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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

 

On the rate of convergence to the normal distribution of the number of false solutions of a system of nonlinear random Boolean equations


Authors: V. I. Masol and S. Ya. Slobodyan
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 76 (2007).
Journal: Theor. Probability and Math. Statist. 76 (2008), 117-129
MSC (2000): Primary 60C05, 15A52, 15A03
Published electronically: July 16, 2008
MathSciNet review: 2368744
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove a theorem on the limit normal distribution (as $ n \to \infty$) of the number of false solutions of a system of nonlinear equations with independent random coefficients belonging to the field GF(2). We assume that every equation contains at least one coefficient for which the probability that it attains the value 1 is close to  $ \frac{1}{2}$; the number of equations $ N$ and the number of unknowns $ n$ are such that $ n-N \to \infty$ as $ n \to \infty$; the system has a solution containing $ \rho (n)$ units and $ \rho (n) \to \infty $ as $ n \to \infty $.


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Additional Information

V. I. Masol
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: vimasol@ukr.net

S. Ya. Slobodyan
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: sv_yaros@rambler.ru

DOI: http://dx.doi.org/10.1090/S0094-9000-08-00736-9
PII: S 0094-9000(08)00736-9
Keywords: Nonlinear random Boolean equations, the limit normal distribution, the number of false solutions
Received by editor(s): March 22, 2006
Published electronically: July 16, 2008
Article copyright: © Copyright 2008 American Mathematical Society