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

中文翻译：

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阈值函数数的渐近性和随机{±1}-矩阵的奇异性概率

摘要 关于阈值函数的个数$$P(2,n)$$和随机的奇异概率$${{\mathbb{P}}_{n}}$$ ( $$n \times n$ $ ) $${\text{\{ }} \pm 1{\text{\} }}$$ - 矩阵成立。得到以下渐近线： $$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 .$$