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	Title	Source	Topics	Written	Found	Rel.	Status
	A Matrix-Theoretic Exact Formula for Counting Primes in Intervals Between Consecutive Odd Squares	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article presents a matrix-theoretic exact formula to count primes in intervals between consecutive odd squares, defining matrix multiplicity and proving an identity that calculates the number of primes in each interval without primality testing. The formula confirms at least one prime exists in such intervals up to $1.37 \times 10^{17}$, establishing a combinatorial condition equivalent to this property. Open full article →
	A probabilistic bijection between twenty-vertex configurations with a free west boundary and Gelfand-Tsetlin patterns avoiding three equal entries in a row	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study establishes a probabilistic bijection between twenty-vertex configurations on quadrangular domains with a free west boundary and Gelfand-Tsetlin patterns that avoid having three consecutive equal entries, revealing a correspondence explained by the alignment of their enumeration formulas. This connection is significant as it links two distinct combinatorial structures through a novel mapping where the west boundary of vertex configurations corresponds to the bottom row of Gelfand-Tsetlin patterns; specifically, when the west boundary is fixed, it aligns with the pattern's bottom row being $(1, 2, \ldots, n)$. Open full article →
	Quasi-Polish spaces and spaces of filters in second-order arithmetic	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The paper formalizes quasi-Polish spaces and their equivalent representations, such as UF spaces and $\mathbf{\Pi}_2^0$ subspaces of $\mathcal{P}(\mathbb{N})$, within second-order arithmetic to conduct a reverse mathematical analysis of the transitions between these structures. This work matters as it provides foundational insights into the logical strength required for equivalences among different representations of quasi-Polish spaces, enhancing understanding in descriptive set theory and topology. Open full article →
	Hecke monoids, their homomorphisms and parabolicity	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study explores homomorphisms in Hecke monoids, focusing on the properties and classification of parabolic and injective homomorphisms, revealing a rich structure of parabolic homomorphisms and classifying locally injective connected homomorphisms between classical types, which unexpectedly provides insight into all homomorphisms between Hecke monoids. Open full article →
	Conformally Invariant Besov Spaces on Chord-Arc Domains	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article introduces Besov-type spaces on simply connected domains, proving that on quasidisks, first-order Besov spaces are isomorphic to higher-order counterparts, which preserve conformal quasi-invariance. This characterization of chord-arc domains through the isomorphism between first-order and boundary Besov spaces extends previous Dirichlet space ($p=2$) results to all $1 < p < \infty$. Open full article →
	Topological complexity sequences of groups	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article defines the topological complexity sequence for groups based on their Milnor constructions, providing an intrinsic refinement that applies to groups with infinite cohomological dimensions. This sequence is shown to be weakly increasing and unbounded for such groups. For a finite group of even order, the asymptotic behavior of this sequence has been determined. Open full article →
	Compactness of products and commutators of inner projections	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The paper characterizes the compactness of products and commutators of inner projections in Hardy spaces over unit disks and polydisks, using Douglas algebra for single-variable cases. It identifies a rigidity phenomenon where on the bidisc, the product of two inner projections is compact only if it has finite rank, contrasting with triviality in higher-dimensional polydiscs. Open full article →
	Large deviations for maximum local time of simple random walk in dimensions $d\ge 3$	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study provides precise asymptotic probabilities for upward and downward deviations of the maximum local time of simple random walks on $\mathbb{Z}^d$ for dimensions $d\ge 3$, both in discrete- and continuous-time settings, with Gumbel-type distributions identified at the logarithmic scale. The research matters as it completes the understanding of local time deviations through a loop-pruning construction that proves matching discrete-time lower bounds. A concrete detail is the derivation of sharp continuous-time asymptotics for downward deviations alongside establishing discrete-time upper bounds. Open full article →
	Geometric bounds for Steklov and weighted Neumann eigenvalues on Euclidean domains	arXiv math	📐 Math, 🫖 Rendering	2026-07-07	2026-07-07	1.00	reviewed
The study establishes sharp upper bounds for the first two nonzero Steklov eigenvalues in Euclidean spaces of dimension $d \geq 7$, normalized by volume and boundary measure, and derives strict upper bounds for dimensions $3 \leq d \leq 6$. Additionally, it extends previous results to provide strict upper bounds for all higher Steklov eigenvalues on planar simply connected domains with continuous boundaries. Open full article →
	Asymptotic theory and first-order bias of the Wallace--Freeman estimator	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The Wallace--Freeman estimator is shown to be equivalent to a penalized likelihood criterion with a $n^{-1}$ penalty weight in regular parametric models, establishing its asymptotic properties within standard theory; this equivalence also reveals an explicit $O(n^{-1})$ bias difference from the maximum likelihood estimator. The findings are illustrated for the Weibull model, where the Wallace--Freeman penalty modifies the leading bias of the shape parameter's estimation. Open full article →
	Cohomological invariants of hermitian forms that detect hyperbolicity	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
Researchers constructed cohomological invariants for hermitian forms using unramified cohomology groups that detect hyperbolicity, applicable across any type (orthogonal, symplectic, unitary) and over fields with any characteristic; this method shows that hermitian pairs over quaternion algebras with trivial classical invariants are hyperbolic in fields of separable dimension 3. Open full article →
	Semi-Cosimplicial Hilbert Spaces with Isometric Coface Operators	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article introduces and develops semi-cosimplicial Hilbert spaces with isometric coface operators, linking to non-commutative probability theory through the concept of spreadability. This work explores applications in areas such as cohomology, representation theory, and graph decomposition, offering a new framework for understanding distributional symmetry. A key detail involves the classification and extensions of semi-cosimplicial sets with injective coface maps, providing foundational insights into their structure and potential theoretical applications. Open full article →
	Proof of Sun's conjectures on hyperbolic cosine series via the Eisenstein--Lambert method	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article proves two conjectures by Zhi-Wei Sun on hyperbolic cosine Lambert series, including the evaluation of a series $S_m$ for integers $m \geq 0$, where specific values for $S_0$ and $S_1$ are derived as $S_0=\frac{1}{12}$ and $S_1=\frac{1}{2\pi^2}$, with $S_m=0$ for $m>1$. The proof employs the Eisenstein--Lambert method, leveraging modular transformations for Eisenstein series. Open full article →
	Duality for Delsarte's extremal problem on compact Gelfand pairs	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study explores Delsarte-type extremal problems for positive definite functions on compact Gelfand pairs using infinite-dimensional linear programming, with implications for Turán and Delsarte problems in number theory, sphere packing, and statistics. The paper establishes and proves a strong duality statement for these problems. Notably, the research includes compact Abelian groups as a specific case. Open full article →
	The group identification problem for $p$-groups of small order	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
Research identifies effective group-theoretic invariants and develops an algorithm to distinguish among non-isomorphic p-groups, specifically applied to the $10,494,213$ groups of order $2^9$. Notably, 56 pairs of these groups are particularly challenging to differentiate using invariants. Open full article →
	On the stable Hopf invariant	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article presents a simplified approach to the stable Hopf invariant, offering elementary proofs for key formulas including the Cartan Formula. It extends these results to the stable category of $\pi$-spaces when $\pi$ is a discrete group, enhancing applicability and understanding of the invariant's uniqueness. Open full article →
	Comparison theorems for the extreme eigenvalues of a random symmetric matrix	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The paper establishes a comparison theorem showing that the maximum eigenvalue of a sum of independent random symmetric matrices is dominated by that of a matching Gaussian random matrix, strengthening previous results and providing corollaries for minimum eigenvalues and spectral norms. This methodology enhances existing bounds on eigenvalues in various fields including spectral graph theory and quantum information theory, and provides the first complete proof for the injectivity properties of sparse random dimension reduction maps conjectured by Nelson & Nguyen in 2013. Open full article →
	Bi-twisted conjugacy in finite groups	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
Researchers offer two methods to calculate the number of bi-twisted conjugacy classes in finite groups—one through irreducible characters and another using ordinary conjugacy classes—highlighting new equalities and inequalities for Reidemeister numbers. The study also establishes connections between bi-twisted conjugacy, representation theory, and fixed-point free automorphisms. One concrete detail includes the derivation of sharp inequalities for Reidemeister numbers related to these conjugacy classes. Open full article →
	Riemann-Wirtinger integrals on the product of two one-dimensional complex tori	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article introduces the Riemann-Wirtinger integral extended to the product of two one-dimensional complex tori, generalizing its application beyond a single complex torus. This extension is significant for understanding the structure of associated twisted cohomology groups and deriving related differential equations. The study derives a specific system of differential equations satisfied by this new form of Riemann-Wirtinger integral. Open full article →
	Pure extension of the theta divisor over the moduli space of abelian varieties	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article extends the theta divisor over the moduli space of abelian varieties, showing that its pure weight 2 extension differs from the Zariski closure by a tropicalization of the Riemann theta function; this work applies Moret-Bailly's "key formula" to derive a universal formula for the Néron-Tate height of a point. Open full article →
	Algebraic realization of stable Poincar\'e-Reeb graphs	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study establishes that any finite graph with a good orientation and vertices of degree 1 or 3 can be realized as the Poincaré-Reeb graph of a stable algebraic domain in $\mathbb{R}^n$ for $n \geq 2$, extending to graphs with vertices of degree 2 when $n \geq 3$. This work uses advanced algebraic approximation techniques, including recent extensions over $\mathbb{Q}$ by Ghiloni and the author, to achieve these realizations. Open full article →
	Parity of $k$-differentials in genus zero and one	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article determines the spin parity of $k$-differentials with specified zero and pole orders on Riemann surfaces of genus zero and one, confirming a previously conjectured result. This achievement resolves Conjecture A.10 by reformulating it using Jacobi symbols, and the proof was formalized in Lean/Mathlib by AxiomProver. Open full article →
	A Baire Category Approach to Besicovitch's Theorem and Measure Regularity	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article demonstrates that Besicovitch's Theorem can be proven within the subsystem $\mathsf{ACA}_0$ by reformulating its proof using a Baire Category argument, and shows that the witnessing subset is computable from one jump of the original set. This equivalence between the Baire Category Theorem for Closed Sets ($\mathsf{BCTC}$) and $\mathsf{ACA}_0$ provides new insights into measure regularity properties and their computational complexity. Open full article →
	Positive characteristic analogues of finite algebraic numbers	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article introduces $\mathcal{P}^0_{\mathcal{A}_K}$ as a positive characteristic analogue of Rosen's ring of finite algebraic numbers over the rational function field $K=\mathbb{F}_q(\theta)$, foundational properties of which are studied to extend number-theoretic concepts into function fields. This extension is significant for bridging number theory and function field arithmetic. The study provides a new framework for analyzing algebraic structures in positive characteristic settings, exemplified through the examination of properties over $\mathbb{F}_q(\theta)$. Open full article →
	Nearly Gorenstein and almost symmetric properties in shifted numerical semigroups	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study proves that properties of being nearly Gorenstein or almost symmetric are preserved from $M_n$ to $M_{n+r_k}$ for sufficiently large $n$, by relating their pseudo-Frobenius elements and correcting a previous literature error. Explicit formulas for the Frobenius and pseudo-Frobenius numbers of $M_{n+r_k}$ have also been derived, offering new insights into shifted numerical semigroups' structure. Open full article →
	The $\ell$-modular local theta correspondence in type II and partial permutations	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The paper calculates multiplicities in the $\ell$-modular local theta correspondence for type II over a non-archimedean field, showing these multiplicities are governed by symmetric group actions on partial permutations. Unlike complex coefficients, significant multiplicities can occur here; if $d$ is the order of the residue cardinality and the rank of involved general linear groups is bounded above by $d\ell$, Pieri's Formula provides explicit algorithms to predict correspondence behavior. Open full article →
	The Wiener Wintner and Return Times Theorem Along the Primes	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article extends the Wiener-Wintner Theorem to arithmetic sequences specifically along prime times, proving convergence for a given measure-preserving transformation and functions in $L^p(X)$ spaces. This extension bridges classical Fourier analysis with combinatorial number theory and ergodic theory, utilizing U^3 theory in its proof, which includes novel $U^3$-estimates for Heath-Brown models of the von Mangoldt function. Open full article →
	Central polynomials of minimal degree for matrices	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
Formanek's conjecture on the minimal degree of central polynomials for $n\times n$ matrix algebras over fields of characteristic 0 is confirmed for $n\leq 3$, with the paper focusing on methods to search for such polynomials, particularly proving no central polynomials in two variables exist for $4\times 4$ matrices at degrees $\leq 12$. Open full article →
	On the sizes of the maximal prime powers divisors of factorials	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article proves that for any prime $p$, there exists a threshold integer $n_0$ such that the maximal power of any larger prime $q > p$ dividing $n!$ is less than that of $p$ for all $n \ge n_0$, establishing a dominance order among prime powers in factorials. For twin primes $p$ and $q = p + 2$, the minimal such threshold $n_0$ is specifically $(p^2+p)/2$. Open full article →
	Tameness of actions on finite rank median algebras	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study establishes that for every finite-rank median algebra $X$, the rank of $X$ equals the independence number of all median-preserving maps to $[0,1]$, with a similar equality in the compact topological case for continuous median-preserving maps. This leads to a generalized Helly selection principle and proves that continuous actions by median automorphisms on compact finite-rank median algebras are dynamically tame, applicable to systems satisfying Sarnak's Möbius disjointness conjecture in metrizable cascade cases. Open full article →
	Zeros of Polynomials in Derivatives of Automorphic $L$-functions	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study establishes an asymptotic formula for the number of nontrivial zeros of polynomials in derivatives of automorphic $L$-functions up to a height $T$, determining its main term based on dimensions, arithmetic conductors, and differentiation orders. It also demonstrates that almost all nontrivial zeros lie near the critical line $\operatorname{Re}(s)=1/2$ under specific conditions. Open full article →
	The number of edges of a symmetric edge polytope	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article establishes a sharp lower bound for the number of edges of a symmetric edge polytope based on elementary graph invariants and characterizes graphs that meet this bound; it also explores how the h*-polynomial of these polytopes behaves under edge deletion, connecting to a conjecture by Ohsugi and Tsuchiya. Open full article →
	Large deviations for invariant measure of stochastic Allen-Cahn equation with inhomogeneous boundary conditions and multiplicative noise	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study validates a small noise large deviation principle for invariant measures of a one-dimensional stochastic Allen-Cahn equation under inhomogeneous Dirichlet boundary conditions and multiplicative noise, overcoming challenges due to weak dissipation. The dynamics converge to the unique minimizer of the Ginzburg-Landau energy functional, with the invariant measure $\mu_\epsilon$ concentrating exponentially around this minimizer as $\epsilon$ approaches zero. A key detail is the use of L. Simon's convergence theorem to establish these results in a Sobolev space framework. Open full article →
	Set theory, logic, and homeomorphism groups of manifolds	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article explores the interplay between set theory axioms and the first-order properties of homeomorphism groups of manifolds, showing that under V=L, these groups are first-order rigid with conjugacy classes determined by type, while under projective sets having the Baire property, noncompact connected manifolds can have elementarily equivalent but non-homeomorphic groups. Notably, infinitary $L_{\omega_1\omega}$ formulas determine both conjugacy classes of homeomorphisms and homeomorphism types of manifolds. Open full article →
	Carath\'eodory number of homogeneous convex cones	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study characterizes homogeneous convex cones where the rank equals the Carathéodory number, linking this property to selfduality when it applies to both the closure and dual cone of the homogeneous convex cone. It also identifies that only spectrahedral cones associated with homogeneous chordal graphs can be sparse and homogeneous. Open full article →
	Cyclic polynomials in Dirichlet-type Spaces of the unit bidisk	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The paper solves Torkinejad Ziarati's open problem by confirming that the polynomial $2-z_1-z_2$ is cyclic in Dirichlet-type space $D_\alpha$ for $\frac{3}{2} < \alpha \leq 2$, thereby completing the characterization of cyclic polynomials in these spaces. Open full article →
	Complexity and curvature of pairs of Burch modules and ideals	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The paper explores the complexity and curvature of pairs involving Burch modules and ideals, unifying and extending previous findings on their extremal properties and Ext/Tor vanishing characteristics. A key new element in the proofs involves demonstrating that the Burch property remains independent under embedding, particularly for Burch modules of depth zero. Open full article →
	Ehrhart Theory over Abelian Group Rings	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article introduces a unified framework for Ehrhart theory using Abelian group rings to encode richer algebraic data than traditional counts, proving that key results like the Brion theorem extend to this setting. This framework enables the derivation of $q$-enumerative and weighted theories as consequences of a single mechanism, enhancing the algebraic versatility of lattice point enumeration. Open full article →
	Non-decomposable Lagrangian cobordisms between Legendrian knots	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
Researchers have constructed a family of non-decomposable Lagrangian cobordisms of any genus $g>0$ between stabilized Legendrian knots in the standard contact three-sphere, utilizing Livingston's estimates to prove their non-decomposability. Open full article →
	Hermite's approach to Abelian integrals revisited	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article presents a new criterion for the linear independence of values of Lauricella hypergeometric series $F_D$ with rational parameters, extending Hermite's work on Abelian integrals to both complex and $p$-adic settings. This extension is achieved through explicit Padé-type approximations to solutions of reducible Jordan-Pochhammer differential equations, with a key innovation being the proof of non-vanishing determinants associated with these approximants. Open full article →
	Non-jumping densities of 3-uniform hypergraphs	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The paper introduces a method to identify non-jumping densities specifically for 3-uniform hypergraphs by utilizing patterns, contradicting Erdős's conjecture that all densities are jumps. This matters as it advances the understanding of Turán densities in hypergraph theory. The authors provide new examples of non-jumps for $r = 3$ as a result of their method. Open full article →
	Linear independence of values of hypergeometric functions and arithmetic Gevrey series	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study establishes new linear independence results for values of generalized hypergeometric functions ${}_pF_q$ at multiple algebraic points over various number fields, using a uniform construction of Padé approximants and a novel non-vanishing argument for Hermite-type Wronskians, extending previous single-point findings to multi-points in both complex and $p$-adic settings. Open full article →
	A local Lorentzian Ferrand-Obata theorem for conformal vector fields	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study proves that a conformal vector field on a closed, real-analytic Lorentzian manifold either has a locally isometric flow or results in a metric that is everywhere conformally flat, addressing a local version of the Lorentzian Lichnerowicz conjecture. This work improves upon previous normal forms for conformal vector fields by focusing on essential linearizable singularities and utilizing global arguments dependent on compactness assumptions. Open full article →
	Universal frame set for rational functions	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study proves the existence of a universal set $\Lambda$ with an upper Beurling density less than $1+\varepsilon$ that generates a frame in $L^2(\mathbb{R})$ for any rational function $g$, highlighting its significance in constructing frames from rational functions without specific adjustments. This matters because it provides a generalized method to create frames for rational functions of bounded degree, enhancing the efficiency and applicability of frame theory in signal processing and harmonic analysis. Open full article →
	Roth's Theorem in Super Smooth Numbers	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article demonstrates that Roth's theorem on arithmetic progressions holds for sets of $y$-smooth numbers $\mathcal{S}(N,y)$ up to $N$, defined as super smooth when $y=\log^KN$ for a large constant $K$. This extends Harper’s previous work, which established the theorem under less stringent conditions. Open full article →
	The Second Moment of $\mathrm{GL}_4 \times \mathrm{GL}_2$ $L$-functions at Special Points	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The paper offers an alternative proof for Chandee and Li's result on the second moment of $\mathrm{GL}_4 \times \mathrm{GL}_2$ special $L$-values, utilizing a more conceptually direct method that avoids detecting 'Eisenstein--Kloosterman' cancellation and does not employ Poisson summation. Open full article →
	A density counterpart of the Scheepers covering property	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
Researchers introduce a density counterpart of the Scheepers covering property $\bigcup_{\mathrm{fin}}(\mathcal O,\Omega)$, which is significant for understanding its relations with known combinatorial density properties. This new property is shown to be equivalent to $M$-separability under the Near Coherence of Filters principle established by Blass and Weiss. Open full article →
	Ring stacks conjecturally related to the stacks $BT_n^{G,\mu}$	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
Researchers have defined new ring stacks using sheared Witt vectors, proposing multiple models for these structures; this work conjectures a relationship between these ring stacks and the n-truncated Barsotti-Tate group stacks, including their Shimura analogs. This matters as it could provide a novel framework for understanding and potentially classifyingBarsotti-Tate groups through ring stacks. One concrete detail is the use of sheared Witt vectors in defining the new ring stacks. Open full article →
	Modular forms for chromatic homotopy: Supersingular congruences	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article proves Larson's conjecture within Behrens' program, providing a criterion for when modular forms attached to the divided beta family in the Adams-Novikov spectral sequence can be represented by powers of the discriminant form $\Delta^t$, specifically for primes $p \geq 5$. The proof relies on showing that for any prime $\ell \neq p$, the value of the modular function $V_\ell(\Delta)/\Delta$ at each supersingular point of $X_0(\ell)$ is a $(p^2-1)/12$-th root of unity. Open full article →
	Nordhaus--Gaddum type bounds for the complement rank	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The paper establishes Nordhaus--Gaddum type bounds for the complement rank of a graph, proving that for any graph $G$, the product and sum of the complement ranks of $G$ and its complement $\overline{G}$ have specific lower bounds ($n$ and $n+1$, respectively), with conditions for equality. The authors also construct examples demonstrating that the upper bounds for these measures, $n^2$ for the product and $2n$ for the sum, are tight for graphs of order $n \ge 4$. Open full article →
	The Schwarz lemma for holomorphic and minimal disks at the boundary	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article proves a Boundary Schwarz lemma for holomorphic disks on the unit ball in $\mathbb{C}^n$ and extends this to minimal conformal disks at the boundary using Forstnerič and Kalaj's work, establishing key results in complex analysis and differential geometry. This matters as it advances understanding of geometric properties of holomorphic and minimal disks. The proof utilizes a Schwarz lemma for harmonic maps which are conformal at a point from Forstnerič and Kalaj’s 2024 paper. Open full article →
	Catalan structures arising from pattern-avoiding Stoimenow matchings and other Fishburn objects	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The paper solves a problem posed by Bevan et al., identifying subsets of Stoimenow matchings counted by Catalan numbers through five solutions involving pattern-avoiding matchings; it also proves that matchings avoiding four infinite families of generalized forbidden patterns are equinumerous, linking these structures to other Fishburn objects like ascent sequences and permutations. Open full article →
	Quasi-Trefftz spaces for a first-order formulation of the Helmholtz equation	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article introduces quasi-Trefftz methods for solving first-order differential systems, specifically developing discrete quasi-Trefftz spaces and their bases without relying on auxiliary scalar equations. This method represents a novel approach compared to traditional decoupling techniques that use second-order scalar equations. A key detail is the focus on computational aspects in constructing these bases. Open full article →
	Bowen's formula and the difference between random iterated function systems and random recursive constructions	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The paper demonstrates that the difference in independence structures between random iterated function systems and random recursive constructions affects the validity of Bowen's formula, with the formula failing for certain random iterated function systems but always holding for random recursive constructions. This distinction impacts dimension theory, showing concrete implications of structural differences between these models. Open full article →
	Full characterisation of Painlev\'e V asymptotics and nonlinear monodromy-Stokes structure	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study provides a comprehensive characterization of Painlevé V equation asymptotics within a right half-plane near infinity, identifying all possible solutions linked to monodromy data across the entire monodromy manifold. This includes new truncated solutions along imaginary axes and elliptic asymptotics in generic directions, complementing previous work by Andreev and Kitaev. The research also outlines a nonlinear monodromy-Stokes structure that describes changes in monodromy data as part of solution expressions during analytic continuation. Open full article →
	Minimal Bounded Speedups of Toeplitz Flows	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study shows that minimal bounded speedups of Toeplitz flows do not necessarily result in other Toeplitz flows, detailing methods to ascertain if the outcome remains Toeplitz. Sufficient conditions are provided to ensure the minimal bounded speedup results in a Toeplitz flow, highlighting nuances in dynamical system transformations. Open full article →
	An Arithmetic Characterization of 2-Generated Numbers	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article establishes an arithmetic criterion for identifying 2-generated numbers based on their prime factorization, where a number $n$ is a 2-generated number if every group of order $n$ can be generated by two elements. This characterization is significant as it provides a clear condition to determine the generation property of groups solely through the lens of number theory. For instance, all square-free numbers are identified as 2-generated under specific conditions related to their prime factors. Open full article →
	A higher arithmetic on the ordinals	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The article introduces an infinite sequence of binary operations extending both standard ordinal arithmetic and natural number hyperoperations. This unification is significant for advancing understanding in transfinite arithmetic. The study offers a new framework that could potentially simplify complex ordinal computations by aligning them with more familiar hyperoperation structures. Open full article →
	On a remark of Serre	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
Researchers extend Serre's work to identify primes $p$ where the maximum Hasse bound for points on an elliptic curve over $\mathbb{F}_{p^5}$ is not met, listing all such instances up to $10^{70}$. This exploration includes heuristics and similar analyses for curves of genus 2 and 3. Open full article →
	Hodge Splittings and Einstein 4-manifolds	arXiv math	📐 Math	2026-07-07	2026-07-07	1.00	reviewed
The study explores pairs of Riemannian metrics $(g,h)$ on oriented 4-manifolds where $g$'s curvature tensor preserves $h$'s Hodge splitting, extending the Einstein condition to a broader class when $h$ is conformal to $g$. This extension satisfies a generalized Hitchin-Thorpe inequality that collapses to the classical form under conformality. A specific example on $\#_5\mathbb{CP}^2$ violates the Hitchin-Thorpe inequality, demonstrating the existence of manifolds admitting metrics satisfying this broader condition but not Einstein metrics. Open full article →

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Ars Technica	rss	active	ok: 20 seen / 0 new	60m
arXiv cs.GR	rss	active	ok: 24 seen / 0 new	60m
arXiv math	rss	active	ok: 1134 seen / 0 new	60m
Game Developer	rss	active	ok: 50 seen / 0 new	60m
Hacker News (front page)	api	active	ok: 50 seen / 2 new	60m
Hacker News (Show/Ask HN)	api	active	ok: 50 seen / 10 new	60m
Lobsters	rss	active	ok: 25 seen / 2 new	60m
Quanta Magazine	rss	active	ok: 5 seen / 0 new	60m
r/gamedev	rss	active	ok: 25 seen / 3 new	60m
Slashdot	rss	active	ok: 15 seen / 0 new	60m
Terence Tao blog	rss	active	ok: 10 seen / 0 new	60m
a16z	rss	off	unverified	60m
GDC Vault	scrape	off	deferred	60m
IndieHackers	rss	off	disabled	60m
Inigo Quilez	rss	off	unverified	60m
Ke-Sen Huang SIGGRAPH index	scrape	off	deferred	60m
LinkedIn	api	off	inactive	60m
Real-Time Rendering blog	rss	off	blocked	60m
Shadertoy blog	rss	off	unverified	60m

Coordinator: sac-vm-containers3 · Inference: SAC-DSK-003 (voice/drafts) · no cloud LLM at runtime