A threshold effect for systems of random equations of a special form

Kolchin, V. F.; Khokhlov, V. I.

doi:10.1515/dma.1995.5.5.425

Cited by 10 publications

(21 citation statements)

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“…The studies initiated in [15] were continued in [49][50][51][52]. Although the latter analyze SREIS systems over GF(q) whose matrices A Nn of coefficients in each row have no more than r non-zero elements, the theorems proved here (similar to Theorem 1) yield the following result pertaining also to CCSRE systems over GF(q): relation (8) is true in the field GF(q).…”

Section: Definition 15 Ms a Nnmentioning

confidence: 93%

“…Kolchin [45] introduced the concepts of a critical set and the independence of critical sets of a matrix over GF(2). Later, he generalized them in [49,50] to a matrix over an arbitrary field GF(q). These concepts appeared rather efficient in the analysis of characteristics of random systems in general.…”

Section: Certainly Compatible Systems Of Random Equations (Ccsre)mentioning

confidence: 99%

“…The studies into the threshold effect for the mathematical expectation of the total number of critical sets MS A Nn ( )) , started in [15] for random Boolean matrices A Nn with no more than r non-zero elements in each row, were continued in [49,51,52] for the matrices A Nn , of similar structure, over the fields GF(p) (2 < p is a prime number) and GF(q). The paper [52] contains the most general result obtained in this area.…”

Section: The Systemmentioning

confidence: 99%

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Systems of Random Equations over Finite Algebraic Structures

Levitskaya

2005

Cybern Syst Anal

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519.21The results on systems of random equations over finite algebraic structures are reviewed. Basic definitions, concepts, and problems in this field are presented.Keywords: system of random equations over finite algebraic structure (finite field, finite ring, finite Abelian group), Boolean system of equations, certainly compatible system of random equations, system of random equations with independent left-and right-hand sides, system of random equations with distorted right-hand sides.This review of the results from the studies on systems of random equations over finite algebraic structures is timed to the 70th anniversary of Igor N. Kovalenko, a prominent Ukrainian mathematician, Doctor of engineering, physics, and mathematics, Academician of the National Academy of Sciences of Ukraine. He may rightfully be considered a founder of this complex and fascinating division in discrete mathematics. His fundamental and mathematically beautiful study [33] (1967) on the invariance of the limit behavior of the number of solutions of one class of systems of random equations over a GF(2) field became a basis for disciples of I. N. Kovalenko's school in extending invariance theory to random systems over arbitrary finite fields and finite rings. Over nearly 40 years since the paper [33] was published, the theory of systems of random equations over finite algebraic structures has been enriched with new problems, results, and methods. To efficiently develop this theory, we today need to sort out and systematize everything accumulated in this area. The author of the present review, a follower of I. N. Kovalenko, has tried to do this with gratitude and deep respect for the teacher.The studies by V. L. Goncharov on random substitutions and series in random (0,1)-sequences, published in the 40s of the last century, have initiated a systematic analysis of combinatory objects using methods of probability theory. Thus, they promoted the development of a new division in discrete mathematics, probabilistic combinatorial analysis, where the theory of systems of random equations over finite algebraic structures is a relatively small subdivision, despite its important applications in the theory of finite state automata, the theory of coding, the theory of random graphs, cryptography, etc. there is an explanation of this. The use of random systems as probability-theoretic models of combinatorial processes occurring in information-protection systems has vetoed publications on this subject in the available scientific literature for many years. The situation began to change for the better only in the early 90s of the last century. Indeed, a large number of papers on random systems over finite algebraic structures have appeared in the last 10 to 14 years in large mathematical journals. It is pleasant to point out that whereas this subjects was considered especially "Russian" when the review [40,41] devoted to random linear systems over finite algebraic structures was written (since the corresponding publications were mainly due to Soviet ma...

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Section: Definition 15 Ms a Nnmentioning

confidence: 93%

Section: Certainly Compatible Systems Of Random Equations (Ccsre)mentioning

confidence: 99%

Section: The Systemmentioning

confidence: 99%

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Systems of Random Equations over Finite Algebraic Structures

Levitskaya

2005

Cybern Syst Anal

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“…Let Ν, Γ -4 oo and T/N -> a. Then for any fixed integers r > 3 and q > 3 there exists a constant oc r such that ES(A r ) -> 0 if a < oc r , and ES(A r ) -> <*> if a > cc r .The research was supported by the Russian Foundation for Basic Research, grants 96-01-00338 and 96-15-96092.We recall the notion of critical sets which was introduced in [3] (see also [1,2]) for the field GF(2) and extended to finite fields in [5].We consider a Γ χ TV matrix A = \\a t j\\ in the finite field GF(q) with q elements and denote its rows by a t = (a t \,...,a tN ], t = Ι,.,.,Γ.The rows with numbers t\,...,t m and weights 81 . .…”

mentioning

confidence: 99%

“…We recall the notion of critical sets which was introduced in [3] (see also [1,2]) for the field GF(2) and extended to finite fields in [5].…”

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A threshold effect for systems of random equations in finite fields

Kolchin¹

1999

Discrete Mathematics and Applications

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We consider the system of equations in GF( f = 1, · · · > 7\ are independent identically distributed random variables taking the values 1,...,N with equal probabilities, the coefficients <2j ,... ,α£ , t = 1,... ,Γ, are independent identically distributed random variables independent of /i(r),... ,//·(/), ί = 1,... ,Γ, and taking the non-zero values from GF( a. Then for any fixed integers r > 3 and q > 3 there exists a constant oc r such that ES(A r ) -> 0 if a < oc r , and ES(A r ) -> <*> if a > cc r .The research was supported by the Russian Foundation for Basic Research, grants 96-01-00338 and 96-15-96092.We recall the notion of critical sets which was introduced in [3] (see also [1,2]) for the field GF(2) and extended to finite fields in [5].We consider a Γ χ TV matrix A = \\a t j\\ in the finite field GF(q) with q elements and denote its rows by a t = (a t \,...,a tN ], t = Ι,.,.,Γ.The rows with numbers t\,...,t m and weights 81 . . . , e m taking non-zero values from GF(qr) form a critical set Β -{t\ , . . . , t m \ ε\ , . . . , e m } if the sum is the zero vector.For critical sets we define the multiplication by a non-zero α G GF(q) setting for5 = {ii,.

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