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Matrix generators for the general and special linear groups, the symplectic groups and the general and special unitary groups over finite fields. For the most part the generators have been obtained by translating Steinberg's generators for…
The purpose of this paper is two-fold: we present some matrix inequalities of log-majorization type for eigenvalues indexed by a sequence; we then apply our main theorem to generalize and improve the Hua-Marcus' inequalities. Our results…
We are concerned with the problem of designing large families of subsets over a common labeled ground set that have small pairwise intersections and the property that the maximum discrepancy of the label values within each of the sets is…
This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and obtain…
We describe a general approach for computing generators for elimination ideals associated with matrix and hypermatrix spectral decomposition constraints. We derive from these generators iterative procedures for approximating the spectral…
In quasi-Monte Carlo methods, generating high-dimensional low discrepancy sequences by generator matrices is a popular and efficient approach. Historically, constructing or finding such generator matrices has been a hard problem. In…
Kinser developed a hierarchy of inequalities dealing with the dimensions of certain spaces constructed from a given quantity of subspaces. These inequalities can be applied to the rank function of a matroid, a geometric object concerned…
Generalized Matrix Chains (GMCs) are products of matrices where each matrix carries features (e.g., general, symmetric, triangular, positive-definite) and is optionally transposed and/or inverted. GMCs are commonly evaluated via sequences…
The rapid advancement of large language models (LLMs) has led to a surge in both model supply and application demands. To facilitate effective matching between them, reliable, generic and efficient benchmark generators are widely needed.…
In the monotone integer dualization problem, we are given two sets of vectors in an integer box such that no vector in the first set is dominated by a vector in the second. The question is to check if the two sets of vectors cover the…
Benchmarks are the de facto standard for tracking progress in large language models (LLMs), yet static test sets can rapidly saturate, become vulnerable to contamination, and are costly to refresh. Scalable evaluation of open-ended items…
Transformers are a type of neural network that have demonstrated remarkable performance across various domains, particularly in natural language processing tasks. Motivated by this success, research on the theoretical understanding of…
Equiangular Algorithm generates a set of equiangular normalized vectors with given angle {\theta} using a set of linearly independence vectors in a real inner product space, which span the same subspaces. The outcome of EA on column vectors…
We consider the problem of minimizing the makespan on batch processing identical machines, subject to compatibility constraints, where two jobs are compatible if they can be processed simultaneously in a same batch. These constraints are…
Dealing with large numbers of symmetries is often problematic. One solution is to focus on just symmetries that generate the symmetry group. Whilst there are special cases where breaking just the symmetries in a generating set is complete,…
Generative adversarial networks (GAN) are a powerful subclass of generative models. Despite a very rich research activity leading to numerous interesting GAN algorithms, it is still very hard to assess which algorithm(s) perform better than…
Over a field of characteristic 0, the algebra of invariants of several $n\times n$ matrices under simultaneous conjugation by $GL_n$ is generated by traces of products of generic matrices. In this paper we have found, in terms of…
This note describes a technique for generating large non-singular matrices with blocks of full rank. Our motivation to construct such matrices arises in the white-box implementation of cryptographic algorithms with S-boxes.
Column generation is often used to solve multi-commodity flow problems. A program for column generation always includes a module that solves a linear equation. In this paper, we address three major issues in solving linear problem during…
This paper presents a parallel algorithm for finding the smallest eigenvalue of a particular form of ill-conditioned Hankel matrix, which requires the use of extremely high precision arithmetic. Surprisingly, we find that commonly-used…