Band 20, Heft 5-6

We obtain two-sided bounds and asymptotic formulas for the number of Boolean functions of n variables which are approximated by affine or linear Boolean functions with a given accuracy.

The paper is devoted to the solution of an actual problem of identification (estimation of parameters and hypotheses testing) for the so-called binary Markov chain of order s with r partial connections subjected to additive distortions.

We consider conditional random graphs of Internet type under the condition that the number of edges of the graph is known and the degrees of the vertices have no mathematical expectation. We prove limit theorems for the maximum vertex degree and the number of vertices of a given degree.

An estimate of the probability of localisation of the diameter of a scale-free random graph is given; this estimate improves the estimate obtained in 2004 by B. Bollobás and O. Riordan. Some possible applications of such graphs to analysis of complex stochastic networks are mentioned.

We investigate properties of a class of plateaued Boolean functions with support of the spectrum defined by a recursive class of matrices. For such supports of the spectrum, we find a precise number of functions with a given support. We also show that the set of these functions is the equivalence class of a function with this support of spectrum with respect to the group of shifts.

We consider a mapping of a discrete circle into itself preserving the order inside triples of points. We describe the structure of the graph of the mapping and investigate the distributions of the number and lengths of its cycles.

In this paper we consider the problem of expressibility of an element of the complete lattice equipped with the closure operation. We find conditions under which this problem has a solution in the form of finite lower neighbourhood. The most interesting case of the complete lattice ℬ( P ) of ordered (by inclusion) subsets of the set P is considered separately. It is shown that existence of a finite lower neighbourhood for each generated closed element of this lattice implies finitariness of the operation of closure in it. In the case of the lattice ℬ( P ) with finitary closure, we find constructive sufficient conditions for existence of a finite lower neighbourhood of its element. We thus generalise Kuznetsov's theorem on functional completeness. The case of the Galois closure induced by some Galois correspondence between lattices of subsets is studied separately. In this case we find necessary and sufficient conditions for existence of a finite lower neighbourhood of an element of the lattice. This generalises, along with Kuznetsov's theorem, Yablonskii's theorem on predicate-describable classes of functions of finite-valued logic. Along with the expressibility problems, we consider the possibility to determine closed elements of the lattice ℬ( P ) in particular, elements of lower neighbourhoods, by finite forbidding sets under some preordering of the set P , and in the case of the Galois closure, by finite descriptions. The theorems we proved are accompanied by examples of their usage.

Let u be a linear recurring sequence of maximal period over the ring Z p n and 𝑣 be a pseudo-random sequence over the field Z p obtained by multiplying the highest coordinate sequence of u by some polynomial. In this paper we analyse possibilities and ways to restore u from a given 𝑣. A short survey of earlier results is given.

We consider the uniform id -decomposition of functions of the class P k of functions of k -valued logic over the class of structural homogeneous functions and the class D k of homogeneous functions generated by the dual discriminator d . We find the degrees of the uniform id -decomposition of the class P k over the classes and D k and give the methods of construction of homogeneous functions over which the uniform id -decomposition of the class P k is realised.

In the algebra of linear integer-coefficient polynomials, we give a criterion of completeness in terms of precomplete classes and find the cardinality of bases.

We study tests for circuits realising disjunction. In a circuit, the inversion fault is allowed at the output of precisely one functional element. For any functionally complete basis B it is found that for any n ≥ 3 the disjunction of n variables can be realised by a circuit over B which admits a unitary diagnostic test consisting of at most two tuples. It is proved that this result cannot be improved in the general case.

The bent functions are the Boolean functions of an even number of variables which are at the maximum possible distance from the set of all affine functions. In this paper, it is shown that each isometric mapping of the set of Boolean functions of n variables to itself preserving the class of bent functions is a combination of an affine transformation of coordinates and a shift by an affine function. It is proved that the affine functions are precisely all Boolean functions which are at the maximum possible distance from the class of bent functions.

The objective of this research is to study the complexity of multiplication in commutative group algebras over arbitrary fields of prime characteristic. In order to solve this problem, we suggest a method to find the structure of group algebras which allows us to use the Alder–Strassen theorem to obtain lower bounds and the Bläser theorem describing all algebras of minimal rank to obtain upper bounds.

We investigate the conditions under which the sum of independent Markov sequences on a finite abelian group G is also a simple homogeneous Markov chain on the group G with some matrix of transition probabilities. The considered problems concern the well-known procedure of consolidation of states of Markov chains. In this paper we develop a method based on the reduction of the initial problem to the solution of a system of special form of nonlinear equations over group algebras. We obtain new conditions under which sums of Markov chains on an arbitrary abelian group G = Z m are Markov chains, and necessary and sufficient conditions under which a sum of independent realisations of the initial Markov chains is also a simple homogeneous Markov chain.

For the number S n of repetition-free Boolean functions of n variables in the basis B 1 , it is proved that, as n → ∞, S n ~ cn –3/2 α n n !, where c and α are some constants.