arXiv preprints from January 1, 2026 through September 5, 2026 — 02:28:17 EST
We prove the Lonely Runner Conjecture for fourteen runners by a computer-assisted extension of the finite-checking framework of Sungkawichai and Trakulthongchai. With one runner stationary, their thirteen-runner result supplies the induction input, and their reduction leaves finitely many modular calculations indexed by primes. We...
The Calderón-Zygmund theory establishes the boundedness of singular integral operators on $L^p$ spaces for $1 < p < \infty$, yet it encounters a failure at the endpoint $p = 1$. While radial counterexamples in $\mathbb{R}^n$ are well-documented, Pan-Shao-Wang-Wu \cite{psww2026} has showed that every nonconstant holomorphic function...
Reliable epidemic monitoring often requires inferring regional infection burden and transmission heterogeneity from noisy, spatially aggregated, and potentially sparse surveillance data. Agent-based models (ABMs) are attractive for this task because they represent individual behavior, contact heterogeneity, and localized...
We study optimal sampling recovery in reproducing kernel Hilbert spaces (RKHS) in the uniform norm. For every RKHS with bounded kernel, we establish new comparisons between linear sampling widths and Gelfand widths that overcome the known square-root gap, without requiring a measure or a Christoffel-type condition. Our bounds rely on...
Let $X$ be a smooth complex projective variety of dimension $n$, and let $L$ be an ample Cartier divisor. We prove that $K_X+mL$ is globally generated for every integer $m\geq\lceil C_0n\rceil$, where $C_0=1.77629\ldots$ is an explicit constant. In particular, $K_X+2nL$ is globally generated. Our main input is a new estimate for the...
Fraenkel's conjecture asserts that a partition of the integers into at least three Beatty sequences with distinct moduli has the binary densities $1,2,4,\ldots,2^{m-1}$, normalized by $2^m-1$. We prove the conjecture through a dimension-free intermediate statement: every such partition contains a component of density at least 1/3....
In this note, we prove the existence of smooth complete admissible hypersurfaces in hyperbolic space with constant higher order mean curvature and a prescribed asymptotic boundary at infinity. Unexpectedly, our result holds for a large part of indices in the subcritical range where the concavity inequality loses its effectiveness. We...
For $n\in\mathbb{N}$, we consider the set of lattice paths from $(0,0)$ to $(n,n)$ using only unit north and east steps. Given a point $B$ to be avoided, we ask: at which point $A$ on the grid with corners $(0,0)$ and $(n,n)$, different from the endpoints, does the largest number of $B$-avoiding lattice paths pass through? We show...
We construct a bounded pseudoconvex complete Reinhardt domain $Ω$ with smooth boundary in $\mathbb{C}^3$ such that the $\bar\partial$-Neumann operator $N_1$ is compact although $bΩ$ contains an analytic disc and thus also fails Catlin's Property $(P)$ and McNeal's Property $(\tilde P)$. This example solves an open problem on...
In this paper, we study an Erlang-type loss system with energy-constrained servers. Each server is equipped with a finite battery and becomes temporarily unavailable for service when its energy is depleted, entering a charging phase before returning to operation upon full recharge. Customers who arrive to find all servers unavailable...
Let $H\in C^\infty(\mathbb R^n\times\mathbb T^n)$ be a periodic Tonelli Hamiltonian with critical value $c$. For each $k\in\mathbb N$, let $u_k$ be the normalized minimizer of the variational functional introduced by Evans[7], \[ I_k[w]=\int_{\mathbb T^n} e^{kH(Dw,x)}\,dx, \qquad \int_{\mathbb T^n}w\,dx=0. \] If $u_\infty$ is a...
Let $R$ be a commutative $\Q$-algebra and let $HR$ denote its Hurwitz series ring. We introduce a one-parameter convolution $\hconv{t}$ on $HR$ for which the specialization $t=-1$ is the intrinsic Hurwitz product. Positive integral parameters act on finite-support truncations and reproduce finite free convolution in coefficient...
This paper establishes necessary and sufficient conditions for locally flat knotted surfaces in simply-connected $4$-manifolds to be equivalent. For surfaces with knot group $\mathbb{Z}_d$, we extend results of Lee-Wilczynski from spheres to surfaces of arbitrary genus; the surfaces are permitted to be nonorientable and have boundary....
We prove that a coaction of a Hopf algebra on another preserving the latter's associative-algebra structure and comultiplication automatically also preserves the counit, antipode, and tensor-interchange map and in fact factors through the coaction dual to an action of an affine group scheme by Hopf-algebra automorphisms. This...
We extend the Dong-Song stability theorem for the Riemannian positive mass theorem to higher dimensional spin manifolds. More precisely, for a sequence of complete asymptotically flat spin n-manifolds with nonnegative scalar curvature and ADM masses tending to zero, the exterior regions obtained by excising domains whose boundary...
For every $n\ge3$, we show, by a family of explicit entire real-analytic solutions, that, for every subcritical phase and every $0<α\le1$, oscillation-based interior $C^α$ estimates fail for the special Lagrangian equation in $\mathbb{R}^n$. In fact, this family has no common interior modulus of continuity.
We establish improved multiplicative bounds relating the Rényi entropies of different orders for weighted sums of independent Bernoulli random variables. In particular, we prove a logarithmic bound between the zeroth-order and infinity-order Rényi entropies, which yields a polynomial improvement over the square-root bound of Jain,...
We exhibit three conics with 144 tritangent circles, disproving the conjectured maximum of 136.
We derive a stable limit theorem for stochastic differential equations driven by multiplicative $α$-stable processes. A key ingredient is the $L^1$-exponential contractivity estimate for the SDEs. The limiting process is a non-degenerate symmetric $α$-stable process with an averaged Lévy measure.
Additive functionals $\overline{J}_t=\frac{1}{t}\int_0^tU(X_s)\circ dX_s$ of Stratonovich-type recently attracted much attention in the context of inference of thermodynamic properties of complex systems from observations $U$ of individual fluctuating paths $(X_s)_{0\le s\le t}$, whereby $X_0$ is initiated from some general measure....
Generalized semi-infinite programs (GSIPs) arise in robust optimal control whenever the admissible uncertainty depends on the state or controls. Existing GSIP methods either impose restrictive structural assumptions or require global optimization that scales poorly to control problems. We present a general framework that reformulates...
Classical chaining controls an indexed stochastic process through a single worst-case bound, which can obscure substantial variation across the index set. We establish the first simultaneous pointwise majorization theory for Banach-valued processes with sub-Weibull or two-metric mixed-tail increments. For an anchored sub-Weibull...
The present research is devoted to the nonparametric estimation of a density-dependent drift coefficient in a multivariate McKean--Vlasov diffusion from independent observations at a common time, as well as the stationary density. Under certain assumptions on the (known) potential, we reduce the problem to the one-dimensional one and...
Hilbert's Tenth Problem over the integers was solved negatively by Y. Matiyasevich in 1970. In this paper, we prove that there is no algorithm to determine for any polynomial equation $P(z_1,\ldots,z_{10})=0$ (with integer coefficients and ten unknowns) whether it has integer solutions. This improves the previous 11 unknowns theorem.
We rigorously justify the quasineutral limit for the warm-ion Euler--Poisson system with Maxwell--Boltzmann electrons on the cylinder $\R\times\T$. The reference dynamics are governed by a globally smooth expanding planar simple wave of the effective Euler system, connecting distinct neutral far-field states. For every finite order...