Clay weekly context brief for the Maths category (ISO week 2026-W33).

Clay tracks publications from the Maths feed list. Below are recent items from this category, each with its source and a short description of what the publication covers when one is available in the source feed.

Recent publications:
1. Dihedral Rigidity for Convex Polytopes by Smooth Approximation
   Source: math.DG (Differential Geometry)
   Link: https://arxiv.org/abs/2608.06320
   Following Brendle's smooth approximation approach, we give a proof of Gromov's dihedral rigidity conjecture for convex polytopes.
2. Oscillatory motion to collision and infinity in the Earth-Moon restricted three body problem
   Source: math.DS (Dynamical Systems)
   Link: https://arxiv.org/abs/2608.05400
   We present a computer assisted proof of the existence of the oscilla- tory motion to collision and to infinity in the Earth-Moon planar circular restricted three body problem.
3. Nearly optimal Kolmogorov widths under holomorphic mappings
   Source: math.FA (Functional Analysis)
   Link: https://arxiv.org/abs/2608.05581
   This paper establishes essentially optimal asymptotic bounds for Kolmogorov widths under holomorphic mappings between complex Banach spaces.
4. The Gregory function and its completed Gregory transform
   Source: math.GM (General Mathematics)
   Link: https://arxiv.org/abs/2608.05189
   We study the entire interpolation \[ \mathcal{G}(z)=\int_0^1 \binom{x}{z}\,dx \] of the Gregory coefficients.
5. A special kind of topology on $C(X)$ lying between the point-open topology and the topology of uniform convergence
   Source: math.GN (General Topology)
   Link: https://arxiv.org/abs/2608.05694
   Let $X$ be a topological space.
6. Centralizers of sofic approximations of Kazhdan groups
   Source: math.GR (Group Theory)
   Link: https://arxiv.org/abs/2608.05362
   We prove that a Kazhdan group admitting a sofic embedding into a metric ultraproduct of symmetric groups with a centralizer that acts ergodically on the associated Loeb probability space is locally embeddable in finite groups (LEF).
7. Lorenz Links that are not Horseshoe and Rossler Links
   Source: math.GT (Geometric Topology)
   Link: https://arxiv.org/abs/2608.05562
   We compare the periodic orbit types of the Lorenz, horseshoe, and R\"ossler systems through their associated templates.
8. Paracomplete Probabilities
   Source: math.LO (Logic)
   Link: https://arxiv.org/abs/2507.04312
   This paper presents an advance in the direction of working with probabilities in a paracomplete setting using Logics of Formal Undeterminedness (LFUs).
9. On the synthesis of complete two-dimensional second-gradient continua: Tri-pantographic fabrics
   Source: math-ph (Mathematical Physics)
   Link: https://arxiv.org/abs/2608.05524
   We introduce a notion of completeness for two-dimensional second-gradient elastic continua and propose a microstructural route toward its synthesis.
10. Structured Tensor Approximation from Lateral Slice Sampling via Basis and Manifold Priors
   Source: math.NA (Numerical Analysis)
   Link: https://arxiv.org/abs/2608.05463
   In this work, we consider a structured tensor approximation problem, where only a limited number of lateral slices are observed.
11. Explicit Algebraic Numbers all whose Integer Parts of Powers are Composite
   Source: math.NT (Number Theory)
   Link: https://arxiv.org/abs/2608.05309
   We present several explicit classes of algebraic integers $\alpha>1$ such that $\lfloor \alpha^n\rfloor$ is composite for all but finitely many $n$.
12. Relative modular categories from $\mathfrak{osp}(2 \vert 2n)$
   Source: math.RT (Representation Theory)
   Link: https://arxiv.org/abs/2608.06333
   We construct relative modular categories from the weighted representation theory of the unrolled quantum group of the orthosymplectic Lie superalgebra $\mathfrak{osp}(2 \vert 2n)$.
13. Asymptotics of Titchmarsh--Weyl functions near the real axis andan application to the KdV hierarchy
   Source: math.SP (Spectral Theory)
   Link: https://arxiv.org/abs/2608.06034
   We establish high-energy asymptotic expansions of Titchmarsh--Weyl functions for one-dimensional Schr\"odinger and Dirac operators in regions whose boundaries approach the spectrum at a prescribed polynomial rate.
14. The graded Betti numbers are not tropical invariants
   Source: math.AC (Commutative Algebra)
   Link: https://arxiv.org/abs/2608.06284
   We show that the graded Betti numbers are not invariants of the tropicalisation of a homogeneous ideal, or the tropicalisation of a projective algebraic scheme over a valued field.
15. Elliptic matroids and modular curves
   Source: math.AG (Algebraic Geometry)
   Link: https://arxiv.org/abs/2608.05299
   For $n\geq 4$, let $T_n$ be the rank-3 matroid on $\mathbb{Z}/n\mathbb{Z}$ whose bases are the three-element non-zero-sum subsets.
16. Unstable Manifolds of Stratified Euler Equations
   Source: math.AP (Analysis of PDEs)
   Link: https://arxiv.org/abs/2608.05370
   We consider a spectrally unstable steady state $(\rho_0,v_0)$ of the incompressible stratified Euler equations on a class of $d$-dimensional domains.
17. Whitehead's theorem for minimal finite models
   Source: math.AT (Algebraic Topology)
   Link: https://arxiv.org/abs/2608.06176
   We investigate the extent to which Whitehead's theorem remains valid for minimal finite models.
18. Degree-uniform regions for Gautschi's conjecture on subrange Jacobi polynomials
   Source: math.CA (Classical Analysis and ODEs)
   Link: https://arxiv.org/abs/2608.05963
   Let $\pi_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\leq 1$, with respect to the Jacobi weight $(1-x)^\alpha(1+x)^\beta$, where $-1<\alpha<\beta$.
19. Fuzzy latin squares and balanced permutation pattern statistics
   Source: math.CO (Combinatorics)
   Link: https://arxiv.org/abs/2608.05335
   A latin square of order $n$ can be viewed as a partition of the $n \times n$ all-ones matrix into permutation matrix summands.
20. Quantalic lambda-calculus and additive disjunction
   Source: math.CT (Category Theory)
   Link: https://arxiv.org/abs/2608.06120
   Motivated by the need to reason about case statements quantitatively, we extend quantalic linear lambda-calculus with additive disjunction.
21. An Orlicz variational formula for David-type Beltrami equations
   Source: math.CV (Complex Variables)
   Link: https://arxiv.org/abs/2608.05618
   Let $\mathcal{U} \Subset \mathbb{C}$ be fixed, $\Phi(s)=e^s-s-1$, $F(\nu)=\frac{\nu}{2+|\nu|}$, and let $f^{F(\nu)}$ denote the principal solution of the corresponding Beltrami equation.
22. The Brunn--Minkowski inequality for the Gaussian measure
   Source: math.DG (Differential Geometry)
   Link: https://arxiv.org/abs/2608.05390
   Let $\gamma_n$ be the standard Gaussian measure on $\mathbb{R}^n$, $n\ge2$, and let $\alpha_\gamma(n)$ be the largest number for which \[ \gamma_n(\lambda K+(1-\lambda)L)^{\alpha_\gamma(n)} \ge \lambda\gamma_n(K)^{\alpha_\gamma(n)} +(1-\lambda)\gamma_n(L)^{\alpha_\gamma(n)} \] holds for all convex bodies $K,L\subset\mathbb{R}^n$ containing the origin and all $\lambda\in[0,1]$.
23. Hyperbolicity and Ergodicity in Randomly Perturbed Billiard Systems
   Source: math.DS (Dynamical Systems)
   Link: https://arxiv.org/abs/2608.05484
   In this paper, we investigate the Lyapunov exponents and hyperbolicity of random billiard systems under small stochastic perturbations.
24. A positive solution to the generalized Busemann--Petty problem with subspace dimensions $2$ and $3$
   Source: math.FA (Functional Analysis)
   Link: https://arxiv.org/abs/2608.05621
   The generalized Busemann-Petty problem asks: If $K$ and $L$ are origin-symmetric convex bodies in $\mathbb{R}^n$, and $\operatorname{vol}_m(K\cap E)$ is no larger than $\operatorname{vol}_m(L\cap E)$ for every $m$-dimensional subspace $E$, $1<m<n$, does it follow that $\operatorname{vol}_n(K)\leq \operatorname{vol}_n(L)$?
25. Some Generalizations of the Bridge and Torch Problem
   Source: math.GM (General Mathematics)
   Link: https://arxiv.org/abs/2608.05190
   For the classic bridge and torch problem with crossing times $\left\{1,\ldots,n\right\}$, we derive a closed-form expression for the optimal crossing time $T\left(n\right)$ using a recurrence relation derived from the problem's optimal substructure, thus obtaining \[T\left(n\right)=\frac{n^2}{4}+3n-5+\frac{\left(-1\right)^n-1}{8}\] which holds for all $n\ge 2$.
26. Basic zero-dimensional spaces: a unifying framework for continuity and openness
   Source: math.GN (General Topology)
   Link: https://arxiv.org/abs/2608.06285
   Following a suggestion in a paper by Ern\'e, Picado and Pultr, we establish the category of basic zero-dimensional spaces and their basic continuous maps.
27. Inverse ambiguous maps on infinite groups
   Source: math.GR (Group Theory)
   Link: https://arxiv.org/abs/2608.06191
   Let $G$ be a group.

Sources in this brief: math-ph (Mathematical Physics); math.AC (Commutative Algebra); math.AG (Algebraic Geometry); math.AP (Analysis of PDEs); math.AT (Algebraic Topology); math.CA (Classical Analysis and ODEs); math.CO (Combinatorics); math.CT (Category Theory); math.CV (Complex Variables); math.DG (Differential Geometry); math.DS (Dynamical Systems); math.FA (Functional Analysis); math.GM (General Mathematics); math.GN (General Topology); math.GR (Group Theory); math.GT (Geometric Topology); math.LO (Logic); math.NA (Numerical Analysis); math.NT (Number Theory); math.RT (Representation Theory); math.SP (Spectral Theory).
Selected 27 of 250 available items for this weekly brief.