Clay weekly context brief for the Maths category (ISO week 2026-W44).

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. Failure of noncommutative Rota's theorem with $L\log L$ endpoint
   Source: math.FA (Functional Analysis)
   Link: https://arxiv.org/abs/2610.31560
   We construct a factorizable Markov map $T$ on a hyperfinite finite von Neumann algebra and a positive element $x\in L\log L$ for which the sequence $T^n(T^*)^n x$ does not converge bilaterally almost uniformly. This answers a question of Anantharaman-Delaroche \cite{AD2006}.
2. Facets of nondegeneracy: central type factor groups, Heisenberg representations and Hadamard matrices
   Source: math.OA (Operator Algebras)
   Link: https://arxiv.org/abs/2610.11785
   Given a pair of complementary subgroups $P$ and $N$ of a finite group $G$, we first discuss how the same type of functions $\mathbb E\colon P\times N\to \mathbb T$, which we call bi-$1$-cocycles, appear naturally in the study of projective representations of $G$ of Heisenberg type and second cohomology of $G$ and of its dual quantum group $\widehat G$.
3. Diophantine Equations over $\mathbb Z$: Universal Bounds and Parallel Formalization
   Source: math.NT (Number Theory)
   Link: https://arxiv.org/abs/2506.20909
   This paper explores multiple closely related themes: bounding the complexity of Diophantine equations over the integers and developing mathematical proofs in parallel with formal theorem provers.
4. Sketching Outside the Hilbert Space for the Infinite-Dimensional Randomised SVD
   Source: math.NA (Numerical Analysis)
   Link: https://arxiv.org/abs/2610.31333
   In randomised numerical linear algebra, common choices for distributions of sketch vectors $\omega \in \mathbb R^n$ are isotropic, so that $\mathbb E[\omega \omega^T]=I_n$.
5. Leafrians, Poisson structures, and Frobenius splittings on moduli spaces
   Source: math.AC (Commutative Algebra)
   Link: https://arxiv.org/abs/2610.31619
   The Poisson $T$-Pfaffians of Bott--Samelson varieties, $G/U$, and $G/B$ produce Frobenius splittings, but only by a local, coordinate-bound argument.
6. Relative Hitchin fibrations and endoscopy for Unitary Friedberg--Jacquet varieties
   Source: math.AG (Algebraic Geometry)
   Link: https://arxiv.org/abs/2610.30366
   We prove a relative fundamental lemma for the unitary symmetric variety $GL_n(k')\backslash U_{2n}$, proving a conjecture of Leslie, Xiao, and Zhang.
7. Scattering of plane waves and a modified Pompeiu criterion
   Source: math.AP (Analysis of PDEs)
   Link: https://arxiv.org/abs/2610.30329
   In this paper we prove that plane waves always scatter from bounded, strictly convex planar inhomogeneities with constant material parameters.
8. Landfall Geometry
   Source: math.AT (Algebraic Topology)
   Link: https://arxiv.org/abs/2610.30993
   We study the negative gradient flow of analytic maps $f:X\to\R$ defined on compact analytic Riemannian manifolds without imposing any nondegeneracy or transversality assumptions.
9. Explicit Measurable Counterexamples to Finite-Difference Equations
   Source: math.CA (Classical Analysis and ODEs)
   Link: https://arxiv.org/abs/2610.30583
   We study finite-difference operators of the form \[ D(f,h)=\sum_{k=1}^n a_k f(b_kh), \] where the coefficients may be complex and the nodes are real.
10. Exact Realization of Edge and Induced Three-Vertex Subgraph Counts
   Source: math.CO (Combinatorics)
   Link: https://arxiv.org/abs/2610.30166
   The study of three-vertex subgraph densities has developed along two complementary lines: exact finite realization in large graphs of a given order, and asymptotic approximate realization by sequences of graphs.
11. The symmetric skew monoidal structure of Gray-categories
   Source: math.CT (Category Theory)
   Link: https://arxiv.org/abs/2610.31410
   In a previous paper, we constructed a closed skew monoidal structure on the category of Gray-categories and Gray-functors whose internal hom Psd(A, B) captures pseudomaps and pseudo-transformations.
12. Comprehensive Resolution of the Gauss-Fourier Conjecture for the First Kind of Zeros
   Source: math.CV (Complex Variables)
   Link: https://arxiv.org/abs/2610.30620
   For over two centuries, the esteemed mathematicians Gauss and Fourier conjectured that each real critical point of the rational function $p^{'}(s)/p(s)$ is associated with a pair of non-real zeros of the real polynomial $p(s)$.
13. Quermassintegral inequalities for capillary hypersurfaces in a horoball
   Source: math.DG (Differential Geometry)
   Link: https://arxiv.org/abs/2610.30771
   In this paper, we study the quermassintegral inequalities for capillary hypersurfaces in a horoball in hyperbolic space.
14. Collinear motions in constant curvature spaces, and coplanar motions in $\mathbb{R^4}$
   Source: math.DS (Dynamical Systems)
   Link: https://arxiv.org/abs/2610.30567
   A collinear motion of an $n$-body problem is one in which the $n$-bodies always lie along some, possibly moving, geodesic during their evolution.
15. A General Variable Neighborhood Search for Logistic Network Flow Design with Nonlinear Edge Costs and Transshipment Capacities
   Source: math.OC (Optimization and Control)
   Link: https://arxiv.org/abs/2610.30139
   We study the problem of routing multiple commodities through a logistic transport network so as to minimise the joint cost of vehicle movements and of transshipment (reload) operations at intermediate storages.
16. A note on functional non-central limit theorem for periodic regime-switching stable-like processes
   Source: math.PR (Probability)
   Link: https://arxiv.org/abs/2610.30342
   In this paper, we establish a functional non-central limit theorem for regime-switching stable-like processes with periodic coefficients.
17. On adjoint double shuffle relations and adjoint reduction relations
   Source: math.QA (Quantum Algebra)
   Link: https://arxiv.org/abs/2610.30720
   In this paper, we give a positive answer to Jarossay's question under the adjoint reduction relations.
18. Classification of compatible left-symmetric conformal algebras on solvable Lie conformal algebras of rank two
   Source: math.RA (Rings and Algebras)
   Link: https://arxiv.org/abs/2610.30471
   In this paper, we give a complete classification of compatible left-symmetric conformal algebra structures on solvable Lie conformal algebras which are free of rank two as $\mathbb{C}[\partial]$-modules up to isomorphism.
19. Lower bounds on Steklov eigenfunctions for a class of curvilinear polygons
   Source: math.SP (Spectral Theory)
   Link: https://arxiv.org/abs/2610.31387
   Steklov eigenfunctions, $\{\phi_{\sigma_j}\}_{j=0}^\infty$, are the eigenfunctions of the Dirichlet to Neumann map on a bounded domain with eigenvalue $\sigma_j$.
20. Positive-measure intersections of Bedford-McMullen carpets under similitudes
   Source: math.MG (Metric Geometry)
   Link: https://arxiv.org/abs/2610.09277
   Let $K=K(n,m,\Gamma)$ be a Bedford-McMullen carpet with $n>m\geq2$ and $1 0$ has a diagonal or anti-diagonal orthogonal part $O$, without any arithmetic restrictions on the defining bases.
21. Cut elimination for the alternation-free modal mu-calculus
   Source: math.LO (Logic)
   Link: https://arxiv.org/abs/2510.11293
   We present a syntactic cut-elimination procedure for the alternation-free fragment of the modal mu-calculus.
22. Hybrid Multiport Receivers for Slow Fluid Antenna Multiple Access
   Source: math.IT (Information Theory)
   Link: https://arxiv.org/abs/2605.06958
   We propose a novel receiver architecture that preserves the performance benefits of multiport selection in fluid-antenna systems while requiring only a very small number of radio-frequency (RF) chains.
23. The Limits of Arithmetical Pluralism: Incompleteness, Large Cardinals, and Graded Non-Pluralism
   Source: math.HO (History and Overview)
   Link: https://arxiv.org/abs/2609.22797
   G\"odelian incompleteness yields arithmetical sentences \(A\) such that \(\mathsf{PA}+A\) and \(\mathsf{PA}+\neg A\) are both consistent.
24. Spin and scalar curvature on punctured four manifolds
   Source: math.GT (Geometric Topology)
   Link: https://arxiv.org/abs/2609.37948
   We show that a closed spin four-manifold with nonzero signature, even after being punctured along a finite graph, does not admit a complete metric with uniformly positive scalar curvature.
25. Weighted Power Means on Upper Triangular Matrix Groups
   Source: math.GR (Group Theory)
   Link: https://arxiv.org/abs/2610.24821
   We study weighted power means on the Lie group $\operatorname{UT}_n$ of real upper triangular matrices with positive diagonal entries.
26. Formalization of Langlands's Second Main Lemma for Local Epsilon Factors
   Source: math.GM (General Mathematics)
   Link: https://arxiv.org/abs/2609.30301
   We describe the Lean 4 formalization of Langlands's Second Main Lemma for local epsilon factors over nonarchimedean local fields, in mixed and equal characteristic.
27. Fefferman--Chang-type atomic decompositions for product Hardy spaces associated with Schr\"odinger operators
   Source: math.FA (Functional Analysis)
   Link: https://arxiv.org/abs/2610.30459
   We establish a geometric (Fefferman--Chang-type) atomic characterization of the two-parameter Hardy space associated with a pair of Schr\"odinger operators $L_i=-\Delta+V_i$ on $\mathbb R^{n_i}$, $i=1,2$.
28. Integral representations of quantum R\'enyi divergences on von Neumann algebras
   Source: math.OA (Operator Algebras)
   Link: https://arxiv.org/abs/2608.21214
   We prove an integral representation of the $\alpha$-$z$ R\'enyi divergences for normal states on arbitrary von Neumann algebras.
29. A partial result towards the Chowla--Milnor conjecture
   Source: math.NT (Number Theory)
   Link: https://arxiv.org/abs/2505.12687
   The Chowla--Milnor conjecture predicts the linear independence of certain Hurwitz zeta values.
30. Numerical reconstruction of polygonal obstacles by diffracted waves
   Source: math.NA (Numerical Analysis)
   Link: https://arxiv.org/abs/2610.31271
   We present a numerical method for recovering rigid polygonal obstacles in two dimensions, based on the diffraction-based inverse scattering theory of Bao, Chen, Lu, and Xiong.

Sources in this brief: 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.GR (Group Theory); math.GT (Geometric Topology); math.HO (History and Overview); math.IT (Information Theory); math.LO (Logic); math.MG (Metric Geometry); math.NA (Numerical Analysis); math.NT (Number Theory); math.OA (Operator Algebras); math.OC (Optimization and Control); math.PR (Probability); math.QA (Quantum Algebra); math.RA (Rings and Algebras); math.SP (Spectral Theory).
Selected 30 of 355 available items for this weekly brief.