Invariance theorems for a class of systems of random nonlinear equations over an arbitrary finite ring with left unity

Levitskaya, A. A.

doi:10.1007/s10559-008-9066-0

Cited by 4 publications

(15 citation statements)

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“…This article is a natural extension of the investigations presented in [8,9]. Its results generalize the corresponding results of these works.…”

supporting

confidence: 76%

“…In [8,9], homogeneous nonlinear systems of random equations are investigated, i.e., a priori solvable systems whose right sides b i = 0, i n s = -1,…”

mentioning

confidence: 99%

“…It is proved in [9] that, for systems (1) and (2) over an arbitrary finite ring with left unity, where b i = 0, i n s = -1,…”

mentioning

confidence: 99%

“…As well as in [8,9], we begin the probabilistic analysis of systems (1) and (2) with the computation of factorial…”

mentioning

confidence: 99%

“…To compute M ( ) n n r , we use the approach proposed in [9] for the computing the moments of the number of solutions of homogeneous systems of random nonlinear equations. We now rewrite the right side of formula (7) in the form [9], for moduli from K r , the concepts of expansibility and nonexpansibility are introduced and a statement connected with them is proved.…”

mentioning

confidence: 99%

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Solving the problem of invariance of probabilistic characteristics for a priori solvable systems of random nonlinear equations over a finite commutative ring with unity

Levitskaja

2010

Cybern Syst Anal

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A class of a priori solvable systems of random non-linear equations over a finite commutative ring with unity is considered. The questions of the bounds of the invariance domains for the limit factorial moments and, accordingly, the limit distribution of the number of solutions that are different from a fixed solution to a given system and also the geometrical structure of these solutions are investigated. Keywords: system of random nonlinear equations over a finite commutative ring with unity (over a finite field), factorial moment of the number of solutions of a system, distribution of the number of solutions of a system.The invariance problem is a key problem in the modern theory of systems of random equations over finite algebraic structures. Its essence lies in imposing constraints on the distribution of coefficients of a system (when the number of its unknowns tends to infinity) under which probabilistic characteristics of the system (its moments and the distribution of the number of its solutions) remain invariable as, for example, in the case of the uniform distribution of its coefficients. First fundamental results connected with the invariance of the limit behavior of characteristics of random linear homogeneous systems over the field GF( ) 2 are obtained by I. N. Kovalenko in [1]. They may be thought of as foundational for developing a new line of investigation of systems of random equations that has led to the creation of the invariance theory for systems of random equations over finite algebraic structures. Theoretical implications of this theory are considered in more detail in [2].This work is devoted to the solution of invariance problems for probabilistic characteristics of a class of nonlinear random systems. The subjects of investigation are the so-called a priori solvable systems of random nonlinear equations over an arbitrary commutative ring R (with unity) of cardinality | | R = m and of the following form:

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“…This article is a natural extension of the investigations presented in [8,9]. Its results generalize the corresponding results of these works.…”

supporting

confidence: 76%

“…In [8,9], homogeneous nonlinear systems of random equations are investigated, i.e., a priori solvable systems whose right sides b i = 0, i n s = -1,…”

mentioning

confidence: 99%

“…It is proved in [9] that, for systems (1) and (2) over an arbitrary finite ring with left unity, where b i = 0, i n s = -1,…”

mentioning

confidence: 99%

“…As well as in [8,9], we begin the probabilistic analysis of systems (1) and (2) with the computation of factorial…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Solving the problem of invariance of probabilistic characteristics for a priori solvable systems of random nonlinear equations over a finite commutative ring with unity

Levitskaja

2010

Cybern Syst Anal

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The rank of sparse random matrices

Coja-Oghlan

Ergür

Gao

et al. 2022

Random Struct Algorithms

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We determine the asymptotic normalized rank of a random matrix A over an arbitrary field with prescribed numbers of nonzero entries in each row and column. As an application we obtain a formula for the rate of low-density parity check codes. This formula vindicates a conjecture of Lelarge ( 2013). The proofs are based on coupling arguments and a novel random perturbation, applicable to any matrix, that diminishes the number of short linear relations.

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Works of the Kiev school of theoretical cryptography

Savchuk

2010

Cybern Syst Anal

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Invariance theorems for a class of systems of random nonlinear equations over an arbitrary finite ring with left unity

Cited by 4 publications

References 1 publication

Solving the problem of invariance of probabilistic characteristics for a priori solvable systems of random nonlinear equations over a finite commutative ring with unity

Solving the problem of invariance of probabilistic characteristics for a priori solvable systems of random nonlinear equations over a finite commutative ring with unity

The rank of sparse random matrices

Works of the Kiev school of theoretical cryptography

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