Vol 216, No 4 (2016)

We consider a polygonal plate weakened by n cyclic symmetric holes in a stressed state whose plasticity region contains only contours of holes and is not spread inside the plate. We construct solutions of this problem and obtain the equation of unknown contours of holes.

In the present paper, we consider the “bulky knots” and “bulky links” that appear after cutting of generalized Möbius–Listing GML₄ⁿ bodies (with corresponding radial cross sections square) along different generalized Möbius–Listing surfaces GML₂ⁿ situated in it. The aim of this article is to examine the number and geometric structure of independent objects that appear after such a cutting process of GML₄ⁿ bodies. In most cases, we are able to count the indices of the resulting mathematical objects according to the known tabulation for knots and links of small complexity.

The tight-binding model of quantum particle on a honeycomb lattice is investigated in the presence of homogeneous magnetic field. The one-particle Hamiltonian is expressed in terms of the generators of the quantum group U_q(sl₂). The corresponding Harper equation is rewritten as a system of two coupled functional equations in the complex plane. The system is shown to exhibit certain symmetry that allows one to resolve the entanglement, and the basic single equation determining the eigenvalues and eigenstates is obtained. Equations specifying the roots of eigenstates in the complex plane are found.

We consider the statics case of the theory of linear thermoelasticity with microtemperatures materials. The representation formula of a general solution of the homogeneous system of differential equations obtained in this paper is expressed by means of four harmonic and three metaharmonic functions. These formulas are very convenient and useful in many particular problems for domains with concrete geometry. Here we demonstrare an application of these formulas to the III type boundary value problem for a half-space. Uniqueness theorems are proved. Solutions are obtained in quadratures.

In this paper, we describe an educational social network. The mathematical model of this social network is proposed. We consider the following tools in the social network: instant messages, blocks, wiki-libraries, video conferences, document repositories, repository of training materials, and electronic courses.

We give a characterization of weakly subgaussian random elements that are γ-subgaussian in infinite-dimensional Banach and Hilbert spaces.

For a topological vector space (X, τ), we consider the family LCT(X, τ) of all locally convex topologies defined on X, which give rise to the same continuous linear functionals as the original topology τ. We prove that for an infinite-dimensional reflexive Banach space (X, τ), the cardinality of LCT(X, τ) is at least \( \mathfrak{c} \).

Among operation research problems, there exist practical problems that can be formulated as network models. Of the results of research of previous years, more than half of the problems on mathematic programming can be realized via network modeling. One of the important problems of network modeling is finding the shortest distance and its realization on a computer. Unlike the known algorithm of finding the shortest distance when we must adopt the formal sum U₁ = 0 and minimizing sum on the following integrations, in the case discussed in this work we maximize the multiplication product and accordingly in the beginning we adopt the formal multiplication product of U₁ = 1. In this work, one problem is solved via two different forms and methods, and an example of its corresponding realization on computer is provided.