Vol 37, No 3 (2016)

In present paper we proved that the operator generated by the differential expression of second order with fractional derivative in lower terms, does not generate associated functions and that the system of eigenfunctions of this operator forms a basis in L₂(0, 1).

In this paper we show the use of commutative operads in public-key cryptography. Commutative operads were introduced by S.N. Tronin in 2006. They are a special case of algebraic operads and a natural generalization of commutative algebraic theories. We consider some cryptographic protocols based on commutative operads. For the protocol of the creation a common secret key, we describe and investigate its implementation and cryptographic security in particular cases.

The article presents and generalizes the results on some performance indicators of genetic algorithm developed by authors and applied to effective search queries and selection of relevant results after document subject search. It is shown that the developed technology expands opportunities of semantic search and increases the number of the found relevant results. In particular, we made an effort to show the ability of the developed algorithm to achieve the neighborhood of the fitness function in a finite number of steps, to provide higher precision of search in comparison with the well-known search engines of the Internet as well as to provide the acceptable semantic relevance of the found documents.

We characterize the fuzzy left (resp. right) ideals, the fuzzy ideals and the fuzzy prime (resp. semiprime) ideals of an ordered Γ-groupoid M in terms of level subsets and we prove that the cartesian product of two fuzzy left (resp. right) ideals of M is a fuzzy left (resp. right) ideal of M × M, and the cartesian product of two fuzzy prime (resp. semiprime) ideals of M is a fuzzy prime (resp. semiprime) ideal of M × M. As a result, if μ and σ are fuzzy left (resp. right) ideals, ideals, fuzzy prime or fuzzy semiprime ideals of M, then the nonempty level subsets (μ × σ)_t are so.

The biharmonic equation formulation of the Stokes flow problem for multiquadrics method is developed. The main advantage of the approach is the iteration free method to find the solution. The numerical method is applied for the problem of steady incompressible fluid flow past a cylinder in the periodic cell of the Kuwabara model. The comparison with known analytical solution and the analysis of absolute and relative errors show that proposed approach gives satisfactory accuracy. The nonmonotonic dependence of the relative errors on the shape parameter typical for multiquadrics method is observed.

May soon be invented quantum computer and some known cryptosystems (such as RSA) will be threatened breaking. This paper is aimed at establishing necessary conditions for the maximum possible algebraic generalization of the classical RSA algorithm. We substitute ideals of a Dedekind ring for integers. Ideals in Dedekind rings allow the unique decomposition into a product of maximal ideals, but may not be the principal ideals. Also we define Euler’s ϕ-function for ideal of a Dedekind ring and describe some properties of this function. We hope that our proposed method will help to develop algorithms for encryption, which is hard to crack using a quantum computer.

The paper proves a theorem on polytopal fans and face polytopes that can be treated as an intermediate value theorem for face polytopes. According to this theorem if all fans F_S obtained from a fan F by replacing one of its cones K with a subdivision S of K in some set H are polytopal, then the fan F is polytopal as well. Moreover, if P_S, S ∈ H, are arbitrary face polytopes of the fans F_S, then some positive combination of P_S, S ∈ H, is a face polytope of the fan F. The reverse of the theorem is not true.

A rightmodule M over a ring R is said to be ADSif, for every decomposition M = S ⊕ T and every complement T' of S, we have M = S ⊕ T'. In this article, we study and provide several new characterizations of new class of essential modules and generalization of ADS modules. We prove that M is semisimple if and only if every module in σ[M] is generalized ADS if and only if every generalized ADS module in σ[M] is M-injective.

Solutions of two problems are proposed based on the Tschirnhausen transform. The first problem is connected with the construction of minimal polynomials of the numbers of the form tan²(π/n) by means of the Tschirnhausen transform for all natural n > 2. The second problem consists in finding the exact roots of the equation x³ − 7x − 7 = 0. A solution of the problem is obtained from the fact that the roots of the equation produce the cyclotomic field Q₇. Examples of construction of minimal polynomials are provided.

The mechanisms of transition to turbulence in the middle wake behind an obstacle on the channel wall for a laminar flow have been studied. Using DNS, it has been shown that the transition to turbulence is mainly caused by the interaction of paired helical vortices generated at the side walls of the channel behind the rib. If the transverse size of these vortices reaches a half of the channel width, their contact causes periodic generation and separation of smaller vortex structures in the interaction area; here, the middle wake behind the obstacle is characterized by transition to turbulence. If the channel width is sufficiently large, the flow remains laminar. The influence of the distance between the side walls on the laminar-turbulent transition for Reynolds numbers ranging from 1500 to 3000.