arXiv preprints from January 1, 2026 through September 19, 2026 — 01:20:47 EST
We study the nonlocal continuity equation \[ \partial_tμ_b =\operatorname{div}\!\left( μ_b\nabla\log\bigl((I-b^2Δ)^{-1}μ_b\bigr) \right) \] on a closed connected Riemannian manifold. For smooth strictly positive initial data, we prove that as $b \to 0$, its global solution converges to heat flow $μ(t)$ at the sharp, uniform-in-time...
Let $A_1,\ldots,A_n$ be independent $d \times d$ real symmetric Gaussian random matrices, and consider the linear operator $A(x) = n^{-1/2}\sum_{i=1}^n x_i A_i$, $x\in \mathbb{R}^n$. We construct an iterative algorithm in the Approximate Message Passing family which iterates over $A$ and its adjoint $A^*$, and establish a state...
Given two finite essential adjacency matrices $A$ and $B$, Hazrat's graded classification conjectures posit that an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism of $K_0$ groups implies graded Morita equivalence of the Leavitt path algebras of $A$ and $B$, while the pointed version predicts a graded isomorphism of the...
The 1-2-3 Conjecture states that for every graph without isolated edges, there exists an edge-weighting from $\{1,2,3\}$ such that adjacent vertices receive distinct sums of weights on their incident edges. In the sequence variant, adjacent vertices are to be distinguished by the sequences of weights on their incident edges. In this...
It is known that the free $p$-convex Banach lattice on a Banach space $X$ can be represented as a space of functions on the unit ball of $X^*$. In this way, it gives rise to certain related (larger) lattices. To investigate such lattices, we introduce a new tool, related to operators into $X$. This tool is then used to (i) determine...
Restriction to the full cusp divisor of a modular curve defines a secondary invariant whose rational indeterminacy comes from a single global modular form. For every integral weakly holomorphic level-one modular form $h$ of weight divisible by four, we prove that the imported value $[h/2]$ vanishes at every nontrivial $Γ_0(N)$ level....
We answer in the negative the question of Thomas, recorded by Conley, Conley--Marks--Unger, and Conley--Tamuz, of whether every locally finite Borel graph admits a Borel unfriendly partition. Our counterexample has unbounded degree and is closed on a zero-dimensional Polish space; its connectedness relation is hyperfinite, and its...
In this paper, we prove that continuous $2$-convex solutions of the $σ_{2}$ equation with positive continuous density on bounded strictly mean-convex $C^{3}$ domains with $C^{3}$ boundary values belong to $W^{2,p}$ for every $1<p<\infty$ in every dimension. If, in addition, the density is $C^α$, $0<α<1$, the solutions belong to...
We study a random Schrödinger operator with a potential distributed according to a Poisson process. Asymptotic expansions for traces of resolvents in the limit of small disorder are derived. Explicit estimates for the expansion coefficients are given and we show that their infinite volume limits are finite as the spectral parameter...
We classify entire stable solutions with quadratic energy growth in the $4$-dimensional abelian Higgs model. More precisely, we prove that such solutions are holomorphic with respect to some complex structure, namely they satisfy the first order equations introduced by Bradlow, and consequently their nodal sets are holomorphic curves....
We investigate the persistence of most probable paths through the Onsager--Machlup functional for multidimensional stochastic differential equations driven by fractional Brownian motion with time-dependent diffusion coefficients and Hurst parameter $H\in(1/4,1)$. Under suitable structural and variational conditions, deterministic...
We investigate the maximization problem for the family of mixed-norm Strichartz inequalities for the Schrödinger equation, $\|e^{-itΔ/2}f\|_{L_t^qL_{\boldsymbol{x}}^r(\mathbb{R}^{1+d})}\le C_{q,r}\lVert f\rVert_{L^2(\mathbb{R}^d)}$, with $2/q+d/r=d/2$, $q,r\geq 2$, and thus $r\leq 2d/(d-2)$ if $d\geq 3$. We show that, in low...
We prove a delocalisation bound for eigenvectors of Wigner matrices with the precise relationship between the size of the largest entry and the decay exponent of the probability. We also prove that the largest entry of an individual eigenvector and the largest entry of all eigenvectors are both Gumbel distributed. The proof is based...
We prove that any homeomorphism of the two-torus homotopic to a Dehn twist map with irrational bounded vertical deviations admits an irrational circle factor. This establishes a rigidity property previously studied by Kocsard, who proved this implication under the additional assumption of a small-wandering-domain hypothesis. We...
We compare eight variants of a three-variable pollution-based tourism model obtained by assigning the same discrete delay to different terms. The delay may enter three slots: the visitor factor in the deterrence term, pollution perception, or visitation-driven growth and reinvestment. We call a placement feasible when every...
We show the existence of counterexamples to a four-decade-old conjecture attributed to Stanley concerning the unimodality of $h^*$-polynomials of IDP polytopes. As additional applications of our main constructions, we also disprove a conjecture by Brenti on the log-concavity of $h^*$-polynomials of Gorenstein IDP polytopes, and a...
In this paper, we study certain categories of labeled finite rooted ordered trees over a fixed set of labels where each label is equipped with an arity: a fixed number of children that the vertex with the given label must have. Equivalently, these are expression trees for operations in a free non-symmetric operad. A morphism between...
We study the distribution of the number of double deficiencies (DD) in permutations of length n avoiding one or two patterns of length 3. Using structural decompositions of these avoidance classes--together with a lattice-path decomposition in the 321-avoiding case--we derive functional equations and convolution-type recurrences that...
We study the information encoded by difference equations for the local Gromov-Witten theory of contractible rational curves in Calabi-Yau threefolds. Starting from the Bryan-Katz-Leung multiple-cover decomposition, we organize all nonzero primitive degrees on an exceptional ray by a single central-difference operator associated with...
We study lower and upper box dimension prints for bounded subsets of \(\mathbb R^n\), defined by weighted covering numbers for independently oriented rectangular boxes with prescribed ordered side-length bounds. The limits range over all eccentricities, including unbounded aspect ratios. For every non-empty bounded set, we identify...
We characterize the standard modules of $\GL_{2m}(\C)$ that are distinguished by $\GL_m(\HH)$. Let $δ_1,\dots, δ_{2m}$ be characters of $\mathbb{C}^\times$. Assume that $S = δ_1 \times \cdots \times δ_{2m}$ is a standard module of $\GL_{2m}(\mathbb{C})$. For each $i$, define ${δ_i^*} (z) = δ_i(\overline{z})^{-1}$ for $ z \in...
We resolve two problems left open by Johnson and Kollár in their paper Arcology: Question 21 and the openness issue arising in Example 14.
We study an inextensible open filament with free ends in a planar Stokes fluid. The system reduces to a third-order nonlocal curvature equation coupled to an elliptic equation for the tension. We prove local well-posedness for nearly critical initial data in supported Sobolev spaces $\widetilde H^s$, $-1/2<s\le0$, satisfying the...
We prove that the best constant in the one-dimensional Lieb-Thirring inequality with exponent one is $4/(3\sqrt{3}π)$, confirming the Lieb-Thirring conjecture in this case.
This paper presents a unified invariant-domain-preserving (IDP) framework for hybrid discretizations of hyperbolic conservation laws, including active flux and PAMPA methods, in which cell averages are updated conservatively while cell-boundary point values evolve under a possibly non-conservative operator. The main challenge is to...