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:
\(P(2,n)\sim 2\left( {\begin{array}{*{20}{c}} {{{2}^{n}} - 1} \\ n \end{array}} \right)\quad {\text{and}}\quad {{P}_{n}}\sim {{n}^{2}} \times {{2}^{{1 - n}}}\quad n \to \infty .\)
Similar content being viewed by others
REFERENCES
L. Schläfli, Gesammelte Mathematische Abhandlungen I (Birkhäuser, Basel, 1950), pp. 209–212.
T. M. Cover, IEEE Trans. Electron. Comput. 14 (3), 326–334 (1965). https://doi.org/10.1109/PGEC.1965.264137
S. Muroga, IEEE Trans. Electron. Comput. 14 (2), 136–148 (1965). https://doi.org/10.1109/PGEC.1965.263958
S. Yajima and T. Ibaraki, IEEE Trans. Electron. Comput. 14 (6), 926–929 (1965). https://doi.org/10.1109/PGEC.1965.264090
Yu. A. Zuev, Sov. Math. Dokl. 39, 512–513 (1989). Zbl 0693.94010
A. M. Odlyzko, J. Combin. Theory Ser. A 47, 124–133 (1988). https://doi.org/10.1016/0097-3165(88)90046-5
T. Zaslavsky, Facing up to Arrangements: Face-Count Formulas for Partitions of Space by Hyperplanes (Am. Math. Soc., Providence, R.I., 1975).
A. A. Irmatov, Discrete Math. Appl. 3 (4), 429–432 (1993). https://doi.org/10.1515/dma-1993-0407
J. Komlós, Stud. Sci. Math. Hungar. 2, 7–21 (1967). MR0221962
J. Komlós, Manuscript (1977), in Random Graphs, Ed. by B. Bollobás (Academic, New York, 1985), pp. 347–350.
J. Kahn, J. Komlós, and E. Szemerédi, J. Am. Math. Soc. 8 (1), 223–240 (1995). https://doi.org/10.1090/S0894-0347-1995-1260107-2
T. Tao and V. Vu, J. Am. Math. Soc. 20 (3), 603–628 (2007). https://doi.org/10.1090/S0894-0347-07-00555-3
J. Bourgain, V. H. Vu, and P. M. Wood, J. Funct. Anal. 258, 559–663 (2010). https://doi.org/10.1016/j.jfa.2009.04.016
K. Tikhomirov, Ann. Math. 191 (2), 593–634 (2020). https://doi.org/10.4007/annals.2020.191.2
A. A. Irmatov, Acta Appl. Math. 68, 211–226 (2001). https://doi.org/10.1023/A:1012087813557
Funding
This work was supported in part by the Russian Foundation for Basic Research, grant no. 18-01-00398 A.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562420030096