Volume 23, Issue 2

- In this work the solvability of the equational theory of commutative medial n-ary groupoids is established by means of an algorithm that decides the satisfiability of the identity u = v in the variety of commutative medial n-ary groupoids.

- Achievable lower and upper bounds for the stability radius of the vector version of the problem on a maximum cut in a graph is obtained in the case where the parameter space is endowed with the Hölder metric. This work was supported by the Belorussian Republican Foundation for Fundamental Research, project F11K-095.

- Markov chains with embeddings as steganography models are considered. Statistical estimators of the model parameters based on frequencies and correlation statistics are constructed and analysed. A polynomial algorithm for likelihood function computing is developed. The maximum likelihood estimators based on this algorithm are constructed. The results of computer experiments are presented.

- The proof of a well-known inequality which may be used to derive quadratic lower bounds for the complexity of Π-circuits (or, the same, formulae over the basis {&,V-}) for many Boolean functions is significantly simplified. This work was financially supported by the Program of Fundamental Research “Algebraic and Combinatorial Methods of Mathematical Cybernetics” of the Mathematical Sciences Department of the Russian Academy of Sciences (project “Synthesis and Complexity of Control Systems”).

- The structure of a finite group whose subgroups are all either nilpotent or Hall subgroups is established.

- The paper is concerned with the implementation of the fast multiplication of binary polynomials and polynomial matrices on 64-bit computers. A method is proposed for an efficient implementation of the fast Fourier transform with the use of the addition and multiplication operations modulo 2 64 subject to a bound on the size of the problem. Comparative performance results are given showing substantial advantages of the implementation of the algorithm proposed over the function of multiplication of binary polynomials from the NTL mathematical library for multiplication of polynomials of degree exceeding 2 25 and multiplication of square polynomial matrices of degree at least 2 14 and of size at least 32 × 32.

- A subset A of a group G is called a difference if there exists a subset B ⊆ G such that A = B − B, where B − B = {x − y | x, y ∈ B}. The asymptotic behavior of the logarithm of the number of differences in groups of prime order is studied. This work was supported by the Russian Foundation for Basic Research, grant 13-01- 00958a.

- We study modules which are characteristic submodules in their injective hulls. The author is supported by the Russian Foundation for Basic Research, project 08-01- 00693-a

- We investigate distributions of values of read-once quasigroup formulae when the variables are replaced by random independent identically distributed elements of the quasigroup. For general read-once formulae of increasing complexity or depth conditions under which of the distributions of values of the formulae converge to the uniform distribution are put forward, and weaker conditions for the convergence ‘in the mean’ is proved also. This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 11-01-00508) and by the Programme of the Division of Mathematics Sciences of the Russian Academy of Sciences “Algebraic and Combinatorial Methods of Mathematical and New Generation of Information Systems” (the project “Problems of Optimal Synthesis of Control Systems”).