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Irmatov, Anwar

doi:10.1023/a:1012087813557

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2023

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Cited by 10 publications

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“…The theorem 7 of [9] is also true for any finite subset H ⊂ RP n . It asserts that the number of collections ( w i 1 , .…”

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confidence: 89%

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Singularity of $\{\pm 1\}$-matrices and asymptotics of the number of threshold functions

Irmatov¹

2020

Preprint

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Two results concerning the number of threshold functions P (2, n) and the probability P n that a random n×n Bernoulli matrix is singular are established. We introduce a supermodular function η ⋆ n : 2 RP n f in → Z ≥0 , defined on finite subsets of RP n , that allows us to obtain a lower bound for P (2, n) in terms of P n+1 . This, together with L. Schläfli's famous upper bound, give us asymptoticsAlso, the validity of the long-standing conjecture concerning P n is proved:

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“…The theorem 7 of [9] is also true for any finite subset H ⊂ RP n . It asserts that the number of collections ( w i 1 , .…”

mentioning

confidence: 89%

“…Taking into account the results of the papers [6], [7], it is possible to show (see [9]) that the absolute value of the Möbius function |µ(0 L H , u)| is equal to the dimension of the reduced homology group of the complex K H∩u with coefficients in an arbitrary field F:…”

mentioning

confidence: 99%