Volume 31, Issue 4

Let Pol k be the set of all functions of k -valued logic representable by a polynomial modulo k , and let Int ( Pol k ) be the family of all closed classes (with respect to superposition) in the partial k -valued logic containing Pol k and consisting only of functions extendable to some function from Pol k . Previously the author showed that if k is the product of two different primes, then the family Int ( Pol k ) consists of 7 closed classes. In this paper, it is proved that if k has at least 3 different prime divisors, then the family Int ( Pol k ) contains an infinitely decreasing (with respect to inclusion) chain of different closed classes.

We use generating functions to account for alphabetic points (or the lack thereof) in compositions and words. An alphabetic point is a value j such that all the values to its left are not larger than j and all the values to its right are not smaller than j . We also provide the asymptotics for compositions and words which have no alphabetic points, as the size tends to infinity. This is achieved by the construction of upper and lower bounds which converge to each other, and in the latter case by probabilistic arguments.

We prove an analogue of Toyoda–Belousov theorem on the structure of medial n -quasigroups for the case of strongly dependent n -ary operations.

The probabilistic characteristics of the graph of k -fold iteration of uniform random mapping are studied. Formulas for the distribution of the length of the aperiodicity segment of an arbitrary vertex with some restrictions are calculated. We obtain exact expressions for the probabilities that two arbitrary vertices belong to the same connected component, that an arbitrary vertex belongs to the preimage set of another vertex and that there exists a collision in the considered graph.

We consider the class F n , k consisting of n -ary Boolean functions that take the value one on exactly k input tuples. For small values of k the class F n , k is splitted into subclasses, and for every subclass we find the asymptotics of the Shannon function of circuit implementation in the basis {x&y,x‾} $ \{x\&y,\overline x\} $ (or in the basis {x∨y,x‾}) $ \{x\vee y,\overline x\}) $ ; the weights of the basic gates are arbitrary strictly positive numbers.

In this first part of the paper we find the asymptotic formulas for the probabilities of large deviations of the sequence defined by the random difference equation Y n +1 = A n Y n + B n , where A 1 , A 2 , … are independent identically distributed random variables and B n may depend on {(Ak,Bk),0⩽k<n} $ \{(A_k,B_k),0\leqslant k \lt n\} $ for any n ≥1. In the second part of the paper this results are applied to the large deviations of branching processes in a random environment.

A generalized scheme of allocation of n particles into ordered cells (components). Some statements containing sufficient conditions for the weak convergence of the number of components with given cardinality and of the total number of components to the negative binomial distribution as n → ∞ are presented as hypotheses. Examples supporting the validity of these statements in particular cases are considered. For some examples we prove local limit theorems for the total number of components which partially generalize known results on the convergence of this distribution to the normal law.