Volume 26, Issue 2

A decomposable Galton - Watson process with two types of particles is considered. Particles of the first type produce equal random numbers of particles of both types, particles of the second type produce particles of the second type only. Under the condition that the total number of the first type particles is equal to N the functional limit theorems are proved for the numbers of particles of both types existing at times of the orders of N$\sqrt N $, of N and of the intermediate orders.

We consider the classes of linear automata functions over finite fields with composition (superposition and feedback) operation and describe an algorithm that decides whether a finite set of functions from such class is complete. Thus we generalize the result that was known for the case of linear automata functions over prime finite fields.

Let X 0 , X 1 , … be a simple ergodic Markov chain with a finite set of states and ξ̃ n, k ( s ) be a number of runs of k -fold repetitions of strings having length s . Estimates of accuracy of the Poisson approximation for the distribution of ξ n, k ( s ) in the sequence X 0 , X 1 , …, X n + s −1 are obtained, these estimates are uniform over k .

A polarized polynomial form (PPF) (modulo k ) is a modulo k sum of products of variables x 1 , . . . , x n or their Post negations, where the number of negations of each variable is determined by the polarization vector of the PPF. The length of a PPF is the number of its pairwise distinct summands. The length of a function f ( x 1 , . . . , x n )of k -valued logic in the class of PPFs is the minimum length among all PPFs realizing the function. The paper presents a sequence of symmetric functions f n ( x 1 , . . . , x n )of three-valued logic such that the length of each function f n in the class of PPFs is not less than ⌊3 n +1 /4⌋, where ⌊a⌋ denotes the greatest integer less or equal to the number a . The complexity of a system of PPFs sharing the same polarization vector is the number of pairwise distinct summands entering into all of these PPFs. The complexity LkPPF(F) $L_k^{{\rm{PPF}}}(F)$ of a system F ={ f 1 ,..., f m } of functions of k -valued logic depending on variables x 1 ,..., x n in the class of PPFs is the minimum complexity among all systems of PPFs { p 1 ,..., p m }such that all PPFs p 1 ,..., p m share the same polarization vector and the PPF p j realizes the function f j , j = 1,..., m . Let LkPPF(m,n) = maxFL2PPF(F) $L_k^{{\rm{PPF}}}(m,n)\, = \,\mathop {\max }\limits_F L_2^{{\rm{PPF}}}(F)$, where F runs through all systems consisting of m functions of k -valued logic depending on variables x 1 ,..., x n . For prime values of k it is easy to derive the estimate LkPPF(m,n) ≤ kn $L_k^{{\rm{PPF}}}(m,n)\, \le \,{k^n}$. In this paper it is shown that LkPPF(m,n) = 2n $L_k^{{\rm{PPF}}}(m,n)\, = \,{2^n}$ and LkPPF(m,n) = 3n $L_k^{{\rm{PPF}}}(m,n)\, = \,{3^n}$ for all m ≥ 2, n = 1, 2, . . . Moreover, it is demonstrated that the estimates remain valid when consideration is restricted to systems of symmetric functions only.

Let an urn contain balls of white and black colors. After each step with probabilities equal to 12 ${1 \over 2}$ either the number of white balls is increased by the number of black balls or the number of black balls is increased by the number of white balls. Formulas for the first two moments of the total number of balls in an urn are derived and it is shown that the limiting distribution function of the proportion of the number of white balls in an urn coincides with the Minkowski number-theoretic function.