Clay weekly context brief for the Maths category (ISO week 2026-W39).

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. Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers II
   Source: math.AC (Commutative Algebra)
   Link: https://arxiv.org/abs/2609.20585
   Let $K$ be a Henselian discretely valued field with excellent ring of integers $\mathcal{O}_K$ and algebraically closed residue field $k$.
2. Balancedness of Normal Bundles of Rational Curves in Grassmannians
   Source: math.AG (Algebraic Geometry)
   Link: https://arxiv.org/abs/2609.19381
   In projective space $\mathbb{P}^r$, the normal bundle of a general rational curve of any degree $d \geq r$ is balanced except over fields of characteristic 2.
3. The unique solvability of strong solution to the multi-dimensional nonhomogeneous incompressible two-phase magnetohydrodynamic model
   Source: math.AP (Analysis of PDEs)
   Link: https://arxiv.org/abs/2609.19205
   The present paper studies the initial-boundary value problem of the nonhomogeneous incompressible two-phase magnetohydrodynamic model with Landau potential in a bounded smooth domain in $\mathbb{R}^d$($d = 2, 3$).
4. New families of moment-angle manifolds diffeomorphic to connected sums of products of spheres
   Source: math.AT (Algebraic Topology)
   Link: https://arxiv.org/abs/2609.19945
   We prove that the moment-angle manifold $\mathcal Z_{\mathcal K}$ is diffeomorphic to a connected sum of products of spheres when $\mathcal K$ is a starshaped (in particular, polytopal) 3-dimensional simplicial sphere with exactly two missing edges that are not adjacent to each other.
5. Rigorous analysis of giant magnetic vortex strings
   Source: math.CA (Classical Analysis and ODEs)
   Link: https://arxiv.org/abs/2609.19411
   This paper studies axially symmetric Abrikosov-Nielsen-Olesen vortices in the quartic Abelian Higgs model as the magnetic flux becomes arbitrarily large.
6. Lean-Certified Infinite Counterexamples to Written on the Wall II Conjecture 194
   Source: math.CO (Combinatorics)
   Link: https://arxiv.org/abs/2609.19195
   For a finite simple graph G, let alpha(G) denote its independence number and let l_avg(G) = (1 / |V(G)|) sum_{v in V(G)} alpha(G[N_G(v)]) be the average independence number of its open neighbourhoods.
7. Target Geometry in Prescribed-Value Schwarz Lemmas for Harmonic Maps
   Source: math.CV (Complex Variables)
   Link: https://arxiv.org/abs/2609.19609
   We study harmonic maps \(F:\D\to G\) into bounded domains in real Hilbert spaces, prescribing \(F(0)\) and \(dF_0(\R^2)\) when \(dF_0\) is nonzero and conformal.
8. Umbilic slopes and cubic Weingarten surfaces
   Source: math.DG (Differential Geometry)
   Link: https://arxiv.org/abs/2609.19188
   We study umbilic slopes and the global classification of cubic Weingarten surfaces.
9. Sobolev mixing constants and compactness in rational moduli space
   Source: math.DS (Dynamical Systems)
   Link: https://arxiv.org/abs/2609.19301
   We study uniformity of Sobolev mixing estimates for rational maps and uniformly quasiregular mappings.
10. Gabor Frame Regions and Non-Frame Obstructions for an even Rational Window
   Source: math.FA (Functional Analysis)
   Link: https://arxiv.org/abs/2609.19232
   We study the Gabor frame properties of the rational window $$ g(x)=\frac{x^2-1}{(x^2+1)(x^2+4)(x^2+9)}, $$ whose poles occur in symmetric pairs.
11. Counting Truchet Tile Balls
   Source: math.GM (General Mathematics)
   Link: https://arxiv.org/abs/2607.24843
   A formula is established that counts the number of different balls that can be made by decorating the pentagons and hexagons of a classic football with Truchet-like patterns.
12. A Quasicontinuous Domain Without an Interval Retract
   Source: math.GN (General Topology)
   Link: https://arxiv.org/abs/2609.20284
   We give negative answers to two questions of X.Xu concerning the occurrence of the unit interval in nonquasialgebraic quasicontinuous domains.
13. Explicit equational bases for the power semirings of $S_7$
   Source: math.GR (Group Theory)
   Link: https://arxiv.org/abs/2609.19957
   For every semigroup $S$, the set $\mathcal{P}(S)$ of all subsets of $S$ and the set $\mathcal{P}^{+}(S)$ of all nonempty subsets of $S$ form additively idempotent semirings under set-theoretic union and elementwise multiplication, called the full and nonempty power semirings of $S$, respectively.
14. Detecting fibered strongly quasi-positive links
   Source: math.GT (Geometric Topology)
   Link: https://arxiv.org/abs/2004.02233
   We prove that an $n$-component fibered link $L$ in $S^3$ is strongly quasi-positive if and only if $\tau(L)=g_3(L)+n-1$, where $g_3(L)$ denotes the Seifert genus and $\tau$ is the Ozsv\'ath-Szab\'o concordance invariant.
15. An expository review of the Chebyshev-Sylvester method in prime number theory
   Source: math.HO (History and Overview)
   Link: https://arxiv.org/abs/2512.02466
   This paper gives a self-contained expository and computational account of the elementary methods developed by P.L.
16. Binary Deletion Channel Capacity to Within One Hundredth of a Bit
   Source: math.IT (Information Theory)
   Link: https://arxiv.org/abs/2609.19412
   The exact capacity of the binary deletion channel remains unknown despite decades of work on achievable rates and converse bounds.
17. Existence of bases implies the axiom of choice, a foundation-free proof
   Source: math.LO (Logic)
   Link: https://arxiv.org/abs/2609.20140
   We prove that, in Zermelo--Fraenkel set theory with the axiom of Foundation removed, the statement that every vector space has a basis implies the Axiom of Choice, concluding that the classical equivalence between $\AC$ and the existence of bases does not require the Axiom of Foundation.
18. The spectral $\zeta$-function of Sturm--Liouville operators with $N$ generalized point potentials
   Source: math-ph (Mathematical Physics)
   Link: https://arxiv.org/abs/2609.19322
   This work analyzes the spectral zeta function associated with regular and singular Sturm--Liouville operators endowed with $N$ generalized point potentials.
19. Wideband Directional $\mathcal{H}^2$-Matrix Compression for the Electric Field Integral Equation with Geometry-Adaptive Cluster Trees
   Source: math.NA (Numerical Analysis)
   Link: https://arxiv.org/abs/2609.19431
   We present an efficient wideband construction of directional $\mathcal{H}^2$-matrices for the electrical field integral equation that, in contrast to existing constructions, supports not only box trees but also geometry-adaptive cluster trees.
20. Endoscopic description of the local Langlands correspondence for $\mathrm{G}_2$
   Source: math.NT (Number Theory)
   Link: https://arxiv.org/abs/2609.19439
   We prove that the $L$-packets for $p$-adic $\mathrm{G}_2$ constructed by Gan and Savin satisfy the endoscopic character identities.
21. TV between Bernoulli products, up to constants
   Source: math.PR (Probability)
   Link: https://arxiv.org/abs/2609.19222
   We obtain efficiently computable upper and lower bounds for the total variation distance between two product Bernoulli measures whose ratio is bounded by an absolute constant.
22. Braided Hopf algebroids and Lie algebroids
   Source: math.QA (Quantum Algebra)
   Link: https://arxiv.org/abs/2609.19563
   We construct braided Hopf algebroids in a braided monoidal category over a field $k$ and study its properties using graphical representation.
23. Piecewise Symmetric Tensors
   Source: math.RA (Rings and Algebras)
   Link: https://arxiv.org/abs/2607.04712
   Every square matrix is uniquely the sum of a symmetric matrix and a skew-symmetric matrix.
24. Putnam-Fuglede commutativity and the range-kernel orthogonality of an elementary operator
   Source: math.SP (Spectral Theory)
   Link: https://arxiv.org/abs/2608.18125
   Given Hilbert space commuting operators $T, S \in \mcl(H)$, such that $T$ is $w$-hyponormal with $\ker T \subseteq \ker T^*$ and $S$ is normal operator.
25. On triviality of $\mathbb{A}^2$-forms admitting a nontrivial $\mathbb{G}_a$-action
   Source: math.AC (Commutative Algebra)
   Link: https://arxiv.org/abs/2603.26289
   T.
26. A Quadratically Enriched Pushforward for Deligne--Mumford Stacks
   Source: math.AG (Algebraic Geometry)
   Link: https://arxiv.org/abs/2609.19423
   We define a quadratically enriched pushforward for proper morphisms between smooth, proper Deligne--Mumford stacks over an arbitrary field using Grothendieck--Serre duality.
27. The Cauchy problem for logarithmic Schr{\"o}dinger equation with a perturbation of power law nonlinearity
   Source: math.AP (Analysis of PDEs)
   Link: https://arxiv.org/abs/2609.19217
   In this paper, we study the Cauchy problem for the nonlinear Schr{\"o}dinger equation with a nonlinearity combining a logarithmic term and a local nonlinearity which has polynomial growth.
28. Representation stability of string links and manifold links
   Source: math.AT (Algebraic Topology)
   Link: https://arxiv.org/abs/2609.20470
   We prove representation stability for rational cohomology of spaces of string links and manifold links in Euclidean space as the number of links grows.
29. Product-profile anti-concentration for block-structured multi-affine polynomials
   Source: math.CA (Classical Analysis and ODEs)
   Link: https://arxiv.org/abs/2609.19473
   We establish product-profile anti-concentration, density, and Remez estimates for multi-affine polynomials on the cube.
30. A Strongly Aperiodic Monotile in Three Dimensions
   Source: math.CO (Combinatorics)
   Link: https://arxiv.org/abs/2609.19214
   Socolar and Taylor asked for a single, simply connected three-dimensional prototile that forces nonperiodicity by shape alone, admitting no weakly nonperiodic tiling; the Schmitt-Conway-Danzer biprism and the three-dimensional Socolar-Taylor tile admit screw motions or a periodic stacking direction.

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.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.HO (History and Overview); math.IT (Information Theory); math.LO (Logic); math.NA (Numerical Analysis); math.NT (Number Theory); math.PR (Probability); math.QA (Quantum Algebra); math.RA (Rings and Algebras); math.SP (Spectral Theory).
Selected 30 of 348 available items for this weekly brief.