论文标题
$ \ {\ pm 1 \} $ - 矩阵和阈值函数数量的渐近学的奇异性
Singularity of $\{\pm 1\}$-matrices and asymptotics of the number of threshold functions
论文作者
论文摘要
有关阈值函数数量$ p（2，n）$的两个结果和概率$ {\ mathbb p} _n $，即建立了一个随机的$ n \ times n $ bernoulli矩阵。我们介绍了一个超模块化函数$η^{\ bigstar} _n：2^{{{\ bf rp}^n} _ {fin} \ to \ m athbb {z} _ {\ geq 0}，$在$ {\ bf rp rp rp rp rp rp rp rp rp rp rp rp rp rp rp rp rp r. p rp rp rp（ $ {\ mathbb p} _ {n+1}的条款。 p} _n $已证明：$$ \ mathbb {p} _n \厚sim（n-1）^22^{1-n}，\ quad n \ to \ to \ infty。
Two results concerning the number of threshold functions $P(2, n)$ and the probability ${\mathbb P}_n$ that a random $n\times n$ Bernoulli matrix is singular are established. We introduce a supermodular function $η^{\bigstar}_n : 2^{{\bf RP}^n}_{fin} \to \mathbb{Z}_{\geq 0},$ defined on finite subsets of ${\bf RP}^n,$ that allows us to obtain a lower bound for $P(2, n)$ in terms of ${\mathbb P}_{n+1}.$ This, together with L.Schläfli's famous upper bound, give us asymptotics $$P(2, n) \thicksim 2 {2^n-1 \choose n},\quad n\to \infty.$$ Also, the validity of the long-standing conjecture concerning ${\mathbb P}_n$ is proved: $$\mathbb{P}_n \thicksim (n-1)^22^{1-n}, \quad n\to \infty .$$
