MAY-WIGNER STABILITY THEOREM. Let B be an n x n matrix with nC (0 < C < 1) randomly located nonzero entries, each chosen independently from a symmetric distribution with variance a. Let A = B I, and let P(a, n, C) be the probability that the corresponding differential system (I) has a stable equilibrium at 0. Let e > 0. Then P(a, n, Q —• 1 as n —• °° provided anC < 1 e; conversely, P(a, nt C) —...

Recent developments in cosmology indicate that every history having a nonzero probability is realized in infinitely many distinct regions of spacetime. Thus, it appears that the universe contains infinitely many civilizations exactly like our own, as well as infinitely many civilizations that differ from our own in any way permitted by physical laws. We explore the implications of this conclusi...

We consider recurrent analog neural nets where each gate is subject to Gaussian noise, or any other common noise distribution whose probability density function is nonzero on a large set. We show that many regular languages cannot be recognized by networks of this type, for example the language begins with , and we give a precise characterization of those languages which can be recognized. This...

Given a field F , a scalar λ ∈ F and a matrix A ∈ F, the twisted centralizer code CF (A, λ) := {B ∈ F | AB − λBA = 0} is a linear code of length n. When A is cyclic and λ 6= 0 we prove that dimCF (A, λ) = deg(gcd(cA(t), λcA(λt))) where cA(t) denotes the characteristic polynomial of A. We also show how CF (A, λ) decomposes, and we estimate the probability that CF (A, λ) is nonzero when |F | is f...

We consider a simple method to compress a sparse (nn) random matrix into a table of size O(n) which works in expected linear time O(n). The worst case time to access entries in the compressed table is O(1). The compression scheme is based on a random greedy algorithm which places every row of a matrix with n nonzero entries into a table of size n. This minimal table size is achieved with high p...

In this paper‎, ‎the‎ ‎authors prove that a strictly Kähler-Berwald manifold with‎ ‎nonzero constant holomorphic sectional curvature must be a‎ Kähler manifold‎. 

let $r$ be a reversible ring which is $alpha$-compatible for an endomorphism $alpha$ of $r$ and $f(x)=a_0+a_1x+cdots+a_nx^n$ be a nonzero skew polynomial in $r[x;alpha]$. it is proved that if there exists a nonzero skew polynomial $g(x)=b_0+b_1x+cdots+b_mx^m$ in $r[x;alpha]$ such that $g(x)f(x)=c$ is a constant in $r$, then $b_0a_0=c$ and there exist nonzero elements $a$ and $r$ in $r$ such tha...