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" @@ -7,4779 +7,12 @@ http://www.rssboard.org/rss-specification en-us - Fri, 23 Jan 2026 05:00:01 +0000 + Sat, 24 Jan 2026 05:00:02 +0000 rss-help@arxiv.org - Fri, 23 Jan 2026 00:00:00 -0500 + Sat, 24 Jan 2026 00:00:00 -0500 Saturday Sunday - - 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/ - Carlos E. Cadenas R., Yorman J. Mendoza N - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hubert Kalf - - - 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/ - Edwige Tolla - - - 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/ - Hin Chung Henry Tsang - - - 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://creativecommons.org/licenses/by/4.0/ - Heinz H. Bauschke, Walaa M. Moursi - - - 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/ - Martin Geller, Terry Lyons - - - 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/ - Adam LaClair, Jason McCullough - - - 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/ - Arman Fazeli, Mohammad M. Mansour, Ziyuan Zhu, Louay Jalloul - - - 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/ - Tiberiu Dumitrescu - - - 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/ - Morten L\"uders, Elia Fiammengo - - - 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://creativecommons.org/licenses/by/4.0/ - Alex Wright - - - 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/ - D. Russell Luke, Johannes-Carl Schnebel, Mathias Staudigl, Juan Peypouquet, Siqi Qu - - - 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/ - Alex Fink, Navid Nabijou, Rob Silversmith - - - 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/ - Ben Mills - - - 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 - math.AG - math.RT - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Martin de Borbon, Dmitri Panov - - - 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 - math.MG - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Silouanos Brazitikos, Minas Pafis - - - 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/ - Stephan Baier - - - 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/ - Shi Feng - - - 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://creativecommons.org/licenses/by/4.0/ - Emily Heath, Coy Schwieder, Shira Zerbib - - - 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/ - Dion Gijswijt. Jan van Neerven - - - 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/ - Amalendu Krishna, Subhadip Majumder - - - 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/ - Achilleas Karras, Benne de Weger - - - 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/ - Alessandro Neri, Ferdinando Zullo - - - 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/ - Alina Iacob - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Oliver Bachtler - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Danil Pavlinov, Svetlana Zhilina - - - 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/ - Tyler Kann, Matthieu R. Bloch, Shrinivas Kudekar, Ruediger Urbanke - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alexander Omelchenko - - - 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/ - F\'elix Kahane, Minmin Wang - - - 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://creativecommons.org/licenses/by/4.0/ - Ni An, Bhola Nath Saha, Bidyut Sanki - - - 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://creativecommons.org/licenses/by/4.0/ - J. H. Ram\'irez Gonz\'alez, Gustavo O. Carvalho, F\'abio P. Machado - - - 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://creativecommons.org/publicdomain/zero/1.0/ - Alejandro Gonz\'alez Nevado - - - 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://creativecommons.org/licenses/by/4.0/ - Felipe Atenas, Minh N. Dao, Matthew K. Tam - - - 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://creativecommons.org/licenses/by/4.0/ - Marcello Lanfranchi - - - 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/ - Jun-Yong Park - - - 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/ - Tamara Kucherenko, Marco L\'opez, Christian Wolf - - - 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/ - Quoc-Anh Tran - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Daniel R. Johnston, Simon N. Thomas, Jonathan P. Sorenson, Jonathan E. Webster - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kristyna Kramer, Jakub Krasensky - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kevin Coulembier, Andrew Snowden - - - 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/ - Chenyang Xu, Ziquan Zhuang - - - 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://creativecommons.org/licenses/by/4.0/ - Nonlinear Anal., Real World Appl. 56, Article ID 103161, 18 p. (2020) - Philip Korman, Dieter S. Schmidt - - - 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://creativecommons.org/licenses/by/4.0/ - 10.1080/00255572.2025.2594896 - Mathematical Gazette, published online 20 January, 2026 - Eric L. Grinberg, Mehmet Z. Orhon - - - 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://creativecommons.org/licenses/by/4.0/ - Amita Malik, Rishabh Sarma - - - 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 - stat.ML - stat.TH - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Yi Yu, Yubo Hou, Yinchong Wang, Nan Zhang, Jianfeng Feng, Wenlian Lu - - - 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://creativecommons.org/licenses/by/4.0/ - Jonghyeon Ahn - - - 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-sa/4.0/ - Scott Powers, Sivaramakrishnan Ramani, Jacob Hahn, Andrew J. Schaefer - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Ali Ebadi - - - 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/ - Masaki Kashima, Riste \v{S}krekovski, Rongxing Xu - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1007/s11139-025-01257-6 - The Ramanujan Journal (2025) 68:108 - Kwangwoo Lee - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mikiya Kametaka, Tatsuki Kawakami - - - 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/ - Zechun Hu, Renming Song, Li Wang - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Shin Satoh, Kodai Wada - - - 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://creativecommons.org/licenses/by-nc-sa/4.0/ - Bihan Chatterjee, Sigr\'un Andrad\'ottir, Hayriye Ayhan - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Hamidreza Abin, Mahdi Zinati, Amin Gohari, Mohammad Hossein Yassaee, Mohammad Mahdi Mojahedian - - - 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/ - Yi Chen, Yatao Hu, Ming Li, Chong Han - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Junying Chen, Ruixiang Xing - - - 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://creativecommons.org/licenses/by/4.0/ - Manujith K. Michel, Varadharaj R. Srinivasan - - - 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/ - Nelson Schuback - - - 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/ - Aiswarya T, Srijanani Anurag Prasad - - - 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 - math.FA - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Duv\'an Cardona - - - 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://creativecommons.org/licenses/by/4.0/ - Fan Gao, Runze Wang - - - 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/licenses/by/4.0/ - M. P. Thejitha, James A. Sellers, S. N. Fathima - - - 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-nd/4.0/ - Xin Chen, Chunfeng Cui, Deren Han, Liqun Qi - - - 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 - math-ph - math.MP - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Myong-Hwan Ri - - - 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/ - Kazuhiko Kurano - - - 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/ - Xin Cao, Xiaochun Fang, Jianchao Wu - - - 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://creativecommons.org/licenses/by/4.0/ - Anthony Lee - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Katrina Barron, Justine Fasquel, Florencia Orosz Hunziker, Gaywalee Yamskulna - - - 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.CO - math.IT - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gunank Jakhar, Gowtham R. Kurri, Suryajith Chillara, Vinod M. Prabhakaran - - - 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/ - Eduardo Garcia-Rio, Rosalia Rodriguez-Gigirey, Ramon Vazquez-Lorenzo - - - 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/ - Ruoyu Diao, Yu-Hong Dai, Liwei Zhang - - - 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-nc-nd/4.0/ - Toshiyuki Kobayashi, Michael Pevzner - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - 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 - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Reo Tsuboya - - - 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-nc-nd/4.0/ - Joan Porti - - - 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/ - Felicia Servina Djuang, Indah Emilia Wijayanti, Yeni Susanti - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jos\'e L. Ansorena, Alejandro Marcos - - - 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/ - Shutong Hou, Mourad Sini, Haibing Wang - - - 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 - math.AP - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Guofang Wang, Wei Wei, Chao Xia, Xuwen Zhang - - - 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/ - Antonio Sasaki (CMA, PSL), Sophie Demassey (CMA, PSL), Valentina Sessa (CMA, PSL) - - - 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://creativecommons.org/licenses/by-nc-nd/4.0/ - Danijel Grahovac, Magdalena Miki\'c - - - 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/ - Euan Aitken - - - 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/ - Said Hamad\`ene (LMM), Ibtissam Hdhiri - - - 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://creativecommons.org/licenses/by-nc-nd/4.0/ - Mar\'ia Jes\'us Carro, Alberto Salguero-Alarc\'on - - - 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/ - Yurui Tang, Xingzhi Zhan - - - 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://creativecommons.org/licenses/by/4.0/ - Kuntal Banerjee, Anubrato Bhattacharyya, Krishnendu Gongopadhyay, Subhamoy Mondal - - - 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/ - Thea Li, Vladimir Zamdzhiev - - - $C^\ast$-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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chaitanya J. Kulkarni - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yan Rybalko - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Bharat Talwar, Prahlad Vaidyanathan, Stefan Wagner - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Mourad Choulli, Hiroshi Takase - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Luca Brandolini, Alessandro Monguzzi, Matteo Monti - - - 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/ - A. Schmidt, A. Vdberg, A. Petit - - - 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 - math.CT - math.MP - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Grigorios Giotopoulos, Igor Khavkine, Hisham Sati, Urs Schreiber - - - 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-nc-sa/4.0/ - Hanno von Bergen, Reinhard Diestel - - - 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/ - Chahat Ahuja - - - 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/ - David Johansson, Yavar Kian - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Marzieh Hasannasab, Larissa Kaldewey, Frederic Schoppert - - - 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/ - Zhen Huan - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xiaowen Li, Ming Mei - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Pramod Singh, Prasad Krishnan, Arti Yardi - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Doan Huu Hieu, Vo Minh Tam, Nguyen Duy Cuong - - - 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/ - Claudio Bonanno, Sharvari Neetin Tikekar - - - 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/ - 10.1007/s11075-026-02315-w - Numerical Algorithms (2026) - Miguel A. Pi\~nar - - - 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/ - Lei Chen, Alice Devillers, Luke Morgan, Friedrich Rober - - - 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 - cs.AI - cs.LG - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Chen Xu - - - 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-nc-nd/4.0/ - Dani Kaufman, Carmen Alves Sabin - - - 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/ - Jinlu Li, Yanghai Yu - - - 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/ - Ko Honda, Yin Tian, Tianyu Yuan - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yuchen Liao, Wenyi Zhang - - - 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/publicdomain/zero/1.0/ - Ye Ren - - - 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 - cs.CC - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kai Wang - - - 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/ - Yaakov Malinovsky - - - 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/ - Eliran Subag - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Serge Gratton, Philippe L. Toint - - - 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/ - Filip Jonsson Kling, Samuel Lundqvist - - - 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://creativecommons.org/licenses/by/4.0/ - Madhusraba Sinha, Jan Stratmann - - - 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://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 - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yuri Soga - - - 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/ - David Aulicino, Howard Masur, Huiping Pan, Weixu Su - - - 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/ - Mario Bravo, Roberto Cominetti, Jongmin Lee - - - 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 - math-ph - math.MP - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Joel Nathe, Ant\^onio S\'a Barreto - - - 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/ - Yunan Wang, Chuxiong Hu, Zhao Jin - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Chunyang Hu - - - 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/ - Ziemowit Kostana, Jaros{\l}aw Swaczyna, Agnieszka Widz - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Salah G. Elgendi - - - 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.IT - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jiancheng An, Chau Yuen, Marco Di Renzo, Mehdi Bennis, Merouane Debbah, Lajos Hanzo - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zhixin Chen, Yixuan Huang, Zhengze Ji, Jie Yang, Shi Jin - - - 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/ - Tianyu Fang, Mengyuan Ma, Markku Juntti, Nhan Thanh Nguyen - - - 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 - math.OC - stat.TH - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Omar Al-Ghattas - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by/4.0/ - Nam Van Tran - - - 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/ - Barinder S. Banwait, Xiaoyu Huang - - - 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/ - Edward McDonald, Michael Ruzhansky, Serikbol Shaimardan, Kanat Tulenov - - - 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/ - Claudia Kirch, Hedvika Rano\v{s}ov\'a, Martin Wendler - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Jose Manuel Garcia Calcines, Jose Antonio Vilches Alarcon - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Kehan Shi - - - 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/ - Giacomo Graziani - - - 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/ - Marcel Scherer - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Laihao Ding, Xiaolan Hu, Suyun Jiang - - - 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/ - Francesco Antonio Denisi, Claudio Onorati, Francesca Rizzo, Sasha Viktorova - - - 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 - math.CT - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Johannes Flake, Robert Laugwitz, Sebastian Posur - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Linus R\"osler - - - 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/ - Teng Zhang - - - 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-nc-nd/4.0/ - Jong In Han - - - 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/ - Robert Luko\v{t}ka - - - 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/ - Xavier Caruso, Florian F\"urnsinn - - - 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/ - Zhibin Wang, Jiahong Wu, Ning Zhu - - - 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/ - Erkao Bao, Robi Huq, Shengzhen Ning - - - 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://creativecommons.org/licenses/by/4.0/ - J\"org Schr\"oder, Maximilian Vorwerk - - - 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-nc-nd/4.0/ - Guohui Song, Congzhi Xia - - - 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://creativecommons.org/licenses/by-nc-nd/4.0/ - Lixin Shen, Guohui Song - - - 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 - quant-ph - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Fabio Bagarello - - - 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/ - Justin Trias - - - 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://creativecommons.org/licenses/by/4.0/ - Diego C\'espedes, Sebasti\'an Donoso - - - 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/ - Justin Trias - - - 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://creativecommons.org/licenses/by/4.0/ - 10.3934/cac.2024009 - Communication on Analysis and Computation 2024, Volume 2, Issue 2: 180 - 198 - Vladimir Kryzhniy - - - 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://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) - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Tim Heib, David Edward Bruschi - - - 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/ - Damian Sercombe - - - 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/ - Emmanuel Abbe, Colin Sandon, Oscar Sprumont - - - 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/ - Aritro Pathak - - - 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://creativecommons.org/licenses/by-nc-nd/4.0/ - Ali Khalesi, Ahmad Tanha, Derya Malak, Petros Elia - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jorge Fari\~na-Asategui, Santiago Radi - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alix Deleporte, Jean Lagac\'e, Marc Rouveyrol - - - 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/ - Luca Di Persio, Matteo Garbelli, Adrian Zalinescu - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Antonio Lacopo - - - 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/ - Itahisa Barrios-Cubas, Matteo Bonforte, Mar\'ia del Mar Gonz\'alez, Clara Torres-Latorre - - - 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/ - Przemys{\l}aw Ohrysko - - - 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/ - Sebasti\'an Donoso, Sovanlal Mondal, Vicente Saavedra-Araya - - - 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.CV - Fri, 23 Jan 2026 00:00:00 -0500 - new - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Gregorio Vettori - - - 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-nc-sa/4.0/ - El Mehdi Er Raqabi, Kevin Dalmeijer, Pascal Van Hentenryck - - - 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/ - 10.1109/LWC.2023.3238851 - Alessandro Traspadini, Marco Giordani, Giovanni Giambene, Michele Zorzi - - - 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/ - Liang Chen - - - 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 - - - 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/ - Laura Shou, Jeet Shah, Matthew Lerner-Brecher, Amol Aggarwal, Alexei Borodin, Victor Galitski - - - 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.IT - 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/ - Andreas Bauer, Seth Lloyd - - - 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 - - - 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 - math.NA - stat.ML - Fri, 23 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Alex Goessmann, Janina Sch\"utte, Maximilian Fr\"ohlich, Martin Eigel - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Shoshana Chipman, Brent Doiron - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Jakub Antczak, James Montgomery, Ma{\l}gorzata O'Reilly, Zbigniew Palmowski, Richard Turner - - - 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/ - Saptarshi Roy, Alessandro Rinaldo, Purnamrita Sarkar - - - 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 - math.MP - Fri, 23 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Nobuyuki Sawado, Yuichiro Shimazaki - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Imran Ali Khan, Saif Khan Mohammed, Ronny Hadani, Ananthanarayanan Chockalingam, Robert Calderbank, Anton Monk, Shachar Kons, Shlomo Rakib, Yoav Hebron - - - 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/ - Tobias Boege, Antony Della Vecchia, Marina Garrote-L\'opez, Benjamin Hollering - - - 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/ - Dennis Dobler, Alina Schenk, Matthias Schmid - - - 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 - - - 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 - math.IT - Fri, 23 Jan 2026 00:00:00 -0500 - cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Aravindh Krishnamoorthy, Robert Schober, Harald Haas - - - 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 - - - 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/ - Jingfu Peng, Yuhong Yang - - - 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 - - - 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 - - - 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 - - - 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/ - Zirui Niu, Giordano Scarciotti, Alessandro Astolfi - - - 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 - math.QA - math.RT - Fri, 23 Jan 2026 00:00:00 -0500 - cross - http://creativecommons.org/licenses/by/4.0/ - Du Pei - - - 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/ - Mike Winkler - - - 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/ - 10.1007/s00222-026-01407-7 - Bin Gui - - - 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/ - Moad Abudia, Tejasvi Channagiri, Joel A. Rosenfeld, Rushikesh Kamalapurkar - - - 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 - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Siddhartha Sarkar - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Antonio Alarcon, Francisco J. Lopez - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1007/s13366-025-00807-9 - Robert Wilms - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - J. E. Paguyo - - - 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/ - 10.1016/j.jfa.2026.111361 - Hang Chen - - - 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/ - David Ellis, Guy Kindler, Noam Lifshitz - - - 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/ - L\'eonard Guetta, Lyne Moser, Maru Sarazola, Paula Verdugo - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Harry Sapranidis Mantelos - - - 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/ - Hank Chen, Florian Girelli - - - 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/ - Xiaoguang Zou, Xiang Gao, Chuangchuang Kang, Jiafeng L\"u - - - 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/ - 10.1112/jlms.70431 - Journal of the London Mathematical Society 113:1 (2026) e70431 - Matteo Capoferri, Dmitri Vassiliev - - - 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/ - Journal of Differential Equations 422 (2025) - Yuxuan Zhou, Wenming Zou - - - 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/ - Tom Gannon, Harold Williams - - - 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 - - - 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/ - Lukas Nakamura - - - 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 - - - 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/ - Sizhong Zhou, Yuli Zhang, Tao Zhang, Hongxia Liu - - - 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/ - Damian Sercombe - - - 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/ - Rapha\"el Danchin (LAMA) - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - William Q. Erickson, Markus Hunziker - - - 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/ - Yuhei Suzuki - - - 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 - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Teodor Rotaru, Fran\c{c}ois Glineur, Panagiotis Patrinos - - - 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/ - Matthew Fickus, Joseph W. Iverson, John Jasper, Dustin G. Mixon - - - 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/ - Ruowei Li, Florentin M\"unch - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Manuel Aprile, Samuel Fiorini, Gwena\"el Joret, Stefan Kober, Micha{\l} T. Seweryn, Stefan Weltge, Yelena Yuditsky - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Yuxuan Zhou, Wenming Zou - - - 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/ - Masoud Kamgarpour, GyeongHyeon Nam, Bailey Whitbread, Stefano Giannini - - - 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/ - Houshan Fu - - - 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/ - Joachim Jelisiejew, Ritvik Ramkumar, Alessio Sammartano - - - 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/ - James Hyde, Yash Lodha - - - 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 - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Song-Ren Fu, Yongyi Yu, Philipp Zimmermann - - - 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 - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Wolfram Bauer, Yawei Wei, Xiaodong Zhou - - - 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 - - - $N=1$ 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/ - Thomas Creutzig, Robert McRae, Florencia Orosz Hunziker, Jinwei Yang - - - 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 - - - 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/ - 10.1016/j.apm.2026.116775 - S. D. M. de Jong, A. Brugnoli, R. Rashad, Y. Zhang, S. Stramigioli - - - 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 - - - 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/ - Ivailo Hartarsky, Lyuben Lichev - - - 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 - - - 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 - - - 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/ - Andrii Ilienko, Ilya Molchanov, Tommaso Vison\`a - - - 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 - math.AP - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/publicdomain/zero/1.0/ - Yi Gao, Kui Wang - - - 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 - - - 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 - - - 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/ - Daniel \'Alvarez - - - 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/ - Riccardo W. Maffucci - - - 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 - math.CV - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Abraham del Valle Rodr\'iguez - - - 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/ - Roy Zhao - - - 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 - - - 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/ - Judith Ludwig, Christian Merten - - - 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 - - - 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/ - Lide Cai, Junqing Chen, Yanpeng Gao - - - 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/ - Byungchul Cha, Dong Han Kim, Deokwon Sim - - - 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 - - - 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/ - 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 - - - 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 - math-ph - math.MP - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Nicola Miele, Alessia Nota, Juan J. L. Vel\'azquez - - - 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/ - Gleb Smirnov, Roman Vershynin - - - 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/ - Di Wu, Zhuoyin Dai, Yong Zeng - - - 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/ - Sarah Eberle-Blick, Henrik Garde, Nuutti Hyv\"onen - - - 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 - - - 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/ - M. Belianovich, G. E. Fasshauer, A. Narayan, V. Shankar - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Yasutada Oohama, Bagus Santoso - - - 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/ - Nuno J. Alves, Jan Haskovec - - - 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/ - Tingting Feng, Yong Zhang, Zhitao Zhang - - - 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/ - Leizhen Bao, Fang Li - - - 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 - - - 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 - - - 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 - - - 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 - 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 - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Lukas Vandeputte - - - 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/ - Tobias Fritz, Nicol\'as Rivera - - - 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/ - Mathilde Andr\'e, F\'elix Foutel-Rodier, Emmanuel Schertzer - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Jian-hao Kang, Zhun Gou, Nan-jing Huang - - - 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/ - Alex McDonald, Micah Nguyen - - - 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 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 - - - 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/ - Jayme Vaz - - - 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/ - Alvan Arulandu, Daniel Gottschalk, Thomas Payne, Alexander Richardson, Thomas Weighill - - - 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 - 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 - - - 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 - - - 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/ - Zhaolin Li - - - 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/ - Atsushi Miki, Toshiyasu Matsushima - - - 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/ - Milena Radnovi\'c, Ruzzel Ragas - - - 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 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 - math.DG - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Hiroshi Hirai - - - 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/ - Takayuki Koike - - - 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 - math.CV - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Jixiang Fu, Shing-Tung Yau, Dekai Zhang - - - 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/ - Nicole Spillane (CMAP), Pierre Matalon (CMAP), Daniel B Szyld - - - 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/ - Jounglag Lim - - - 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/ - Jan Rataj, Ludek Zajicek - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Abdelouahab Khelifati - - - 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/ - Johann Verwee - - - 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/ - Claudio A. Buzzi, Daniel Panazzolo, Paulo R. da Silva - - - 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/ - James M Parks - - - 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/ - Martin Klazar - - - 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/ - Frank Gilson - - - 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 - math.ST - stat.TH - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Behzad Aalipur - - - 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/ - Kousuke Kuto, Kazuhiro Oeda - - - 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/ - Jiahao Jiang - - - 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/ - Andrey Piatnitski, Vladimir Sloushch, Tatiana Suslina, Elena Zhizhina - - - $A_3$-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/ - Ambrus P\'al, Gereon Quick - - - 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 - - - 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/ - Myriam Ouna\"ies - - - 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 - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Xizhi Liu, Jie Ma, Tianming Zhu - - - 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 - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Anton Alexa - - - 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/ - Seung-Jo Jung, Morihiko Saito - - - 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 - cs.NA - Fri, 23 Jan 2026 00:00:00 -0500 - replace - http://creativecommons.org/licenses/by/4.0/ - Wenzhong Zhang, Zheyuan Hu, Wei Cai, George EM Karniadakis - - - 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/ - Tristan Simas - - - 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/ - Hirofumi Ota, Masaaki Imaizumi - - - 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 - 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 - - - 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/ - Ruben Carpenter, Colin Defant, Noah Kravitz - - - 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/ - Thai Nguyen, Mitja Stadje - - - 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/ - 10.1088/1361-6382/adb537 - Class. Quantum Grav. 42, 065017 (2025) - Nikodem Pop{\l}awski - - - 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/ - Hugo Melchers, Joost Prins, Michael Abdelmalik - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Zhirui Chen, Vincent Y. F. Tan - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by-nc-sa/4.0/ - Diego Guti\'errez-Oribio, Ioannis Stefanou - - - 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 - math.FA - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 10.1007/s10472-026-10003-7 - V.E. Ismailov, Ann. Math. Artif. Intell. (2026) - Vugar Ismailov - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Dragos-Patru Covei - - - 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 - math.IT - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Erfan Delfani, Agapi Mesodiakaki, Leandros Tassiulas, Nikolaos Pappas - - - 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.bio-ph - physics.chem-ph - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Jakob Deser, Raphael R\"omer, Niklas Boers, Christian Kuehn - - - 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 - cs.NA - math.NA - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - S. Ill\'esov\'a, V. Nov\'ak, T. Bezd\v{e}k, C. Possel, M. Beseda - - - 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 - math-ph - math.MP - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - 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 - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Son T. Duong, Tho Le-Ngoc - - - 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/ - Mario Tuci, Lennart Bastian, Benjamin Dupuis, Nassir Navab, Tolga Birdal, Umut \c{S}im\c{s}ekli - - - 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-ph - math.MP - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Xing M. Wang - - - 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.ST - stat.AP - stat.ME - stat.TH - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Lei Qian, Wu Su, Yanqi Huang, Song Xi Chen - - - 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 - cs.DS - cs.NA - math.NA - physics.chem-ph - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Christopher Kang, Yuan Su - - - 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.TH - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by-nc-nd/4.0/ - Simon Mack, Morten Overgaard, Dennis Dobler - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Himanshu Badhani, Dhanuja G. S., Siddhartha Das - - - 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 - math.OC - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Shaocong Ma, Heng Huang - - - 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://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Guillaume Geoffroy (UCBL, ICJ, AGL) - - - 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.AT - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Yu Chen, Hongwei Lin - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Sudhi Sharma Padillath Vasudevan - - - 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 - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://arxiv.org/licenses/nonexclusive-distrib/1.0/ - Cesare Tronci - - - 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 - math.OC - stat.ML - Fri, 23 Jan 2026 00:00:00 -0500 - replace-cross - http://creativecommons.org/licenses/by/4.0/ - Sofiane Tanji, Samuel Vaiter, Yassine Laguel - - - 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-nc-sa/4.0/ - Wenzhi Gao, Chang He, Madeleine Udell - - - 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/ - Robert L. Bray - - - 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://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) -