Estimates for the probability that a system of random equations is solvable in a given set of vectors over the field $\text {\bf {GF}}(3)$
Authors:
V. I. Masol and L. O. Romashova
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 87 (2013), 135-152
DOI:
https://doi.org/10.1090/S0094-9000-2014-00909-6
Published electronically:
March 21, 2014
MathSciNet review:
3241451
Full-text PDF Free Access
Abstract |
References |
Additional Information
Abstract: Let $P_n$ be the probability that a second order system of nonlinear random equations over the field $\mathbf{GF}(3)$ has a solution in a given set of vectors, where $n$ is the number of unknowns in the system. A necessary and sufficient condition is found for $P_n\to 0$ as $n\to \infty$. Some rates of convergence to zero are found and some applications are described.
References
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- V. I. Masol and L. A. Romaschova, Uniqueness conditions for the solution of an inhomogeneous system of nonlinear random equations over the field $GF(3)$, Kibernet. Sistem. Anal. 46 (2010), no. 2, 23–36 (Russian, with Russian summary); English transl., Cybernet. Systems Anal. 46 (2010), no. 2, 185–199. MR 2921497, DOI https://doi.org/10.1007/s10559-010-9197-y
- K. A. Rybnikov, Vvedenie v kombinatornyĭ analiz, 2nd ed., Moskov. Gos. Univ., Moscow, 1985 (Russian). MR 812275
- William Feller, An introduction to probability theory and its applications. Vol. I, 3rd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0228020
- Albert N. Shiryaev, Problems in probability, Problem Books in Mathematics, Springer, New York, 2012. Translated by Andrew Lyasoff. MR 2961901
References
- V. A. Kopyttsev and V. G. Mikhailov, Poisson-type theorems for the number of special solutions of a random linear inclusion, Diskret. Matem. 22 (2010), no. 2, 3–21; English transl. in Discrete Math. Appl. 22 (2010), no. 2, 191–211. MR 2730124 (2011m:60216)
- V. I. Masol and L. A. Romashova, Uniqueness conditions for the solution of an inhomogeneous system of nonlinear random equations over the field $GF(3)$, Kibernet. Sistem. Analiz (2010), no. 2, 23–36; English transl. in Cybernet. Systems Anal. 46 (2010), no. 2, 185–199. MR 2921497
- K. A. Rybnikov, Introduction to Combinatorial Analysis, second edition, Moscow University, Moscow, 1985. (Russian) MR 812275 (87b:05002)
- W. Feller, An Introduction to Probability Theory and its Applications, third edition, vol. 1, John Wiley & Sons, Inc., New York–London–Sydney, 1968. MR 0228020 (37:3604)
- A. N. Shiryaev, Problems in Probability, “MCNMO”, Moscow, 2006; English transl., Springer, New York, 2012. MR 2961901
Additional Information
V. I. Masol
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine
L. O. Romashova
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4E, Kiev 03127, Ukraine
Email:
deezee@ukr.net
Keywords:
System of nonlinear random equations,
probability that a system is solvable,
rate of convergence,
a field containing three elements
Received by editor(s):
July 4, 2011
Published electronically:
March 21, 2014
Article copyright:
© Copyright 2014
American Mathematical Society
