Band 28, Heft 2

The paper is concerned with complications of linear recurrent sequences over the field GF ( q ) and the ring GR ( q n , p n ) with interconnected recurrent relations. The cross-correlation function between cycles of given sequences is estimated.

The phenomenon of reduction of the set of permutations H arising in theorems on the weak convergence of the number of pairs of H -equivalent tuples in a segment of an indecomposable finite Markov chain to discrete distributions of the Poisson type is investigated.

The paper is concerned with subsets I of the residue group Z d in which the difference of any two elements is not relatively prime to d . The class of such subsets is denoted by U ( d ), the class of sets from U ( d ) of cardinality r is denoted by U ( d , r ). The present paper gives formulas for evaluation or estimation of | U ( d )| and | U ( d , r )|.

Computational capabilities of register machines with counters is investigated. It is shown that the class of functions that are strictly computable on register machines with counters agrees with that of total recursive functions.

Let A be a commutative arithmetical ring. It is proved that the ring A has Krull dimension if and only if every factor ring of A is finite-dimensional and does not have idempotent proper essential ideals.

A two-type critical decomposable branching process with discrete time 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. Assuming that the offspring distributions of particles of both types may have infinite variance, the asymptotic behavior of the tail distribution of the random variable Ξ 2 , the total number of the second type particles ever born in the process is found. Limit theorems are proved describing (as N → ∞) the conditional distribution of the amount of the first type particles in different generations given that either Ξ 2 = N or Ξ 2 > N .

Let 𝓧 𝓝 be a set consisting of N elements and F 1 , F 2 , … be a sequence of random independent equiprobable mappings 𝓧 𝓝 → 𝓧 𝓝 . For a subset S 0 ⊂ 𝓧 𝓝 , | S 0 | = n , we consider a sequence of its images S t = F t (… F 2 ( F 1 ( S 0 ))…), t = 1, 2 … The conditions on n , t , N → ∞ under which the distributions of image sizes | S t | are asymptotically connected with the standard normal distribution are presented.