arXiv preprints from January 1, 2026 through September 22, 2026 — 03:14:12 EST
We construct a natural chain map from the Kontsevich graph complex to the rational singular chain complex of $B\mathrm{Diff}_\partial(D^{2k})$ when the dimension $2k$ is sufficiently large, generalizing Goussarov and Habiro's theories of surgery on 3-valent graphs in 3-manifolds. Our construction can be considered as a topological...
The McKay correspondence establishes a bijection between the cohomology of a minimal resolution and the irreducible representations of a finite subgroup $Γ\subset \text{SU}(2)$. While traditional proofs rely on static algebraic isomorphisms, we propose a dynamical framework grounded in gauge theory and Morse-Bott theory. We analyze an...
This paper presents a rigorous numerical framework for computing multiple solutions of semilinear elliptic problems by spatiotemporal high-index saddle dynamics (HiSD), which extends the traditional HiSD to the continuous-in-space setting, explicitly incorporating spatial differential operators. To enforce the Stiefel manifold...
Non-asymptotic central limit theorem (CLT) rates play a central role in modern machine learning and operations research. In this paper, we study CLT rates for multivariate dependent data in Wasserstein-$p$ ($W_p$) distance, for general $p\ge 1$. We focus on two fundamental dependence structures that commonly arise in practice: locally...
The twisted product of functions on $R^{2N}$ is extended to a $*$-algebra of tempered distributions which contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant under the Fourier transformation. The regularity properties of the twisted product are...
In this paper we establish an asymptotic formula for the signless Laplacian spectral radius of a $2$-dimensional simplicial complex with given $2$-th Betti number. Furthermore, we characterize the $2$-dimensional simplicial complex that achieves the maximum signless Laplacian spectral radius among all-dimensional simplicial complex...
Pseudo-cones serve as the noncompact counterpart of convex bodies in convex geometry. This paper establishes a necessary and sufficient condition for the existence of solutions to the Orlicz-Gauss image problem for pseudo-cones and further demonstrates its connection to spherical optimal transport. Our approach combines the...
We introduce tropical Kummer quartic surfaces in tropical projective $3$-space as the images of certain principally polarized tropical abelian surfaces under tropical theta functions of second order. We study some of their properties, showing that they are included in the tropicalizations of Kummer quartic surfaces defined over...
We establish the global existence of weak entropy solutions for 1D isentropic gas dynamics with general pressure laws ($γ> 1$). To address vacuum degeneracy, we introduce a novel structural regularization via a "Synchronized Dual Translation" strategy. This approach offers a distinct advantage over the flux-modification method of Lu...
The paper studies a general norm minimization problem on a product of normed vector spaces. We establish dual necessary and sufficient optimality conditions and derive explicit formulas for the corresponding solution sets. These formulas are obtained under the assumption that one optimal solution together with its associated dual...
Flow cones of a directed acyclic graph admit a family of unimodular triangulations given by Danilov, Karzanov, and Koshevoy (DKK) whose normal fans are related to (generalizations) of the associahedron and permutahedron. A correspondence between these triangulations for certain graphs and maximal cones of a $g$-vector fan of a gentle...
The Birkhoff Ergodic Theorem establishes pointwise convergence for integrable observables, but for $f\notin L^1$, no normalization yields almost sure convergence. This paper investigates trimmed ergodic sums, where the largest observations are removed, for observables with polynomial tails $¶(f>t)\asymp t^{-1/α}$ in exponentially...
In this paper, we prove Bernstein type theorems for entire convex graphical hypersurfaces with zero Gaussian curvature in both Euclidean and Minkowski context. A supplementary example illustrates that zero Gaussian convex spacelike hypersurfaces are not necessary hyperplanes without additional conditions. We show that a zero Gaussian...
We give a short, self-contained analytic proof of the existence of self-similar Dirichlet forms on pillow-type carpets, a family of infinitely ramified fractals that includes the Sierpiński carpet.
We systematically compute improved asymptotic rank bounds for tensors. Using numerical implicitization, we implement the geometric framework of Kaski and Michałek across all computationally feasible cases. By detecting the absence of low-degree vanishing polynomials on secant varieties, we obtain new asymptotic rank bounds that...
It is found that the wave functions of the Gross-Pitaevskii equation (GPE) often vary significantly in different spatial regions, with some components exhibiting sharp variations while others remain smooth. Solving the GPE on a single mesh, even with adaptive refinement, can lead to excessive computational costs due to the need to...
We establish two-sided weighted integrability estimates, often referred to as a norm equivalence result, for stochastic differential equations (SDEs) with locally Lipschitz coefficients. As a key ingredient in our approach, we also derive an SDE satisfied by the inverse stochastic flow under reduced regularity assumptions in the...
We generalize the notion of mod $p^m$ singular Siegel modular forms of $p$-rank $r$ to the vector-valued case and we show that also in this case a congruence mod $(p-1)p^{m-1}$ between the scalar weight and the $p$-rank must hold. In some sense our proof is even simpler than the one we gave previously in the scaler valued case.
In this paper, we present a complete characterization of mutual-visibility sets in trees. It is shown that a subset $S$ is a mutual-visibility set of a tree $T$ if and only if it coincides with the set of leaves of the Steiner subtree $T\langle S\rangle$. For trees containing branch vertices, the notion of legs is introduced, and an...
Maximal quantum $f$-divergences, defined via the commutant Radon--Nikodym derivative, form a fundamental class of distinguishability measures for quantum states associated with operator convex functions. In this paper, we study the optimization of maximal quantum $f$-divergences along unitary orbits of two quantum states. For any...
The edge-to-edge tilings of the sphere by congruent polygons, where all edges are straight, have been completely classified. We classify the curvilinear version of the similar triangular tilings, where the edges may not be straight, and find that these are the modifications of the straight triangular tilings.
The algebraic formulation of the derivation and integration related by the First Fundamental Theorem of Calculus (FFTC) gives rise to the notion of differential Rota-Baxter algebra. The notion has a remarkable list of categorical properties, in terms of the existence of (co)extensions of differential and Rota-Baxter operators, of the...
Monkeypox is a viral disease belonging to the smallpox family. Although it has milder symptoms than smallpox in humans, it has become a global threat in recent years, especially in African countries. Initially, incidental immunity against monkeypox was provided by smallpox vaccines. However, the eradication of smallpox over time and...
We consider time-harmonic electromagnetic scattering by a cluster of hybrid dielectric-plasmonic dimers in $\mathbb{R}^3$. Each dimer consists of a high-contrast dielectric nanoparticle and a moderately contrasting plasmonic nanoparticle separated by a subwavelength distance. The cluster is assumed to contain many such dimers whose...
We consider time-harmonic incompressible flow with a spatially resolved complex viscosity field $μ^*(\mathbf{x},ω)$ and, at fixed forcing frequency $ω>0$, its constitutive phase texture $\varphi(\mathbf{x})=\argμ^*(\mathbf{x},ω)$. In three-dimensional domains periodic in a spanwise direction $z$, $z$-dependence of $μ^*$ converts...