arXiv preprints from January 1, 2026 through September 22, 2026 — 12:09:05 EST
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.
Complex 3D spatial text to image generation requires models to convert natural language into stable visual geometry, not merely semantic appearance. Existing prompt-driven or layout-conditioned methods improve controllability, but often lack an optimizable and verifiable spatial intermediary before visual sampling. As a result, object...
We study weighted fair division of indivisible mixed manna under additive valuations. First, we resolve the general existence open question for weighted envy-freeness up to one item (WEF1), and show that every instance with arbitrary positive entitlements admits a complete WEF1 allocation computable in polynomial time. We then show...
We present SG-AMP, integrating robust depth completion with input-conditioned uncertainty, persistent panoptic mapping, plant scene-graph reasoning, and semantics-aware active view-motion planning. Beyond inspecting uncertain observed regions, the scene graph explicitly hypothesizes unobserved pepper--peduncle attachments and directs...
Extracting structured fields from hundreds of millions of documents annually remains costly in regulated industries: bespoke OCR cascades cover only a fraction of workflows, privacy rules preclude external models, and existing open-source VLMs that clear quality thresholds cost more to serve than human annotation. We present a...
How to divide a fixed annotation budget between supervised fine-tuning (SFT) and reinforcement learning (RL) during LLM post-training remains an open problem. Existing work characterizes only broad trends (e.g., SFT dominates in low-data regimes), lacks a principled allocation framework, and does not examine whether the optimal ratio...
Data-residency constraints force enterprises to self-host LLMs, but continuous adoption of newer models without decommissioning their predecessors expands the serving fleet, fragmenting a finite GPU pool. We consolidate traffic from over 200 internal applications onto a single model by closing quality gaps identified through...
Introduction: Clinical research informatics (CRI) platforms support biomedical discovery by integrating advanced computational tools into research workflows. Emerging technologies such as spatial omics and AI-enabled imaging expand research capabilities but introduce complex interfaces that increase cognitive burden and alter...
Vision-Language Models (VLMs) provide useful priors for interactive decision-making, but using them directly as policies is expensive and brittle: they must be queried at every step, do not improve from environment interaction, and can repeat systematic errors. We study how to learn a cheap autonomous policy from an online, expensive,...
Large language models (LLMs) struggle to classify text into taxonomies with many semantically similar labels, as the distinctions are domain-specific and not captured by pre-training. To handle large label spaces, a common approach retrieves top-$K$ candidate labels by embedding similarity and prompt the LLM to choose among them....
Semiconductor supply chains face escalating risks from geopolitical tensions, geographic concentration, and rapid technological shifts, yet no scalable system continuously extracts, structures, and prioritizes risk intelligence from public corporate disclosures. We present an end-to-end pipeline that retrieves corporate documents for...
We present H3-World, an efficient framework that turns the 33B MiniMax-H3 video generator into an interactive world model. Our key finding is that, as large video generators become more capable, language is emerging as a natural interface for control. MiniMax-H3, for example, already supports zero-shot control of character behavior...