Volume 28, Issue 5

A two-type decomposable branching process is considered in which particles of the first type may produce at the death moment offspring of both types while particles of the second type may produce at the death moment offspring of their own type only. The reproduction law of the first type particles is specified by a random environment. The reproduction law of the second type particles is one and the same for all generations. A limit theorem is proved describing the conditional distribution of the number of particles in the process at time nt , t ∈ (0,1], given the survival of the process up to moment n → ∞.

The conditions under which the nonextincting trajectories of a discrete time bounded branching process with probability 1: either only finitely many times hit the upper boundary, either infinitely often hit the upper boundary, or coincide with the upper boundary after some random moment

Let all vertices of a complete q -ary tree of finite height be independently and equiprobably labeled by the elements of some finite alphabet. We consider the numbers of pairs of identical tuples of labels on chains of subsequent vertices in the tree. Exact formulae for the expectations of these numbers are obtained, convergence to the compound Poisson distribution is proved. For the size of cluster composed by pairs of identically labeled chains we also obtain exact formula for the expectation.

When investigating the complexity of implementing Boolean functions, it is usually assumed that the basis inwhich the schemes are constructed and the measure of the complexity of the schemes are known. For them, the Shannon function is introduced, which associates with each Boolean function the least complexity of implementing this function in the considered basis. In this paper we propose a generalization of such a Shannon function in the form of an upper bound that is taken over all functionally complete bases. This generalization gives an idea of the complexity of implementing Boolean functions in the “worst” bases for them. The conceptual content of the proposed generalization is demonstrated by the example of a conjunction.

A critical Bellman-Harris branching process { Z ( t ), t ≥ 0} with finite variance of the offspring number is considered. Assuming that 0 < Z ( t ) ≤ φ ( t ), where either φ ( t ) = o ( t ) as t → ∞ or φ ( t ) = at , a >0, we study the structure of the process where Z ( s , t ) is the number of particles in the initial process at moment s which either survive up to moment t or have a positive number of descendants at this moment.

Let X N be a set of N elements and F 1 , F 2 ,… be a sequence of random independent equiprobable mappings X N → N . For a subset S 0 ⊂ X N , | S 0 |= m , we consider a sequence of its images S t = F t (… F 2 ( F 1 ( S 0 ))…), t =1,2… An approach to the exact recurrent computation of distribution of | S t | is described. Two-sided inequalities for M {| S t ||| S 0 |= m } such that the difference between the upper and lower bounds is o ( m )for m , t , N → ∞, mt = o ( N ) are derived. The results are of interest for the analysis of time-memory tradeoff algorithms.

Necessary conditions for power commuting in a finite-dimensional algebra over a field are presented.