arXiv preprints from January 1, 2026 through September 8, 2026 — 17:08:09 EST
We show that a set function $ν$ satisfies the gross substitutes property if and only if its homogeneous generating polynomial $Z_{q,ν}$ is a Lorentzian polynomial for all positive $q \le 1$, answering a question of Eur-Huh. We achieve this by giving a rank 1 upper bound for the distance matrix of an ultrametric tree, refining a...
We generalize the global construction of code loops introduced by Nagy, which is based on the connection between Moufang loops and groups with triality. This follows from the construction of a nilpotent group $G_n$ of class 3 with triality and $2n$ generators, based on embeddings of $G_n$ into direct products of copies of $G_3$. In...
We state and prove the spectral expansion of the theta series attached to the Rankin-Selberg spherical variety $(\mathrm{GL}_{n+1} \times \mathrm{GL}_n)/\mathrm{GL}_n$. This is a key result towards the fine spectral expansion of the Jacquet-Rallis trace formula. Our expansion is written in terms of regularized Rankin--Selberg periods...
Let $(X,ω)$ be a translation surface whose Veech group $Γ$ is a lattice. We prove that the generic orbit of the group of affine homeomorphisms of $(X,ω)$ can be used to approximate each point of $X$ with Diophantine precision. The proof utilizes an induced $SL_2(\mathbb{R})$-action on a fiber bundle $Y$ whose base is...
This article proposes new perspectives for developing derivative based numerical algorithms, supported by the introduction of a generalized derivative operators. It demonstrates that these operators have the potential to enhance and extend existing derivativebased numerical methods. To this end, two iterative derivative driven methods...
We develop energy-conserving numerical methods for a two-dimensional hyperbolic approximation of the Serre-Green-Naghdi equations with variable bathymetry and either periodic or reflecting boundary conditions. The hyperbolic formulation avoids the costly inversion of an elliptic operator present in the classical model. Our schemes...
We study a quasilinear chemotaxis system of singular type, where the diffusion operator is given by $Δu^m$ with $0<m<1$, corresponding to the fast diffusion regime, and where the chemotactic drift is nonlinear. Since Hölder continuity constitutes the optimal regularity class for weak solutions to the porous medium equation, we...
We study the accumulation of zeros of a polynomial arising from the enumeration of edge-coloured graphs along certain limit curves. The polynomial is a variant of an edge-chromatic polynomial, which specialises to the partition function of the ferromagnetic Ising model on a random regular graph. We call this accumulation behaviour a...
By modifying a construction of Abe and Tange, we exhibit arbitrarily large families of Lagrangian slice disks with Weinstein deformation equivalent exteriors. This answers a Lagrangian version of a question of Hitt and Sumners. We raise other open questions related to Lagrangian slice disks and their exteriors.
Artificial intelligence has advanced rapidly through large neural networks trained on massive datasets using thousands of GPUs or TPUs. Such training can occupy entire data centers for weeks and requires enormous computational and energy resources. Yet the optimization algorithms behind these runs have not kept pace. Most large scale...
The $p_1$ model plays a fundamental role in modeling directed networks, where the reciprocal effect parameter $ρ$ is of special interest in practice. However, due to nonlinear factors in this model, how to estimate $ρ$ efficiently is a long-standing open problem. We tackle the problem by the cycle count approach. The challenge is, due...
We prove that $Γ_0(2^n)$ ($n\ge2$) has a Farey sequence $\{e_i\}$ such that $e_i \le 2^{n-1}$ for all $e_i$. The above upper bound is optimal, and there exists a unique $j$ such that $e_j= 2^{n-1} $. For each $e_i$, there exists a unique $a_i$ such that $\{ a_i/e_i\}\cup \{\infty\}$ is the set of ideal vertices of a fundamental domain...
The notion of sectional-hyperbolicity is a weakened form of hyperbolicity introduced for vector fields in order to understand the dynamical behavior of certain higher-dimensional systems such as the multidimensional Lorenz attractor. In this paper we address the questions proposed in [\emph{Math. Z.}, \textbf{298} (2021), 469-488] and...
Distributionally Robust (DR) optimization aims to certify worst-case risk within a Wasserstein uncertainty set. Current certifications typically rely either on global Lipschitz bounds, which are often conservative, or on local gradient information, which provides only a first-order approximation. This paper introduces a novel...
Double Boolean algebras (dBas), introduced by Wille, are based on twenty-three identities. We present a simplified axiom system, the D-core algebra, and prove it is equivalent to Wille's original definition. This reduction allows improved structural results, including a refined Boolean representation theorem showing fewer conditions...
We introduce and study a three-sign cancellation hypernumber system $H$ which extends the real field by adjoining a third sign $Λ$. The underlying set is $H=\{0\}\cup\{+,-,Λ\}\times\mathbb{R}_{>0}$, with a single-valued multiplication and a hyperaddition $\boxplus$ designed to encode cancellation phenomena between positive and...
The present paper studies the existence of valuative interpolation on the local ring of an irreducible analytic subvariety at singular points. We firstly develop the concepts and methods of Zhou weights and Tian functions near singular points of irreducible analytic subvarieties. By applying these tools, we establish the necessary and...
This paper constructs solutions to linear and nonlinear Schrödinger-type equations in two and three spatial dimensions that exhibit prescribed, extraordinary gradient amplification and localization. For any finite time interval $[0,T]$, any prescribed collection of $n\in\mathbb{N}$ distinct points on $\partial D$, where $D$ is the...
Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) is a well-established stabilization technique for affine nonlinear systems. However, its application is generally hindered by the requirement of solving a set of partial differential equations (PDEs), i.e., the so-called matching equation. This paper introduces...
We prove certain polynomial relations between the values of complex irreducible characters of general finite symmetric groups. We use it to find some sets of conjugacy classes such that no finite symmetric group has a complex irreducible character that vanishes at every class in the set. In particular, we show that if $n$ satisfies...
Let $F_n$ be the free group of rank $n$, and let $ρ_{ab}:\mathrm{Aut}(F_n)\to \mathrm{GL}_n(\mathbb Z)$ be the map induced by the natural projection $F_n\to\mathbb Z^n$. It is a long-standing open problem whether the subgroup of $\mathrm{IA}$-automorphisms $\mathrm{IA}_n=\mathrm{Ker}ρ_{ab}$ is finitely presented for $n\geq 4$. In this...
We investigate the two-dimensional Muskat problem with a nonlinear elastic interface, for both one-phase and two-phase scenarios. Following the framework developed by Nguyen [35,36], we demonstrate that the problem is locally well-posed in $H^s$ for $s\geq 2$ for arbitrary initial data. Furthermore, for the one-phase case and the...
This work investigates the performance of the final iterate produced by stochastic gradient descent (SGD) under temporally dependent data. We consider two complementary sources of dependence: $(i)$ martingale-type dependence in both the covariate and noise processes, which accommodates non-stationary and non-mixing time series data,...
We provide a method for formulating superintegrable magnetic geodesic flows on reductive homogeneous spaces $M=G/A$, with $G$ a compact semisimple Lie group and $A$ a closed subgroup of $G$. In the twisted cotangent bundle $(T^*M,ω_\varepsilon)$, with $ω_\varepsilon=ω_{\mathrm{can}}+\varepsilon\,π^*ω_{\mathrm{KKS}}$ being the...
The Vietoris-Rips complex, denoted $R_β(X)$, of a metric space $(X,d)$ at scale $β$ is an abstract simplicial complex where each $k$-simplex corresponds to $(k+1)$ points of $X$ within diameter $β$. For any abstract simplicial complex $K$ with the vertex set $K^{(0)}$ a Euclidean subset, its shadow, denoted $S(K)$, is the union of the...