Том 218, № 6 (2016)

We consider an elasticity and plasticity problem for a square plate weakened by a square hole with rounded vertices. The plasticity region contains only the unknown rounded part of the hole contour and is not spread inside the plate. We contract the solutions and obtain the equation of the unknown part of the hole contour when the normal displacement is constant on the linear parts of the boundary and there is no normal stress on the unknown part of the hole contour.

This paper is devoted to the study of groups from the category M of R-power groups. We examine problems on the commutation of the tensor completion with basic group operations and on the exactness of the tensor completion. Moreover, we introduce the notion of a variety and obtain a description of abelian varieties and some results on nilpotent varieties of A-groups. We prove the hypothesis on irreducible coordinate groups of algebraic sets for the nilpotent R-groups of nilpotency class 2, where R is a Euclidean ring. We state that the analog to the Lyndon result for the free groups (see [10]) holds in this case, whereas the analog to the Myasnikov–Kharlampovich result fails.The paper is dedicated to partial R-power groups which are embeddable to their A-tensor completions. The free R-groups and free R-products are described with usual group-theoretical free constructions.

The notion of a W-exponential Hall group is introduced and the limiting theorems on w-complete groups are proved.

In the present paper, the Bitsadze–Samarski boundary-value problem is considered for a quasi-linear differential equation of first order on the plane and the existence and uniqueness theorem for a generalized solution is proved; the necessary (in the linear case) and sufficient optimality conditions for optimal control problems are found. The optimal control problem is posed, where the behavior of control functions is described by elliptic-type equations with Bitsadze–Samarski nonlocal boundary conditions. The necessary and sufficient optimality conditions are obtained in the form of the Pontryagin maximum principle and the solution existence and uniqueness theorem is proved for the conjugate problem. Nonlocal boundary-value problems and conjugate problems are solved by the algorithm, which reduces nonlocal boundary value problems to a sequence of Dirichlet problems. The numerical method of solution of an optimal control problem by the Mathcad package is presented.

In this paper, we present some new results on the question whether a Lie group can be represented as the multiplication group of a three-dimensional topological loop. We deal with the classes of quasi-simple Lie groups and nilpotent Lie groups.

In this paper, we introduce a novel polynomial-time algorithm to compute graph invariants based on the idea of a modified random walk on graphs. Though not proved to be a full graph invariant yet, our method gives the right answer for the graph instances other well-known methods could not compute (such as special Fürer gadgets and point-line incidence graphs of finite projective planes of higher degrees).

A purely implicit three-layer semi-discrete scheme of second-order approximation is considered in the Banach space. The three-layer semi-discrete scheme is reduced by means of the disturbance algorithm to two two-layer schemes. Using the solutions of these schemes, we construct an approximate solution of the initial problem. The stability of the constructed scheme is proved.

Connections between certain types of categories (logoses and toposes) and intuitionistic predicate logic was established in 1960–1970 by Lowvere. The possibility of extending this connection to some types of modal logics by using the internal structure of categories of particular type (logos) was also established. Category-theoretical constructs were hence used as one of the possible semantic interpretations of intuitionistic logic. This interpretation has also included intuionistic modal logics using different semantical tools such as adjoint pair of functors. In this paper, we discuss one of the possible extension of intuitionistic logic.

We prove that m principal singular vectors of a matrix X_d constructed on the basis of a time series, contained periodical deterministic components with additive white noise, have equal pseudospectra and their pseudo-spectral structure is identical to that of the time series. The structures of pseudo-spectra of the rest singular vectors differ from the structures of pseudo-spectra of the principal vectors and the time series. It is shown that the time series allow one to increase the resolving capacity and to improve the statistical stability of spectral estimation.

We propose a setting for De Donder–Hamilton field theory in jet bundles, generalizing the usual multisymplectic formalism. Using a reformulation of Hamilton theory for the family of local Lagrangians related to a global Euler–Lagrange form, we construct a dual Hamiltonian bundle and corresponding Legendre maps, linking a Lagrangian system on a jet bundle with a canonical Hamiltonian system on the affine dual. Our approach significantly extends the family of regular variational problems that can be treated directly within a dual Hamiltonian formalism, thus avoiding the necessity to use the Dirac constraint formalism.

In this paper, we present a general procedure for solving of homogeneous equations that describe penetration and diffusion of X-rays in plane geometry. Starting from Van Kampen’s and Case’s observation that it suffices that “solutions” be distributions, elementary solutions of a homogeneous equation are found. We also prove that general solutions can be obtained by superposition of elementary solutions.

The notion of a weakly locally compact topological abelian group introduced in this paper generalizes the notion of a fibrous topological abelian group studied by N. Ya. Vilenkin.

Since in the class of locally compact topological abelian groups we distinguish classes of compact and discrete topological groups, in the class of weakly locally compact topological abelian groups we distinguish classes of quasicompact and quasidiscrete groups which are dual in the sense of Pontryagin’s theory of characters. We prove here that the group of characters of a weakly locally compact topological abelian group is weakly locally compact and construct universal groups for weakly locally compact groups.

The paper presents a full description of idempotent elements of the semigroup of binary relations B_X(D), which are defined by semilattices of the class Σ₃(X, 8). For the case where X is a finite set and Z₇ = Ø, we derive formulas for calculating the number of idempotent elements of the respective semigroup.

A complete semigroup of binary relations is defined by semilattices of the class Σ₂(X, 8). A description of idempotent elements of this semigroup is given. For the case where X is a finite set and Z₇ ∩ Z₆ ≠ ∅, formulas are derived by calculating the number of idempotent elements of the semigroup.