arXiv preprints from January 1, 2026 through July 21, 2026 — 02:18:11 EST
A main conjecture in the field of Positive Geometry states that amplituhedra, which are certain semi-algebraic sets in the Grassmannian, are positive geometries. It is motivated by examples showing that the canonical forms of certain amplituhedra compute scattering amplitudes in particle physics. Beyond a small number of special...
We give concentration inequalities in Wasserstein distance for the empirical measure of a sequence of independent and identically distributed random variables with values in a Polish space E. These inequalities involve the covering dimension of the support of the distribution of the variables. More precisely, we obtain a complete...
Using the theory of evolutionary equations, we consider abstract differential equations including non-local integral operators. After providing a condition for the well-posedness of the addressed equation we consider a numerical method of approximating its solution. We provide convergence proofs under conditions on the kernel of the...
We revisit the problem of rigorously and deterministically finding elements of large order in the multiplicative group of integers modulo a natural number $N$. Solving this problem is an essential step in several recent deterministic algorithms for factoring $N$, including the currently fastest ones. In 2018, the second author gave an...
A cut in a graph $G$ is called a {\em bond} if both parts of the cut induce connected subgraphs in $G$, and the {\em bond polytope} is the convex hull of all bonds. Computing the maximum weight bond is an NP-hard problem even for planar graphs. However, the problem is solvable in linear time on $(K_5 \setminus e)$-minor-free graphs,...
We study jumping lines loci of logarithmic bundles associated with finite sets of points in the projective plane. Using the interpolation matrix introduced in [DMTG25], we describe these loci as the zero sets of explicit determinants depending on parameters $(d,m)$ determined by the number of points. We show that for points in general...
In non-Hausdorff topology, many spaces exhibit significant separation properties, such as sober spaces, well-filtered spaces and d-spaces. These properties serve to fundamentally classify T0 topological spaces. In this paper, we introduce and study a new class of topological spaces called co-Noetherian spaces, which can refine the...
We construct degeneration limits of vertex operators for the Virasoro algebra. Our method relies on the rearranged expansion of compositions of vertex operators together with their integral representations. Using this framework, we obtain a vertex operator between Verma modules of rank $r+1$ as a degeneration of a composition of two...
The increased volatility of markets and the pressing need for resource sustainability are driving supply chains towards more agile, distributed, and dynamic designs. Motivated by the Physical Internet initiative, we introduce the Dynamic Stochastic Modular and Mobile Capacity Planning (DSMMCP) problem, which fosters hyperconnectivity...
Recently, Xu proposed a strongly well-filtered space in [24] and systematically investigated some of its properties and characterizations. In this paper, we introduce a new class of T0-spaces called S*-well-filtered spaces, which is strictly larger than the class of strongly well-filtered spaces. First, we establish some connections...
In this paper, we consider the construction of irreducible representations of finite pattern groups in terms of Panov's associative polarization, which is a finite-field analogue of Kirillov's orbital method. Using this construction, first, we are able to classify the irreducible representations of the unipotent radical of the...
The space of parametric b-measures endowed with appropriate topologies is introduced to define a new class of generalized ODEs given by parametric b-measures. This framework offers a new approach for dealing with precompact families of Carathéodory ODEs using nonautonomous dynamical systems techniques. An application to the study of...
We propose the use of de Rham cohomology of special fibers of Shimura varieties to formulate a geometric version of the weight part of Serre's conjecture. We conjecture that this formulation is equivalent to the one using Serre weights and the étale cohomology of Shimura varieties. We prove this equivalence for generic weights and...
For an $\E$-ring spectrum $R$ and a map $f:X\to Pic(R)$ of spaces, the Thom spectrum $\T f$ is a comodule over $R\otimes\Si X$. In this paper we study the topological coHochschild homology of $R\otimes\Si X$ with coefficient $\T f$. More concretely, for a simply connected space $X$, we will give a filtration on $\mathrm{coTHH}^R(\T...
We construct a new family of mod $p$ weight shifting differential operators on Hodge type Shimura varieties at hyperspecial level. First we construct basic theta operators, labelled by positive roots, that generalize Katz's theta operator for modular forms. Secondly we construct theta linkage maps, these are operators between...
For a finite group $G$, let $ψ(G)$ be the sum of the orders of its elements, and define the corresponding normalized sum as $ψ'(G) := ψ(G)/ψ(\mathcal{C}_{|G|})$, where $\mathcal{C}_{|G|}$ is the cyclic group of the same order as $G$. Inspired by analogous criteria for the classes of soluble, supersoluble, and nilpotent groups, our...
We introduce the rank-nullity ring of a matroid $M$, which is a subring of the Chow ring of the permutahedral toric variety. This subring contains the tautological Chern classes of $M$, a fact we deduce from a highly symmetric formula for these classes. When the matroid $M$ is a uniform matroid, the rank-nullity ring coincides with...
In this article, we study the ergodic problem associated to viscous Hamilton-Jacobi equation where the diffusion is governed by the censored fractional Laplacian, a nonlocal elliptic operator restricted to a bounded domain $Ω\subset \mathbb{R}^N$. We restrict ourselves to the case in which the nonlinear gradient term has a scaling...
A graph is said to be globally rigid in $d$-dimensional space if almost all of its embeddings are unique up to isometries. If a graph has enough automorphisms to send any of its vertices into any other, then it is called vertex-transitive. We show that, in any dimension, highly regular vertex-transitive graphs are globally rigid,...
Let $p(n)$ denote the number of partitions of a natural number $n$. As $ n \to \infty$, the $n$th root of $p(n)$ tends to $1$, which is related to the Cauchy--Hadamard test for power series. Andrews also discovered an elementary proof. Sun conjectured that this happens in a certain way for $n\geq 6$: \begin{equation*} \sqrt[n]{p(n)} >...
Let $F$ be a Parseval frame in a Hilbert space and let $E$ be a set of real numbers. From these data, we construct an operator $H_{E,e}$ and a positive operator-valued measure (POVM) $F_{E,e}$. This paper investigates in detail the relationship between the operator $H_{E, e}$ and the POVM $F_{E,e}$. Our results extend the classical...
We show that a vast class of finitary fragments of geometric logic admit a form of Craig interpolation property. In doing so, we provide a new dictionary to import technology from algebraic logic to categorical logic.
We study model-free policy learning for discrete-time mean-field control (MFC) problems with finite state space and compact action space. In contrast to the extensive literature on value-based methods for MFC, policy-based approaches remain largely unexplored due to the intrinsic dependence of transition kernels and rewards on the...
For an integer $n\geq1$, consider a random partition $Π_{n}$ of $\{1,\ldots,n\}$ into $K_{n}$ partition sets with $K_{r,n}$ partition subsets of size $r=1,\ldots,n$, and assume $Π_{n}$ distributed according to the Ewens-Pitman model with parameters $α\in]0,1[$ and $θ>-α$. Although the large-$n$ asymptotic behaviors of $K_{n}$ and...
In this article, we demonstrate that for any positive integer $n$, the knot surgery $4$-manifold $E(n)_K$ has a handle decomposition without $1$- and $3$-handles. Here, $K$ represents either a fibered two-bridge knot $C(2ε_1, 2ε_2,\cdots, 2ε_{2g})$ ($ε_i \in \{ 1, -1\}$) in Conway's notation or a Stallings knot $K_m$ ($m \in \mathbb{Z}$).