arXiv preprints from January 1, 2026 through September 7, 2026 — 14:59:35 EST
This review outlines the major approaches to modelling opinion formation and manipulation in mathematics and computer science. Key tools such as ordinary and partial differential equations, stochastic models, control theory, and interaction protocols are introduced and compared as methods for describing manipulation. The review is...
Assume that $n = 2k$ potential roommates each have an ordered preference of the $n-1$ others. A stable matching is a perfect matching of the $n$ roommates in which no two unmatched people prefer each other to their matched partners. In their seminal 1962 stable marriage paper, Gale and Shapley noted that not every instance of the...
In this work, we design and analyze a Discrete de Rham (DDR) scheme for a contact mechanics problem involving fractures along which a model of Tresca friction is considered. Our approach is based on a mixed formulation involving a displacement field and a Lagrange multiplier, enforcing the contact conditions, representing tractions at...
We prove that the reduced cross-sectional algebra of a Fell bundle with the approximation property over an inverse semigroup is exact if and only if the unit fiber of the Fell bundle is exact. This generalizes a recent result of the first-named author for actions of second countable locally compact Hausdorff groupoids on separable...
We construct an $\ll^2$-solution (also known as a weakly low solution) to ${\mathrm{D}^2}$ within ${\mathrm{B}Σ^0_{3}}$ and prove the $\ll^2$-basis theorem for $\mathrm{RT}^2$ over ${\mathrm{B}Σ^0_{3}}$. The $\ll^2$-basis theorem is a variant of the low basis theorem, which has recently received focus in the context of the first-order...
We completely describe inhomogeneous properly embedded almost symmetric submanifolds of Euclidean space as certain unions of parallel symmetric submanifolds of the ambient Euclidean space.
The dynamics of the left translation semigroup $\{T_t\}_{t \geq 0}$ on weighted $L^p$ spaces over a directed metric tree $L(G)$ is investigated. Necessary and sufficient conditions on the weight family $ρ$ for the strong continuity of the semigroup are provided. Furthermore, hypercyclicity and weak mixing properties are characterized...
This paper is an introduction to Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries. It consists of essentially two parts that could be read independently. In the first we introduce non-obtuse polyhedra in the spherical, Euclidean, and hyperbolic spaces, and prove various fundamental theorems originated from Andreev,...
We study a weak form of formality for differential graded algebras, called $A_3$-formality, for the cohomology of pro-p Demushkin groups at odd primes p. We show that the differential graded $\mathbb{F}_p$-algebras of continuous cochains of Demushkin groups with q-invariant not equal 3 are $A_3$-formal, whereas Demushkin groups with...
This work addresses an enumeration problem on weighted bi-colored plane trees with prescribed vertex data, with all vertices labeled distinctly. We give a bijection proof of the enumeration formula originally due to Kochetkov, hence affirmatively answer a question of Adrianov-Pakovich-Zvonkin. The argument is purely combinatorial and...
We show that the holonomy group of a connected Riemannian locally symmetric space (not necessarily complete) without local flat factor is compact and has finite index in its normalizer in the orthogonal group.
We give a self-contained introduction to linear algebraic and semialgebraic groups over real closed fields, and we generalize several key results about semisimple Lie groups to algebraic and semialgebraic groups over real closed fields. We prove that a torus in a semisimple algebraic group is maximal $\mathbb{R}$-split if and only if...
Most numerical methods for time integration use real-valued time steps. Complex time steps, however, can provide an additional degree of freedom, as we can select the magnitude of the time step in both the real and imaginary directions. We show that specific paths in the complex time plane lead to expanded stability regions, providing...
In this paper, we construct canonical extensions of principal $\mathcal{G}^c$- (and $M^c$-)bundles on toroidal compactifications of integral canonical models of abelian-type Shimura varieties with hyperspecial levels.
In this note, we consider contractible loops of contactomorphisms that are positive over some non-empty closed subset of a contact manifold. Such closed subsets are called immaterial. We argue that the complement of a Reeb-invariant immaterial subset can be seen as big in contact geometric terms. This is supported by two results: one...
We compute extension groups in the category of duals of $p$-adic Banach space representations of $\mathrm{GL}_2(\mathbb{Q}_p)$. Focusing on representations arising from the $p$-adic local Langlands correspondence for generic Galois representations, we classify these extensions completely. These results are then applied to prove the...
Among the remarkable properties shared with convex cocompact representations, Anosov representations admit cocompact domains of discontinuity in flag varieties. For representations produced by embedding Fuchsian representations into higher rank Lie groups, these domains are known to admit fiber bundle structures and the structure...
We present a MATLAB package for the solution of multiple scattering problems, coupling Trefftz Discontinuos Galerkin methods for Helmholtz scattering with the T-matrix method. We rely on the TMATROM package to numerically approximate the T-matrices and deal with multiple scattering problem, providing a framework to handle scattering...
We introduce asymptotic-Möbius (AM) maps, a large-scale analogue of quasi-Möbius maps tailored to geometric group theory. AM-maps capture coarse cross-ratio behavior for configurations of points that lie far apart, providing a notion of "conformality at infinity" that is stable under quasi-isometries, compatible with scaling limits,...
The cave polynomial of a polymatroid was recently introduced and used to study the syzygies of polymatroidal ideals. We study the combinatorial relationships between three formulas for the cave polynomial. As an application, we interpret the Snapper polynomial in terms of these three formulas.
Let $H$ be a polynomial automorphism of $\mathbb{C}^2$ of positive entropy and degree $d \ge 2$. We prove that the escaping set $U^+$ (or equivalently, the non-escaping set $K^+$), of $H$ is rigid under the action of holomorphic automorphisms of $\mathbb{C}^2$. Specifically, every holomorphic automorphism of $\mathbb{C}^2$ that...
Two families $\mathcal{F}$ and $\mathcal{G}$ are cross-intersecting if every set in $\mathcal{F}$ intersects every set in $\mathcal{G}$. The covering number $τ(\mathcal{F})$ of a family $\mathcal{F}$ is the minimum size of a set that intersects every member of $\mathcal{F}$. In 1992, Frankl and Tokushige determined the maximum of...
Inspired by a parabolic system of Li-Yuan-Zhang and the continuity equation of La Nave-Tian, we study a system of elliptic equations for a Kähler metric $ω$ and a closed $(1, 1)$-form $α$. Assuming a uniform estimate for $ω$, we prove higher order estimates and smooth convergence to a cscK metric coupled to a harmonic $(1, 1)$-form. A...
We study completion of partial matrices with nonnegative entries to matrices of nonnegative rank at most $r$ for some $r \in \mathbb{N}$. Most of our results are for $r \leq 3$. We show that a partial matrix with nonnegative entries has a nonnegative rank-1 completion if and only if it has a rank-1 completion. This is not true in...
We study an optimal claim reporting problem in a bonus-malus setting. We assume, that the insurance contract consists of two regimes, where reporting a claim leads to a transition to a higher-premium regime, whereas remaining claim-free for a prespecified time period results in a shift to the lower premium regime. The insured can...