arXiv preprints from January 1, 2026 through September 6, 2026 — 08:01:46 EST
An ordered $r$-uniform matching of size $n$ is a collection of $n$ pairwise disjoint $r$-subsets of a linearly ordered set of $rn$ vertices. For $n=2$, such a matching is called an $r$-pattern, as it represents one of $\tfrac12\binom{2r}r$ ways two disjoint edges may intertwine. Given a set $\mathcal{P}$ of $r$-patterns, a...
We study LU-type factorizations of the infinitesimal generator of a birth--death process on $\mathbb{N}_0$. Our goal is to characterize those factorizations whose Darboux transformations (that is, inverting the order of the factors) yield new infinitesimal generators of birth--death processes. Two types are considered: lower--upper...
The search for the optimal shape parameter for Radial Basis Function (RBF) kernel approximation has been an outstanding research problem for decades. In this work, we establish a theoretical framework for this problem by leveraging a recently established theory on sharp direct, inverse and saturation statements for kernel based...
Time-dependent structures often appear in differential geometry, particularly in the study of non-autonomous differential equations on manifolds. One may study the geodesics associated with a time-dependent Riemannian metric by extremizing the corresponding energy functional, but also through the introduction of a more general concept...
For compactly supported $μ\in \mathcal{P}(\mathrm{SL}(2,\mathbb{R}))$ satisfying strong irreducibility and proximality, we obtain a formula for the Frostman dimension of the associated Furstenberg measure. We also describe the left neighbourhood of 0 for which the classical transfer operators defined by Le Page have a spectral gap on...
We prove a splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature. Additionally, we extend the first variation formula to spaces with any timelike curvature bound, either from above or below, and different from 0.
Recently, versions of neural networks with infinite-dimensional affine operators inside the computational units (``neural operator'' networks) have been applied to learn solutions to differential equations. To enable practical computations, one employs finite-dimensional encoding/decoding theorems of the following kind: every...
We consider the equality of the values of the $n$th and $k$th elementary symmetric polynomials of $n$ not necessarily distinct positive integers. For $k < n$, we prove that this equation always has a solution, but only finitely many solutions. Furthermore, we consider the equality of the values of the $n$th and $(n-2)$th elementary...
This paper is a sequel to arXiv:2511.01024 (Base 1), where an axiomatic framework for angles and the foundations of difference-angle geometry were introduced. In difference-angle geometry, where the difference of slopes of lines is treated as a primary angular quantity (the difference angle), we reconstruct the focal structure of...
We prove some classification results for tight contact structure in the 3-space, -ball and -sphere that are invariant with respect to some arbitrary involution, that is conjugated to the standard rotation around the x-axis. Unlike the classical scenario, a new integral torsion appears, dictating a splitting between equivalence...
Causality is one of the fundamental structures of spacetimes, determining the possible behaviour and propagation of physical information. Causal structure can be analysed through the various modal logics it induces. The modal logics for the chronological and causal relations of the archetypal Minkowski spacetime have been classified....
The article establishes a rigorous shift operator calculus for stationary time series modeling, addressing a certain gap in the literature. It provides proofs of existence and isometry for the transfer function operators $f(B)$ and $f(T)$ where $B$ is the bilateral shift operator and $T$ is the unilateral shift operator for different...
In this short note, we study the spectrum of prime Serre ideals of global representations for noetherian families. In particular, we prove that the spectrum of prime Serre ideals of finitely generated VI-modules is homeomorphic to N^{*}, the one-point compactification of N, which differs from the Balmer spectrum of derived VI-modules....
We establish a lower bound for the principal $p-$frequency $λ_{1,p}(Ω)$ on a bounded domain $Ω$ in a non-compact Riemannian manifold of dimension $n.$ Under the assumption that the Ricci curvature satisfies $\operatorname{Ric} \geq (n-1)K$ with $K<0,$ we prove that $λ_{1,p}(Ω) > \barλ_{D,K,n}$, where $D$ is the diameter of $Ω$ and...
We study non-linear traces of Choquet type on AF algebras. Building on the characterization of Choquet traces on matrix algebras due to Nagisa--Watatani, we generalize the construction to arbitrary unital AF algebras. We show that there is a one-to-one correspondence between such traces and increasing functions on the dimension scale,...
Allowing for adversarial contamination and heavy tails, we study testing whether the mean of a high-dimensional random vector equals zero. Because standard max-tests based on sample averages are highly non-robust, we propose a max-test based on quantile-winsorized observations. The test controls asymptotic size under adversarial...
The processes of simultaneous coagulation and Ostwald ripening of particles in the concluding stage of phase transformation are considered. We solve the integro-differential system of Smoluchowski-type kinetic and mass balance equations using a computationally efficient numerical algorithm based on low-rank matrices. We compare our...
Let $X$ be a smooth projective curve over a finite field of characteristic $p$. We describe and implement a practical algorithm for computing the $p$-divisible group $Jac(X)[p^\infty]$ via computing its Dieudonné module, or equivalently computing the Frobenius and Verschiebung operators on the first crystalline cohomology of $X$. We...
Ferromagnetic exponential random graph models (ERGMs) are random graph models under which the presence of certain small structures (such as triangles) is encouraged; they can be constructed by tilting an Erdős--Rényi model by the exponential of a particular nonlinear Hamiltonian. These models are mixtures of metastable wells which...
Genetic algorithms are high-level heuristic optimization methods which enjoy great popularity thanks to their intuitive description, flexibility, and, of course, effectiveness. The optimization procedure is based on the evolution of possible solutions following three mechanisms: selection, mutation, and crossover. In this paper, we...
The semisimple module categories over a braided fusion category $\mathcal{C}$ form a connected fusion 2-category $\text{Mod}(\mathcal{C})$. Its Drinfeld center $\mathcal{Z}(\text{Mod}(\mathcal{C}))$ is a braided fusion 2-category. To any braided fusion 2-category, Johnson-Freyd and Reutter arXiv:2105.15167v3 [math.QA] have associated...
We show that the (toric) local height of a toric variety with respect to a semipositive torus-invariant singular metric is given by the integral of a concave function over a compact convex set. This generalizes a result of Burgos, Philippon, and Sombra for the case of continuous metrics and answers a question raised by Burgos, Kramer,...
We introduce classes of supergraphs and superpermutations with novel universal graphon and permuton limiting objects whose construction involves the two-parameter Poisson-Dirichlet process introduced by Pitman and Yor (1997). We demonstrate the universality of these limiting objects through general invariance principles in a...
The goal of this note is to explore, from a geometric and probabilistic point of view, the dynamics of cone structures adapted to open book decompositions. This is inspired by the picture which arises in the study of the circular restricted three body problem (CR3BP). This yields geometric obstructions to reaching a point from another...
Graph burning is a discrete-time process on graphs where vertices are sequentially activated and burning vertices cause their neighbours to burn over time. In this work, we focus on a dynamic setting in which the graph grows over time, and at each step we burn vertices in the growing grid $G_n = [-f(n),f(n)]^2$. We investigate the set...