Probabilistic properties of systems of random linear equations over finite algebraic structures
type: Item
instance of: scholarly article
title: Probabilistic properties of systems of random linear equations over finite algebraic structures
zbMATH Open document ID: 0855.60059
DOI: 10.1007/BF01125544
author: A. A. Levitskaya
author: I. M. Kovalenko
published in: Cybernetics and Systems Analysis
publication date: 1995-08-06T00:00:00Z
review text: This is a simple review of the former Soviet Union's probabilists' works on the limiting properties of random matrices and systems of random linear equations over finite algebraic structures. The first research works in this field were initiated by D. Slepian, P. Erdös and A. Rényi in 1955 and 1963, then followed by G. V. Balakin, V. F. Kolchin and the authors. Let \({\mathbf A} = (a_{ij})_{1 \leq i \leq N,1 \leq j \leq n}\), \(N \leq n\), be a random matrix, where \(a_{ij}\) are jointly independent random variables taking values in a finite field or ring. Then the asymptotic distributions of the linear rank of \({\mathbf A}\) and the number of solutions of (homogeneous or non-homogeneous) systems of linear equations with random coefficient matrix \({\mathbf A}\) are considered. The invariance theory has been developed. It means under the restrictions that the probabilities of \(a_{ij}\) are bounded away from zero in some manner the limit properties are invariant to a distribution of the matrix \({\mathbf A}\). Many results have been collected in ``Selected problems of probabilistic combinatorics'' (Kiev, 1986) by the authors and \textit{M. N. Sovchuk}.
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 11T99
Mathematics Subject Classification ID: 60C05
zbMATH DE Number: 780307
zbMATH Keywords: random matrix
zbMATH Keywords: system of random linear equations
zbMATH Keywords: limiting distribution
MaRDI profile type: Publication
cites work: Q5534677
cites work: Q5576597
cites work: Q4365540
cites work: Q5600562
cites work: Q3896298
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full work available at URL: https://doi.org/10.1007/bf01125544
OpenAlex ID: W2054931452
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description: scientific article; zbMATH DE number 780307