arXiv preprints from January 1, 2026 through September 22, 2026 — 08:04:36 EST
We propose a novel approach for studying small-time asymptotics of the fractional heat content of $C^2$ non-characteristic domains in Carnot groups. Denoting the sub-Laplacian operator by $\mathcal{L}$, the fractional heat content of a bounded domain $Ω$ is defined as $Q^{(α)}_Ω(t)=\int_Ωu_α(x,t) dx$, where $u_α$ is the solution to...
Let $F$ be a $k\times \ell$ (0,1)-matrix. A matrix is simple if it is a (0,1)-matrix with no repeated columns. A (0,1)-matrix $A$ is said to have a $F$ as a configuration if there is a submatrix of $A$ which is a row and column permutation of $F$. In the language of sets, a configuration is a trace. Let $\mathrm{Avoid}(m,F)$ be all...
A paradefinite logic is a logic that can serve as the underlying logic for theories that are inconsistent or incomplete. A well-known paradefinite logic is Belnap-Dunn logic. Various expansions of Belnap-Dunn logic have been studied in the literature. In this note, it is argued that the most natural paradefinite logic relative to...
We study the total variation distance under two information-erasing maps on inhomogeneous Bernoulli product measures: summation and homogenization. While summation is a Markov kernel and hence satisfies the usual data processing inequality, homogenization -- which maps each Bernoulli parameter to the cumulative mean -- is not....
We study pattern densities in binary sequences, finding optimal limit sequences with fixed pattern densities.
We work in the context of a complete totally transcendental theory $T = T^{eq}$. We consider the prime model $M_{A}$ over a set $A$. For intermediate sets $B$ with $A\subseteq B \subseteq M_{A}$ which are normal ($Aut(M_{A}/A)$-invariant) and ``minimal" we give a full Galois correspondence between intermediate definably closed sets...
We show that a generalised sparse grid combination technique which combines multi-variate extrapolation of finite difference solutions with the standard combination formula lifts a second order accurate scheme on regular meshes to a fourth order combined sparse grid solution. In the analysis, working in a general dimension, we...
The tautological $\mathbb{Q}$-subalgebra $\mathsf{R}^*(\mathcal{A}_g) \subset \mathsf{CH}^*(\mathcal{A}_g)$ of the Chow ring of the moduli space of principally polarized abelian varieties is generated by the Chern classes of the Hodge bundle. There is a canonical $\mathbb{Q}$-linear projection operator $\mathsf{taut}:...
We introduce a two-parameter framework that refines several classical graph invariants by imposing higher-order constraints along bounded-length geodesics. For integers $k,d\ge1$, a vertex set is called $k,d$-independent if every shortest path of length at most $d$ contains fewer than $k$ vertices of the set, giving rise to...
We study a class of parabolic equations in non-divergence form with measurable coefficients that exhibit singular and/or degenerate behavior governed by weights in the $A_{1+\frac{1}{n}}$-Muckenhoupt class. Under a smallness assumption on a weighted mean oscillation of the weights, we establish weighted $W^{2,\varepsilon}$-estimates...
We establish standard zero-free regions with no exceptional Landau-Siegel zeros for Rankin-Selberg $L$-functions and triple product $L$-functions in several new families for which modularity is not yet known.
Representations of vertex operator algebras $V$ (VOAs) have numerous applications, including the construction of sheaves of conformal blocks on moduli spaces of curves. For a $V$-module $W = \oplus W_d$, a sequence of associative algebras $\mathfrak{A}_d$ acts on each graded component $W_d$. When these $d$th-mode transition algebras...
We consider the two-dimensional time-harmonic transmission problem for an impedance-matched (ρ= 1) right-angle penetrable wedge at refractive index ratio ν= \sqrt{2}, in the integrable lemniscatic configuration (θ_w ,ν,ρ) = (π/4,\sqrt{2},1). Starting from Sommerfeld spectral representations, the transmission conditions on the two...
The saturation property for Littlewood--Richardson coefficients was established by Knutson and Tao in 1999. In 2004, Kirillov conjectured that the saturation property extends to Schubert coefficients. We disprove this conjecture in a strong form, by showing that it fails for a large family of instances. We also discuss computational...
We present results on the trace regularity of the stress vector on the boundary of an elastic solid satisfying the time-dependent, displacement-traction problem for the Navier equations of linear elasticity in a bounded domain of $\mathbb{R}^3$. Specifically, the solid's displacement is subject to Dirichlet- and Neumann-type...
We develop a method to compute scattering amplitudes for the Helmholtz equation in variable, unbounded media with possibly long-range asymptotics. Combining Penrose's conformal compactification and Melrose's geometric scattering theory, we formulate the time-harmonic scattering problem on a compactified manifold with boundary and...
We study the Ergodic Properties of Random Walks in stationary ergodic environments without uniform ellipticity under a minimal assumption. There are two main components in our work. The first step is to adopt the arguments of Lawler to first prove a uniqueness principle. We use a more general definition of environments...
In Evans and Hendel's recent proof of an outstanding conjecture on the resistance distances of a family of linear 3-trees, a key technique in the proof was calculating the recursion satisfied by a family of determinants. The underlying algorithm employed to prove the conjecture converged (i.e., terminated) in the particular case...
Let $K$ be a number field of degree $d$ so that $K/\mathbb Q$ is a Galois extension. The {\it normal basis theorem} states that $K$ has a $\mathbb Q$-basis consisting of algebraic conjugates, in fact $K$ contains infinitely many such bases. We prove an effective version of this theorem, obtaining a normal basis for $K/\mathbb Q$ of...
We conjecture that a unital C*-algebra is a W*-algebra if and only if each of its maximal abelian self-adjoint subalgebras is a W*-algebra; this is a space-free analogue of a known result due to G.K. Pedersen. Our main result is a proof that this conjecture holds for finite C*-algebras if and only if every $2$-quasitrace on a unital...
We introduce and study the maximal-variation locus in families and moduli spaces of projective curves, defined via conductor-level balancing of meromorphic differentials on the normalization. This notion captures precisely when the space of canonical differentials behaves with the expected dimension under degeneration. We prove...
The performance of eigenvalue problem solvers (eigensolvers) depends on various factors such as preconditioning and eigenvalue distribution. Developing stable and rapidly converging vectorwise eigensolvers is a crucial step in improving the overall efficiency of their blockwise implementations. The present paper is concerned with the...
Olsson showed in [Ols25] that if $\mathcal{X} \to X$ is a $\mathbf{G}_m$-gerbe over a smooth projective variety over an algebraically closed field $k$ such that the Brauer class of $\mathcal{X}$ has order prime to the characteristic of $k$, then the homomorphism of $k$-group algebraic spaces $\operatorname{Aut}^0_{\mathcal{X}} \to...
This work was intended to be all about, and only about, hypergeometric 3F2(1). The initial goal was to revisit many identities from the literature that have been derived over the years and show that they can be obtained in a simpler way armed, with only a minimum of elementary identities. That goal has been achieved as a (patient)...
We answer a question of Arant, Kechris and Lutz by showing that there is a Borel graphable equivalence relation with no Borel graphing of diameter less than 3. More specifically, we prove that there is an equivalence relation with a Borel graphing of diameter at most 4 but no Borel graphing of diameter less than 3. Our proof relies on...