, input variables may be continuous, integer, or categorical. A variable is described as meta if its value governs the presence of other decreed variables, enabling the modeling of conditional and hierarchical structures. We further introduce the concept of partially-decreed variables, whose activation depends on contextual conditions. - To capture these inter-variable hierarchical relationships, we introduce design space graphs, combining principles from feature modeling and graph theory. This allows the definition of general hierarchical domains suitable for describing complex system architectures. - Our framework defines hierarchical distances and kernels to enable surrogate modeling and optimization on hierarchical domains. We demonstrate its effectiveness on complex system design problems, including a neural network and a green-aircraft case study. Our methods are available in the open-source Surrogate Modeling Toolbox (SMT 2.0). - oai:arXiv.org:2506.22621v3 - cs.LG - math.OC - stat.ML - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - 10.1007/s00158-026-04249-2 - Paul Saves, Edward Hall\'e-Hannan, Jasper Bussemaker, Youssef Diouane, Nathalie Bartoli - - - Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systems - https://arxiv.org/abs/2507.07222 - arXiv:2507.07222v3 Announce Type: replace-cross -Abstract: The Koopman operator provides a principled framework for analyzing nonlinear dynamical systems through linear operator theory. Recent advances in dynamic mode decomposition (DMD) have shown that trajectory data can be used to identify dominant modes of a system in a data-driven manner. Building on this idea, deep learning methods such as VAMPnet and DPNet have been proposed to learn the leading singular subspaces of the Koopman operator. However, these methods require backpropagation through potentially numerically unstable operations on empirical second moment matrices, such as singular value decomposition and matrix inversion, during objective computation, which can introduce biased gradient estimates and hinder scalability to large systems. In this work, we propose a scalable and conceptually simple method for learning the top-$k$ singular functions of the Koopman operator for stochastic dynamical systems based on the idea of low-rank approximation. Our approach eliminates the need for unstable linear-algebraic operations and integrates easily into modern deep learning pipelines. Empirical results demonstrate that the learned singular subspaces are both reliable and effective for downstream tasks such as eigen-analysis and multi-step prediction. - oai:arXiv.org:2507.07222v3 - cs.LG - cs.NA - math.DS - math.NA - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Minchan Jeong, J. Jon Ryu, Se-Young Yun, Gregory W. Wornell - - - Exact CHY Integrand Construction Using Combinatorial Neural Networks and Discrete Optimization - https://arxiv.org/abs/2508.02248 - arXiv:2508.02248v2 Announce Type: replace-cross -Abstract: Constructing a rational CHY integrand that realizes prescribed physical pole constraints is a discrete inverse problem whose combinatorial complexity grows with multiplicity. We encode the pole hierarchy through generalized pole degrees $K(A)$ (channels $s_A$), defined as signed internal-edge counts associated with particle subsets in a colored integrand graph. Additivity under integrand multiplication together with the elementary face recursion on the subset lattice expresses all higher-channel $K(A)$ as linear functions of the two-particle data $\{K(s_{ij})\}$ and reduces the inverse step to a mixed-integer linear feasibility problem. The subset lattice provides a fixed dependency graph for deterministic message passing with forward evaluation and backward residual propagation; this computation is parameter-free and involves no training. In factorial-rescaled variables $\widetilde K(A)=(|A|-2)!\,K(A)$, every local update is integral, so propagation is exact in the rescaled recursion variables and does not rely on numerical reconstruction. We further organize generalized integrand graphs by an $n$-regular grading under multiplication, where degree-zero (0-regular) factors act as M\"obius-invariant insertions that can be decomposed into four-point cross ratios. We illustrate the construction at six and eight points, including pick-pole selection and higher-order pole reduction. - oai:arXiv.org:2508.02248v2 - hep-th - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Simeng Li, Yaobo Zhang - - - Matrix-Free Two-to-Infinity and One-to-Two Norms Estimation - https://arxiv.org/abs/2508.04444 - arXiv:2508.04444v2 Announce Type: replace-cross -Abstract: In this paper, we propose new randomized algorithms for estimating the two-to-infinity and one-to-two norms in a matrix-free setting, using only matrix-vector multiplications. Our methods are based on appropriate modifications of Hutchinson's diagonal estimator and its Hutch++ version. We provide oracle complexity bounds for both modifications. We further illustrate the practical utility of our algorithms for Jacobian-based regularization in deep neural network training on image classification tasks. We also demonstrate that our methodology can be applied to mitigate the effect of adversarial attacks in the domain of recommender systems. - oai:arXiv.org:2508.04444v2 - cs.LG - cs.NA - math.NA - stat.ML - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Askar Tsyganov, Evgeny Frolov, Sergey Samsonov, Maxim Rakhuba - - - Generalization of anomaly formula for time reversal symmetry in (2+1)D abelian bosonic TQFTs - https://arxiv.org/abs/2508.04990 - arXiv:2508.04990v3 Announce Type: replace-cross -Abstract: We study time-reversal symmetry in $(2+1)$D abelian bosonic topological phases. Time-reversal anomalies in such systems are classified by $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry-protected topological (SPT) phases in $(3+1)$D, and can be diagnosed via partition functions on manifolds such as $\mathbb{RP}^4$ and $\mathbb{CP}^2$. These partition functions are related by the anomaly formula \begin{equation*} - Z(\mathbb{RP}^4)\, Z(\mathbb{CP}^2) = \theta_{\mathcal{M}}, \end{equation*} where $\theta_\mathcal{M}$ is the Dehn twist phase associated with the crosscap state. - Meanwhile, the existence of gapped boundaries is constrained by so-called higher central charges $\xi_n$, which serve as computable invariants encoding obstruction data. Motivated by the known relation $Z(\mathbb{CP}^2) = \xi_1$, we propose a generalization of the anomaly formula that involves both the higher central charges $\xi_n$ and a new time-reversal invariant $\eta_n$. Introducing a distinguished subset $\mathcal{M}^n \subset \mathcal{A}$ of anyons, we establish the relation \begin{equation*} - \eta_n \cdot \xi_n = \frac{\sum_{a \in \mathcal{M}^n} \theta(a)^n}{\left| \sum_{a \in \mathcal{M}^n} \theta(a)^n \right|}, \end{equation*} which generalizes the known anomaly formula. - We analyze the algebraic structure of $\mathcal{M}^n$, derive consistency relations it satisfies, and clarify its connection to the original anomaly formula. - oai:arXiv.org:2508.04990v3 - hep-th - cond-mat.str-el - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - 10.1093/ptep/ptaf155 - Prog. Theor. Exp. Phys. 2025, 123B02 (2025) - Ippo Orii - - - Commuting integrable maps from a deformed D$_4$ cluster algebra - https://arxiv.org/abs/2508.05270 - arXiv:2508.05270v2 Announce Type: replace-cross -Abstract: In this paper we revisit an integrable map of the plane which we obtained recently as a two-parameter family of deformed mutations in the cluster algebra of type D$_4$. The rational first integral for this map defines an invariant foliation of the plane by level curves, and we explain how this corresponds to a rational elliptic surface of rank 2. This leads us to construct another (independent) integrable map, commuting with the first, such that both maps lift to compositions of mutations in an enlarged cluster algebra, whose underlying quiver is equivalent to the one found by Okubo for the $q$-Painlev\'e VI equation. The degree growth of the two commuting maps is calculated in two different ways: firstly, from the tropical (max-plus) equations for the d-vectors of the cluster variables; and secondly, by constructing the minimal space of initial conditions for the two maps, via blowing up $\mathbb{P}^1 \times \mathbb{P}^1$. - oai:arXiv.org:2508.05270v2 - nlin.SI - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - A. N. W. Hone, W. Kim, T. Mase - - - Bias correction for Chatterjee's graph-based correlation coefficient - https://arxiv.org/abs/2508.09040 - arXiv:2508.09040v2 Announce Type: replace-cross -Abstract: Azadkia and Chatterjee (2021) recently introduced a simple nearest neighbor (NN) graph-based correlation coefficient that consistently detects both independence and functional dependence. Specifically, it approximates a measure of dependence that equals 0 if and only if the variables are independent, and 1 if and only if they are functionally dependent. However, this NN estimator includes a bias term that may vanish at a rate slower than root-$n$, preventing root-$n$ consistency in general. In this article, we (i) analyze this bias term closely and show that it could become asymptotically negligible when the dimension is smaller than four; and (ii) propose a bias-correction procedure for more general settings. In both regimes, we obtain estimators (either the original or the bias-corrected version) that are root-$n$ consistent and asymptotically normal. - oai:arXiv.org:2508.09040v2 - stat.ME - econ.EM - math.ST - stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mona Azadkia, Leihao Chen, Fang Han - - - Optimal Hamiltonian for a quantum state with finite entropy - https://arxiv.org/abs/2508.16575 - arXiv:2508.16575v3 Announce Type: replace-cross -Abstract: We consider the following task: how for a given quantum state $\rho$ to find a grounded Hamiltonian $H$ satisfying the condition $\mathrm{Tr} H\rho\leq E_0<+\infty$ in such a way that the von Neumann entropy of the Gibbs state $\gamma_H(E)$ corresponding to a given energy $E>0$ be as small as possible. - We show that for any mixed state $\rho$ with finite entropy and any $E>0$ there exists a solution $H(\rho,E_0,E)$ of the above problem (unique in the non-degenerate case) which we call optimal Hamiltonian for the state $\rho$. Explicit expressions for $H(\rho,E_0,E)$, $\gamma_{H(\rho,E_0,E)}(E)$ and $S(\gamma_{H(\rho,E_0,E)}(E))$ are obtained. Analytical properties of the function $E\mapsto S(\gamma_{H(\rho,E_0,E)}(E))$ are explored. Several examples are considered. - We also consider a modification of the above task in which arbitrary Hamiltonians (not necessarily grounded) are considered. - The basic application motivated this research is described. As examples, new semicontinuity bounds for the von Neumann entropy and for the entanglement of formation are obtained and briefly discussed (with the intention to give a detailed analysis in a separate article). - oai:arXiv.org:2508.16575v3 - quant-ph - cs.IT - math-ph - math.IT - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - M. E. Shirokov - - - A predictive modular approach to constraint satisfaction under uncertainty -- with application to glycosylation in continuous monoclonal antibody biosimilar production - https://arxiv.org/abs/2508.16803 - arXiv:2508.16803v4 Announce Type: replace-cross -Abstract: The paper proposes a modular-based approach to constraint handling in process optimization and control. This is partly motivated by the recent interest in learning-based methods, e.g., within bioproduction, for which constraint handling under uncertainty is a challenge. The proposed constraint handler, called predictive filter, is combined with an adaptive constraint margin and a constraint violation cost monitor to minimize the cost of violating soft constraints due to model uncertainty and disturbances. The module can be combined with any controller and is based on minimally modifying the controller output, in a least squares sense, such that constraints are satisfied within the considered horizon. The proposed method is computationally efficient and suitable for real-time applications. The effectiveness of the method is illustrated through a realistic case study of glycosylation constraint satisfaction in continuous monoclonal antibody biosimilar production using Chinese hamster ovary cells, employing a metabolic network model consisting of 23 extracellular metabolites and 126 reactions. In the case study, the average constraint-violation cost is reduced by more than 60% compared to the case without the proposed constraint-handling method. - oai:arXiv.org:2508.16803v4 - eess.SY - cs.SY - math.OC - q-bio.QM - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.jprocont.2026.103632 - Journal of Process Control, Volume 158, 2026, 103632, ISSN 0959-1524 - Yu Wang, Xiao Chen, Hubert Schwarz, V\'eronique Chotteau, Elling W. Jacobsen - - - Quantum spatial best-arm identification via quantum walks - https://arxiv.org/abs/2509.05890 - arXiv:2509.05890v2 Announce Type: replace-cross -Abstract: Quantum reinforcement learning has emerged as a framework combining quantum computation with sequential decision-making, and applications to the multi-armed bandit (MAB) problem have been reported. The graph bandit problem extends the MAB setting by introducing spatial constraints, yet quantum approaches remain limited. We propose a quantum algorithmic framework for best-arm identification in graph bandits, termed Quantum Spatial Best-Arm Identification (QSBAI), which is applicable to general graph structures. The method employs quantum walks to encode superpositions over graph-constrained actions, extending amplitude amplification and generalizing the Quantum BAI algorithm via Szegedy's walk framework. This establishes a link between Grover-type search and reinforcement learning tasks with structural restrictions. We focus our theoretical analysis on complete and bipartite graphs, deriving the maximal success probability of identifying the best arm and the time step at which it is achieved. Our results highlight the potential of quantum walks to accelerate exploration in constrained environments and extend the applicability of quantum algorithms for decision-making. - oai:arXiv.org:2509.05890v2 - quant-ph - cs.AI - cs.LG - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Tomoki Yamagami, Etsuo Segawa, Takatomo Mihana, Andr\'e R\"ohm, Atsushi Uchida, Ryoichi Horisaki - - - Vanishing of the $H^3$ obstruction for time-reversal symmetry in (2+1)D abelian bosonic TQFTs - https://arxiv.org/abs/2509.07368 - arXiv:2509.07368v2 Announce Type: replace-cross -Abstract: In $(2+1)$-dimensional topological quantum field theories (TQFTs), the action of a global symmetry group on the anyon system is one of the central topics of research. Owing to the subtle categorical nature of anyons, a global symmetry acting on them is generally realized in a projective manner. Symmetry fractionalization encodes this projective realization. The obstruction to defining symmetry fractionalization is captured by a cohomology class, known as the $H^3$ obstruction, whose nontriviality signals a failure to define symmetry fractionalization consistently. In this short note, we prove that the $H^3$ obstruction for time-reversal symmetry always vanishes in abelian bosonic TQFTs. - oai:arXiv.org:2509.07368v2 - hep-th - cond-mat.str-el - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ippo Orii + Anton Alexa - Robustness of quantum algorithms: Worst-case fidelity bounds and implications for design - https://arxiv.org/abs/2509.08481 - arXiv:2509.08481v2 Announce Type: replace-cross -Abstract: Errors occurring on noisy hardware pose a key challenge to reliable quantum computing. Existing techniques such as error correction, mitigation, or suppression typically separate the error handling from the algorithm analysis and design. In this paper, we develop an alternative, algorithm-centered framework for understanding and improving the robustness against errors. For a given quantum algorithm and error model, we derive worst-case fidelity bounds which can be efficiently computed to certify the robustness. We consider general error models including coherent and (Markovian) incoherent errors and allowing for set-based error descriptions to address uncertainty or time-dependence in the errors. Our results give rise to guidelines for robust algorithm design and compilation by optimizing our theoretical robustness measure. We demonstrate the practicality of the framework with numerical results on algorithm analysis and robust optimization, including the robustness analysis of a 50-qubit modular adder circuit. - oai:arXiv.org:2509.08481v2 - quant-ph - cs.SY - eess.SY - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross + Factoriality of normal projective varieties + https://arxiv.org/abs/2601.13151 + arXiv:2601.13151v3 Announce Type: replace +Abstract: For a normal projective variety $X$, the $\bf Q$-factoriality defect $\sigma(X)$ is defined to be the rank of the quotient of the group of Weil divisors by the subgroup of Cartier ones. We prove a slight improvement of a topological formula of S.G. Park and M. Popa asserting that $\sigma(X)=h^{2n-2}(X)-h^2(X)$ by assuming only 2-semi-rationality, that is, $R^k\pi_*{\mathcal O}_{\widetilde{X}}=0$ for $k=1,2$, instead of rational singularities for $X$, where $\pi:\widetilde{X}\to X$ is a desingularization with $h^k(X):=\dim H^k(X,{\bf Q})$ and $n:=\dim X>2$. Our proof generalizes the one by Y. Namikawa and J.H.M. Steenbrink for the case $n=3$ with isolated hypersurface singularities. We also give a proof of the assertion that $\bf Q$-factoriality implies factoriality if $X$ is a local complete intersection whose singular locus has at least codimension three. (This seems to be known to specialists in the case $X$ has only isolated hypersurface singularities with $n=3$ using Milnor's Bouquet theorem.) These imply a slight improvement of Grothendieck's theorem in the projective case asserting that $X$ is factorial if it is a local complete intersection whose singular locus has at least codimension three and at general points of its components of codimension three, $X$ has rational singularities and is a $\bf Q$-homology manifold. In the hypersurface singularity case, the last condition means that any spectral number of a transversal slice to the singular locus is greater than 1, and is not an integer, that is, 1 is not an eigenvalue of the Milnor monodromy. + oai:arXiv.org:2601.13151v3 + math.AG + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Julian Berberich, Tobias Fellner, Robert L. Kosut, Christian Holm + Seung-Jo Jung, Morihiko Saito - Learning to Solve Optimization Problems Constrained with Partial Differential Equations - https://arxiv.org/abs/2509.24573 - arXiv:2509.24573v2 Announce Type: replace-cross -Abstract: Partial differential equation (PDE)-constrained optimization arises in many scientific and engineering domains, such as energy systems, fluid dynamics and material design. In these problems, the decision variables (e.g., control inputs or design parameters) are tightly coupled with the PDE state variables, and the feasible set is implicitly defined by the governing PDE constraints. This coupling makes the problems computationally demanding, as it requires handling high dimensional discretization and dynamic constraints. To address these challenges, this paper introduces a learning-based framework that integrates a dynamic predictor with an optimization surrogate. The dynamic predictor, a novel time-discrete Neural Operator (Lu et al.), efficiently approximate system trajectories governed by PDE dynamics, while the optimization surrogate leverages proxy optimizer techniques (Kotary et al.) to approximate the associated optimal decisions. This dual-network design enables real-time approximation of optimal strategies while explicitly capturing the coupling between decisions and PDE dynamics. We validate the proposed approach on benchmark PDE-constrained optimization tasks inlacing Burgers' equation, heat equation and voltage regulation, and demonstrate that it achieves solution quality comparable to classical control-based algorithms, such as the Direct Method and Model Predictive Control (MPC), while providing up to four orders of magnitude improvement in computational speed. - oai:arXiv.org:2509.24573v2 + Deep Neural networks for solving high-dimensional parabolic partial differential equations + https://arxiv.org/abs/2601.13256 + arXiv:2601.13256v2 Announce Type: replace +Abstract: The numerical solution of high dimensional partial differential equations (PDEs) is severely constrained by the curse of dimensionality (CoD), rendering classical grid--based methods impractical beyond a few dimensions. In recent years, deep neural networks have emerged as a promising mesh free alternative, enabling the approximation of PDE solutions in tens to thousands of dimensions. This review provides a tutorial--oriented introduction to neural--network--based methods for solving high dimensional parabolic PDEs, emphasizing conceptual clarity and methodological connections. We organize the literature around three unifying paradigms: (i) PDE residual--based approaches, including physicsinformed neural networks and their high dimensional variants; (ii) stochastic methods derived from Feynman--Kac and backward stochastic differential equation formulations; and (iii) hybrid derivative--free random difference approaches designed to alleviate the computational cost of derivatives in high dimensions. For each paradigm, we outline the underlying mathematical formulation, algorithmic implementation, and practical strengths and limitations. Representative benchmark problems--including Hamilton--Jacobi--Bellman and Black--Scholes equations in up to 1000 dimensions --illustrate the scalability, effectiveness, and accuracy of the methods. The paper concludes with a discussion of open challenges and future directions for reliable and scalable solvers of high dimensional PDEs. + oai:arXiv.org:2601.13256v2 + math.NA cs.LG - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross + cs.NA + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Yusuf Guven, Vincenzo Di Vito, Ferdinando Fioretto + Wenzhong Zhang, Zheyuan Hu, Wei Cai, George EM Karniadakis - Effective Free Energy Landscapes and Black Hole Thermodynamic Phase Transitions - https://arxiv.org/abs/2509.25039 - arXiv:2509.25039v2 Announce Type: replace-cross -Abstract: A recent interesting development in the dynamics of black hole phase transitions has been the so-called Gibbs free energy landscape approach. In this formalism, it is assumed that there exists a canonical ensemble of a series of black hole spacetimes with arbitrary horizon radius at a given ensemble temperature. An off-shell Gibbs free energy is defined for every spacetime state in the ensemble, with the horizon radius treated as the order parameter. The minima (maxima) of this function correspond to the various stable (unstable) black hole states. This off-shell Gibbs free energy is then treated as a classical effective drift potential of an associated Fokker-Planck equation used to study the stochastic dynamics of black hole phase transition under thermal fluctuations. Additive noise, which is independent of the black hole size, is assumed in obtaining the Fokker-Planck equation. In this work we extend the previous treatment by considering the effects of multiplicative noise, namely, noise that could scale with black hole size. This leads to an effective free energy function that can be used to study the modification of the thermodynamic phase transition of a black hole system. It is realized that it is generally difficult to form black holes under a multiplicative noise, - unless the effective and the original free energy become extremal at the same horizon radius. For this latter situation some theoretical noise profiles which are monotonically increasing/deceasing functions of the horizon radius are considered. It is found that stronger noise disfavors the formation of black hole - oai:arXiv.org:2509.25039v2 - gr-qc - cond-mat.stat-mech - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross + Identification capacity and rate-query tradeoffs in classification systems + https://arxiv.org/abs/2601.14252 + arXiv:2601.14252v2 Announce Type: replace +Abstract: We extend classical rate-distortion theory to a discrete classification setting with three resources: tag rate $L$ (bits of storage per entity), identification cost $W$ (queries to determine class membership), and distortion $D$ (misidentification probability). We prove an information barrier: when distinct classes share identical attribute profiles (i.e., the attribute-profile map $\pi$ is not injective on classes), zero-error identification from attribute queries alone is impossible. We characterize the unique Pareto-optimal zero-error point in the $(L,W,D)$ tradeoff space: a nominal tag of length $L=\lceil\log_2 k\rceil$ bits for $k$ classes yields $W=O(1)$ and $D=0$. Without tags ($L=0$), zero-error identification requires $W=\Omega(d)$ attribute queries, where $d$ is the distinguishing dimension; in the worst case $d=n$ (the ambient attribute count), giving $W=\Omega(n)$. In the presence of attribute collisions, any tag-free scheme incurs $D>0$. Conversely, in any information-barrier domain, any scheme achieving $D=0$ requires $L\ge \log_2 k$ bits; this is tight. We show minimal sufficient query sets form the bases of a matroid, so the distinguishing dimension is well-defined, connecting to zero-error source coding via graph entropy. We instantiate the theory to type systems, databases, and biological taxonomy. All results are machine-checked in Lean 4 (6000+ lines, 0 sorry). + oai:arXiv.org:2601.14252v2 + cs.IT + cs.PL + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - Phys. Scr. 101 (2026) 025006 - Choon-Lin Ho + Tristan Simas - Consistent kinetic modeling of compressible flows with variable Prandtl numbers: Double-distribution quasi-equilibrium approach - https://arxiv.org/abs/2510.04197 - arXiv:2510.04197v2 Announce Type: replace-cross -Abstract: A consistent kinetic modeling and discretization strategy for compressible flows across all Prandtl numbers and specific heat ratios is developed using the quasi-equilibrium approach within two of the most widely used double-distribution frameworks. The methodology ensures accurate recovery of the Navier-Stokes-Fourier equations, including all macroscopic moments and dissipation rates, through detailed hydrodynamic limit analysis and careful construction of equilibrium and quasi-equilibrium attractors. Discretization is performed using high-order velocity lattices with a static reference frame in a discrete velocity Boltzmann context to isolate key modeling aspects such as the necessary requirements on expansion and quadrature orders. The proposed models demonstrate high accuracy, numerical stability and Galilean invariance across a wide range of Mach numbers and temperature ratios. Separate tests for strict conservation and measurements of all dissipation rates confirm these insights for all Prandtl numbers and specific heat ratios. Simulations of a thermal Couette flow and a sensitive two-dimensional shock-vortex interaction excellently reproduce viscous Navier-Stokes-Fourier-level physics. The proposed models establish an accurate, efficient and scalable framework for kinetic simulations of compressible flows with moderate supersonic speeds and discontinuities at arbitrary Prandtl numbers and specific heat ratios, offering a valuable tool for studying complex problems in fluid dynamics and paving the way for future extensions to the lattice Boltzmann context, by application of correction terms, as well as high-Mach and hypersonic regimes, employing target-designed reference frames. - oai:arXiv.org:2510.04197v2 - physics.flu-dyn - math-ph - math.MP - nlin.CG - physics.comp-ph - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross + Finite-Sample Inference for Sparsely Permuted Linear Regression + https://arxiv.org/abs/2601.14872 + arXiv:2601.14872v2 Announce Type: replace +Abstract: We study a linear observation model with an unknown permutation called \textit{permuted/shuffled linear regression}, where responses and covariates are mismatched and the permutation forms a discrete, factorial-size parameter. The permutation is a key component of the data-generating process, yet its statistical investigation remains challenging due to its discrete nature. We develop a general statistical inference framework on the permutation and regression coefficients. First, we introduce a localization step that reduces the permutation space to a small candidate set building on recent advances in the repro samples method, whose miscoverage decays polynomially with the number of Monte Carlo samples. Then, based on this localized set, we provide statistical inference procedures: a conditional Monte Carlo test of permutation structures with valid finite-sample Type-I error control. We also develop coefficient inference that remains valid under alignment uncertainty of permutations. For computational purposes, we develop a linear assignment problem computable in polynomial time and demonstrate that, with high probability, the solution is equivalent to that of the conventional least squares with large computational cost. Extensions to partially permuted designs and ridge regularization are further discussed. Extensive simulations and an application to air-quality data corroborate finite-sample validity, strong power to detect mismatches, and practical scalability. + oai:arXiv.org:2601.14872v2 + math.ST + cs.LG + stat.ME + stat.ML + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://creativecommons.org/licenses/by/4.0/ - R. M. Str\"assle, S. A. Hosseini, I. V. Karlin + Hirofumi Ota, Masaaki Imaizumi - Finite elements and moving asymptotes accelerate quantum optimal control -- FEMMA - https://arxiv.org/abs/2510.04798 - arXiv:2510.04798v2 Announce Type: replace-cross -Abstract: Quantum optimal control is central to designing spin manipulation pulses. Gradient-based pulse optimization can be facilitated by either accelerating gradient evaluation or enhancing the convergence rate. In this work, we accelerated single-spin optimal control by combining the finite element method with the method of moving asymptotes. By treating discretized time as spatial coordinates, the Liouville - von Neumann equation was reformulated as a linear system, efficiently yielding a joint solution of the spin trajectory and control gradient. The method of moving asymptotes, relying on the ensemble fidelities and gradients, achieves rapid convergence for a target fidelity of 0.995. - oai:arXiv.org:2510.04798v2 - physics.chem-ph + From carbon management strategies to implementation: Modeling and physical simulation of CO2 pipeline infrastructure -- a case study for Germany + https://arxiv.org/abs/2601.15090 + arXiv:2601.15090v2 Announce Type: replace +Abstract: Carbon capture and storage or utilization (CCUS) will play an important role to achieve climate neutrality in many economies. Pipelines are widely regarded as the most efficient means of CO2 transport; however, they are currently non-existent. Policy-makers and companies need to develop large-scale infrastructure under substantial uncertainty. Methods and analyses are needed to support pipeline planning and strategy development. This paper presents an integrated method for designing CO2 pipeline networks by combining energy system scenarios with physical network simulation. Using Germany as a case study, we derive spatially highly resolved CO2 balances to develop a dense-phase CO2 pipeline topology that follows existing gas pipeline corridors. The analyzed system includes existing sites for cement and lime production, waste incineration, carbon users, four coastal CO2 hubs, and border crossing points. We then apply the multiphysical network simulator MYNTS to assess the technical feasibility of this network. We determine pipeline diameters, pump locations, and operating conditions that ensure stable dense-phase transport. The method explicitly accounts for elevation and possible impurities. The results indicate that a system of about 7000 km pipeline length and a mixed normed diameter of DN700 on main corridors and of DN500/DN400 on branches presents a feasible solution to connect most sites. Investment costs for the optimized pipeline system are calculated to be about 17 billion Euros. The method provides a reproducible framework and is transferable to other countries and to European scope. + oai:arXiv.org:2601.15090v2 math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Mengjia He, Yongbo Deng, Burkhard Luy, Jan G. Korvink + Fri, 23 Jan 2026 00:00:00 -0500 + replace + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Mehrnaz Anvari, Marius Neuwirth, Okan Akca, Luna L\"utz, Simon Lukas Bussmann, Tobias Fleiter, Bernhard Klaassen - Weyl symmetry without the traceless condition - https://arxiv.org/abs/2510.09957 - arXiv:2510.09957v3 Announce Type: replace-cross -Abstract: We show that the requirement that the trace of the stress-energy tensor of matter must vanish if invariance under Weyl transformations is a symmetry of a given gravitational theory is not universal. This requirement holds whenever the masses of timelike fields are constant parameters that are not transformed by conformal transformations, or when the energy density of perfect fluids transforms as $\rho\rightarrow\Omega^{-3}\rho$. In contrast, if the masses of timelike fields are point-dependent quantities transforming under conformal transformations as $m\rightarrow\Omega^{-1}m$, and the energy density of perfect fluids transforms as $\rho\rightarrow\Omega^{-4}\rho$, the Weyl symmetry does not require the vanishing of the trace of the matter SET. In consequence, any matter fields, regardless of whether the trace of their stress-energy tensor vanishes or not, can be coupled to gravity. The phenomenological and physical consequences of the novel result are drawn. - oai:arXiv.org:2510.09957v3 - gr-qc - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross + On the number of permutation-twisted dot products + https://arxiv.org/abs/2601.15276 + arXiv:2601.15276v2 Announce Type: replace +Abstract: For distinct real numbers $a_1, \ldots, a_n$ and distinct real numbers $b_1, \ldots, b_n$, consider the sum $S=\sum_{i=1}^n a_i b_{\pi(i)}$ as $\pi$ ranges over the permutations of $[n]$. We show that this sum always assumes at least $\Omega(n^3)$ distinct values, which is optimal. This ``support'' bound complements recent work of Do, Nguyen, Phan, Tran, and Vu, and of Hunter, Pohoata, and Zhu on the anticoncentration properties of $S$ when $\pi$ is chosen uniformly at random. + oai:arXiv.org:2601.15276v2 + math.CO + Fri, 23 Jan 2026 00:00:00 -0500 + replace http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Israel Quiros + Ruben Carpenter, Colin Defant, Noah Kravitz - TCitH- and VOLEitH-based Signatures from Restricted Decoding - https://arxiv.org/abs/2510.11224 - arXiv:2510.11224v2 Announce Type: replace-cross -Abstract: Threshold-Computation-in-the-Head (TCitH) and VOLE-in-the-Head (VOLEitH), two recent developments of the MPC-in-the-Head (MPCitH) paradigm, have significantly improved the performance of digital signature schemes. This work embeds the restricted decoding problem within these frameworks: we propose a structurally simple modeling that achieves competitive signature sizes. Specifically, by instantiating the restricted decoding problem with the same hardness assumption underlying CROSS, we reduce sizes by more than a factor of two compared to the NIST submission. Moreover, we observe that ternary full-weight decoding, closely related to the hardness assumption underlying WAVE, is a restricted decoding problem. Using ternary full-weight decoding, we obtain signature sizes comparable to the smallest MPCitH-based candidates in the NIST competition. - oai:arXiv.org:2510.11224v2 - cs.CR - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + Utility maximization under endogenous pricing + https://arxiv.org/abs/2005.04312 + arXiv:2005.04312v5 Announce Type: replace-cross +Abstract: We study the expected utility maximization problem of a large investor who is allowed to make transactions on tradable assets in an incomplete financial market with endogenous permanent market impacts. The asset prices are assumed to follow a nonlinear price curve quoted in the market as the utility indifference curve of a representative liquidity supplier. Using generalized subgradients, we show that optimality can be fully characterized via a system of coupled forward-backward stochastic differential equations (FBSDEs) which corresponds to a non-linear backward stochastic partial differential equation (BSPDE). We show existence of solutions to the optimal investment problem and the FBSDEs in the case where the driver function of the representative market maker grows at least quadratically or the utility function of the large investor falls faster than quadratically or is exponential. Furthermore, we derive smoothness results for the existence of solutions of BSPDEs. Examples are provided when the market is complete, the driver is positively homogeneous or the utility function is exponential. + oai:arXiv.org:2005.04312v5 + q-fin.MF + math.OC + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sebastian Bitzer, Michele Battagliola, Antonia Wachter-Zeh, Violetta Weger + Thai Nguyen, Mitja Stadje - Unsupervised Constitutive Model Discovery from Sparse and Noisy Data - https://arxiv.org/abs/2510.13559 - arXiv:2510.13559v2 Announce Type: replace-cross -Abstract: Recently, unsupervised constitutive model discovery has gained attention through frameworks based on the Virtual Fields Method (VFM), most prominently the EUCLID approach. However, the performance of VFM-based approaches, including EUCLID, is affected by measurement noise and data sparsity, which are unavoidable in practice. The statistical finite element method (statFEM) offers a complementary perspective by providing a Bayesian framework for assimilating noisy and sparse measurements to reconstruct the full-field displacement response, together with quantified uncertainty. While statFEM recovers displacement fields under uncertainty, it does not strictly enforce consistency with constitutive relations. In this work, we integrate statFEM with unsupervised constitutive model discovery in the EUCLID framework, yielding statFEM-EUCLID. The framework is demonstrated for isotropic hyperelastic materials. The results show that this integration reduces sensitivity to noise and data sparsity, while ensuring that the reconstructed fields remain consistent with both equilibrium and constitutive laws. - oai:arXiv.org:2510.13559v2 - cs.CE - cs.NA - math.NA - Wed, 21 Jan 2026 00:00:00 -0500 + Black holes in the expanding Universe + https://arxiv.org/abs/2405.16673 + arXiv:2405.16673v3 Announce Type: replace-cross +Abstract: The McVittie metric does not describe a physical black hole in an expanding Universe because the curvature scalar and pressure at its event horizon are infinite. We show that extending this metric to an inhomogeneous scale factor, which depends on both the time and radial coordinate, removes those infinities by imposing at the horizon the constancy of the Hubble parameter and a particular constraint on the gradient of the scale factor. We consider a special case of this metric, and show that the Hubble parameters at the event horizons of all centrally symmetric black holes are equal to the same constant $H_\textrm{hor}=(\Lambda/3)^{1/2}$. Because of this equality and the equivalence to the Kottler metric near the horizon, black holes do not grow with the Universe expansion. + oai:arXiv.org:2405.16673v3 + gr-qc + math-ph + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1016/j.cma.2025.118722 - Vahab Knauf Narouie, Jorge-Humberto Urrea-Quintero, Fehmi Cirak, Henning Wessels + 10.1088/1361-6382/adb537 + Class. Quantum Grav. 42, 065017 (2025) + Nikodem Pop{\l}awski - Gauss Principle in Incompressible Flow: Unified Variational Perspective on Pressure and Projection - https://arxiv.org/abs/2510.22925 - arXiv:2510.22925v2 Announce Type: replace-cross -Abstract: Following recent work (Gonzalez and Taha 2022; Peters and Ormiston 2025), this manuscript clarifies what the Gauss-Appell principle determines in incompressible, inviscid flow and how it connects to classical projection methods. At a fixed time, freezing the velocity and varying only the material acceleration leads to minimization of a quadratic subject to acceleration-level constraints. First-order conditions yield a Poisson-Neumann problem for a reaction pressure whose gradient removes the non-solenoidal and wall-normal content of the provisional residual, i.e. the Leray-Hodge projection. Thus, Gauss-Appell enforces the instantaneous kinematic constraints and recovers Euler at the instant. In steady flows, this principle -- on its own -- cannot select circulation or stagnation points because these are properties of the velocity state, not the instantaneous acceleration correction. The principle only determines the reaction pressure for an already-specified velocity field. The impressed/reaction pressure bookkeeping can be supplemented with orthogonality conventions that separate prescribed conservative forcing (if any) from the reaction enforcing the constraints. This variational viewpoint also yields a simple computational diagnostic: the minimized Appellian equals a L-2 norm of the reaction-pressure gradient which vanishes for constraint-compatible updates and grows with the magnitude of divergence and wall-flux mismatch. The goal of this note is simply to lend more clarity to the application of the Gauss principle, and to connect it concretely to well known concepts including potential flow theory, recent variational approaches and projection algorithms. - oai:arXiv.org:2510.22925v2 - physics.flu-dyn - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 + Neural Green's Operators for Parametric Partial Differential Equations + https://arxiv.org/abs/2406.01857 + arXiv:2406.01857v5 Announce Type: replace-cross +Abstract: This work introduces a paradigm for constructing parametric neural operators that are derived from finite-dimensional representations of Green's operators for linear partial differential equations (PDEs). We refer to such neural operators as Neural Green's Operators (NGOs). Our construction of NGOs preserves the linear action of Green's operators on the inhomogeneity fields, while approximating the nonlinear dependence of the Green's function on the coefficients of the PDE using neural networks. This construction reduces the complexity of the problem from learning the entire solution operator and its dependence on all parameters to only learning the Green's function and its dependence on the PDE coefficients. Furthermore, we show that our explicit representation of Green's functions enables the embedding of desirable mathematical attributes in our NGO architectures, such as symmetry, spectral, and conservation properties. Through numerical benchmarks on canonical PDEs, we demonstrate that NGOs achieve comparable or superior accuracy to Deep Operator Networks, Variationally Mimetic Operator Networks, and Fourier Neural Operators with similar parameter counts, while generalizing significantly better when tested on out-of-distribution data. For parametric time-dependent PDEs, we show that NGOs that are trained on a single time step can produce pointwise-accurate dynamics in an auto-regressive manner over arbitrarily large numbers of time steps. For parametric nonlinear PDEs, we demonstrate that NGOs trained exclusively on solutions of corresponding linear problems can be embedded within iterative solvers to yield accurate solutions, provided a suitable initial guess is available. Finally, we show that we can leverage the explicit representation of Green's functions returned by NGOs to construct effective matrix preconditioners that accelerate iterative solvers for PDEs. + oai:arXiv.org:2406.01857v5 + cs.LG + cs.NA + math.NA + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://creativecommons.org/licenses/by/4.0/ - Karthik Duraisamy + Hugo Melchers, Joost Prins, Michael Abdelmalik - Cyclic Counterfactuals under Shift-Scale Interventions - https://arxiv.org/abs/2510.25005 - arXiv:2510.25005v2 Announce Type: replace-cross -Abstract: Most counterfactual inference frameworks traditionally assume acyclic structural causal models (SCMs), i.e. directed acyclic graphs (DAGs). However, many real-world systems (e.g. biological systems) contain feedback loops or cyclic dependencies that violate acyclicity. In this work, we study counterfactual inference in cyclic SCMs under shift-scale interventions, i.e., soft, policy-style changes that rescale and/or shift a variable's mechanism. - oai:arXiv.org:2510.25005v2 - cs.AI + On the Exponential Convergence for Offline RLHF with Pairwise Comparisons + https://arxiv.org/abs/2406.12205 + arXiv:2406.12205v2 Announce Type: replace-cross +Abstract: We consider the problem of offline reinforcement learning from human feedback (RLHF) with pairwise comparisons proposed by Zhu et al. (2023), where the implicit reward is a linear function of an unknown parameter. Given an offline dataset, our objective consists in ascertaining the optimal action for each state, with the ultimate goal of minimizing the {\em simple regret}. We propose an algorithm, \underline{RL} with \underline{L}ocally \underline{O}ptimal \underline{W}eights or {\sc RL-LOW}, which yields an exponential form of simple regret of $\exp ( - \Omega(n/H) )$ where $n$ is the number of data samples and $H$ denotes an instance-dependent hardness quantity that depends explicitly on the suboptimality gap of each action. Furthermore, we derive a first-of-its-kind instance-dependent lower bound in offline RLHF with pairwise comparisons. Interestingly, we observe that the lower and upper bounds on the simple regret match order-wise in the exponent, demonstrating order-wise optimality of our {\sc RL-LOW}. In view of privacy considerations in practical applications, we also extend {\sc RL-LOW} to the setting of $(\varepsilon,\delta)$-differential privacy and show, somewhat surprisingly, that the hardness parameter $H$ is unchanged in the asymptotic regime as $n$ tends to infinity; this underscores the inherent efficiency of {\sc RL-LOW} in terms of preserving the privacy of the observed rewards. Given our focus on establishing instance-dependent bounds of exponential convergence, our research fills the research gap in existing studies that concentrate on establishing worst-case regrets of {\em inverse polynomial convergence} (e.g., $\widetilde{O}(\frac{1}{\sqrt{n}})$) for offline RLHF with pairwise comparisons. + oai:arXiv.org:2406.12205v2 cs.LG + cs.AI + cs.IT + math.IT math.ST stat.ML stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross - http://creativecommons.org/licenses/by/4.0/ - Saptarshi Saha, Dhruv Vansraj Rathore, Utpal Garain + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Zhirui Chen, Vincent Y. F. Tan - A Theory of Saving under Risk Preference Dynamics - https://arxiv.org/abs/2511.03142 - arXiv:2511.03142v3 Announce Type: replace-cross -Abstract: Empirical evidence shows that wealthy households have substantially higher saving rates and markedly lower marginal propensity to consume (MPC) than other groups. Existing theory cannot account for this pattern unless under restrictive assumptions on returns, discounting, and preferences. This paper develops a general theory of optimal savings with preference shocks, allowing risk aversion to vary across states and over time, and shows that incorporating such heterogeneity in risk attitudes fundamentally reshapes the asymptotic dynamics of consumption and saving. In particular, zero asymptotic MPCs (100% asymptotic saving rates) arise under markedly weaker conditions than in existing theory. Strikingly, such outcomes occur whenever there is a positive probability that agents become less risk averse in the future. Therefore, the vanishing MPC emerges as a generic feature rather than a knife-edge result of the optimal savings model, offering a more theoretically robust and empirically consistent account of the saving behavior of wealthy households. - oai:arXiv.org:2511.03142v3 - econ.TH + Third-quantized master equations as a classical Ornstein-Uhlenbeck process + https://arxiv.org/abs/2408.11893 + arXiv:2408.11893v3 Announce Type: replace-cross +Abstract: Third quantization is used in open quantum systems to construct a superoperator basis in which quadratic Lindbladians can be turned into a normal form. From it follows the spectral properties of the Lindbladian, including eigenvalues and eigenvectors. However, the connection between third quantization and the semiclassical representations usually employed to obtain the dynamics of open quantum systems remains opaque. We introduce an alternative basis for third quantization that bridges this gap between third quantization and the $Q$ representation by projecting the master equation onto a superoperator coherent-state basis. The equation of motion reduces to a multidimensional complex Ornstein-Uhlenbeck process. + oai:arXiv.org:2408.11893v3 + quant-ph + cond-mat.quant-gas + math-ph + math.MP + physics.class-ph + Fri, 23 Jan 2026 00:00:00 -0500 + replace-cross + http://creativecommons.org/licenses/by-nc-nd/4.0/ + 10.1103/ntv6-jzvb + Phys. Rev. A 112, 063724 (2025) + L\'eonce Dupays + + + Robust Output Tracking for Induced Seismicity Mitigation in Underground Reservoirs Governed by a Nonlinear 3D PDE-ODE System + https://arxiv.org/abs/2412.06327 + arXiv:2412.06327v3 Announce Type: replace-cross +Abstract: This paper presents a robust output-feedback controller for induced seismicity mitigation in geological reservoirs described by a coupled 3D PDE-ODE model. The controller is a MIMO Super-Twisting design, producing a continuous control signal and requiring minimal model information, while accommodating parameter uncertainty and spatial heterogeneity. Two operational outputs are regulated simultaneously: regional pressures and seismicity rates computed over reservoir sub-regions. Closed-loop properties are established via explicit bounds on the solution and its time derivative for both the infinite-dimensional dynamics and the nonlinear ODE system, yielding finite-time or exponential convergence of the tracking errors. The method is evaluated on a Groningen gas-field case study in two scenarios: gas production while not exceeding the intrinsic seismicity of the region, and combined production with CO$_2$ injection toward net-zero operation. Simulations demonstrate accurate tracking of pressure and seismicity targets across regions under significant parameter uncertainty, supporting safer reservoir operation without sacrificing production objectives. + oai:arXiv.org:2412.06327v3 + eess.SY + cs.SY math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Qingyin Ma, Xinxi Song, Alexis Akira Toda + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Diego Guti\'errez-Oribio, Ioannis Stefanou - Conformal Prediction-Driven Adaptive Sampling for Digital Water Twins - https://arxiv.org/abs/2511.05610 - arXiv:2511.05610v2 Announce Type: replace-cross -Abstract: Digital Twins (DTs) for Water Distribution Networks (WDNs) require accurate state estimation with limited sensors. Uniform sampling often wastes resources across nodes with different uncertainty. We propose an adaptive framework combining LSTM forecasting and Conformal Prediction (CP) to estimate node-wise uncertainty and focus sensing on the most uncertain points. Marginal CP is used for its low computational cost, suitable for real-time DTs. Experiments on Hanoi, Net3, and CTOWN show 33--34\% lower demand error than uniform sampling at 40\% coverage and maintain 89.4--90.2\% empirical coverage with only 5--10\% extra computation. - oai:arXiv.org:2511.05610v2 + On shallow feedforward neural networks with inputs from a topological space + https://arxiv.org/abs/2504.02321 + arXiv:2504.02321v2 Announce Type: replace-cross +Abstract: We study feedforward neural networks with inputs from a topological space (TFNNs). We prove a universal approximation theorem for shallow TFNNs, which demonstrates their capacity to approximate any continuous function defined on this topological space. As an application, we obtain an approximative version of Kolmogorov's superposition theorem for compact metric spaces. + oai:arXiv.org:2504.02321v2 cs.LG - cs.AI - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + math.FA + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross - http://creativecommons.org/licenses/by-sa/4.0/ - Mohammadhossein Homaei, Mehran Tarif, Pablo Garcia Rodriguez, Andres Caro, Mar Avila + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + 10.1007/s10472-026-10003-7 + V.E. Ismailov, Ann. Math. Artif. Intell. (2026) + Vugar Ismailov - Direction-of-Arrival and Noise Covariance Matrix joint estimation for beamforming - https://arxiv.org/abs/2511.10639 - arXiv:2511.10639v4 Announce Type: replace-cross -Abstract: We propose a joint estimation method for the Direction-of-Arrival (DoA) and the Noise Covariance Matrix (NCM) tailored for beamforming applications. Building upon an existing NCM framework, our approach simplifies the estimation procedure by deriving an quasi-linear solution, instead of the traditional exhaustive search. Additionally, we introduce a novel DoA estimation technique that operates across all frequency bins, improving robustness in reverberant environments. Simulation results demonstrate that our method outperforms classical techniques, such as MUSIC, in mid- to high-angle scenarios, achieving lower angular errors and superior signal enhancement through beamforming. The proposed framework was also fared against other techniques for signal enhancement, having better noise rejection and interference canceling capabilities. These improvements are validated using both theoretical and empirical performance metrics. - oai:arXiv.org:2511.10639v4 - eess.AS - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + New Insights into Population Dynamics from the Continuous McKendrick Model + https://arxiv.org/abs/2504.21103 + arXiv:2504.21103v3 Announce Type: replace-cross +Abstract: This article presents a comprehensive study of the continuous McKendrick model, which serves as a foundational framework in population dynamics and epidemiology. The model is formulated through partial differential equations that describe the temporal evolution of the age distribution of a population using continuously defined birth and death rates. In this work, we provide rigorous derivations of the renewal equation, establish the appropriate boundary conditions, and perform a detailed analysis of the survival functions. The central result demonstrates that the population approaches extinction if and only if the net reproduction number $R_{n}$ is strictly less than unity. We present two independent proofs: one based on Laplace transform techniques and Tauberian theorems, and another employing a reformulation as a system of ordinary differential equations with eigenvalue analysis. Additionally, we establish the connection between the deterministic framework and stochastic process formulations, showing that the McKendrick equation emerges as the fluid limit of an individual-based stochastic model. + oai:arXiv.org:2504.21103v3 + q-bio.PE + math.DS + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross - http://creativecommons.org/licenses/by/4.0/ - Vitor Gelsleichter Probst Curtarelli, Stephan Paul, Anderson Wedderhoff Spengler + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Dragos-Patru Covei - Forgetting-MarI: LLM Unlearning via Marginal Information Regularization - https://arxiv.org/abs/2511.11914 - arXiv:2511.11914v3 Announce Type: replace-cross -Abstract: As AI models are trained on ever-expanding datasets, the ability to remove the influence of specific data from trained models has become essential for privacy protection and regulatory compliance. Unlearning addresses this challenge by selectively removing parametric knowledge from the trained models without retraining from scratch, which is critical for resource-intensive models such as Large Language Models (LLMs). Existing unlearning methods often degrade model performance by removing more information than necessary when attempting to ''forget'' specific data. We introduce Forgetting-MarI, an LLM unlearning framework that provably removes only the additional (marginal) information contributed by the data to be unlearned, while preserving the information supported by the data to be retained. By penalizing marginal information, our method yields an explicit upper bound on the unlearn dataset's residual influence in the trained models, providing provable undetectability. Extensive experiments confirm that our approach outperforms current state-of-the-art unlearning methods, delivering reliable forgetting and better preserved general model performance across diverse benchmarks. This advancement represents an important step toward making AI systems more controllable and compliant with privacy and copyright regulations without compromising their effectiveness. - oai:arXiv.org:2511.11914v3 - cs.AI - cs.CL - cs.CR + Semantics-Aware Unified Terrestrial Non-Terrestrial 6G Networks + https://arxiv.org/abs/2505.01796 + arXiv:2505.01796v2 Announce Type: replace-cross +Abstract: The integration of Terrestrial and Non-Terrestrial Networks (TN-NTNs), introduced in 5G, is advancing toward a unified and seamless network of networks in Sixth-Generation (6G). This evolution markedly increases the volume of generated and exchanged data, imposing stringent technical and operational requirements along with higher cost and energy consumption. Consequently, efficient management of data generation and transmission within this unified architecture has become essential. In this article, we investigate semantics-aware information handling in unified TN-NTNs, where data communication between distant TN nodes is enabled via an NTN. We consider an Internet of Things (IoT) monitoring system in which status updates from a remote Energy Harvesting (EH) device are delivered to a destination monitor through a network of Low Earth Orbit (LEO) satellites. We leverage semantic metrics, such as Query Version Age of Information, which collectively capture the timeliness, relevance, and utility of information. This approach minimizes the transmission of stale, uninformative, or unusable information, thereby reducing the volume of data that must be transmitted and processed. The result is a substantial reduction in energy consumption and data exchange within the network-achieving up to 73% lower energy-charging requirements and fewer transmission demands than the state of the art-without compromising the conveyed information. + oai:arXiv.org:2505.01796v2 + cs.NI cs.IT - cs.LG math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Shizhou Xu, Yuan Ni, Stefan Broecker, Thomas Strohmer + Erfan Delfani, Agapi Mesodiakaki, Leandros Tassiulas, Nikolaos Pappas - Empirical Quantum Advantage in Constrained Optimization from Encoded Unitary Designs - https://arxiv.org/abs/2511.14296 - arXiv:2511.14296v3 Announce Type: replace-cross -Abstract: We introduce the Constraint-Enhanced Quantum Approximate Optimization Algorithm (CE-QAOA), a shallow, constraint-aware ansatz that operates inside the one-hot product space [n]^m, where m is the number of blocks and each block is initialized in an n-qubit W_n state. We give an ancilla-free, depth-optimal encoder that prepares W_n using n-1 two-qubit rotations per block, and a two-local block-XY mixer that preserves the one-hot manifold and has a constant spectral gap on the one-excitation sector. At the level of expressivity, we establish per-block controllability, implying approximate universality per block. At the level of distributional behavior, we show that, after natural block and symbol permutation twirls, shallow CE-QAOA realizes an encoded unitary 1-design and supports approximate second-moment (2-design) behavior; combined with a Paley-Zygmund argument, this yields finite-shot anticoncentration guarantees. - Algorithmically, we wrap constant-depth sampling with a deterministic feasibility checker to obtain a polynomial-time hybrid quantum-classical solver (PHQC) that returns the best observed feasible solution in O(S n^2) time, where S is a polynomial shot budget. We obtain two advantages. First, when CE-QAOA fixes r >= 1 locations different from the start city, we achieve a Theta(n^r) reduction in shot complexity even against a classical sampler that draws uniformly from the feasible set. Second, against a classical baseline restricted to raw bitstring sampling, we show an exp(Theta(n^2)) minimax separation. In noiseless circuit simulations of traveling salesman problem instances with n in {4,...,10} locations from the QOPTLib benchmark library, we recover the global optimum at depth p = 1 using polynomial shot budgets and coarse parameter grids defined by the problem size. - oai:arXiv.org:2511.14296v3 - cs.ET - cs.DM + Chaotic Kramers' Law: Hasselmann's Program and AMOC Tipping + https://arxiv.org/abs/2505.18904 + arXiv:2505.18904v2 Announce Type: replace-cross +Abstract: In bistable dynamical systems driven by Wiener processes, the widely used Kramers' law relates the strength of the noise forcing to the average time it takes to see a noise-induced transition from one attractor to the other. We extend this law to bistable systems forced by fast chaotic dynamics, which we argue is in some cases a more realistic modeling approach than unbounded noise forcing. Transitions similar to the noise-driven case can only occur if the amplitude of the chaotic forcing is large enough. If this is the case, in our numerical example - a reduced-order model of the Atlantic Meridional Overturning Circulation (AMOC) - we observe the chaotic Kramers' law to hold even when the chaotic forcing is far from the stochastic limit. We discuss the limitations of the chaotic Kramers' law, how to address the numerical issues associated with the timescale separation, and give a possible explanation for the dynamics of recently found AMOC collapses and recoveries in complex climate models. + oai:arXiv.org:2505.18904v2 + nlin.CD math-ph + math.DS math.MP - physics.app-ph - quant-ph - Wed, 21 Jan 2026 00:00:00 -0500 + physics.bio-ph + physics.chem-ph + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://creativecommons.org/licenses/by/4.0/ - Chinonso Onah, Roman Firt, Kristel Michielsen + Jakob Deser, Raphael R\"omer, Niklas Boers, Christian Kuehn - Folded optimal transport and its application to separable quantum optimal transport - https://arxiv.org/abs/2512.01722 - arXiv:2512.01722v3 Announce Type: replace-cross -Abstract: We introduce folded optimal transport, as a method to extend a cost or distance defined on the extreme boundary of a convex to the whole convex, related to convex extension. This construction broadens the framework of standard optimal transport, found to be the particular case of the convex being a simplex. Relying on Choquet's theory and standard optimal transport, we introduce the folded Kantorovich cost and folded Wasserstein distances, and study their induced metric properties. We then apply the construction to the quantum setting, and obtain an actual separable quantum Wasserstein distance on the set of density matrices from a distance on the set of pure states, closely related to the semi-distance of Beatty and Stilck-Franca [4], and of which we obtain a variety of properties. We also find that the semiclassical Golse-Paul [16] cost writes as a folded Kantorovich cost. Folded optimal transport therefore provides a unified framework for classical, semiclassical and separable quantum optimal transport. - oai:arXiv.org:2512.01722v3 + Numerical Optimization Strategies for the Variational Hamiltonian Ansatz in Noisy Quantum Environments + https://arxiv.org/abs/2505.22398 + arXiv:2505.22398v4 Announce Type: replace-cross +Abstract: The prevalence of variational methods in near-term quantum computing makes optimizer choice critical, yet selection is frequently intuition-based. We therefore present a systematic benchmark of eight classical optimization algorithms for variational quantum chemistry using the truncated Variational Hamiltonian Ansatz. Performance is evaluated on H$_2$, H$_4$, and LiH in both full and active-space representations under noiseless and finite-shot sampling noise. Sampling noise substantially reshapes cost landscapes, induces wandering near minima, and flips optimizer rankings: gradient-based methods perform best in noiseless simulations, whereas population-based optimizers, particularly CMA-ES, show greater robustness under finite-shot noise. Optimizer performance is strongly problem dependent: Hartree-Fock initialization aids small systems, but its advantage diminishes with system size. Also, we observe that finite shot sampling frequently violates the lower bound given by the variational principle, a principle that cannot be strictly held in the presence of noise. By exploiting the guaranteed convergence of Evolution Strategies to a steady state distribution defined by the noise floor, we utilize the symmetry of these violations to achieve energy estimation precision beyond the intrinsic sampling limit. + oai:arXiv.org:2505.22398v4 quant-ph - math-ph - math.FA - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Thomas Borsoni - - - Generalised 4d Partition Functions and Modular Differential Equations - https://arxiv.org/abs/2512.02107 - arXiv:2512.02107v2 Announce Type: replace-cross -Abstract: We prove the equivalence of a class of generalised Schur partition functions $\mathcal Z_G(q;\alpha)$ of 4d $\mathcal N=2$ superconformal gauge theories to contour integral representations of vector-valued modular forms of the type that arise in 2d rational conformal field theories (RCFT). Concretely, we consider the $USp(2N)$ theory with $2N+2$ fundamental hypermultiplets and analytically prove that $\mathcal Z_{USp(2N)}(q;\alpha)$ satisfies an order-$(N+1)$ modular linear differential equation (MLDE) with vanishing Wronskian index, explaining how the parameter $\alpha$ of the former determines the parameters of the latter. Several connections are made to characters of RCFTs including unitary ones. We then propose a two-parameter extension $\mathcal Z_{USp(2N)}(q;\alpha,\beta)$ of the generalised Schur partition function. Finally, we relate the $\alpha=-k$ specialisation to quantum monodromy traces ${\rm Tr}\,M^k$ and formulate a conjecture linking their $k$-dependence to MLDEs. - oai:arXiv.org:2512.02107v2 - hep-th - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 + cs.NA + math.NA + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://creativecommons.org/licenses/by/4.0/ - A. Ramesh Chandra, Sunil Mukhi, Palash Singh + S. Ill\'esov\'a, V. Nov\'ak, T. Bezd\v{e}k, C. Possel, M. Beseda - Interplay between Standard Quantum Detailed Balance and Thermodynamically Consistent Entropy Production - https://arxiv.org/abs/2512.06707 - arXiv:2512.06707v2 Announce Type: replace-cross -Abstract: We demonstrate that, for a quantum Markovian semigroup on a finite-dimensional Hilbert space, if it satisfies the standard quantum detailed balance condition, its generator admits a special representation that yields a vanishing entropy production rate. Conversely, if the generator admits a special representation adhering to the condition of thermodynamic consistency and leading to a vanishing entropy production rate, then the corresponding quantum Markovian semigroup must satisfy the standard quantum detailed balance condition. In this context, we adopt the definition of entropy production rate that is motivated by the physics literature and standard for thermodynamically consistent Lindbladians. - oai:arXiv.org:2512.06707v2 + Erasure cost of a quantum process: A thermodynamic meaning of the dynamical min-entropy + https://arxiv.org/abs/2506.05307 + arXiv:2506.05307v4 Announce Type: replace-cross +Abstract: The erasure of information is fundamentally an irreversible logical operation, carrying profound consequences for the energetics of computation and information processing. We investigate the thermodynamic costs associated with erasing (and preparing) quantum processes. Specifically, we analyze an arbitrary bipartite unitary gate acting on logical and ancillary input-output systems, where the ancillary input is always initialized in the ground state. We focus on the adversarial erasure cost of the reduced dynamics -- that is, the minimal thermodynamic work cost to erase the logical output of the gate for any logical input, assuming full access to the ancilla but no access to any purifying reference of the logical input state. We determine that this adversarial erasure cost is directly proportional to the negative min-entropy of the reduced dynamics, thereby giving the dynamical min-entropy a clear operational meaning. The dynamical min-entropy can take positive and negative values, depending on the underlying quantum dynamics. The negative value of the erasure cost implies that the extraction of thermodynamic work is possible instead of its consumption during the process. A key foundation of this result is the quantum process decoupling theorem, which quantitatively relates the decoupling ability of a process with its min-entropy. This insight bridges thermodynamics, information theory, and the fundamental limits of quantum computation. + oai:arXiv.org:2506.05307v4 quant-ph - cond-mat.stat-mech math-ph math.MP - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xin-Hai Tong, Kohei Yoshimura, Tan Van Vu, Naruo Ohga + 10.1088/2058-9565/ae34e2 + Quantum Science and Technology, vol. 11, no. 1, page 015038, January 2026 + Himanshu Badhani, Dhanuja GS, Swati Choudhary, Vishal Anand, Siddhartha Das - Heuristics for Combinatorial Optimization via Value-based Reinforcement Learning: A Unified Framework and Analysis - https://arxiv.org/abs/2512.08601 - arXiv:2512.08601v2 Announce Type: replace-cross -Abstract: Since the 1990s, considerable empirical work has been carried out to train statistical models, such as neural networks (NNs), as learned heuristics for combinatorial optimization (CO) problems. When successful, such an approach eliminates the need for experts to design heuristics per problem type. Due to their structure, many hard CO problems are amenable to treatment through reinforcement learning (RL). Indeed, we find a wealth of literature training NNs using value-based, policy gradient, or actor-critic approaches, with promising results, both in terms of empirical optimality gaps and inference runtimes. Nevertheless, there has been a paucity of theoretical work undergirding the use of RL for CO problems. To this end, we introduce a unified framework to model CO problems through Markov decision processes (MDPs) and solve them using RL techniques. We provide easy-to-test assumptions under which CO problems can be formulated as equivalent undiscounted MDPs that provide optimal solutions to the original CO problems. Moreover, we establish conditions under which value-based RL techniques converge to approximate solutions of the CO problem with a guarantee on the associated optimality gap. Our convergence analysis provides: (1) a sufficient rate of increase in batch size and projected gradient descent steps at each RL iteration; (2) the resulting optimality gap in terms of problem parameters and targeted RL accuracy; and (3) the importance of a choice of state-space embedding. Together, our analysis illuminates the success (and limitations) of the celebrated deep Q-learning algorithm in this problem context. - oai:arXiv.org:2512.08601v2 - stat.ML - cs.LG - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + Distance-Domain Degrees of Freedom in Near-Field Region + https://arxiv.org/abs/2507.01227 + arXiv:2507.01227v3 Announce Type: replace-cross +Abstract: Extremely large aperture arrays operating in the near-field regime unlock additional spatial resources, which can be exploited to simultaneously serve multiple users even when they share the same angular direction. This work investigates the distance-domain degrees of freedom (DoF), defined as the DoF when a user varies only its distance to the base station and not the angle. To obtain the distance-domain DoF, we study a line-of-sight (LoS) channel with a source representing a base station and an observation region representing users, where the source is a large two-dimensional transmit (Tx) array with arbitrary shape and the observation region is an arbitrarily long linear receive (Rx) array with collinearly aligned elements located at different distances from the Tx array. We assume that both the Tx and Rx arrays have continuous apertures with an infinite number of elements and infinitesimal spacing, which establishes an upper bound for the distance-domain DoF in the case of a finite number of elements. First, we analyze an ideal case where the Tx array is a single piece and the Rx array is on the broadside of the Tx array. By reformulating the channel as an integral operator with a Hermitian convolution kernel, we derive a closed-form expression for the distance-domain DoF via the Fourier transform. Our analysis shows that the distance-domain DoF is predominantly determined by the extreme boundaries of both the Tx and Rx arrays rather than their detailed interior structure. We further extend the framework to non-broadside configurations by employing a projection method that converts the problem to an equivalent broadside case. Finally, we extend the analytical framework to modular arrays and show the distance-domain DoF gain over a single-piece array under a fixed total physical length. + oai:arXiv.org:2507.01227v3 + eess.SP + cs.IT + math.IT + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross - http://creativecommons.org/licenses/by/4.0/ - Orit Davidovich, Shimrit Shtern, Segev Wasserkrug, Nimrod Megiddo + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Son T. Duong, Tho Le-Ngoc - Chaotic discretization theorems for forced linear and nonlinear coupled oscillators - https://arxiv.org/abs/2512.10565 - arXiv:2512.10565v2 Announce Type: replace-cross -Abstract: We prove the holding of chaos in the sense of Li-Yorke for a family of four-dimensional discrete dynamical systems that are naturally associated to ODEs systems describing coupled oscillators subject to an external non-conservative force, also giving an example of a discrete map that is Li-Yorke chaotic but not topologically transitive. Analytical results are generalized to a modular definition of the problem and to a system of nonlinear oscillators described by polynomial potentials in one coordinate. We perform numerical simulations looking for a strange attractor of the system; furthermore, we perform a bifurcation analysis of the system presenting 1D and 2D bifurcation diagrams, together with spectra of Lyapunov exponents and basins of attraction. - oai:arXiv.org:2512.10565v2 - nlin.CD - math-ph - math.DS - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 + Stability, Complexity and Data-Dependent Worst-Case Generalization Bounds + https://arxiv.org/abs/2507.06775 + arXiv:2507.06775v2 Announce Type: replace-cross +Abstract: Providing generalization guarantees for stochastic optimization algorithms remains a key challenge in learning theory. Recently, numerous works demonstrated the impact of the geometric properties of optimization trajectories on generalization performance. These works propose worst-case generalization bounds in terms of various notions of intrinsic dimension and/or topological complexity, which were found to empirically correlate with the generalization error. However, most of these approaches involve intractable mutual information terms, which limit a full understanding of the bounds. In contrast, some authors built on algorithmic stability to obtain worst-case bounds involving geometric quantities of a combinatorial nature, which are impractical to compute. In this paper, we address these limitations by combining empirically relevant complexity measures with a framework that avoids intractable quantities. To this end, we introduce the concept of \emph{random set stability}, tailored for the data-dependent random sets produced by stochastic optimization algorithms. Within this framework, we show that the worst-case generalization error can be bounded in terms of (i) the random set stability parameter and (ii) empirically relevant, data- and algorithm-dependent complexity measures of the random set. Moreover, our framework improves existing topological generalization bounds by recovering previous complexity notions without relying on mutual information terms. Through a series of experiments in practically relevant settings, we validate our theory by evaluating the tightness of our bounds and the interplay between topological complexity and stability. + oai:arXiv.org:2507.06775v2 + cs.LG + math.AT + stat.ML + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Stefano Disca, Vincenzo Coscia + Mario Tuci, Lennart Bastian, Benjamin Dupuis, Nassir Navab, Tolga Birdal, Umut \c{S}im\c{s}ekli - Quadratic Stability of Entropy Minimizers under Block-Separable Convex Constraints - https://arxiv.org/abs/2512.16192 - arXiv:2512.16192v2 Announce Type: replace-cross -Abstract: We investigate entropy minimization problems for quantum states subject to convex block-separable constraints. Our principal result is a quantitative stability theorem: under a natural confining (fixed-support) hypothesis, if a state has entropy within {\epsilon} of the minimum permitted by the constraint, then it must lie within O({\epsilon}^{1/2}) in trace norm of the set of entropy minimizers. We show that this rate is optimal and cannot be improved uniformly. - The analysis is entirely finite-dimensional and exploits the block-separable structure of the constraint set, which induces a natural decomposition of entropy into a marginal (classical) component and conditional (internal) components. Quadratic stability emerges from the curvature of Shannon entropy on the marginal polytope and of von Neumann entropy on the constrained block states, yielding explicit stability constants determined by the geometry of the constraint. - We further demonstrate that this stability phenomenon cannot be derived from Pinsker-type inequalities or standard entropy continuity bounds, since no reference state is fixed a priori and the entropy minimizer arises intrinsically from the constraint geometry. The framework is abstract and independent of any arithmetic input, and provides a general quadratic stability principle for entropy minimization under structured convex constraints. - oai:arXiv.org:2512.16192v2 + Einstein Electron and Local Branching: Unitarity without Many Worlds --Local Hilbert spaces, boundaries, and quantum nonlocality + https://arxiv.org/abs/2507.16123 + arXiv:2507.16123v2 Announce Type: replace-cross +Abstract: Traditional interpretations of quantum mechanics often present a dichotomy: either the wavefunction collapses upon measurement (Copenhagen), violating unitarity, or the entire universe branches into countless parallel worlds (Many-Worlds), with significant ontological proliferation. The Branched Hilbert Subspace Interpretation (BHSI) resolves this tension by introducing branching strictly within local Hilbert spaces. This framework reinterprets scenarios such as the Einstein 1927 electron-diffraction thought experiment, in which all quantum events are confined to a local Hilbert space, allowing the Born rule to emerge naturally from branch weights. Crucially, BHSI treats branching as a dynamical process tied to information recording. This leads to a testable proposal: a dual-layer experiment in which the particle transit time between layers is shorter than the sensor response time, enabling a direct probe of measurement timing and mismatched or uncommitted outcomes. We argue that a quantum system behaves as a unified whole, an island of coherence, within which unitary branching is confined to the system boundary, without observable correlations with distant, unentangled systems. Finally, we show that quantum nonlocality (e.g., in Bell tests or tunneling) arises naturally from the intrinsic vector-space structure of local Hilbert spaces, rather than from superluminal signaling. + oai:arXiv.org:2507.16123v2 quant-ph - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 + math-ph + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hassan Nasreddine + Xing M. Wang - Deep Legendre Transform - https://arxiv.org/abs/2512.19649 - arXiv:2512.19649v2 Announce Type: replace-cross -Abstract: We introduce a novel deep learning algorithm for computing convex conjugates of differentiable convex functions, a fundamental operation in convex analysis with various applications in different fields such as optimization, control theory, physics and economics. While traditional numerical methods suffer from the curse of dimensionality and become computationally intractable in high dimensions, more recent neural network--based approaches scale better, but have mostly been studied with the aim of solving optimal transport problems and require the solution of complicated optimization or max--min problems. Using an implicit Fenchel formulation of convex conjugation, our approach facilitates an efficient gradient--based framework for the minimization of approximation errors and, as a byproduct, also provides a posteriori estimates of the approximation accuracy. Numerical experiments demonstrate our method's ability to deliver accurate results across different high-dimensional examples. Moreover, by employing symbolic regression with Kolmogorov--Arnold networks, it is able to obtain the exact convex conjugates of specific convex functions. - oai:arXiv.org:2512.19649v2 + Likelihood Matching for Diffusion Models + https://arxiv.org/abs/2508.03636 + arXiv:2508.03636v2 Announce Type: replace-cross +Abstract: We propose a Likelihood Matching approach for training diffusion models by first establishing an equivalence between the likelihood of the target data distribution and a likelihood along the sample path of the reverse diffusion. To efficiently compute the reverse sample likelihood, a quasi-likelihood is considered to approximate each reverse transition density by a Gaussian distribution with matched conditional mean and covariance, respectively. The score and Hessian functions for the diffusion generation are estimated by maximizing the quasi-likelihood, ensuring a consistent matching of both the first two transitional moments between every two time points. A stochastic sampler is introduced to facilitate computation that leverages both the estimated score and Hessian information. We establish consistency of the quasi-maximum likelihood estimation, and provide non-asymptotic convergence guarantees for the proposed sampler, quantifying the rates of the approximation errors due to the score and Hessian estimation, dimensionality, and the number of diffusion steps. Empirical and simulation evaluations demonstrate the effectiveness of the proposed Likelihood Matching and validate the theoretical results. + oai:arXiv.org:2508.03636v2 + stat.ML cs.LG - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Aleksey Minabutdinov, Patrick Cheridito - - - Validation methodology on real data of reversible Kalman Filter for state estimation with Manifold - https://arxiv.org/abs/2512.22126 - arXiv:2512.22126v2 Announce Type: replace-cross -Abstract: This work extends a previous study that introduced an algorithm for state estimation on manifolds within the framework of the Kalman filter. Its objective is to address the limitations of the earlier approach. The reversible Kalman filter was designed to provide a methodology for evaluating the accuracy of existing Kalman filter variants with arbitrary precision on synthetic data. It has favorable numerical properties on synthetic data, achieving arbitrary precision without relying on the small-velocity assumption and depending only on sensor noise. However, its application to real data encountered difficulties related to measurement noise, which was mitigated using a heuristic. In particular, the heuristic involved an event detection step switching between reversible Kalman filter and classical Kalman variant at chosen moments. In the present work, we propose a study of this detection step and propose a methodology to prove at which moment the reversible Kalman approach improves on classical multiplicative variant. - oai:arXiv.org:2512.22126v2 - eess.SY - cs.SY - math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + math.ST + stat.AP + stat.ME + stat.TH + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Svyatoslav Covanov, Cedric Pradalier + http://creativecommons.org/licenses/by/4.0/ + Lei Qian, Wu Su, Yanqi Huang, Song Xi Chen - Les Houches Lecture Notes on Tensor Networks - https://arxiv.org/abs/2512.24390 - arXiv:2512.24390v2 Announce Type: replace-cross -Abstract: Tensor networks provide a powerful new framework for classifying and simulating correlated and topological phases of quantum matter. Their central premise is that strongly correlated matter can only be understood by studying the underlying entanglement structure and its associated (generalised) symmetries. In essence, tensor networks provide a compressed, holographic description of the complicated vacuum fluctuations in strongly correlated systems, and as such they break down the infamous many-body exponential wall. These lecture notes provide a concise overview of the most important conceptual, computational and mathematical aspects of this theory. - oai:arXiv.org:2512.24390v2 - cond-mat.str-el - hep-th - math-ph - math.MP + Quantum matrix arithmetics with Hamiltonian evolution + https://arxiv.org/abs/2510.06316 + arXiv:2510.06316v2 Announce Type: replace-cross +Abstract: The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We develop a suite of methods to perform matrix arithmetics -- with the result encoded in the off-diagonal blocks of a Hamiltonian -- using Hamiltonian evolutions of input operators. We show how to maintain this $\textit{Hamiltonian block encoding}$, so that matrix operations can be composed one after another, and the entire quantum computation takes $\leq 2$ ancilla qubits. We achieve this for matrix multiplication, matrix addition, matrix inversion, Hermitian conjugation, fractional scaling, integer scaling, complex phase scaling, as well as singular value transformation for both odd and even polynomials. We also present an overlap estimation algorithm to extract classical properties of Hamiltonian block encoded operators, analogous to the well known Hadmard test, at no extra cost of qubit. Our Hamiltonian matrix multiplication uses the Lie group commutator product formula and its higher-order generalizations due to Childs and Wiebe. Our Hamiltonian singular value transformation employs a dominated polynomial approximation, where the approximation holds within the domain of interest, while the constructed polynomial is upper bounded by the target function over the entire unit interval. We describe a circuit for simulating a class of sum-of-squares Hamiltonians, attaining a commutator scaling in step count, while leveraging the power of matrix arithmetics to reduce the cost of each simulation step. In particular, we apply this to the doubly factorized tensor hypercontracted Hamiltonians from recent studies of quantum chemistry, obtaining further improvements for initial states with a fixed number of particles. We achieve this with $1$ ancilla qubit. + oai:arXiv.org:2510.06316v2 quant-ph - Wed, 21 Jan 2026 00:00:00 -0500 + cs.DS + cs.NA + math.NA + physics.chem-ph + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross - http://creativecommons.org/licenses/by/4.0/ - Bram Vancraeynest-De Cuiper, Weronika Wiesiolek, Frank Verstraete + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Christopher Kang, Yuan Su - Exact finite mixture representations for species sampling processes - https://arxiv.org/abs/2512.24414 - arXiv:2512.24414v2 Announce Type: replace-cross -Abstract: Discrete random probability measures are central to Bayesian inference, particularly as priors for mixture modeling and clustering. A broad and unifying class is that of proper species sampling processes (SSPs), encompassing many Bayesian nonparametric priors. We show that any proper SSP admits an exact conditional finite-mixture representation by augmenting the model with a latent truncation index and a simple reweighting of the atoms, which yields a conditional random finite-atom measure whose marginalized distribution matches the original SSP. This yields at least two consequences: (i) distributionally exact simulation for arbitrary SSPs, without user-chosen truncation levels; and (ii) posterior inference in SSP mixture models via standard finite-mixture machinery, leading to tractable MCMC algorithms without ad hoc truncations. We explore these consequences by deriving explicit total-variation bounds for the conditional approximation error when this truncation is fixed, and by studying practical performance in mixture modeling, with emphasis on Dirichlet and geometric SSPs. - oai:arXiv.org:2512.24414v2 + Inference in pseudo-observation-based regression using (biased) covariance estimation and naive bootstrapping + https://arxiv.org/abs/2510.06815 + arXiv:2510.06815v2 Announce Type: replace-cross +Abstract: The pseudo-observation method is regularly applied to time-to-event data. However, to date such analyses have relied on not formally verified statements or ad-hoc methods regarding covariance estimation. This paper strives to close this gap in the literature. To begin with, we demonstrate that the usual Huber-White estimator is not consistent for the limiting covariance of parameter estimates in pseudo-observation regression approaches. By confirming that a plug-in estimator can be used instead, we obtain asymptotically exact and consistent tests for general linear hypotheses in the parameters of the model. Additionally, we confirm that naive bootstrapping can not be used for covariance estimation in the pseudo-observation model either. However, it can be used for hypothesis testing by applying a suitable studentization. Simulations illustrate the good performance of our proposed methods in many scenarios. Finally, we obtain a general uniform law of large numbers for U- and V-statistics, as such statistics are central in the mathematical analysis of the inference procedures developed in this work. + oai:arXiv.org:2510.06815v2 stat.ME math.ST - stat.CO stat.TH - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Rams\'es H. Mena, Christos Merkatas, Theodoros Nicoleris, Carlos E. Rodr\'iguez + http://creativecommons.org/licenses/by-nc-nd/4.0/ + Simon Mack, Morten Overgaard, Dennis Dobler - Spin-operator form factors of the critical Ising chain and their finite volume scaling limits - https://arxiv.org/abs/2601.00751 - arXiv:2601.00751v4 Announce Type: replace-cross -Abstract: In this work, we provide a self-contained derivation of the spin-operator matrix elements in the fermionic basis, for the critical periodic Ising chain at a generic system length $N\in 2Z_{\ge 2}$. The approach relies on the near-Cauchy property of certain matrices formed by the Toeplitz symbols in the critical model, and leads to a few square-root products for the leg functions. The square root products allow simple integral representations, that further reduce to the Binet's second integral and its generalization by Hermite, in the finite volume scaling limit. This leads to product formulas for the spin operator matrix elements in the scaling limit, providing explicit expressions for the spin-operator form factors of the Ising CFT in the fermionic basis, that were computed iteratively in Yurov:1991my. They are all rational numbers up to $\sqrt{2}$. We also determine the normalization factor of the spin-operator and show explicitly how the coefficient $G(\frac{1}{2})G(\frac{3}{2})$ appear through a ground state overlap. Moreover, by expanding the spin-spin two point correlator in the fermionic basis, we observed a Fredholm determinant identity that allows to show the convergence of the rescaled two-point correlator to the CFT version on a cylinder. - oai:arXiv.org:2601.00751v4 - hep-th + Thermodynamics of quantum processes: An operational framework for free energy and reversible athermality + https://arxiv.org/abs/2510.12790 + arXiv:2510.12790v3 Announce Type: replace-cross +Abstract: We explore the thermodynamics of quantum processes (quantum channels) by axiomatically introducing the free energy for channels, defined via the quantum relative entropy with an absolutely thermal channel whose fixed output is in equilibrium with a thermal reservoir. This definition finds strong support through its operational interpretations in designated quantum information and thermodynamic tasks. We construct a resource theory of athermality for quantum processes, where free operations are Gibbs preserving superchannels and golden units are unitary channels with respect to absolutely thermal channel having fully degenerate output Hamiltonian. We exactly characterize the one-shot distillation and formation of quantum channels using hypothesis-testing and max-relative entropy with respect to the absolutely thermal channel. These rates converge asymptotically to the channel free energy (up to a multiplicative factor of half the inverse temperature), establishing its operational meaning and proving the asymptotic reversibility of the athermality. We show the direct relation between the resource theory of athermality and quantum information tasks such as private randomness and purity distillation, and thermodynamic tasks of erasure and work extraction. Our work connects the core thermodynamic concepts of free energy, energy, entropy, and maximal extractable work of quantum processes to their information processing capabilities. + oai:arXiv.org:2510.12790v3 + quant-ph cond-mat.stat-mech + hep-th math-ph math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yizhuang Liu - - - Renewal theory for Brownian motion with stochastically gated targets - https://arxiv.org/abs/2601.01588 - arXiv:2601.01588v2 Announce Type: replace-cross -Abstract: There are a wide range of first passage time (FPT) problems in the physical and life sciences that can be modelled in terms of a Brownian particle binding to a reactive surface (absorption). However, prior to absorption, the particle may undergo several rounds of surface attachment (adsorption), detachment (desorption) and diffusion. Alternatively, the surface may be stochastically gated so that absorption can only occur when the gate is open. In both cases one can view each return to the surface as a renewal event. In this paper we develop a renewal theory for stochastically gated FPT problems along analogous lines to previous work on adsorption/desorption processes. We proceed by constructing a renewal equation that relates the joint probability density for particle position and the state of a gate (or multiple gates) to the probability density and FPT density for a totally absorbing (non-gated) boundary. This essentially decomposes sample paths into an alternating sequence of bulk diffusion and instantaneous adsorption/desorption events, which is terminated when adsorption coincides with an open gate. Through a variety of examples, we show how renewal theory provides a general mathematical framework for incorporating stochastic gating into FPT problems. - oai:arXiv.org:2601.01588v2 - cond-mat.stat-mech - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Paul C Bressloff - - - Horizon Activation Mapping for Neural Networks in Time Series Forecasting - https://arxiv.org/abs/2601.02094 - arXiv:2601.02094v3 Announce Type: replace-cross -Abstract: Neural networks for time series forecasting have relied on error metrics and architecture-specific interpretability approaches for model selection that don't apply across models of different families. To interpret forecasting models agnostic to the types of layers across state-of-the-art model families, we introduce Horizon Activation Mapping (HAM), a visual interpretability technique inspired by grad-CAM that uses gradient norm averages to study the horizon's subseries where grad-CAM studies attention maps over image data. We introduce causal and anti-causal modes to calculate gradient update norm averages across subseries at every timestep and lines of proportionality signifying uniform distributions of the norm averages. Optimization landscape studies with respect to changes in batch sizes, early stopping, train-val-test splits, architectural choices, univariate forecasting and dropouts are studied with respect to performances and subseries in HAM. Interestingly, batch size based differences in activities seem to indicate potential for existence of an exponential approximation across them per epoch relative to each other. Multivariate forecasting models including MLP-based CycleNet, N-Linear, N-HITS, self attention-based FEDformer, Pyraformer, SSM-based SpaceTime and diffusion-based Multi-Resolution DDPM over different horizon sizes trained over the ETTm2 dataset are used for HAM plots in this study. NHITS' neural approximation theorem and SpaceTime's exponential autoregressive activities have been attributed to trends in HAM plots over their training, validation and test sets. In general, HAM can be used for granular model selection, validation set choices and comparisons across different neural network model families. - oai:arXiv.org:2601.02094v3 - cs.LG - math.FA - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Krupakar Hans, V A Kandappan + Himanshu Badhani, Dhanuja G. S., Siddhartha Das - Normalized Conditional Mutual Information Surrogate Loss for Deep Neural Classifiers - https://arxiv.org/abs/2601.02543 - arXiv:2601.02543v3 Announce Type: replace-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:2601.02543v3 + Robust Reinforcement Learning in Finance: Modeling Market Impact with Elliptic Uncertainty Sets + https://arxiv.org/abs/2510.19950 + arXiv:2510.19950v2 Announce Type: replace-cross +Abstract: In financial applications, reinforcement learning (RL) agents are commonly trained on historical data, where their actions do not influence prices. However, during deployment, these agents trade in live markets where their own transactions can shift asset prices, a phenomenon known as market impact. This mismatch between training and deployment environments can significantly degrade performance. Traditional robust RL approaches address this model misspecification by optimizing the worst-case performance over a set of uncertainties, but typically rely on symmetric structures that fail to capture the directional nature of market impact. To address this issue, we develop a novel class of elliptic uncertainty sets. We establish both implicit and explicit closed-form solutions for the worst-case uncertainty under these sets, enabling efficient and tractable robust policy evaluation. Experiments on single-asset and multi-asset trading tasks demonstrate that our method achieves superior Sharpe ratio and remains robust under increasing trade volumes, offering a more faithful and scalable approach to RL in financial markets. + oai:arXiv.org:2510.19950v2 cs.LG cs.AI - cs.CV - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + math.OC + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://creativecommons.org/licenses/by/4.0/ - Linfeng Ye, Zhixiang Chi, Konstantinos N. Plataniotis, En-hui Yang + Shaocong Ma, Heng Huang - Fast Surrogate Models for Adaptive Aircraft Trajectory Prediction in En route Airspace - https://arxiv.org/abs/2601.03075 - arXiv:2601.03075v2 Announce Type: replace-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:2601.03075v2 - cs.CE - math.DS - Wed, 21 Jan 2026 00:00:00 -0500 + Cut-free Deductive System for Continuous Intuitionistic Logic + https://arxiv.org/abs/2510.26849 + arXiv:2510.26849v3 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:2510.26849v3 + cs.LO + math.LO + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross - http://creativecommons.org/licenses/by/4.0/ - 10.2514/6.2026-1611 - Nick Pepper, Marc Thomas, Zack Xuereb Conti + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Guillaume Geoffroy (UCBL, ICJ, AGL) - SIGMA: Scalable Spectral Insights for LLM Model Collapse - https://arxiv.org/abs/2601.03385 - arXiv:2601.03385v2 Announce Type: replace-cross -Abstract: The rapid adoption of synthetic data for training Large Language Models (LLMs) has introduced the technical challenge of "model collapse"-a degenerative process where recursive training on model-generated content leads to a contraction of distributional variance and representational quality. While the phenomenology of collapse is increasingly evident, rigorous methods to quantify and predict its onset in high-dimensional spaces remain elusive. In this paper, we introduce SIGMA (Spectral Inequalities for Gram Matrix Analysis), a unified framework that benchmarks model collapse through the spectral lens of the embedding Gram matrix. By deriving and utilizing deterministic and stochastic bounds on the matrix's spectrum, SIGMA provides a mathematically grounded metric to track the contraction of the representation space. Crucially, our stochastic formulation enables scalable estimation of these bounds, making the framework applicable to large-scale foundation models where full eigendecomposition is intractable. We demonstrate that SIGMA effectively captures the transition towards degenerate states, offering both theoretical insights into the mechanics of collapse and a practical, scalable tool for monitoring the health of recursive training pipelines. - oai:arXiv.org:2601.03385v2 + TUN: Detecting Significant Points in Persistence Diagrams with Deep Learning + https://arxiv.org/abs/2512.14274 + arXiv:2512.14274v2 Announce Type: replace-cross +Abstract: Persistence diagrams (PDs) provide a powerful tool for understanding the topology of the underlying shape of a point cloud. However, identifying which points in PDs encode genuine signals remains challenging. This challenge directly hinders the practical adoption of topological data analysis in many applications, where automated and reliable interpretation of persistence diagrams is essential for downstream decision-making. In this paper, we study automatic significance detection for one-dimensional persistence diagrams. Specifically, we propose Topology Understanding Net (TUN), a multi-modal network that combines enhanced PD descriptors with self-attention, a PointNet-style point cloud encoder, learned fusion, and per-point classification, alongside stable preprocessing and imbalance-aware training. It provides an automated and effective solution for identifying significant points in PDs, which are critical for downstream applications. Experiments show that TUN outperforms classic methods in detecting significant points in PDs, illustrating its effectiveness in real-world applications. + oai:arXiv.org:2512.14274v2 + cs.CV cs.LG - math.PR - Wed, 21 Jan 2026 00:00:00 -0500 + math.AT + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross - http://creativecommons.org/licenses/by-sa/4.0/ - Yi Gu, Lingyou Pang, Xiangkun Ye, Tianyu Wang, Jianyu Lin, Carey E. Priebe, Alexander Aue + http://arxiv.org/licenses/nonexclusive-distrib/1.0/ + Yu Chen, Hongwei Lin - The Geometry of the Pivot: A Note on Lazy Pivoted Cholesky and Farthest Point Sampling - https://arxiv.org/abs/2601.03706 - arXiv:2601.03706v3 Announce Type: replace-cross -Abstract: Low-rank approximations of large kernel matrices are ubiquitous in machine learning, particularly for scaling Gaussian Processes to massive datasets. The Pivoted Cholesky decomposition is a standard tool for this task, offering a computationally efficient, greedy low-rank approximation. While its algebraic properties are well-documented in numerical linear algebra, its geometric intuition within the context of kernel methods often remains obscure. In this note, we elucidate the geometric interpretation of the algorithm within the Reproducing Kernel Hilbert Space (RKHS). We demonstrate that the pivotal selection step is mathematically equivalent to Farthest Point Sampling (FPS) using the kernel metric, and that the Cholesky factor construction is an implicit Gram-Schmidt orthogonalization. We provide a concise derivation and a minimalist Python implementation to bridge the gap between theory and practice. - oai:arXiv.org:2601.03706v3 - cs.LG + A Domain Decomposition-based Solver for Acoustic Wave propagation in Two-Dimensional Random Media + https://arxiv.org/abs/2512.23027 + arXiv:2512.23027v2 Announce Type: replace-cross +Abstract: An acoustic wave propagation problem with a log normal random field approximation for wave speed is solved using a sampling-free intrusive stochastic Galerkin approach. The stochastic partial differential equation with the inputs and outputs expanded using polynomial chaos expansion (PCE) is transformed into a set of deterministic PDEs and further to a system of linear equations. Domain decomposition (DD)-based solvers are utilized to handle the overwhelming computational cost for the resulting system with increasing mesh size, time step and number of random parameters. A conjugate gradient iterative solver with a two-level Neumann-Neumann preconditioner is applied here showing their efficient scalabilities. + oai:arXiv.org:2512.23027v2 + cs.CE + cs.DC cs.NA math.NA - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gil Shabat + Sudhi Sharma Padillath Vasudevan - Improved Lower Bounds for Learning Quantum Channels in Diamond Distance - https://arxiv.org/abs/2601.04180 - arXiv:2601.04180v4 Announce Type: replace-cross -Abstract: We prove that learning an unknown quantum channel with input dimension $d_A$, output dimension $d_B$, and Choi rank $r$ to diamond distance $\varepsilon$ requires $ \Omega\!\left( \frac{d_A d_B r}{\varepsilon \log(d_B r / \varepsilon)} \right)$ channel queries when $d_A= rd_B$, and $\Omega\!\left( \frac{d_A d_B r}{\varepsilon^2 \log(d_B r / \varepsilon)} \right)$ channel queries when $d_A\le rd_B/2$. These lower bounds improve upon the best previous $\Omega(d_A d_B r)$ bound by introducing explicit, near-optimal $\varepsilon$-dependence. Moreover, when $d_A\le rd_B/2$, the lower bound is optimal up to a logarithmic factor. The proof constructs ensembles of channels that are well separated in diamond norm yet admit Stinespring isometries that are close in operator norm. - oai:arXiv.org:2601.04180v4 + Madelung hydrodynamics of spin-orbit coupling: action principles, currents, and correlations + https://arxiv.org/abs/2601.10698 + arXiv:2601.10698v2 Announce Type: replace-cross +Abstract: We exploit the variational and Hamiltonian structures of quantum hydrodynamics with spin to unfold the correlation and torque mechanisms accompanying spin-orbit coupling (SOC) in electronic motion. Using Hamilton's action principle for the Pauli equation, we isolate SOC-induced quantum forces that act on the orbital Madelung--Bohm trajectories and complement the usual force terms known to appear in quantum hydrodynamics with spin. While the latter spin-hydrodynamic forces relate to the quantum geometric tensor (QGT), SOC-induced orbital forces originate from a particular current operator that contributes prominently to the spin current. This distinction between force terms reveals two fundamentally different mechanisms generating quantum spin-orbit correlations. Leveraging the Hamiltonian structure of the hydrodynamic system, we also elucidate spin transport features such as the correlation-induced quantum torques and the current shift in the spin Hall effect. This Hall shift leads to a new definition of the transport spin current thereby addressing an open question in spintronics. Finally, we illustrate the framework via the Madelung--Rashba equations for planar SOC configurations and propose a particle-based scheme for numerical implementation. + oai:arXiv.org:2601.10698v2 quant-ph + cond-mat.mes-hall + cond-mat.other math-ph math.MP - Wed, 21 Jan 2026 00:00:00 -0500 + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Aadil Oufkir, Filippo Girardi - - - A Conservative Log-Size Master Equation for Fragmentation PBEs: Jump Transport, Drift--Diffusion Asymptotics, and PSD Inference - https://arxiv.org/abs/2601.06638 - arXiv:2601.06638v2 Announce Type: replace-cross -Abstract: Fragmentation population-balance equations (PBEs) describe how particle size distributions (PSDs) evolve under breakage and daughter fragment redistribution. From a standard self-similar fragmentation class we derive an \emph{exact conservative transport equation in log-size} for the \emph{normalized mass fraction}: a state-dependent \emph{pure-jump} master equation (nonlocal internal-coordinate mass transfer). We also give an explicit Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) factorization whose diagonal sector reproduces this master equation, used here as an \emph{optional} structure-preserving operator representation and constrained parameterization for inverse modeling (rather than a computational necessity). - In a controlled small-jump regime, the nonlocal jump transport reduces to a drift--diffusion (Fokker--Planck) operator in log-size space. Under detailed-balance conditions this operator admits the standard symmetrization to a self-adjoint Schr\"odinger-type spectral problem, enabling compact parametric hypothesis classes for PSD shapes. We then present two inverse routes: (i) time-resolved parametric fitting of transport/spectral parameters, and (ii) a regularized steady-state inversion that reconstructs an effective potential from a measured steady PSD. - To address practical validation, we include numerical benchmarks: forward simulation of the jump transport model (CTMC discretization) and its drift--diffusion reduction, quantitative discrepancy metrics, and inverse parameter recovery on an Airy half-line synthetic benchmark under controlled multiplicative noise. - oai:arXiv.org:2601.06638v2 - cond-mat.stat-mech - math-ph - math.MP - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Juan J. Segura - - - Breaking the Orthogonality Barrier in Quantum LDPC Codes - https://arxiv.org/abs/2601.08824 - arXiv:2601.08824v3 Announce Type: replace-cross -Abstract: Classical low-density parity-check (LDPC) codes are a widely deployed and well-established technology, forming the backbone of modern communication and storage systems. It is well known that, in this classical setting, increasing the girth of the Tanner graph while maintaining regular degree distributions leads simultaneously to good belief-propagation (BP) decoding performance and large minimum distance. In the quantum setting, however, this principle does not directly apply because quantum LDPC codes must satisfy additional orthogonality constraints between their parity-check matrices. When one enforces both orthogonality and regularity in a straightforward manner, the girth is typically reduced and the minimum distance becomes structurally upper bounded. In this work, we overcome this limitation by using permutation matrices with controlled commutativity and by restricting the orthogonality constraints to only the active part of the construction, while preserving regular check-matrix structures. This design circumvents conventional structural distance limitations induced by parent-matrix orthogonality, and enables the construction of quantum LDPC codes with large girth while avoiding latent low-weight logical operators. As a concrete demonstration, we construct a girth-8, (3,12)-regular $[[9216,4612, \leq 48]]$ quantum LDPC code and show that, under BP decoding combined with a low-complexity post-processing algorithm, it achieves a frame error rate as low as $10^{-8}$ on the depolarizing channel with error probability $4 \%$. - oai:arXiv.org:2601.08824v3 - quant-ph - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Kenta Kasai + Cesare Tronci - Kinematic Tokenization: Optimization-Based Continuous-Time Tokens for Learnable Decision Policies in Noisy Time Series - https://arxiv.org/abs/2601.09949 - arXiv:2601.09949v2 Announce Type: replace-cross -Abstract: Transformers are designed for discrete tokens, yet many real-world signals are continuous processes observed through noisy sampling. Discrete tokenizations (raw values, patches, finite differences) can be brittle in low signal-to-noise regimes, especially when downstream objectives impose asymmetric penalties that rationally encourage abstention. We introduce Kinematic Tokenization, an optimization-based continuous-time representation that reconstructs an explicit spline from noisy measurements and tokenizes local spline coefficients (position, velocity, acceleration, jerk). This is applied to financial time series data in the form of asset prices in conjunction with trading volume profiles. Across a multi-asset daily-equity testbed, we use a risk-averse asymmetric classification objective as a stress test for learnability. Under this objective, several discrete baselines collapse to an absorbing cash policy (the Liquidation Equilibrium), whereas the continuous spline tokens sustain calibrated, non-trivial action distributions and stable policies. These results suggest that explicit continuous-time tokens can improve the learnability and calibration of selective decision policies in noisy time series under abstention-inducing losses. - oai:arXiv.org:2601.09949v2 + Fairness-informed Pareto Optimization : An Efficient Bilevel Framework + https://arxiv.org/abs/2601.13448 + arXiv:2601.13448v2 Announce Type: replace-cross +Abstract: Despite their promise, fair machine learning methods often yield Pareto-inefficient models, in which the performance of certain groups can be improved without degrading that of others. This issue arises frequently in traditional in-processing approaches such as fairness-through-regularization. In contrast, existing Pareto-efficient approaches are biased towards a certain perspective on fairness and fail to adapt to the broad range of fairness metrics studied in the literature. In this paper, we present BADR, a simple framework to recover the optimal Pareto-efficient model for any fairness metric. Our framework recovers its models through a Bilevel Adaptive Rescalarisation procedure. The lower level is a weighted empirical risk minimization task where the weights are a convex combination of the groups, while the upper level optimizes the chosen fairness objective. We equip our framework with two novel large-scale, single-loop algorithms, BADR-GD and BADR-SGD, and establish their convergence guarantees. We release badr, an open-source Python toolbox implementing our framework for a variety of learning tasks and fairness metrics. Finally, we conduct extensive numerical experiments demonstrating the advantages of BADR over existing Pareto-efficient approaches to fairness. + oai:arXiv.org:2601.13448v2 cs.LG - cs.AI math.OC - Wed, 21 Jan 2026 00:00:00 -0500 + stat.ML + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://creativecommons.org/licenses/by/4.0/ - Griffin Kearney - - - Discrete versus continuous -- lattice models and their exact continuous counterparts - https://arxiv.org/abs/2601.10184 - arXiv:2601.10184v2 Announce Type: replace-cross -Abstract: We review and study the correspondence between discrete lattice/chain models of interacting particles and their continuous counterparts represented by partial differential equations. We study the correspondence problem for nearest neighbour interaction lattice models as well as for multiple-neighbour interaction lattice models, and we gradually proceed from infinite lattices to periodic lattices and finally to finite lattices with fixed ends/zero Dirichlet boundary conditions. The whole study is framed as systematic specialisation of Fourier analysis tools from the continuous to the discrete setting and vice versa, and the correspondence between the discrete and continuous models is examined primarily with regard to the dispersion relation. - oai:arXiv.org:2601.10184v2 - physics.class-ph - cs.NA - math.NA - physics.comp-ph - Wed, 21 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lorenzo Fusi, Oliver K\v{r}enek, V\'it Pr\r{u}\v{s}a, Casey Rodriguez, Rebecca Tozzi, Martin Vejvoda + Sofiane Tanji, Samuel Vaiter, Yassine Laguel - Hybrid Encryption with Certified Deletion in Preprocessing Model - https://arxiv.org/abs/2601.10542 - arXiv:2601.10542v2 Announce Type: replace-cross -Abstract: Certified deletion allows Alice to outsource data to Bob and, at a later time, obtain a verifiable guarantee that the file has been irreversibly deleted at her request. The functionality, while impossible using classical information alone, can be achieved using quantum information. Existing approaches rely either on one-time pad (OTP) encryption, or on computational hardness assumptions that may be vulnerable to future advances in classical or quantum computing. In this work, we introduce and formalize hybrid encryption with certified deletion in the preprocessing model (pHE-CD) and propose two constructions. Each construction composes an information-theoretic key encapsulation mechanism (iKEM) with a data encapsulation mechanism that provides certified deletion (DEM-CD) security, offering different types of security depending on the security properties of DEM-CD. When DEM-CD is one-time information theoretically secure, the composition provides {\em information-theoretic security} for both encryption and certified deletion. When DEM-CD is computationally secure, the composed construction offers computationally secure (post-quantum) encryption and {\em everlasting certified deletion} where confidentiality is computational up to the point that the deletion certificate is verified, and after successful verification of the certificate, becomes unconditional. That is, successful verification of deletion certificate guarantees that the data has been removed information-theoretically from the adversary's view. Both pHE-CD schemes are for encryption of arbitrarily long messages. Construction 2 is key efficient and uses a DEM-CD that is constructed using quantum coding and AES, providing quantum-safe security for encryption. We discuss our results and directions for future work. - oai:arXiv.org:2601.10542v2 - cs.CR - cs.IT - math.IT - Wed, 21 Jan 2026 00:00:00 -0500 + Small Gradient Norm Regret for Online Convex Optimization + https://arxiv.org/abs/2601.13519 + arXiv:2601.13519v2 Announce Type: replace-cross +Abstract: This paper introduces a new problem-dependent regret measure for online convex optimization with smooth losses. The notion, which we call the $G^\star$ regret, depends on the cumulative squared gradient norm evaluated at the decision in hindsight $\sum_{t=1}^T \|\nabla \ell(x^\star)\|^2$. We show that the $G^\star$ regret strictly refines the existing $L^\star$ (small loss) regret, and that it can be arbitrarily sharper when the losses have vanishing curvature around the hindsight decision. We establish upper and lower bounds on the $G^\star$ regret and extend our results to dynamic regret and bandit settings. As a byproduct, we refine the existing convergence analysis of stochastic optimization algorithms in the interpolation regime. Some experiments validate our theoretical findings. + oai:arXiv.org:2601.13519v2 + stat.ML + cs.LG + math.OC + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross - http://creativecommons.org/licenses/by/4.0/ - Kunal Dey, Reihaneh Safavi-Naini + http://creativecommons.org/licenses/by-nc-sa/4.0/ + Wenzhi Gao, Chang He, Madeleine Udell - Rewriting Systems on Arbitrary Monoids - https://arxiv.org/abs/2601.10564 - arXiv:2601.10564v2 Announce Type: replace-cross -Abstract: In this paper, we introduce monoidal rewriting systems (MRS), an abstraction of string rewriting in which reductions are defined over an arbitrary ambient monoid rather than a free monoid of words. This shift is partly motivated by logic: the class of free monoids is not first-order axiomatizable, so "working in the free setting" cannot be treated internally when applying first-order methods to rewriting presentations. - To analyze these systems categorically, we define $\mathbf{NCRS_2}$ as the 2-category of Noetherian Confluent MRS. We then prove the existence of a canonical biadjunction between $\mathbf{NCRS_2}$ and $\mathbf{Mon}$. - Finally, we classify all Noetherian Confluent MRS that present a given fixed monoid. For this, we introduce Generalized Elementary Tietze Transformations (GETTs) and prove that any two presentations of a monoid are connected by a (possibly infinite) sequence of these transformations, yielding a complete characterization of generating systems up to GETT-equivalence. - oai:arXiv.org:2601.10564v2 - cs.FL - cs.LO - math.CT - Wed, 21 Jan 2026 00:00:00 -0500 + An $\Omega(\log(N)/N)$ Lookahead is Sufficient to Bound Costs in the Overloaded Loss Network + https://arxiv.org/abs/2601.14538 + arXiv:2601.14538v2 Announce Type: replace-cross +Abstract: I study the simplest model of revenue management with reusable resources: admission control of two customer classes into a loss queue. This model's long-run average collected reward has two natural upper bounds: the deterministic relaxation and the full-information offline problem. With these bounds, we can decompose the costs faced by the online decision maker into (i) the \emph{cost of variability}, given by the difference between the deterministic value and the offline value, and (ii) the \emph{cost of uncertainty}, given by the difference between the offline value and the online value. \cite{Xie2025} established that the sum of these two costs is $\Theta(\log N)$, as the number of servers, $N$, goes to infinity. I show that we can entirely attribute this $\Theta(\log N)$ rate to the cost of uncertainty, as the cost of variability remains $O(1)$ as $N \rightarrow \infty$. In other words, I show that anticipating future fluctuations is sufficient to bound operating costs -- smoothing out these fluctuations is unnecessary. In fact, I show that an $\Omega(\log(N)/N)$ lookahead window is sufficient to bound operating costs. + oai:arXiv.org:2601.14538v2 + econ.TH + math.PR + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross http://creativecommons.org/licenses/by/4.0/ - Eduardo Magalh\~aes + Robert L. Bray - Jordan-Segmentable Masks: A Topology-Aware definition for characterizing Binary Image Segmentation - https://arxiv.org/abs/2601.10577 - arXiv:2601.10577v2 Announce Type: replace-cross -Abstract: Image segmentation plays a central role in computer vision. However, widely used evaluation metrics, whether pixel-wise, region-based, or boundary-focused, often struggle to capture the structural and topological coherence of a segmentation. In many practical scenarios, such as medical imaging or object delineation, small inaccuracies in boundary, holes, or fragmented predictions can result in high metric scores, despite the fact that the resulting masks fail to preserve the object global shape or connectivity. This highlights a limitation of conventional metrics: they are unable to assess whether a predicted segmentation partitions the image into meaningful interior and exterior regions. - In this work, we introduce a topology-aware notion of segmentation based on the Jordan Curve Theorem, and adapted for use in digital planes. We define the concept of a \emph{Jordan-segmentatable mask}, which is a binary segmentation whose structure ensures a topological separation of the image domain into two connected components. We analyze segmentation masks through the lens of digital topology and homology theory, extracting a $4$-curve candidate from the mask, verifying its topological validity using Betti numbers. A mask is considered Jordan-segmentatable when this candidate forms a digital 4-curve with $\beta_0 = \beta_1 = 1$, or equivalently when its complement splits into exactly two $8$-connected components. - This framework provides a mathematically rigorous, unsupervised criterion with which to assess the structural coherence of segmentation masks. By combining digital Jordan theory and homological invariants, our approach provides a valuable alternative to standard evaluation metrics, especially in applications where topological correctness must be preserved. - oai:arXiv.org:2601.10577v2 - cs.CV - cs.NA - math.AT - math.NA - Wed, 21 Jan 2026 00:00:00 -0500 + Enhanced posterior sampling via diffusion models for efficient metasurfaces inverse design + https://arxiv.org/abs/2601.15210 + arXiv:2601.15210v2 Announce Type: replace-cross +Abstract: The inverse design of metasurfaces faces inherent challenges due to the nonlinear and highly complex relationship between geometric configurations and their electromagnetic behavior. Traditional optimization approaches often suffer from excessive computational demands and a tendency to converge to suboptimal solutions. This study presents a diffusion-based generative framework that incorporates a dedicated consistency constraint and advanced posterior sampling methods to ensure adherence to desired electromagnetic specifications. Through rigorous validation on small-scale metasurface configurations, the proposed approach demonstrates marked enhancements in both accuracy and reliability of the generated designs. Furthermore, we introduce a scalable methodology that extends inverse design capabilities to large-scale metasurfaces, validated for configurations of up to $98 \times 98$ nanopillars. Notably, this approach enables rapid design generation completed in minute by leveraging models trained on substantially smaller arrays ($23 \times 23$). These innovations establish a robust and efficient framework for high-precision metasurface inverse design. + oai:arXiv.org:2601.15210v2 + physics.optics + math-ph + math.MP + Fri, 23 Jan 2026 00:00:00 -0500 replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Serena Grazia De Benedictis, Amedeo Altavilla, Nicoletta Del Buono + http://creativecommons.org/licenses/by-sa/4.0/ + Mathys Le Grand (Institut des nanotechnologies de Lyon, STMicroelectronics), Pascal Urard (STMicroelectronics), Denis Rideau (STMicroelectronics), Loumi Tr\'emas (STMicroelectronics), Damien Maitre (STMicroelectronics), Louis-Henri Fernandez-Mouron (STMicroelectronics), Adam Fuchs (STMicroelectronics), R\'egis Orobtchouk (Institut des nanotechnologies de Lyon)