arXiv preprints from January 1, 2026 through July 20, 2026 — 02:30:55 EST
A filling pair $(α, β)$ of a surface $S_g$ is a pair of simple closed curves in minimal position such that the complement of $α\cupβ$ in $S_g$ is a disjoint union of topological disks. A filling pair is said to be minimally intersecting if the number of intersections between them, or equivalently, the number of complementary disks, is...
We consider a complete bipartite graph of size $n$ endowed with i.i.d. uniform edge weights and run Prim's Algorithm to obtain a ranking of its vertices. Let $ρ^{(n)}_k$ be the proportion of black vertices among the first $k$ vertices in this ranking. We characterise the limit behaviour of $ρ^{(n)}_k$ as both $n$ and $k$ tend to...
We develop a purely combinatorial framework for the systematic enumeration of knot and link diagrams supported on the thickened torus $T^2\times I$. Using the theory of maps on surfaces, cellular $4$--regular torus projections are encoded by permutation pairs $(α,σ)$, and unsensed projection classes are enumerated completely and...
The orthogonality graph of an Okubo algebra with isotropic norm over an arbitrary field $\mathbb{F}$ is considered. Its connected components are described, and their diameters are computed. It is shown that there exist at most two shortest paths between any pair of vertices, and the conditions under which the shortest path is unique...
In this paper, we present and prove some results in multi-objective optimisation that are considered folklore. For the most part, proofs for these results exist in special cases, but they are used in more general settings since their proofs can be (largely) transferred. We do this transfer explicitly and try to state the results as...
Large language models (LLMs) process and predict sequences containing text to answer questions, and address tasks including document summarization, providing recommendations, writing software and solving quantitative problems. We provide a mathematical framework for LLMs by describing the encoding of text sequences into sequences of...
We consider a (left) coherent ring R. We prove that if the character module of every Ding injective (left) R-module is Gorenstein flat, then the class of Gorenstein flat (right) R-modules, GF, is preenveloping. We show that this is the case when every injective (left) R-module has finite flat dimension. In particular, GF is...
The determinants of modular Collatz graphs and the modular Conway amusical permutation graph are determined, and some interesting number theoretic properties are described.
We show that the non-log version of Kato's ramification filtration on the Brauer group of a separated and finite type regular scheme over a positive characteristic local field coincides with the evaluation filtration. This extends a recent result of Bright-Newton to positive characteristics. Among several applications, we extend some...
The standard proof of the equivalence of Fourier type on \(\mathbb R^d\) and on the torus \(\mathbb T^d\) is usually stated in terms of an implicit constant which can be expressed in terms of the global minimiser of the functions \[f_r(x)=\sum_{m\in\mathbb{Z}}\left|\frac{\sin(π(x+m))}{π(x+m)}\right|^{2r},\qquad x\in [0,1], \ r\ge 1.\]...
We study the generalized Ramsey numbers $f(Q_n, C_{k}, q)$, that is, the minimum number of colors needed to edge-color the hypercube $Q_n$ so that every copy of the cycle $C_{k}$ has at least $q$ colors. Our main result is that for any integers $k,q$ satisfying $k \geq 6$ and $3 \leq q \leq k/2+1$, we have $f(Q_n, C_{k}, q)= o\left(...
We establish analogs of Cheeger's inequality for probability measures with heavy tails. As one of the principal applications, suppose $λ> 3$ and define the (Pareto) probability measure $μ_λ$ on $[1,\infty)$ by $dμ_λ(x) = (λ- 1) x^{-λ}$. Let $μ_λ^n$ denote the product measure of $μ_λ$ on $\mathbb{R}^n$. Then, for any $1$-Lipschitz...
The Sparse Identification of Nonlinear Dynamics (SINDy) framework has been frequently used to discover parsimonious differential equations governing natural and physical systems. This includes recent extensions to SINDy that enable the recovery of discrete delay differential equations, where delay terms are represented explicitly in...
Given a smooth variety together with two smooth subvarieties, we construct, via two successive blowups, smooth varieties admitting small contractions. This generalizes Kawamata's example of small contraction in dimension 4. We also construct the flip of the contraction explicitly. As an application, from products of two del Pezzo...
This paper studies the categorical entropy of autoequivalences of derived categories of Hilbert schemes of points on surfaces and hyperkähler manifolds. One of the central questions about categorical entropy is whether it satisfies a Gromov-Yomdin type formula $h_{\mathrm{cat}}(Φ) = \logρ(Φ)$. We say that $X$ has the Gromov-Yomdin...
In this paper, we present sufficient conditions ensuring that the sum of the image of quadratic functions and the nonnegative orthant is convex. The hidden convexity of the trust-region problem with linear inequality constraints is established under a newly proposed assumption, which is compared with the previous one in [{\it Math....
We establish a ring isomorphism between the derived Hall algebra of the Jordan quiver and the ring of double symmetric functions (i.e., the ring of symmetric polynomials in two sets of countably many variables, invariant under the respective actions of their symmetric groups) with a parameter $t$. This isomorphism maps the derived...
We develop the theory of linear-quadratic (LQ) mean field games (MFGs) in Hilbert spaces with common noise modeled by an infinite-dimensional Wiener process that affects the dynamics of all agents. In the presence of common noise, the mean-field consistency condition is characterized by a system of coupled forward-backward stochastic...
We prove a noncommutative analogue of Minkowski's integral inequality for commuting squares of tracial von Neumann algebras. The inequality implies a necessary condition for a quadruple of graphs to be realized as inclusion graphs of a commuting square of multi-matrix algebras. As a corollary, we obtain a unitary categorification...
In this note, we briefly introduce the background and motivation of the collaborative work [arXiv:2508.20797], and provide an outline of the main results. The latter relates to matrix and higher order scalar differential equations satisfied by certain Hankel and Toeplitz determinants involving I-Bessel functions, or equivalently...
This paper introduces a self-supervised learning framework for approximating the Security-Constrained DC Optimal Power Flow (SC-DCOPF) problem using a parametric linear model. The approach preserves the physical structure of the DC-OPF while incorporating demand-dependent tunable parameters that scale transmission line limits. These...
A ladder determinantal module is an arbitrary direct sum of ideals of maximal minors of a generic ladder matrix. In this article, we give necessary and sufficient conditions for the special fiber ring of such modules to be Gorenstein. These conditions are expressed in terms of data obtained from the underlying matrix.
We compute icanonical basis of the quasi-split rank one modified iquantum group, by obtaining explicit transition matrices among the icanonical basis, monomial basis, and standardized canonical basis; all these bases can be naturally categorified. These transition matrices follow from their counterparts computed in this paper among...
In this paper, we introduce the symmetric multiple Eisenstein series, a variant of the multiple Eisenstein series. As a fundamental result, we show that they satisfy the linear shuffle relation. As a case study, we investigate the vector space spanned by symmetric double Eisenstein series of weight $k$. When $k$ is even, it coincides...
We use the methods introduced by Lue Pan to study the locally analytic vectors of the completed cohomology of Shimura curves associated to an indefinite quaternion algebra $D$ which is ramified at a prime number $p$. Let $D_p^{\times}$ be the group of units of $D$ at $p$. Using $p$-adic uniformization of the quaternionic Shimura...