Clay weekly context brief for the Maths category (ISO week 2026-W40).

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. Long-term behaviour of refracted L\'evy processes in a half-line
   Source: math.PR (Probability)
   Link: https://arxiv.org/abs/2609.29326
   We study the long-term behaviour of refracted L\'evy processes within a half-line, with killing on exiting the domain and at state-dependent rate inside it.
2. A $q$-Weyl Freeness Principle for Nichols Algebras and Pointed Hopf Algebras of Square-Free Dimension
   Source: math.QA (Quantum Algebra)
   Link: https://arxiv.org/abs/2609.29853
   Let $H$ be a pointed Hopf algebra of square-free dimension over an algebraically closed field of characteristic $p>0$.
3. On the Regularity and Clean Properties of Matrices over Dual Numbers
   Source: math.RA (Rings and Algebras)
   Link: https://arxiv.org/abs/2609.29468
   In this paper, we study regularity and clean-type decompositions for matrices over dual numbers.
4. Transmutation operators for the Sturm--Liouville equation in impedance form with an integrable potential
   Source: math.SP (Spectral Theory)
   Link: https://arxiv.org/abs/2609.28772
   In this work, we study a family of transmutation operators for the Sturm--Liouville operator in impedance form with an integrable potential satisfying a smallness condition.
5. Anchored Extra-Proximal Methods: Optimal Higher-Order Methods for Monotone Inclusion Problems
   Source: math.OC (Optimization and Control)
   Link: https://arxiv.org/abs/2609.30212
   We study the deterministic oracle complexity of finding approximate solutions to composite monotone inclusion problems, formed by the sum of a smooth single-valued monotone operator and a maximally monotone set-valued operator, under the tangent-residual criterion.
6. Matrix factorization of quotient singularities of type A and D
   Source: math.AC (Commutative Algebra)
   Link: https://arxiv.org/abs/2609.28824
   In this article, we extend the classical relationship between ADE singularities and their maximal Cohen-Macaulay modules to quotient singularities of types A and D.
7. Reducibility and Diagonal Equality Strata for Finite Jacobi Pencils
   Source: math.AG (Algebraic Geometry)
   Link: https://arxiv.org/abs/2609.28490
   We study reducibility of the characteristic polynomials of a class of finite Jacobi pencils.
8. Generic Structural Stability for Riemann Solutions to $n \times n$ Systems of Hyperbolic Conservation Laws
   Source: math.AP (Analysis of PDEs)
   Link: https://arxiv.org/abs/2609.28714
   This paper proves generic structural stability for Riemann solutions to $n \times n$ systems of hyperbolic conservation laws in one spatial dimension.
9. Equivariant persistence topological complexity
   Source: math.AT (Algebraic Topology)
   Link: https://arxiv.org/abs/2609.29759
   We introduce equivariant persistent analogues of Lusternik-Schnirelmann category, topological complexity, cup length, and zero-divisors cup length for persistent spaces with group actions, and establish their stability.
10. On the Alzer-Berg problem: an optimal Bernstein boundary and a uniqueness conjecture
   Source: math.CA (Classical Analysis and ODEs)
   Link: https://arxiv.org/abs/2609.28707
   We study the two-parameter exponential family associated with the complete-monotonicity problem of Alzer and Berg.
11. Counting almost independent sets in regular graphs
   Source: math.CO (Combinatorics)
   Link: https://arxiv.org/abs/2609.28527
   Kahn proved that, among bipartite $d$-regular graphs on $n$ vertices, the number of independent sets is maximized by a disjoint union of copies of $K_{d,d}$.
12. On the degenerate Weyl problem on isometric immersions
   Source: math.DG (Differential Geometry)
   Link: https://arxiv.org/abs/2609.28905
   This paper is concerned with the Weyl problem, \emph{i.e.}, the existence of isometric immersions or embeddings of two-spheres into the three-dimensional Euclidean space or general ambient three-manifolds.
13. Sterile Insect Technique in a Metapopulation Model.
   Source: math.DS (Dynamical Systems)
   Link: https://arxiv.org/abs/2609.28748
   The sterile insect technique (SIT) is a biological control method that reduces insect pest and disease vector populations through the release of sterile insects (in general only males).
14. A Direct Approach to Rough Paths in Time-Weighted Besov Spaces
   Source: math.FA (Functional Analysis)
   Link: https://arxiv.org/abs/2609.28548
   We extend the Besov rough path framework of Friz and Seeger to paths lying in Besov spaces that are Muckenhoupt-weighted in time.
15. A Measure-Theoretic Derivation of Zadeh Fuzzy Logic
   Source: math.GM (General Mathematics)
   Link: https://arxiv.org/abs/2609.28510
   Zadeh fuzzy logic is normally introduced by assigning scalar membership degrees and postulating standard complement, minimum conjunction, and maximum disjunction.
16. Shading A-polynomials via huge representations of $U_q(\mathfrak{su}_N)$
   Source: math.GN (General Topology)
   Link: https://arxiv.org/abs/2605.22560
   Classical A-polynomials $A(\ell,m)$ define constraints on coordinates $\ell$ and $m$ in $SL(2,\mathbb{C})$ (a complexification of $SU(2)$) character varieties associated to knot complements $S^3\setminus K$.
17. Heat kernels on Cayley graphs of free products of finite groups
   Source: math.GR (Group Theory)
   Link: https://arxiv.org/abs/2609.28774
   We obtain explicit formulas for heat kernels of the normalized Laplacians for two infinite families of Cayley graphs.
18. On symplectic aspects of $SU(2)$ character varieties for punctured surfaces
   Source: math.GT (Geometric Topology)
   Link: https://arxiv.org/abs/2609.28907
   For a surface with an odd number of punctures, the moduli space of flat $SU(2)$ connections with traceless holonomy around each puncture is a symplectic manifold.
19. The spectral relation between irreducible cyclic codes and generalized Paley graphs
   Source: math.IT (Information Theory)
   Link: https://arxiv.org/abs/2609.28546
   Let $p$ be a prime and $\mathbb{F}_q/\mathbb{F}_r$ a finite field extension with $q=p^m$ and $r=p^s$.
20. Finite-coefficient Gersten injectivity fails in ramified mixed characteristic
   Source: math.KT (K-Theory and Homology)
   Link: https://arxiv.org/abs/2608.05005
   Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,3)$ in which $3$ is a uniformizer, and put $A=V[[x,y]]/(3+x^2-y^3)$.
21. Combinatorics of translations of meager and closed measure zero sets
   Source: math.LO (Logic)
   Link: https://arxiv.org/abs/2609.28669
   We study combinatorial properties of translations of meager and closed sets of measure zero.
22. Moir\'e Pattern of Rotated Sierpi\'nski Gaskets
   Source: math.MG (Metric Geometry)
   Link: https://arxiv.org/abs/2609.30235
   We study the moir\'e pattern formed by the intersection of a planar Sierpi\'nski gasket and a copy rotated about its centre.
23. Bi-Oriented Interlaced Matrix Multiplication and an Operator Factorisation of the Quantum Harmonic Oscillator
   Source: math-ph (Mathematical Physics)
   Link: https://arxiv.org/abs/2609.28511
   We introduce a bi-oriented interlaced matrix product on $2\times 2$ matrices over a unital, possibly non-associative algebra, extending a matrix mnemonic for Cayley--Dickson doubling.
24. Error analysis of penalized $H({\rm div})$-conforming finite element methods for surface Stokes problems
   Source: math.NA (Numerical Analysis)
   Link: https://arxiv.org/abs/2609.28729
   In recent years a number of finite element methods have been developed for the surface Stokes equations.
25. Effective logarithmic two-point Chowla bounds in every window
   Source: math.NT (Number Theory)
   Link: https://arxiv.org/abs/2609.28526
   We quantify the entropy-decrement proof of the logarithmically averaged two-point Chowla theorem, uniformly in the affine forms and the averaging window.
26. Extinction and Survival for a Generalized Contact Process with Deterministic Cures
   Source: math.PR (Probability)
   Link: https://arxiv.org/abs/2609.29425
   We study a generalized contact process on \(\mathbb{Z}^d\) parameterized by an infection rate \(\lambda\) and a resetting probability \(p \in [0,1]\), modeling deterministic cure times.
27. The crystal of the categorified quantum group
   Source: math.QA (Quantum Algebra)
   Link: https://arxiv.org/abs/2609.30073
   A categorification of the crystal $B(\widetilde{U}_q(\mathfrak{g}))$ of the modified quantized enveloping algebra associated to a symmetrizable Kac-Moody algebra $\mathfrak{g}$ is given.
28. Failure of Ado-type theorems for preLie and postLie algebras
   Source: math.RA (Rings and Algebras)
   Link: https://arxiv.org/abs/2609.29582
   Over an arbitrary field of characteristic zero, we construct a two-dimensional preLie algebra and a two-dimensional postLie algebra that do not embed into the corresponding induced structures of any finite-dimensional preassociative or postassociative algebra, respectively.
29. Microlocal analysis and spectral gap for complex hyperbolic manifolds
   Source: math.SP (Spectral Theory)
   Link: https://arxiv.org/abs/2609.28853
   We prove the existence of an essential spectral gap for convex cocompact complex hyperbolic manifolds under the hypothesis that the limit set does not contain a complex circle.
30. Fuel-optimal Boost-Back Guidance via Successive Convexification with a Terminal Constraint for ReusableLaunch Vehicles
   Source: math.OC (Optimization and Control)
   Link: https://arxiv.org/abs/2609.28759
   This paper proposes a fuel-optimal boost-back guidance algorithm for reusable launch vehicles using successive convexification (SCvx).
31. Jordan Blocks, Differential Traces, and Colon-Ideal Noncontainment over Noetherian Quotients
   Source: math.AC (Commutative Algebra)
   Link: https://arxiv.org/abs/2609.29516
   Let $k$ be a field of characteristic zero, let $R$ be a commutative $k$-algebra, let $I$ be a proper ideal of $R$, and assume that $A=R/I$ is Noetherian.

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.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.IT (Information Theory); math.KT (K-Theory and Homology); math.LO (Logic); math.MG (Metric Geometry); math.NA (Numerical Analysis); math.NT (Number Theory); math.OC (Optimization and Control); math.PR (Probability); math.QA (Quantum Algebra); math.RA (Rings and Algebras); math.SP (Spectral Theory).
Selected 31 of 379 available items for this weekly brief.