Clay weekly context brief for the Maths category (ISO week 2026-W29). 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. A guide to topological reconstruction on endomorphism monoids and polymorphism clones Source: math.RA (Rings and Algebras) Link: https://arxiv.org/abs/2512.01086 Various spaces of symmetries of a structure are naturally endowed with both an algebraic and a topological structure. 2. Supercongruences Arising from Truncated Appell Series Source: math.NT (Number Theory) Link: https://arxiv.org/abs/2607.04583 In recent years, both Appell series and truncated Appell series have garnered significant research interest among scholars. 3. Solving Hamilton-Jacobi equations by residual minimization of monotone finite-difference discretizations Source: math.NA (Numerical Analysis) Link: https://arxiv.org/abs/2601.21764 We introduce a method for solving Hamilton--Jacobi equations, both inviscid and viscous, by minimizing the squared residuals of monotone finite-difference discretizations on grids of varying resolution. 4. Loop vs. Bernoulli percolation on trees: strict inequality of critical values Source: math-ph (Mathematical Physics) Link: https://arxiv.org/abs/2503.03319 We study loop ensembles on locally finite rooted trees. 5. Wrench-Based Bayesian Pose Estimation via Matrix--Fisher Gaussian Inference Source: math.MG (Metric Geometry) Link: https://arxiv.org/abs/2607.07306 In this paper, a residual-safeguarded local Matrix Fisher--Gaussian (MFG) inference method is developed for wrench-based pose estimation on $\mathrm{SO}(3)\times\mathbb{R}^3$. 6. Fast-Track ODE: A Flipped, Game-Based Summer Differential Equations Course Source: math.HO (History and Overview) Link: https://arxiv.org/abs/2607.09454 This article describes the implementation of a flipped, interactive, and game-based instructional design in an Elementary Differential Equations course taught during a six-week summer session. 7. Periodic symplectic and Hamiltonian diffeomorphisms on irrational ruled surfaces Source: math.GT (Geometric Topology) Link: https://arxiv.org/abs/2411.18580 We investigate when finite-order Hamiltonian diffeomorphisms extend to Hamiltonian circle actions, probing the transition from discrete to continuous symmetry in symplectic topology. 8. Strong truncations and Maximal Ideal Principles Source: math.FA (Functional Analysis) Link: https://arxiv.org/abs/2606.10202 We compare two existence principles for maximal ideals, a classical one for vector lattices with a strong unit and a second, newly introduced one for vector lattices with a strong truncation. 9. Transcendence of simple geodesics on finite modular covers Source: math.DS (Dynamical Systems) Link: https://arxiv.org/abs/2606.08842 The real projective line $\mathbb{R}\mathbf{P}^1$ is the boundary of $\mathbf{HP}=\{z\in \mathbb{C}\colon \Im(z)>0\}$, a model of the hyperbolic plane whose space of geodesics identifies with $\mathcal{G}(\mathbf{HP})=\mathbb{R}\mathbf{P}^1 \times \mathbb{R}\mathbf{P}^1 \setminus \mathrm{diagonal}$. 10. Vanishing theorems for pseudo-effective line bundles Source: math.CV (Complex Variables) Link: https://arxiv.org/abs/2602.13821 In the present paper, we establish a general Kawamata-Viehweg-Koll\'ar-Nadel type vanishing theorem for higher direct images in terms of numerical dimension for closed positive currents on compact K\"ahler manifolds, unifying a number of important vanishing theorems. 11. Change of Enrichment for Monoidal Model Categories Source: math.CT (Category Theory) Link: https://arxiv.org/abs/2607.09612 A classical theorem of category theory says that given an adjunction between monoidal categories with lax monoidal right adjoint, there is an induced adjunction between categories of enriched categories. 12. On $(p,q)$-binomial coefficient ratios for complex parameters Source: math.CA (Classical Analysis and ODEs) Link: https://arxiv.org/abs/2607.08163 We prove local asymptotics for near-central complex $(p,q)$-binomial coefficient moduli ratios allowing an imaginary parameter perturbation of order $n^{-3/4}$ at a $\sqrt{n}$ length scale from the centre. 13. Elementary $\infty$-toposes from type theory Source: math.AT (Algebraic Topology) Link: https://arxiv.org/abs/2512.18891 We prove that every categorical model of dependent type theory with dependent sums and products, intensional identity types and univalent universes presents via its $\infty$-localisation an elementary $\infty$-topos, that is, a finitely complete, locally cartesian closed $\infty$-category with enough univalent universal morphisms. 14. A variational formulation of a Multi-Population Mean Field Games with non-local interactions Source: math.AP (Analysis of PDEs) Link: https://arxiv.org/abs/2408.03118 We propose a MFG model with quadratic Hamiltonian involving $N$ populations. 15. The algebro-geometric aspect of the iterated limit of a quaternary of means of four terms Source: math.AG (Algebraic Geometry) Link: https://arxiv.org/abs/2512.22002 We study the iterated limit of a quaternary of means of four terms through the period map from the family of cyclic fourfold coverings of the complex projective line branching at six points to the three-dimensional complex ball $\mathbb{B}_3$ embedded into the Siegel upper half-space of degree four. 16. On the uniform positivity of $F$-signature under reduction modulo $p$ Source: math.AC (Commutative Algebra) Link: https://arxiv.org/abs/2507.16566 Carvajal-Rojas, Schwede and Tucker asked whether the mod $p$ reductions of a complex klt type singularity have uniformly positive $F$-signature for almost all primes $p$. 17. Combinatorial proof of a permuted basement Macdonald polynomial identity Source: math.QA (Quantum Algebra) Link: https://arxiv.org/abs/2508.20337 A well-known and fundamental property of the Macdonald polynomials $P_\lambda(x;q,t)$ is their invariance under the transformation sending $(q,t)$ to $(q^{-1},t^{-1})$. 18. Large-Scale Portfolio Optimization Problem Under Cardinality Constraint With Enhanced Multi-Objective Evolutionary Algorithms Source: math.OC (Optimization and Control) Link: https://arxiv.org/abs/2607.09566 Decision-making is posing an increasingly formidable challenge to investors because of the growing number of alternatives available in financial markets. 19. Deconstructible classes of modules and stability Source: math.LO (Logic) Link: https://arxiv.org/abs/2512.17799 We show that every deconstructible class of modules with all embeddings, all pure embedding and all RD-embeddings is stable. 20. Function-Correcting Codes for Sum-Rank Metric Source: math.IT (Information Theory) Link: https://arxiv.org/abs/2607.03857 Function-Correcting Codes (FCCs) are a class of codes designed to protect the evaluation of a specific function of a message against channel errors at a higher level than the level of protection for the message, while requiring significantly less redundancy than conventional error-correcting codes. 21. The type I dichotomy for two-step nilpotent locally compact groups Source: math.GR (Group Theory) Link: https://arxiv.org/abs/2509.26212 We address the type I dichotomy for two-step nilpotent locally compact groups. 22. A representation of range decreasing group homomorphisms Source: math.DG (Differential Geometry) Link: https://arxiv.org/abs/2504.17459 The method of range decreasing group homomorphisms was first introduced in \cite{z23} and later generalized in \cite{z25}. 23. Unicyclic Graphs of Arbitrary Girth with the Same Chromatic Symmetric Function Source: math.CO (Combinatorics) Link: https://arxiv.org/abs/2607.09563 An open question asks whether the chromatic symmetric function (CSF) of a graph distinguishes non-isomorphic trees. 24. Differential graded Brauer groups over dg-rings Source: math.RA (Rings and Algebras) Link: https://arxiv.org/abs/2604.08069 Brauer groups of graded rings were defined and studied by Caenepeel and van Oystaeyen. 25. Complete Asymptotic Expansion of the Additive Mertens Sum $S_k(x)$ Source: math.NT (Number Theory) Link: https://arxiv.org/abs/2607.04366 Let $p_1, \dotsc, p_k$ be primes not exceeding $x$ ($k \geqslant 2$), and define the additive Mertens sum \[ S_k(x) = \sum_{p_1 \leqslant x} \cdots \sum_{p_k \leqslant x} \frac{1}{p_1 + \dotsm + p_k}. 26. Bifurcations in Interior Transmission Eigenvalues: Theory and Computation Source: math.NA (Numerical Analysis) Link: https://arxiv.org/abs/2511.11016 The interior transmission eigenvalue problem (ITP) plays a central role in inverse scattering theory and in the spectral analysis of inhomogeneous media. 27. All Hilbert spaces are the same: consequences for generalized coordinates and momenta Source: math-ph (Mathematical Physics) Link: https://arxiv.org/abs/2502.08494 Making use of the simple fact that all separable complex Hilbert spaces of given dimension are isomorphic, we show that there are just six basic ways to define generalized coordinate operators in Quantum Mechanics. 28. Reducing the Large Set Threshold for Oertel's Conjecture on the Mixed-Integer Volume Source: math.MG (Metric Geometry) Link: https://arxiv.org/abs/2411.11864 In 1960, Gr\"{u}nbaum proved that for any convex body $C\subset\mathbb{R}^d$ and every halfspace $H$ containing the centroid of $C$, one has that the volume of $H\cap C$ is at least a $\frac{1}{e}$-fraction of the volume of $C$. 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.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.OC (Optimization and Control); math.QA (Quantum Algebra); math.RA (Rings and Algebras). Selected 28 of 259 available items for this weekly brief.