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 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 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
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
1d ago
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 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.
arXiv Machine Learning
Jun 30
Chamber geometry and specification numbers of Boolean threshold functions
arXiv:2606. 29477v1 Announce Type: cross Abstract: The specification number $\sigma_n(f)$ of a Boolean threshold function $f$ on $n$ variables is the least number of points whose $f$-values determine $f$ uniquely among all threshold functions.
arXiv Machine Learning
Aug 4
Polyatomic Complexes: A topologically-informed learning representation for atomistic systems
arXiv:2409. 15600v3 Announce Type: replace Abstract: A representation of a molecule or material should be invariant to the symmetries of physics, unique, continuous, efficient and general.
arXiv AI
5d ago
Algebraic Decomposition Theory for Transformer Length Generalization
arXiv:2608. 13433v1 Announce Type: cross Abstract: Transformer-based language models are known to sometimes generalize to sequences longer than seen during training, but we lack a precise characterization of which tasks admit length generalization.
arXiv Machine Learning
Jul 27
Shallower ReLU Network Representations via Exact Linear Algebra
arXiv:2607. 21651v1 Announce Type: new Abstract: We prove that the maximum of $n$ real numbers is exactly representable by a ReLU network with two hidden layers for every $n\le 10$.
arXiv AI
Jun 12
Real-rootedness of the Poincar\'e polynomials of $\overline{\mathcal M}_{0,n}$: an AI-assisted proof
arXiv:2605. 29151v2 Announce Type: replace-cross Abstract: We prove real-rootedness for the Poincar\'e polynomial \[ P_n(t)=\sum_{i=0}^{n-3} \dim H^{2i}(\overline{\mathcal M}_{0,n};\mathbb{Q})t^i \] of the Deligne--Mumford moduli space $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational curves, proving a conjecture of Aluffi--Chen--Marcolli.