Volume 29, Issue 4

Centrally essential rings were defined earlier for associative unital rings; in this paper, we define them for rings which are not necessarily associative or unital. In this case, it is proved that centrally essential semiprime rings are commutative. It is proved that all idempotents of a centrally essential alternative ring are central. Several examples of non-commutative centrally essential rings are provided, some properties of centrally essential rings are described.

We consider configuration graphs with N vertices. The degrees of vertices are independent identically distributed random variables having the power-law distribution with positive parameter τ.$\tau .$We study properties of random graphs such that the sum of vertex degrees does not exceed n and the parameter is a random variable uniformly distributed on the interval [a,  b],0<a<b<∞.$\left[ a,\,\,b \right],0<a<b<\infty .$We find limit distributions of the number μr${{\mu }_{r}}$of vertices with degree r for various types of variation of N, n and r.

For a nonhomogeneous polynomial scheme, conditions are found under which the Pearson statistic distributions converge to the distribution of nonnegative quadratic form of independent random variables with the standard normal distribution.

We obtain estimates for the complexity of the implementation of n -place Boolean functions by circuits and formulas built of unbounded fan-in conjunction and disjunction gates and either negation gates or negations of variables as inputs. Restrictions on the depth of circuits and formulas are imposed. In a number of cases, the estimates obtained in the paper are shown to be asymptotically sharp. In particular, for the complexity of circuits with variables and their negations on inputs, the Shannon function is asymptotically estimated as 2⋅2n/2;$2\cdot {{2}^{n/2}};$this estimate is attained on depth-3 circuits.

We consider problems on the convergence of distributions of the total number of components and numbers of components with given volume to the Poisson law. Sufficient conditions of such convergence are given. Our results generalize known statemets on the limit Poisson laws of the number of components (cycles, unrooted and rooted trees, blocks and other structures) in the corresponding generalized of allocation schemes.

The paper is concerned with sets of Bernoulli distributions which are closed under substitutions of independent random variables into Boolean functions from a given set (an algebra of Bernoulli distributions). A description of all finite algebras of Bernoulli distributions is given.