arXiv preprints from January 1, 2026 through September 9, 2026 — 15:52:05 EST
Let $E/\mathbb Q$ be an elliptic curve and for each prime $p$, let $N_p$ denote the number of points of $E$ modulo $p$. The original version of the conjecture of Birch and Swinnerton-Dyer asserts that $\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x) ^{\text{rank}(E(\mathbb Q))}$ as $x \to \infty$. In this paper, we formulate a...
Okuyama--Saito ask whether, for an ordered negative support family at a fake exponent of an $A$-hypergeometric system, intrinsic perturbation inside the relation lattice can produce a strictly smaller coefficient space than ambient perturbation. It can, and we measure the gap by an intrinsic-perturbation obstruction module, a quotient...
Toric varieties can be constructed from rational polytopes, and several invariants of toric varieties can be expressed in terms of the combinatorics of the corresponding polytope. Barthel-Brasselet-Fieseler-Kaup (BBFK) introduced combinatorial intersection cohomology for convex polytopes, which agrees with the intersection cohomology...
For a smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright \overline{\mathbb{B}}^{N}(1)$, we observe that the sharp systolic inequality forces a sharp lower bound on its maximal normal curvature $κ(F)$. In dimensions $m=2,3$, the sharp inequalities of Pu and Bray--Brendle--Eichmair--Neves therefore give $κ(F)^2\ge...
Transport-dominated problems remain challenging for data-driven methods, which often exhibit severe numerical oscillations near shocks or steep gradients due to globally supported basis functions or overly smooth hypothesis spaces. Gegenbauer reconstruction has shown promise in mitigating such oscillations, but its effectiveness...
We study pseudoreflections of geometric origin on Prym varieties of étale double covers. We prove that if the genus of the base curve is $g \geq 4$ then every such pseudoreflection has order 2. We use this result to show that, for $g \geq 5$, a non-trivial finite group $G$ of automorphisms of geometric origin acting faithfully on the...
Lehel's conjecture states that every 2-edge-colouring of K_n admits a partition of its vertex set into two monochromatic cycles. It was proven for sufficiently large n by Łuczak, Rödl, and Szemerédi in 1998, later improved by Allen in 2008, and fully resolved by Bessy and Thomassé in 2010. In this paper, we consider a rainbow...
We study the action of the Fourier transform on the Stokes data of irregular connections on the complex affine line with symmetric irregular classes at infinity, both from the point of view of Stokes filtered local systems and of Stokes local systems, and we show that it is governed by a rich combinatorial structure: (1) Observing...
In this paper, we introduce a new approach for constructing multivariate fractal interpolation functions and $α$-fractal functions associated with multivariate functions. Unlike the existing methods that rely on the Banach contraction principle, here the construction is based on Matkowski and Rakotch contractions. While numerous...
We introduce Hydra-0, a generalist world model conditioned on action flow, which represents robot actions as pixel motion. This shared visual interface enables generalist world modeling and control by learning action consequences across embodiments, tasks, environments, and video-generation backbones. Our best configuration achieves...
Large-scale image generation has benefited from advances in data scale, quality, rebalancing, and recaptioning, yet conventional pipelines typically optimize task-specific datasets in isolation. A central challenge is not only how to curate each task-specific corpus, but also how to organize heterogeneous supervision according to the...
We prove unique ergodicity of the projective process associated with the two dimensional Navier--Stokes equation in vorticity form, with additive diagonal noise acting on every nonzero real Fourier phase and satisfying two sided power law bounds. Consequently, the exact Furstenberg--Khasminskii formula for the top Lyapunov exponent...
A well-known conjecture of Chern, do Carmo, and Kobayashi asserts that, for $n,m \geq 1$, the scalar curvature of a closed, minimally immersed $n$-submanifold of $\mathbb{S}^{n+m}$ with second fundamental form of constant length takes values in a discrete set. This property holds in every codimension when $n \in \{1,2\}$. We disprove...
Conformal symmetry strongly constrains correlation functions. In momentum space, the conformal Ward identities are solved for three-point functions by integrals of a product of three Bessel functions ("triple-$K$ integrals"); more generally, integrals of this type ("multiple-$K$ integrals"), serve as the building blocks of a wide...
Purpose: To develop and evaluate a locally deployed multi-agent AI system for radiology report structuring and quality assurance. Materials and Methods: This retrospective study included 638 radiology reports from CT examinations of the chest, abdomen, and pelvis dictated by 15 board-certified radiologists in 2023 and 2024. A...
Programmable arrays of ultracold atoms are a leading platform for quantum computation and simulation, enabling state-of-the-art implementations of quantum error correction, and analog simulations of Hubbard models that address open problems in condensed matter physics. In these systems, all local control is mediated through precisely...
In this paper, we provide estimators for quantum Rényi and Tsallis entropies with nearly optimal sample complexity. Specifically, for order $α$, dimension $d$, and additive error $\varepsilon$, 1. For $0 < α< 1$, the sample complexity is $O(d^{1+1/α}/\varepsilon^{1/α} + d^{1/α-1}/\varepsilon^{2})$ for Rényi entropy and...
How level-0 players behave and how they are perceived by higher-level players are central questions in the literature on level-k models of boundedly rational strategic reasoning. To study these twin questions, we apply the intra-team communication method developed by Burchardi and Penczynski (2014) to identify level-0 actions and...
We show that the best constant in the weak-type $(1,1)$ bound for the vector Riesz transform on $\mathbb{R}^n$ is at most $2$, independent of the dimension $n$. The proof relies on a new decomposition of the input data involving an obstacle problem for the fractional Laplacian and an associated Lewy-Stampacchia type estimate on an...
Learning algorithms are often used to make decisions in repeated multi-agent environments. When another player understands how a learner adapts from past experience, that player can plan strategically across rounds to influence the learner's future behavior. Recent work shows that optimizing against Replicator Dynamics, the...
Memory-based self-improving agents--those that learn from an online stream of tasks and improve over time by maintaining a textual memory bank--have shown great promise in recent literature. However, the reliability aspects of these methods have been critically overlooked. In this work, we conduct a comprehensive re-evaluation of two...
Neural Quantum States (NQS) provide a powerful neural network-based variational framework for representing many-body wave functions and solving for ground states. Recurrent Neural Networks (RNNs) are particularly promising owing to their relatively low computational cost and their autoregressive property, which enables perfect...
We resolve the admissibility problem for scattering transforms: every Parseval filter bank on $\mathbb{R}^d$ whose low-pass filter is nonvanishing at the origin induces a norm-preserving scattering transform, without requiring any analyticity, frequency-localization, or geometric covering assumptions. Furthermore, for any Sobolev...
High-resolution image editing is increasingly demanded in professional workflows, yet existing diffusion-based models remain constrained to resolutions below 1K due to quadratic attention complexity and prohibitive memory requirements. A prevalent workaround employs a two-stage pipeline: editing at low resolution followed by...
Language model tokenizers are typically selected with minimal evaluation, despite the fact that their design choices directly impact model capabilities. This can be partly attributed to a limited understanding of which tokenizer properties affect which aspects of downstream performance. We introduce TokEval, a framework of tokenizer...