arXiv preprints from January 1, 2026 through July 20, 2026 — 11:53:40 EST
Motivated by the celebrated Lonely Runner Conjecture, we study a variant in which the runners have time-dependent velocities. Let $n \geq 3$ runners start from the same point on the unit circle, where each runner $i\in[n]$ has a locally integrable velocity function $ν_i\in L^1_{\mathrm{loc}}(\mathbb{R}_{>0})$Assume that their...
We establish asymptotic formulas for abelian extensions of global function fields ordered by conductor and subject to prescribed local conditions. Our proof combines harmonic analysis with a theory of frobenian functions over global function fields developed in this paper. We interpret our result via equidistribution on algebraic stacks.
We consider the $hp$-FEM applied to the Helmholtz scattering problem with wavenumber $k$, truncated with a perfectly-matched layer. The scatterer consists of a combination of Dirichlet, Neumann, and penetrable obstacles together with variable coefficients. Provided that the Helmholtz solution operator is polynomially bounded in $k$,...
An $n \times n$ matrix $W$ with exactly $w$ non-zero entries taken from the set of $k^{\rm th}$ complex roots of unity in each row and column satisfying $WW^{\ast} = wI_n$ is a complex generalised weighing matrix $CGW(n,w;k)$. We study such matrices through the centraliser algebras of monomial representations of finite groups. Using...
Let $d \geq 3$, \[ D \subsetneq \{ 0,1,\dots,d-1\}, \qquad |D| \geq 2, \ 0 \in D \] be a finite alphabet, and define the integer Cantor set \begin{align} \mathcal{C} := \mathcal{C}_{D} := \bigcup_{J \geq 0} \Big\{ \sum_{j =0}^J a_j d^j : a_j \in D \Big\}. \end{align} We prove that for any $σ$-finite measure-preserving system,...
Green's popular difference theorem says that for every \(\varepsilon>0\), all sufficiently large primes \(p\), and every set \(A\subseteq\mathbb F_p\) of density \(α\), there exists a nonzero \(d\in\mathbb F_p\) such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x)1_A(x+d)1_A(x+2d) \geq α^3-\varepsilon. \] We show that a stronger...
Let $G = (V,E)$ be a graph of order $n$. A bijection $f : V \rightarrow \{1,2,\cdots,n\}$ is a distance magic labeling of $G$ if there exists a positive integer $k$ such that $\sum_{u \in N(v)}f(u) = k$ for all $v \in V$, where $N(v)$ is the neighborhood of $v$. Any graph which admits a distance magic labeling is called a distance...
In this paper, we study the primitive actions of almost simple groups with socle \(E_7(q)\) or \(E_8(q)\) on an \(s\)-arc-transitive digraph. Our motivation goes back to the question of whether \(s\) is bounded above for finite connected \(G\)-vertex-primitive \(s\)-arc-transitive digraphs that are not directed cycles. The question...
We study the problem of optimal control for dynamical systems near a pitchfork bifurcation, motivated by the role of external cues in guiding symmetry-breaking transitions in cell-fate selection and other natural processes. Using an asymptotic expansion of the optimality conditions obtained from the Pontryagin maximum principle, the...
We study $(2+1)$-dimensional oscillatory integral operators of the form \[ T_λf(x,y)=\int_{\mathbb{R}}e^{iλP(x,y)t^k}ψ(x,y,t)f(t)dt,\qquad k\geq 1, \] where the phase $P$ is a real-analytic function with a critical point at the origin. We establish the sharp $L^2\to L^2$ decay rate of $\frac12\min\{1/h_{P}, 1/k\}$, where $h_{P}$...
We disprove the conjecture that every globally asymptotically stable homogeneous polynomial vector field admits a homogeneous polynomial Lyapunov function. The counterexample is a planar homogeneous cubic polynomial vector field with integer coefficients. It admits no positive definite homogeneous polynomial with nonpositive Lie...
In this paper, we establish an analogue of the Kővari-Sós-Turán Theorem for $\operatorname{GF}(q)$-representable matroids. For $2\leq s\leq t$, we show that if $M$ is a rank-$n$ simple $\operatorname{GF}(q)$-representable matroid having no $M(K_{s,t})$-restriction, then \[ |E(M)|=O_{q,s,t}\bigl(q^{(1-1/s)n}\bigr). \] In particular,...
Inversion sets of permutations in the affine symmetric group $\widetilde{S}_n$ were studied extensively by Björner and Brenti. One of their methods for encoding an inversion set is through an affine inversion graph, which is a certain weighted graph on vertex set $[n]=\{1,2,\ldots,n\}$. Subsequent work by Papi characterized which...
We continue the study of $D(z)$-quadruples in the ring $\mathbb{Z}[\sqrt{-2}]$. Motivated by the earlier classification due to the authors and by the subsequent partial results for the remaining families, we consider the exceptional congruence classes arising in the forms $24a+5+(12b+6)\sqrt{-2}$, $24a+2+(12b+6)\sqrt{-2}$, and...
Reconstructing the location and shape of an unknown impenetrable scatterer from far-field measurements is a fundamental inverse problem in elastic scattering. In this paper, we propose monotonicity-based shape characterization theorems and develop corresponding algorithms for rigid and traction-free impenetrable scatterers. By...
We prove, for all fixed $0 < δ< 1$, and all sufficiently large $n$, that there exists $S \subset [n]$ with $|S| \ge δn$ such that $A + B \not \subset S$ for all ${A, B \subset \mathbb{N}}$ satisfying $$\min\big\{|A|, |B|\big\} \ge \big(3 + o(1)\big) \frac{\log n }{ \log (1 / δ)}.$$ A very recent result of Hernández and Hetzel shows...
A set of points $C \subset \mathbb{R}^n$ is called canonically Ramsey if there is some set of points $S\subset \mathbb{R}^{n'}$ such that any colouring of $S$, using any number of colours, must contain either a monochromatic copy of $C$ or a rainbow copy of $C$. Mao, Ozeki, and Wang introduced this notion, showing that 30-60-90...
Computer-assisted proofs of self-similar singularity formation for fluid equations often rely on numerically constructed approximate profiles. One effective approach to establishing stability of perturbations around a numerically constructed profile is to perform weighted energy estimates with singular weights near the singularity....
We address fundamental challenges in representing and computing $\mathbb{R}^{d}$-valued predictable square-integrable processes over $[0,T]$, collected in the space $\mathcal{H}^2_T(\mathbb{R}^{d})$. These processes are central to continuous-time stochastic control, reinforcement learning, and mathematical finance. Although...
In this article, we extend the theorem of heart\cite{Barwick_2015}, which implies Quillen's dévissage theorem by \cite{Efimov2025}, to generic small stable $\infty$-categories. To be precise, we establish a necessary and sufficient condition under when an exact functor between stable $\infty$-categories induces isomorphisms of...
Suppose that $G$ is a finite solvable group and $V$ is a finite, faithful and completely reducible $G$-module. Let $N$ be a nilpotent subgroup of $G$, then there exits $v \in V$ such that $|\bC_N(v)| \leq (|N|/p)^{1/p}$, where $p$ is the smallest prime divisor of $|N|$.
Motivated by Applied Physics and Photonics studies of random resonators, we study in the stochastic part of this paper random acoustic operators in non-smooth bounded domains $G \subset \mathbb{R}^d$ and introduce m-dissipative impedance boundary conditions containing (eigenfunction) fractional Gaussian fields. The deterministic part...
In this paper we construct nontrivial weak solutions to a class of stationary active scalar equations with a non-odd nonlocal operator in the drift term using a convex integration scheme. We show our solutions lie in $$ \bigcap_{0 < ε< 1} \dot{B}^{-ε}_{\infty,\infty}(\mathbb{T}^d) \cap L^{2-ε}(\mathbb{T}^d) $$ for $d \geq 2$. The key...
We study a generalized defocusing Hartree equation with nonlocal exchange potential and repulsive Hartree--Fock interaction. Using an inverse optimal problem (IOP) approach, we prove the existence and uniqueness of ground state solutions. Additionally, we establish the existence of principal solutions, their continuous dependence on...
We study discrete random fields $\{X_t: t\in \mathbb{Z}^d\}$ parameterized on the $d$-dimensional integer lattice $\mathbb{Z}^d$. For a fixed threshold $u$, the excursion set $\{t \in \mathbb{Z}^d : X_t > u\}$ decomposes into connected components or clusters, whose size, defined as the number of lattice points they contain, are...