Related papers: Constructive Membership Tests in Some Infinite Mat…
We show that if one selects uniformly independently and identically distributed matrices $A_1, \ldots, A_s \in \mathrm{SL}_2(\mathbb{Z})$ from a ball of large radius $X$ then with probability at least $1 - X^{-1 + o(1)}$ the matrices $A_1,…
Given two conjugate mapping classes f and g, we produce a conjugating element w such that |w| < K(|f|+|g|), where |.| denotes the word metric with respect to a fixed generating set, and K is a constant depending only on the generating set.…
A method is given that "inverts" a logic grammar and displays it from the point of view of the logical form, rather than from that of the word string. LR-compiling techniques are used to allow a recursive-descent generation algorithm to…
We consider a programming language based on the lamplighter group that uses only composition and iteration as control structures. We derive generating functions and counting formulas for this language and special subsets of it, establishing…
The cohomology of the degree-$n$ general linear group over a finite field of characteristic $p$, with coefficients also in characteristic $p$, remains poorly understood. For example, the lowest degree previously known to contain nontrivial…
In this paper we study possibilities of using hierarchical reasoning, symbol elimination and model generation for the verification of parametric systems, where the parameters can be constants or functions. Our goal is to automatically…
Every finitely generated self-similar group naturally produces an infinite sequence of finite $d$-regular graphs $\Gamma_n$. We construct self-similar groups, whose graphs $\Gamma_n$ can be represented as an iterated zig-zag product and…
We explore a new way to evaluate generative models using insights from evaluation of competitive games between human players. We show experimentally that tournaments between generators and discriminators provide an effective way to evaluate…
A natural definition of the product of infinite matrices mimics the usual formulation of multiplication of finite matrices with the caveat (in the absence of any sense of convergence) that the intersection of the support of each row of the…
We propose an algorithm which for any recursive group $G$, given by its effectively enumerable generators and recursively enumerable relations, outputs an explicit embedding of $G$ into a finitely presented group directly written by its…
With high probability, among $O(\log n)$ independent randomly selected elements from a finite $n$-dimensional classical group, some pair of elements power to a $2$-element generating set for a naturally embedded classical subgroup of…
In a software product line (SPL), a collection of software products is defined by their commonalities in terms of features rather than explicitly specifying all products one-by-one. Several verification techniques were adapted to establish…
Semantic feature norms, lists of features that concepts do and do not possess, have played a central role in characterizing human conceptual knowledge, but require extensive human labor. Large language models (LLMs) offer a novel avenue for…
Mathematically constructed S-boxes arise from algebraic structures and finite field theory to ensure strong, provable cryptographic properties. These mathematically grounded constructions allow for generation of thousands of S-Boxes with…
Conditional set generation learns a mapping from an input sequence of tokens to a set. Several NLP tasks, such as entity typing and dialogue emotion tagging, are instances of set generation. Seq2Seq models, a popular choice for set…
We investigate whether large language models (LLMs) can generate effective, user-facing explanations from a mathematically interpretable recommendation model. The model is based on constrained matrix factorization, where user types are…
The aim of the article is to show that there are many finite extensions of arithmetic groups which are not residually finite. Suppose $G$ is a simple algebraic group over the rational numbers satisfying both strong approximation, and the…
We consider an arbitrary representation of the additive group over a field of characteristic zero and give an explicit description of a finite separating set in the corresponding ring of invariants.
We count the finitely generated subgroups of the modular group $\textsf{PSL}(2,\mathbb{Z})$. More precisely: each such subgroup $H$ can be represented by its Stallings graph $\Gamma(H)$, we consider the number of vertices of $\Gamma(H)$ to…
We present explicit algorithms for computing structured matrix-vector products that are optimal in the sense of Strassen, i.e., using a provably minimum number of multiplications. These structures include Toeplitz/Hankel/circulant,…