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arXiv:2601.02323v2 Announce Type: replace Abstract: We study the crossing matrix of a braid and introduce a polynomial invariant for braid systems that is invariant under Hurwitz equivalence. As an application to the study of surface braids and surface links, we also define an invariant that can be used as an indicator of the necessity of Euler fusion or fission between braid systems.
arXiv:2601.06895v3 Announce Type: replace Abstract: This paper derives closed-form expressions for four-parameter Euler sums containing generalized harmonic numbers with rational arguments, expressed in terms of the polylogarithm and Lerch transcendent, and reducible (under admissible parameter choices) to Riemann zeta and Hurwitz zeta values. It further obtains closed forms for related polylogarithmic integrals.
arXiv:2601.09553v3 Announce Type: replace Abstract: We establish a non-ergodic version of the Host-Kra-Ziegler structure theorem for measure-preserving $\mathbb{Z}^d$-actions. Our argument reduces the non-ergodic case to the ergodic theorem (for $d\ge 2$ due to Candela and Szegedy) via a measurable selection procedure. We also establish a non-ergodic vertical nilcharacter version of our main result. The non-ergodic version of the Host-Kra-..
arXiv:2601.12818v2 Announce Type: replace Abstract: The Lloyd Theorem of (Sol\'e, 1989) is combined with the Schwartz-Zippel Lemma of theoretical computer science to derive non-existence results for perfect codes in the Lee metric, NRT metric, mixed Hamming metric, and for the sum-rank distance. The proofs are based on asymptotic enumeration of integer partitions. The framework is the new concept of {\em polynomial} weakly metric associati..
arXiv:2601.15576v2 Announce Type: replace Abstract: In this note, we discuss a number of open problems in K-stability theory.
arXiv:2601.18376v3 Announce Type: replace Abstract: We present a nesting-free normal form for the formalism of nested conditions and constraints in the context of finite lattices of subgraphs.
arXiv:2601.19050v3 Announce Type: replace Abstract: We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from $C$ to $E$. We also show that for every genus-$2$ curve $C$, there is an integer $n$ with $1 < n \le 59$ such that there is no minimal degree-$n$ map ..
Persuasive Privacy - arxiv.org - 2 days ago - eng
arXiv:2601.22945v2 Announce Type: replace Abstract: We propose a novel framework for measuring privacy from a Bayesian game-theoretic perspective. This framework enables the creation of new, purpose-driven privacy definitions that are rigorously justified, while also allowing for the assessment of existing privacy guarantees through game theory. We show that pure and probabilistic differential privacy are special cases of our framework, an..
arXiv:2602.01001v2 Announce Type: replace Abstract: In this article, we are interested in finding rational points on certain superelliptic curves.
arXiv:2602.01460v3 Announce Type: replace Abstract: Policy-gradient methods are widely used in reinforcement learning, yet training often becomes unstable or slows down as learning progresses. We study this phenomenon through the noise-to-signal ratio (NSR) of a policy-gradient estimator, defined as the estimator variance (noise) normalized by the squared norm of the true gradient (signal). Our main result is that, for (i) finite-horizon l....
arXiv:2602.02250v2 Announce Type: replace Abstract: Kullback-Leibler (KL) divergence regularization is widely used in reinforcement learning, but it becomes infinite under support mismatch and can degenerate in low-noise regimes. Using a unified information-geometric framework, we introduce KL analogs by replacing the Fisher-Rao geometry in the dynamical formulation of the KL with transport-based geometries, and derive closed-form expressi....
arXiv:2602.02938v2 Announce Type: replace Abstract: From a geometric viewpoint, billiard trajectories and geodesics are related by mutual approximation results. In one direction, it is known that every geodesic curve on the boundary of a smooth convex body can be approximated by a sequence of billiard trajectories inside of it. We establish the other direction by proving that, for Riemannian billiard tables (under mild assumptions), there ..
arXiv:2602.03330v2 Announce Type: replace Abstract: We identify a structural extremal principle governing residual \(L^2\)-norms over operator-ordered covariance envelopes. In contrast to the centered setting, where such quantities reduce to trace expressions involving covariance operators, the non-centered framework generates mixed terms that cannot be recovered from covariance ordering alone. We show that the worst-case squared residual..
arXiv:2602.05682v2 Announce Type: replace Abstract: The $\Delta$-unknotting number for a knot is defined as the minimum number of $\Delta$-moves needed to deform the knot into the trivial knot. It is known that, for positive pretzel knots, the $\Delta$-unknotting number coincides with the second coefficient of their Conway polynomial. In this paper, we compute the $\Delta$-unknotting number for positive pretzel knots. As a consequence of t..
arXiv:2602.09815v4 Announce Type: replace Abstract: The space of all finite non-empty subsets of a topological space $X$, also known as the Ran space of $X$, is weakly contractible for $X$ path connected. We consider subspaces $\mathrm{Ran}_{\leqslant n}(X)$ of the Ran space given by all subsets of $X$ of size at most $n$, and their first homotopy groups. These groups are known to be trivial for $n\geqslant 3$ when $X$ is a path connected ..
arXiv:2602.11051v3 Announce Type: replace Abstract: Consider a discrete-time simple random walk $(X_t)_{t\ge 0}$ on an infinite, connected, locally finite simple graph $G$, and let \[ R_t := |\{X_0,\ldots,X_t\}| \] denote its range. The main result of this revised note is that positive vertex isoperimetry already forces linear expected range, with no bounded-degree assumption: if \[ \iota_V(G) := \inf_{0<|S|<\infty} \frac{|\partial_V S|}{|....
arXiv:2602.13032v2 Announce Type: replace Abstract: We consider (random) strategic interactions in a large population consisting of a variety of players. A rational player chooses actions that maximise certain utility functions, while a behavioural player chooses actions based on preferences such as avoid-the-crowd or follow-the-majority. We specifically study a turn-by-turn dynamic process in which players choose their actions sequentiall....
arXiv:2602.14848v3 Announce Type: replace Abstract: We consider an inverse problem governed by the initial-boundary value problem for the thermo-piezoelectric Kelvin-Voigt dynamical system \[ \left\{ \begin{aligned} \rho(z,t) u_{tt} &= \frac{d}{dz} \big( \Gamma(\Theta) u_{zt} +p_1 u_z +p_2(z,t)\phi_z^0 +p_2(z,t)\chi_z -\beta \Theta \big), \\[1ex] 0 &=-\frac{d}{dz} \big( p_2(z,t)u_z -p_3(z,t)\phi_z^0 -p_3(z,t)\chi_z\big), \\ b(z,t)\Theta_t ....
arXiv:2602.16572v3 Announce Type: replace Abstract: Over an arbitrary commutative ring $R$, we develop a theory of quantum cellular automata. We then use algebraic K-theory to construct a space $\mathbf{Q}(X)$ of quantum cellular automata (QCA) on a given metric space $X$. In most cases of interest, $\pi_0 \mathbf{Q}(X)$ classifies QCA up to quantum circuits and stabilization. Notably, the QCA spaces are related by homotopy equivalences $\..
arXiv:2602.17845v2 Announce Type: replace Abstract: Coron established a homological obstruction to continuous feedback stabilization of nonlinear control systems $\dot{x}=f(x,u)$ with $f \in C(\Omega,\mathbb{R}^n)$ and $f(0,0)=0$, showing that local asymptotic stabilizability implies the induced homomorphism $f_*$ satisfies $f_*\big(H_{n-1}(\Sigma_\epsilon)\big)=H_{n-1}(S^{n-1})$, where $\Sigma_\epsilon:=\Big(\big(\mathbb{B}_\epsilon^{\mat..
arXiv:2602.21640v2 Announce Type: replace Abstract: We study the ground-state of a Fermi gas with short range attrative interactions in one or two dimensions. N fermions are placed in a confining potential, and interact with each other through a negative potential, whose range is larger than the typical distance between particles. We show the convergence of the ground state energy of the Hamiltonian to a Thomas-Fermi energy in the large N ..
arXiv:2602.22767v3 Announce Type: replace Abstract: We prove the persistence of boundary smoothness of vortex patches for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations generalize the Euler equations by including an additional parameter, the Rossby radius $\varepsilon^{-1}$, which modifies the relationship between the streamfunction and the (potential) vorticity. In addition, we prove that solutions of the QGSW eq..
arXiv:2602.23572v2 Announce Type: replace Abstract: SJT reducibility between sets $A,B \subseteq \mathbb N$ is defined by $A \le_{SJT} B$ if for each computable function $h$ that is unbounded and nondecreasing, there is an $h$-bounded uniformly $B$-c.e.\ trace $(T_n)_{n \in \mathbb N} $ such that for each $n$, the value $J^A(n)$ of the jump is in $T_n$, if defined. This reducibility is slightly weaker than Turing reducibility. We study SJT..
arXiv:2602.24219v2 Announce Type: replace Abstract: A statistic can be a function of multiple samples. There is little existing work on asymptotic theory for such statistics when group membership is random. We propose a flexible framework that can handle both deterministic and random membership. We prove some asymptotic properties and apply the framework to the stratified sampling context.
arXiv:2603.00752v4 Announce Type: replace Abstract: In this paper we give a construction for a linear quotient ordering of a class of products of two ideals which have linear quotients. We apply this construction to give a class of modified anticycle graphs whose square and cube have linear quotients.
arXiv:2603.02410v2 Announce Type: replace Abstract: We present a theoretical and numerical dynamical-systems analysis of tubular origami tessellations by identifying the inverse module number, $N^{-1}$, as a perturbation parameter within the framework of Kolmogorov--Arnold--Moser (KAM) theory. In the large-module limit ($N \to \infty$), we show that the conservative dynamics formally converges to an integrable map with a variational struct....
arXiv:2603.05455v3 Announce Type: replace Abstract: This is the third paper in a series, following [FPVa] and [FPVb]. We classify all modular compactifications of the universal Jacobian over $\overline{\mathcal{M}}_{g,n}$, both as stacks and as their relative good moduli spaces. Our main result gives a combinatorial parametrization of compactified universal Jacobian stacks by $V$-functions on a stability domain $\mathbb{D}_{g,n}$ of half-....
arXiv:2603.06033v2 Announce Type: replace Abstract: We prove Oort's conjecture that generically on the supersingular locus of the moduli space of principally polarized abelian varieties of genus g and in characteristic p, the automorphism group of the universal principally polarized abelian variety consists only of $\pm 1$, unless g=2 or 3 and p=2. On the way, we provide an explicit description of the a=1-locus in the Rapoport-Zink space o..
arXiv:2603.06678v2 Announce Type: replace Abstract: Partial Information Decomposition (PID) has become one of the most prominent information-theoretic frameworks for describing the structure and quality of information in complex systems. Despite its widespread utility, there exists no unique solution constraining precisely how a PID should be constructed, leading to a multiverse of different formalisms with different mathematical commitmen....
arXiv:2603.06886v2 Announce Type: replace Abstract: We consider a general round-robin tournament model with equally strong players, where $X_{ij}$ denotes the score of player $i$ against player $j$. We assume that $X_{ij}$ takes values in a countable subset of $[0,1]$ and satisfies $X_{ij}+X_{ji}=1$. We prove that if $k(n)\to\infty$ as $n\to\infty$ and $ \frac{k(n)^2\log\!\bigl(n/k(n)\bigr)}{\sqrt n}\to 0, $ then, with probability tending ..
arXiv:2603.07439v3 Announce Type: replace Abstract: Given any two forests (pseudoforests) with the same degree sequence, we show that one can be transformed into the other by a sequence of 2-switches in such a way that all the intermediate graphs of the transformation are forests (pseudoforests). We also prove that the 2-switch operation perturbs minimally some well-known integer parameters in families of graphs with the same degree sequen..
arXiv:2603.08994v3 Announce Type: replace Abstract: According to the classical Wagstaff heuristic, the probability that a Mersenne number \(M_p=2^p-1\) is prime depends primarily on the size of the exponent \(p\). We investigate whether the divisor structure of \(p-1\) produces detectable secondary variations within this asymptotic framework. We introduce the normalized divisor parameter S(p)=\frac{\log\tau(p-1)}{\log\log p}, which measure....
arXiv:2603.10312v3 Announce Type: replace Abstract: In this paper, we establish the Kobayashi-Hitchin correspondence for nef and big cohomology classes by introducing the notions of adapted closed positive $(1,1)$-currents and adapted Hermitian-Yang-Mills metrics. As applications, we investigate the equality cases of both the Bogomolov-Gieseker inequality for semistable reflexive sheaves with respect to big classes admitting a bimeromorphi..
arXiv:2603.11022v2 Announce Type: replace Abstract: We show that for generic smooth compact initial surfaces the mean curvature flow in $\mathbb{R}^3$ has spherical or nondegenerate neck pinch singularities at the first singular time. In particular the singularities at the first singular time are isolated in spacetime. As an application we give a new approach to constructing a mean curvature flow with surgery for smooth compact initial sur..
arXiv:2603.16111v2 Announce Type: replace Abstract: We study the perturbed Hofstadter $Q$-recursion \[ Q(1)=Q(2)=1,\qquad Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1)^n \quad (n\ge3). \] We investigate its value frequencies and dyadic fluctuation structure. Our first main result is an explicit dyadic frequency law: if $F(s)$ denotes the number of occurrences of the value $2s-1$, then for every $k\ge0$, \[ \{F(s):2^k\le s<2^{k+1}\} = \{3+\nu_2(j):1\le....
arXiv:2603.16698v4 Announce Type: replace Abstract: Recently Watanabe has given an algorithm to compute a bijection, that he calls (quantum) Littlewood-Richardson (LR) map (or quantum LR rule of type AII), between semi-standard Young tableaux of shape a partition with at most $2n$ parts and pairs of tableaux consisting of a symplectic tableau with shape a partition with at most $n$ parts, and a recording tableau of skew-shape given by the ....
arXiv:2603.17805v2 Announce Type: replace Abstract: We prove that if \(B\) is a finite skew brace with \((B,\cdot)\cong S\times T\), where \(S\) and \(T\) are non-abelian simple groups, then \((B,+)\) is not supersolvable.
arXiv:2603.20438v2 Announce Type: replace Abstract: We study the relationship between disturbance decoupling (DD) and H2 optimal control for linear time-invariant (LTI) systems, revealing a fundamental gap between DD subspace constraints and semi-definite program (SDP)-based H2 minimization. We show that DD is equivalent to the existence of zero H2 gain without requiring internal stability, whereas SDP-based H2 minimization strictly optimi....
arXiv:2603.21001v2 Announce Type: replace Abstract: Let G be a primitive permutation group on a finite set Omega. Let p^2 divide |G|, for a prime p. We show that when G is solvable, there exists a subset of Omega whose stabilizer S has the property that 1<|S|_p<|G|_p. We offer a counting argument which should be helpful when G is not solvable.
arXiv:2603.24340v3 Announce Type: replace Abstract: We study the interaction between the group law on an abelian variety and the additive structure induced on its image under a morphism to projective space. Let $A/F$ be a simple abelian variety, $f:A \rightarrow \mathbb{P}^n$ be a morphism which is finite onto its image, and $\Gamma \subseteq A(F)$ be a finite-rank subgroup. We show that for any affine chart $\mathbb{A}^n \subseteq \mathbb..
arXiv:2603.26376v2 Announce Type: replace Abstract: By using a similar pattern of arguments, we show that in three categories the collection of isomorphisms forms a residual subset of the space of morphisms. We first consider surjective continuous mappings on Cantor spaces. Next, we look at measure preserving maps on Polish measure spaces. Finally, we examine continuous, measure preserving maps on Cantor spaces equipped with so-called good..
arXiv:2603.27712v2 Announce Type: replace Abstract: We study a semimartingale optimal transport problem interpolating between the Schr\"odinger bridge and the stretched Brownian motion associated with the Bass solution of the Skorokhod embedding problem. The cost combines an entropy term on the drift with a quadratic penalization of the diffusion coefficient, leading to a stochastic control problem over drift and volatility. We establish a....
arXiv:2603.29923v3 Announce Type: replace Abstract: This paper investigates the scaling limit of the one-dimensional lattice Ising--Kac--Kawasaki dynamics near the critical temperature. Starting from a martingale formulation for the Kac coarse-grained field, we decompose the microscopic dynamics into a discrete conservative drift and a pure-jump Dynkin martingale. Under a critical scaling regime, the nonlinear drift is identified via a con....
arXiv:2603.29998v5 Announce Type: replace Abstract: We provide representations of Euler's constant $\gamma=0.577...$ as series which converge geometrically fast (but use a certain sequence whose computation induces a quadratic cost). The asymptotic oscillations of these coefficients are determined to all orders. A result of independent interest, about sufficient conditions for the validity, in the case of unbounded parameters, for the Tric..
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