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arXiv:2606.02405v1 Announce Type: new Abstract: We prove that the bilinear generating function for Wronskians of Hermite polynomials can be expressed as the classical Mehler kernel multiplied by a polynomial, thereby extending the result of Pupasov-Maksimov for exceptional Hermite polynomials. We establish several properties of the polynomials appearing in this extended version of the Mehler formula and present four conjectures about them.
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Analogues of the Lindel\"of Hypothesis for the Barnes multiple zeta function and related problems
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arxiv.org
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2 days ago
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eng
arXiv:2606.02407v1 Announce Type: new Abstract: For the Lindel\"of Hypothesis concerning the Riemann zeta function $\zeta(s)$, upper bounds as $\Im(s)\to\infty$ have been extensively studied for many years. In particular, the Lindel\"of Hypothesis is one of the most important open problems in analytic number theory. It is also known to be equivalent to certain mean value estimates, which provide a fundamental connection between pointwise u..
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Framed bordism of Lagrangian homotopy spheres via generating functions
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arxiv.org
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2 days ago
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eng
arXiv:2606.02409v1 Announce Type: new Abstract: This note combines a result of B\"okstedt and Waldhausen concerning the so-called derivative map on tubes with the existence theorem for generating functions of tube type for nearby Lagrangian homotopy spheres due to Abouzaid, Courte, Guillermou and Kragh to obtain a restriction on the smooth structure of nearby Lagrangian homotopy spheres. Concretely, if a homotopy $n$-sphere $L$ admits a La....
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On rational representations and rational group algebra of $\operatorname{GL}_2(q)$
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arxiv.org
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2 days ago
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eng
arXiv:2606.02415v1 Announce Type: new Abstract: In this article, we study rational representations of $G=\operatorname{GL}_2(q)$, where $q$ is a prime power. Let $\rho$ be an irreducible representation of $G$ over $\mathbb{Q}$. Then $\rho$ affords the character \[ \Omega(\chi)=m_{\mathbb{Q}}(\chi)\sum_{\sigma\in\operatorname{Gal}(\mathbb{Q}(\chi)/\mathbb{Q})}\chi^{\sigma}, \] for some irreducible complex character $\chi$ of $G$, where $m_{....
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arXiv:2606.02426v1 Announce Type: new Abstract: For the family of Dirichlet stadia $S_t$ whose rectangular part has height $\pi$ and half-length $\pi t/2$, $t \in [1,2]$, we show that for Lebesgue almost every $t$ there exist real eigenfunctions $u_j$ and a smooth mean-zero physical observable $Q$ for which $\langle Q u_j,u_j\rangle$ has a non-zero subsequential limit. Consequently, along the same subsequence, the eigenfunction mass fails ..
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arXiv:2606.02427v1 Announce Type: new Abstract: Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dynamical structure. A model may match solutions while exhibiting incorrect sensitivities, distorted frequency response, spurious mode coupling, or unstable tangent behavior. We introduce a Jacobian-based spectral aud....
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arXiv:2606.02428v1 Announce Type: new Abstract: We determine the logarithmic growth exponents of the $L^p$ norms, $1\le p\le\infty$, of $L^2$-normalized Laplace eigenfunctions on the unit disk, for both Dirichlet and Neumann boundary conditions. We also prove sharp uniform $L^p$ upper and lower bounds for every $L^2$-normalized Dirichlet eigenfunction and every non-constant Neumann eigenfunction $u_{\lambda}$ on the disk. The proof uses st..
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arXiv:2606.02429v1 Announce Type: new Abstract: This note gives a simple algorithm for the following effectivity problem: given a genus $2$ curve $X$ together with a nonconstant map $\pi:X\to E$ to an elliptic curve, determine an elliptic curve $E'$ and a map $\pi':X\to E'$ independent of $\pi$. Equivalently, we compute the complementary elliptic factor in the decomposition of $\operatorname{Jac}(X)$ up to isogeny. While the problem has be..
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Symplectic capacities of $S^1$-invariant dynamically convex domains in $\mathbb{R}^4$
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arxiv.org
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2 days ago
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eng
arXiv:2606.02452v1 Announce Type: new Abstract: In this paper, we prove that all normalized symplectic capacities agree for dynamically convex domains in $\mathbb{C}^2$ that are invariant under any Hamiltonian $S^1$-action isotopic to the Hopf diagonal action. We also give necessary and sufficient conditions for $S^1$-invariant domains to be dynamically convex.
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arXiv:2606.02464v1 Announce Type: new Abstract: We study admission control to an unobservable M/M/1 queue. A memoryless controller can only randomly thin arrivals (random routing, RR). We show that a gated admission (GA) policy, blocking arrivals for a fixed period after each admission, stochastically dominates RR at equal throughput, improving social welfare under any sojourn-based cost. We characterize the welfare-maximizing threshold an..
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arXiv:2606.02468v1 Announce Type: new Abstract: In this thesis we present the combinatorial game theory of the game Rex+. Rex+ is a variant of the game Hex, played on a four sided board made out of hexagons. Both players take turns placing as many stones of their colour as they would like on the board, with the objective being to force the other player to connect their two sides. We describe a new ordering, and present some preliminary res..
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A novel L-shaped refinement chain cuts method for two-stage stochastic programs
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arxiv.org
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2 days ago
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eng
arXiv:2606.02469v1 Announce Type: new Abstract: This paper introduces the L-shaped refinement chain cuts method, a novel approach for solving two-stage stochastic programs. The proposed method integrates the refinement chain of scenarios within the classical L-shaped decomposition framework. In the proposed approach, the full scenario set is partitioned into subgroups at each level of the refinement chain, and one subproblem is solved for ....
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arXiv:2606.02471v1 Announce Type: new Abstract: Using our new notion of folding shuffle algebras, we prove a conjecture of Hernandez on the equality between certain $q$-characters of quantum untwisted affine algebra modules and their twisted counterparts. We generalize this result to the setting of arbitrary quivers with automorphisms, in particular by defining and describing twisted quantum toroidal algebras.
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arXiv:2606.02472v1 Announce Type: new Abstract: We introduce and study a new model of correlated uniform attachment (UA) trees, where correlation is sprinkled throughout the time evolution of the process. In this model, two UA trees are grown in parallel, and at each time step a new node is added to each tree, with an edge between it and a uniformly chosen existing vertex in the respective tree. The two choices of attachment are correlated....
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Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers
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arxiv.org
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2 days ago
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eng
arXiv:2606.02475v1 Announce Type: new Abstract: Classical finite-difference solvers remain reliable tools for partial differential equations, but their efficiency depends on where mesh resolution is placed. Uniform refinement can waste degrees of freedom when solution difficulty is localised near sharp gradients, fronts, oscillations, or constraint-sensitive regions. This paper studies a hybrid strategy in which a physics-informed neural n....
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arXiv:2606.02477v1 Announce Type: new Abstract: In the representation theory of Lorentzian orthogonal groups there are well known arguments as to why the parity operator $\mathcal{P}$ and the time reversal operator $\mathcal{T}$ should be realized as linear and anti-linear operators respectively (Wigner 1932). In this paper it is shown that the only operators satisfying the requisite properties for the spinor representation of the de Sitte..
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arXiv:2606.02478v1 Announce Type: new Abstract: A regular clock map is a regular map of directed spaces from a saturated directed space to the directed circle. We prove that the category of regular clock maps is a small-orthogonality class of the category of clock maps. Hence it is locally presentable. Any geometric realization of precubical sets and of transverse sets gives rise to a regular clock map. Finally, we prove that for the under..
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Construction of Lyapunov Functions for Switched Systems using Meshfree Collocation
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arxiv.org
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2 days ago
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eng
arXiv:2606.02495v1 Announce Type: new Abstract: Switched systems are a family of dynamical systems where a switching rule indicates which system is "switched on". This rule can be dependent on time and/or position in the state space. Stability of switched systems is a property that is often investigated using the existence of one (or multiple) Lyapunov function(s). We develop an algorithm using a scattered approximation method to construct..
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arXiv:2606.02496v1 Announce Type: new Abstract: We introduce the notion of timelike ideal boundary of a Lorentzian length space as the set of asymptotic classes of future or past-directed timelike geodesic rays, a construction complementary to the causal boundary in the sense of Geroch-Kronheimer-Penrose and akin to the concept of ideal boundary of a metric space. We endow such a timelike ideal boundary with a natural cone topology and an ..
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arXiv:2606.02513v1 Announce Type: new Abstract: We provide a classification of neighborly Murai spheres, which implies that all of them are polytopal. Furthermore, we show that each neighborly $d$-sphere with no more than $d+4$ vertices is combinatorially equivalent to a Murai sphere for any $d\geq 1$.
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Weak-strong uniqueness and low Mach number limit for a viscous compressible fluid around a rotating body
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arxiv.org
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2 days ago
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eng
arXiv:2606.02517v1 Announce Type: new Abstract: We study the flow of an isothermal compressible Newtonian fluid around a body that performs a (time-independent) rigid motion. We derive a weak-strong uniqueness principle, and show that in the low Mach number limit, the governing equation is well approximated by the Navier-Stokes equations for incompressible rotating flow. Both results are based on the derivation of a relative energy inequal..
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arXiv:2606.02525v1 Announce Type: new Abstract: In the first part of our paper we give an abstract characterization of matrix convex sets, and compact matrix convex sets. We also extend the real case of this for classical sets (due to Marshall Stone and others) to the complex case. Our approach is in some part via a universal Banach space (resp.\ operator space $X_K$ of an abstract compact convex set (resp.\ matrix convex set) $K$. This tu....
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arXiv:2606.02529v1 Announce Type: new Abstract: Incentive design studies how a central authority can influence strategic agents through payments, subsidies, or taxes, so that individual objectives align with collective welfare. This paper introduces a No-Regret Adaptive Incentive Design (RAID) framework for nonlinear games with continuous action spaces and private agent costs. In this framework, the authority (planner) designs incentives t....
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arXiv:2606.02541v1 Announce Type: new Abstract: For $k$ sufficiently large, we show that there is a three-coloring of the first $2^{k (\log^* k)/4}$ positive integers without any monochromatic $k$-term arithmetic progressions. Thus, the three-color van der Waerden number $w(k;3)$ grows faster than any exponential in $k$. We further prove a new lower bound on multicolor van der Waerden numbers which resolves a problem of Erd\H{o}s and Graha..
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arXiv:2606.02546v1 Announce Type: new Abstract: Given $k$-uniform hypergraphs $G$ and $H$, we say that $G$ is a unique subhypergraph of $H$ if $H$ contains exactly one subhypergraph isomorphic to $G$. For an $n$-vertex $k$-graph $H$, let $f_k(H)$ be the number of non-isomorphic unique subhypergraphs of $H$, normalized by $2^{\binom n k}/n!$, and let $f_k(n)$ be the maximum of $f_k(H)$ over all $n$-vertex $k$-graphs $H$. In the graph case $....
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Conjugacy Problem for Dehn Twists of Free Products of Free Abelian Groups
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arxiv.org
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2 days ago
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eng
arXiv:2606.02558v1 Announce Type: new Abstract: We show solubility of the conjugacy problem for Dehn twist automorphisms of finitely generated free products of free abelian groups.
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Pure UCP Maps on Finite Toeplitz Systems and Quantum Gromov--Hausdorff Convergence
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arxiv.org
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2 days ago
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eng
arXiv:2606.02561v1 Announce Type: new Abstract: We study pure unital completely positive maps on the finite Toeplitz operator system $ T_{d}$ of $d \times d$ Toeplitz matrices. Our first main result gives an explicit characterization of pure UCP maps from $T_{d}$ to $M_n$ in terms of positive $n\times n$ matrix-valued trigonometric polynomials of degree at most $d-1$. This characterization provides a checkable criterion for deciding when a..
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arXiv:2606.02567v1 Announce Type: new Abstract: We show that for any set of $n$ unit vectors $v_1,\ldots,v_n$ in a real Hilbert space and positive numbers $p_1,\ldots,p_n$ satisfying $\sum_j p_j = 1$, there exists a unit vector $u$ such that \[ \sum_{j=1}^n \frac{p_j^2}{\langle v_j, u\rangle^2}\leq 1. \] This inequality is a weighted version of the strong polarization inequality. As immediate corollaries, it yields a polarization inequ..
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arXiv:2606.02570v1 Announce Type: new Abstract: We study stochastic completeness for landmark spaces equipped with Riemannian metrics induced by right-invariant metrics on subgroups of the diffeomorphism group of the shape domain. We extend a previous stochastic completeness result, which only covers the case of exactly two landmarks, to landmark spaces with any number of landmarks. This succeeds the characterization of geodesic completene....
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Stationary Measures and Mean Flux Depending on Multiple Conserved Quantities in a Stochastic Cellular Automaton
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arxiv.org
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2 days ago
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eng
arXiv:2605.28052v1 Announce Type: cross Abstract: We analyze a stochastic 5-neighbor cellular automaton with several conserved quantities, including the particle density. By examining the eigenvalue problem of the associated transition matrix, we derive an explicit formula for the stationary distribution on each irreducible component, in which the weight of each configuration is expressed in terms of the numbers of occurrences of two speci..
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Intrinsic linking of simplicial $n$-complexes in $\mathbb{R}^{2n}$: An additional minimal $n$-complex
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arxiv.org
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2 days ago
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eng
arXiv:2605.28496v2 Announce Type: cross Abstract: For any positive integer $n$, the author previously constructed several minimal simplicial $n$-complexes which necessarily contain a non-splittable two-component link, consisting of an $(n-1)$-sphere and an $n$-sphere, in any embedding into $\mathbb{R}^{2n}$. In this paper, we present an additional simplicial $n$-complex with the same property.
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Optimal Rates for Differentially Private Hypothesis Testing with E-values
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arxiv.org
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2 days ago
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eng
arXiv:2605.28952v2 Announce Type: cross Abstract: E-values have attracted considerable interest in recent years as flexible tools for enabling anytime-valid and adaptive data analysis. Hypothesis testing is at the core of many of these applications, which can often involve private or sensitive data. In this work, we answer a simple but important question: given two distributions $\mathbb{P}$ and $\mathbb{Q}$, what is the maximum achievable....
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A Boolean-Lattice Perspective for All-Loop Two-Site Cosmological Wavefunction
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arxiv.org
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2 days ago
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eng
arXiv:2605.30797v1 Announce Type: cross Abstract: We revisit the shifted-tree decomposition formula proposed in our previous work arXiv:2410.17192 for two-site cosmological wavefunction coefficients. For the two-site bubble-like family at arbitrary loop order, we show that the nontrivial central part of the decomposition reduces to an alternating subset sum over shifted diagonal divisors. This subset sum is naturally organized by the Boole....
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Sum-of-squares certificates for symmetric polynomials on the hypercube: a counterexample to a conjecture of De Klerk and Laurent
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arxiv.org
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2 days ago
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eng
arXiv:2605.31169v1 Announce Type: cross Abstract: This paper studies sum-of-squares (SoS) representations of nonnegative polynomials over the hypercube $[0,1]^n$. De Klerk and Laurent (SIAM J. Optim., 2010) conjectured that the smallest constant $C_n$ such that the polynomial $x_1\cdots x_n +C_n$ is contained in the degree-$n$ truncated quadratic module $M_{n,n}(x_1-x_1^2,\ldots,x_n-x_n^2)$ of the hypercube is $C_n=1/(n(n+2))$, for $n$ eve..
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arXiv:2606.00055v1 Announce Type: cross Abstract: Tensor train (TT) methods have recently gained popularity for accelerating the solving of systems of PDEs. Here, we evaluate the performance of TT methods in the context of geophysical fluid dynamics (GFD) using the shallow water equations and a discretization scheme employed by the ocean component of the Energy Exascale Earth System Model (E3SM). Through a suite of four test cases of incre..
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arXiv:2606.00063v1 Announce Type: cross Abstract: Systems moving in low Reynolds number fluid regimes are known to be governed by a ``motility map'' which linearly relates their shape change rates to they body frame velocity moving through the fluid. A consequence of this is ``Purcell's Scallop Theorem'' -- a locomotion system that undergoes shape changes that follow the same path forward and backward in time (reciprocal body deformations)....
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On the evolution of the concept of probability as a mirror of the evolution of reason
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arxiv.org
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2 days ago
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eng
arXiv:2606.00102v1 Announce Type: cross Abstract: Over the centuries, probability theory has grown from the calculus of games of chance into a central framework for reasoning under uncertainty. This article interprets that evolution not merely as a mathematical history, but as a transformation of rationality itself. From Pascal and Fermat's combinatorial symmetry to the inductive logic of Bayes and Laplace, from Poisson's statistics of eve....
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Recursive Vision Transformer with Dynamic Depth and Width Adjustment for Resource-Efficient Image Semantic Communication
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arxiv.org
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2 days ago
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eng
arXiv:2606.00114v1 Announce Type: cross Abstract: Image semantic communication is a critical component in next-generation wireless communication systems. However, such systems typically suffer from large memory footprints and high computational complexity, making them difficult to deploy on resource-constrained devices. To address these challenges, we propose a vision transformer (ViT)-enabled image semantic communication system. In this s....
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Mathematical framework for perception-driven parameter choice in image denoising
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arxiv.org
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2 days ago
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eng
arXiv:2606.00122v1 Announce Type: cross Abstract: We approach image denoising from a perception-driven perspective: how can we select the parameters that are best suited for human visual perception? We combine research methods in mathematics and psychology to develop a mathematical framework for measuring perceived similarity. We construct a sample set of differently denoised photographs by using the same base image as input data and by tu..
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One Adaptive Trailing Head Can Outperform Many Oblivious Trailing Heads
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arxiv.org
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2 days ago
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eng
arXiv:2606.00127v1 Announce Type: cross Abstract: In the setting of multi-head finite-state dimensions, trailing heads lag behind a leading head, accessing past data to aid a finite-state gambler placing bets on successive bits read by the leading head. Cruz, Glashausser, Li, and Lutz (2026) proved that, for any fixed number of trailing heads, adaptive (data-dependent) movement rules can strictly outperform oblivious (data-independent) mov..
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arXiv:2606.00157v1 Announce Type: cross Abstract: We consider establishing the interpretability theory of deep learning through constructing a corresponding relationship between the renormalization group (RG) method in statistical physics and the training process of deep neural networks (DNNs). We have proved the constructed relationship using the one-dimensional Ising model as the input data. In this paper we generalize our results to the....
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Future global stability of Maxwell-J\"uttner equilibria and vacuum for the massless Boltzmann equation on FLRW spacetimes
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arxiv.org
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2 days ago
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eng
arXiv:2606.00175v1 Announce Type: cross Abstract: In this work we study the general relativistic massless Boltzmann equation on Friedmann-Lema\^itre-Robertson-Walker spacetimes with spatial topology $\mathbb{T}^3$ in the linear and decelerated expanding regimes, where the scale factor is $t^{\mathfrak{q}}$ with $\mathfrak{q}\in [0,1]$. The massless Boltzmann equation on these backgrounds admits non-stationary Maxwell-J\"uttner equilibria o....
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Eigenvalue formulation of Stochastic Inflation and application to large perturbation generating inflationary features
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arxiv.org
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2 days ago
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eng
arXiv:2606.00176v1 Announce Type: cross Abstract: Stochastic inflation is a powerful technique for calculating the probability distribution function (PDF) of large inflationary perturbations, which may collapse to form Primordial Black Holes. The PDF, $P({\cal N})$, of the stochastic number of e-folds, ${\cal N}$, satisfies an adjoint Fokker-Planck Equation. We develop a new self-contained eigenvalue technique which can be used to determin....
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