Band 13, Heft 2

We describe distributions of the lengths of initial, covering, and final runs in binary Markov sequences. For the means and variances, we give exact and asymptotic formulas. We also give the generating functions. We observe that in Markov sequences the probabilities of run lengths do not necessarily decrease as the lengths grow, and hence, the corresponding distributions may be of quite complex form. We investigate conditions under which, due to the Markov property, the probabilities increase as the run lengths do. We consider operator equations which include final runs.

We consider a problem of constructing operators of metric closeness in the n -dimensional cube in the class of automaton circuits of a special form. We study two characteristics of complexity, the spacial and temporal characteristics (the number of elements of the circuit and the time required for calculations realised by the circuit). We suggest a method of constructing the circuits realising such operators with constant running time and optimal in order number of elements.

We study recurring sequences over finite fields sets and the set N = {0, 1, 2, ... }. The complexity of recurring sequences over finite sets is estimated as the complexity of computing on determinate linearly bounded automata. We introduce the notion of a branching recurring sequence. The complexity of branching recurring sequences over finite sets is estimated as the complexity of computing on non-determinate linearly bounded automata. Recurring sequences over the set N simulate computations on multi-tape Minsky machines. We ascertain undecidability of some problems concerning this type of recurring sequences.

Let V T be the T -dimensional linear space over a finite field K , and let B 1 ,..., B m be subsets of V T not containing the zero-point. Let a subspace L be chosen randomly and equiprobably from the set of all n -dimensional linear subspaces of V T . We consider the number µ( B i ) of points in the intersections L ∩ B i , i = 1,..., m . We study the limit behaviour of the distribution of the vector (µ( B 1 ),..., µ.( B m )) as T, n → ∞ and the sets vary in such a way that the means of µ( B i ) tend to finite limits. The field K is fixed. We prove that this random vector has in limit the compound Poisson distribution. Necessary and sufficient conditions for asymptotic independency of the random variables µ( B i ),...,µ( B m ) are derived.

We consider estimates of the function equal to the square-free part of the positive integer argument t . V. G. Sprindzhuk posed the following problem. Is there a constant c > 0 such that the inequality S (( n + 1) ... ( n + k )) < k k is fulfilled for an infinite number of pairs of positive integers n and k such that k < ln c n ? We prove that there exist positive constants c 7 ,..., c 10 such that for n ≥ c 7 In the paper, we obtain several other estimates of the function S ( t ) and discuss some conjectures concerning S ( t ) and derive corollaries of those conjectures.

We investigate the activity of cell circuits, the measure of their complexity, which describes the functioning of such circuits from the energy point of view. For the system K n of all elementary conjunctions of n variables we find the order of the minimal activity as n → ∞. We prove that it is impossible to reach simultaneously the minimal in order activity and complexity of realising the system K n in the class of cell circuits.