arXiv AI
Jul 28
An Exact Counterexample to Carlson's Associated-Prime Depth Conjecture from a Group of Order 128
arXiv:2607. 23732v1 Announce Type: cross Abstract: In Question~3.
Extremal Chowla sets and their linear analogues: A human-AI mathematical investigation using Co-Scientist
arXiv:2607. 24847v1 Announce Type: cross Abstract: We introduce an extremal invariant associated with Chowla-type order conditions in finite groups.
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arXiv Machine Learning
Sep 4
What is Smoothness?
arXiv:2609. 03246v1 Announce Type: cross Abstract: Smoothness of a function on the real line is reflected in the decay of its Fourier transform, which suggests that smoothness of a function in $L^2(G)$ for a group $G$ should mean concentration of the Fourier coefficients at low frequency.
arXiv AI
Aug 10
A Finite E-Group of Nilpotency Class Three
arXiv:2608. 07275v1 Announce Type: cross Abstract: A group is an E-group if every element commutes with each of its endomorphic images.
arXiv Machine Learning
Sep 17
Generic Characteristic-Zero Equivalence Between Derivative B\'ezout Inversion and Multipoint Evaluation
arXiv:2609. 17578v1 Announce Type: cross Abstract: Let $a_1,\ldots,a_m$ be distinct elements of a field $K$, and let $Z(X)=\prod_{i=1}^m (X-a_i)$.
arXiv AI
Jul 28
Exact values and exact upper bounds for families of integers with arithmetic progression intersections (Erd\H{o}s Problem #272)
arXiv:2607. 23004v1 Announce Type: cross Abstract: Let $t(N)$ be the largest $t$ for which there exist distinct sets $A_1,\dots,A_t \subseteq \{1,\dots,N\}$ such that $A_i \cap A_j$ is a nonempty arithmetic progression for all $i \neq j$ (Erdos Problem #272).
arXiv Machine Learning
Sep 4
Relative Prime Factorization and Finite-State Presentations under Fixed Finite-Monoid Observation
The paper investigates exact factorization and canonical presentations of languages relative to a fixed finite‑monoid observation. It shows that unique factorization does not guarantee a finite relative presentation property (FRP) by presenting a 36‑element quotient with infinite valid prime‑return rules, and introduces the stronger finite‑state relative presentation property (FSRP). The authors further define prime‑target left‑division determinism (PTLD), prove its implications for factorization and rule bounds, and provide efficient learning algorithms for the canonical PTLD presentation and FSRP controller.
arXiv Machine Learning
Aug 12
Discrete Coefficients and Open Invariant Covers in the Homology of Ample Groupoids
arXiv:2603. 20861v2 Announce Type: replace-cross Abstract: The homology of an ample groupoid is computed from the complex of compactly supported continuous functions on the nerve.
arXiv Machine Learning
Sep 2
A penalised Saito functional for heuristic search of free line arrangements
arXiv:2604. 02995v3 Announce Type: replace-cross Abstract: We introduce the penalised Saito functional $\mathfrak S_{\lambda,\beta}(\mathcal{A};d_1,d_2)$ for a reduced arrangement $\mathcal{A}$ of $n$ lines and a prescribed pair $d_1+d_2=n-1$.
arXiv Machine Learning
Aug 18
A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results
arXiv:2608. 15558v1 Announce Type: cross Abstract: For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p.
arXiv Computation and Language
Sep 15
Query-Oblivious Coresets for Softmax Attention: Improved Bounds and Efficient Constructions
arXiv:2609.06327v2 Announce Type: replace-cross Abstract: A query-oblivious coreset for a softmax-attention head is a subset of the key-value pairs whose attention output is within $\varepsilon$ of t...
arXiv Machine Learning
Jul 14
The VC dimension of partial concept classes via Radon's theorem
arXiv:2607. 10751v1 Announce Type: new Abstract: Following Alon, Hanneke, Holzman, and Moran (FOCS 2021), we define a partial concept class (PCC) as a family of partial functions \(f: V\to\{0,1,\ast\}\); equivalently, its concepts partition the ground set into black ($f^{-1}(1)$), grey ($f^{-1}(\ast)$), and white parts ($f^{-1}(0)$).
arXiv AI
Jul 28
Computational Experiments in Number Theory
arXiv:2504. 19451v4 Announce Type: replace-cross Abstract: This paper presents two concrete applications of Artificial Intelligence to algorithmic and analytic number theory.