Volume 15, Issue 3

In order to test homogeneity of r independent polynomial schemes with the same number of outcomes N under non-classical conditions where the numbers of trials n d , d = 1, . . . , r , in each of the schemes and the number of outcomes N tend to infinity, we suggest a statistic I ( λ , r ) which is a multidimensional analogue of the statistic I ( λ ) introduced by T. Read and N. Cressie. We obtain conditions of asymptotic normality of the distributions of the statistics I ( λ ) and I ( λ , r ) for an arbitrary fixed integer λ , λ ≠ 0, −1, as N → ∞, n d N −1 → ∞, d = 1, . . . , r . The expressions for the centring and normalising parameters are given in the explicit form for the hypothesis H 0 under which the distributions in these r schemes coincide, and for some class of alternatives close to H 0 .

We find conditions which are sufficient for convergence of the distribution of the number of matches of values of a function considered on tuples of arguments taken from a sequence of independent identically distributed random variables to the Poisson law and estimate the convergence rate. We derive a series of corollaries of this result. In particular, in the equiprobable polynomial scheme we obtain Poisson limit theorems for the number of pairs of non-overlapping tuples with identical frequencies of occurrences of symbols and for the number of pairs of tuples with identical structure.

We find the asymptotic behaviour of probabilities of large deviations of symmetric decomposable statistics in the context of the generalised allocation scheme where the Cramér condition is broken. We consider the case of the so-called small samples.

The strict avalanche criterion was introduced by Webster and Tavares while studying some cryptographic functions. We say that a binary function ƒ( x ), x ∈ V n , satisfies this criterion if replacing any coordinate of the vector x by its complement changes the values of ƒ( x ) exactly in a half of cases. In this paper we establish an upper bound for the number of such functions for n large enough.

We investigate the dependence of the length of a simple conditional adjustment experiment for an automaton on local transformations of the transition diagram and outputs of the automaton. We give upper and lower bounds for the Shannon function of the length of experiments if any p arrows in the Moore diagram of the automaton can be redirected and the values of the output function in any p points can be changed.

On the base of equation calculus, we define the operator of equational closure. We give examples of equationally complete systems and equationally closed classes. We find the cardinality of the set of equationally precomplete classes and give criteria of equational completeness. We present all equationally closed classes of Boolean functions.

We study representations of Boolean functions by formulas and describe in terms of remainder functions the classes of repetition-free Boolean functions in the pre-elementary bases

In this paper, we present upper and lower linear bounds for the Shannon function for length of checking tests for repetition-free functions in the basis {0, 1, &, ∨, ¬}.

We consider the Touchard C -polynomials closely related to the cycle indices of symmetric groups and introduce the so-called M -polynomials. We demonstrate that the C - and M -polynomials constitute a quasi-orthogonal system. For the partition polynomials under consideration, we give a series of recurrence relations.