Band 16, Heft 4

We describe the logic approach to pattern recognition; its key notion is a test. Analysing the tests allows us to construct functionals characterising the pattern, as well as procedures to compute them. We present qualitative and quantitative properties of tests, functionals, and recognition procedures. Solutions of a series of known problems are also given.

We give necessary and sufficient conditions of equiprobability at a finite step for limit distributions of products of random variables whose values lie in a finite group and whose distributions are defined by a Markov chain such that the uniform limit distribution is concentrated on the corresponding cosets of the finite group by some its conjugate subgroups.

We present several recursive procedures for calculating exact (non-limit) distributions of decomposable statistics in the multinomial scheme. In order to obtain recurrence relations, we make use of the generating functions of the statistics under consideration. In the general case, the method of calculating reduces to the algorithm suggested by A. Zubkov in 1996 which is presented in the language of Markov chains.

We consider two random hypergraphs with n vertices and M = M ( n ) edges among which M i = M i ( n ) edges consist of i non-ordered vertices, i = 0,1, . . . , m , M = M 0 + M 1 + . . . + M m . The choice of vertices for each edge is realised by equiprobable sampling with replacement from n possible vertices for the former random hypergraph and by that without replacement for the latter one. We investigate the distribution of the numbers of subgraphs isomorphic to given subgraphs as n → ∞, M = M ( n ). The notion of degree and balanced subgraphs are extended to nonhomogeneous hypergraphs. The limit multivariate Poisson theorem for the numbers of strictly balanced subgraphs of equal degrees is obtained. A threshold function for presence of a subgraph isomorphic to an arbitrary finite hypergraph is constructed. Such results have been obtained for random graphs by P. Erdős, A. Rényi, B. Bollobás, and for homogeneous hypergraphs (that is, for M = M m ) by the author.

Let on the set S n,N of all different permutations of degree n with N cycles the uniform distribution be given. We obtain estimates of the rate of convergence of the distribution of the number of cycles of given length in a random permutation of S n,N to the limit distributions as n, N → ∞ in such a way that either n/N → 1 or n/N → ∞.

We prove that the affine rank of any plateaued function with spectrum support of cardinality 16 is equal to 4, 5 or 6. For any positive integer h , we consider plateaued functions with spectrum support of cardinality 4 h , give bounds for the affine rank of such functions and construct functions with affine rank equal to any possible value from 2 h to 2 h +1 −2.

A functional graph is a digraph describing the action of a function on a set. The endomorphisms of such graphs are of interest in group theory: the centraliser of an element of the semigroup of all mappings of a set into itself coincides with the set of all endomorphisms of a digraph corresponding to this element. In this study, the functional graphs with a countable set of vertices are considered. We introduce the notion of lineals and calculate the cardinality of the set of endomorphisms of functional graphs which are not lineals.