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Asymptotics of the Number of Threshold Functions and the Singularity Probability of Random {±1}-Matrices

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Abstract

Two results concerning the number \(P(2,n)\) of threshold functions and the singularity probability \({{\mathbb{P}}_{n}}\) of random (\(n \times n\)) \({\text{\{ }} \pm 1{\text{\} }}\)-matrices are established. The following asymptotics are obtained:

\(P(2,n)\sim 2\left( {\begin{array}{*{20}{c}} {{{2}^{n}} - 1} \\ n \end{array}} \right)\quad {\text{and}}\quad {{P}_{n}}\sim {{n}^{2}} \times {{2}^{{1 - n}}}\quad n \to \infty .\)

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Funding

This work was supported in part by the Russian Foundation for Basic Research, grant no. 18-01-00398 A.

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Correspondence to A. A. Irmatov.

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Translated by I. Ruzanova

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Irmatov, A.A. Asymptotics of the Number of Threshold Functions and the Singularity Probability of Random {±1}-Matrices. Dokl. Math. 101, 247–249 (2020). https://doi.org/10.1134/S1064562420030096

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