Том 101, № 5-6 (2017)

The solution of the Riemann–Hilbert problem for an analytic function in a canonical domain for the case in which the data of the problem is piecewise constant can be expressed as a Christoffel–Schwartz integral. In this paper, we present an explicit expression for the parameters of this integral obtained by using a Jacobi-type formula for the Lauricella generalized hypergeometric function F_D^(N). The results can be applied to a number of problems, including those in plasma physics and the mechanics of deformed solids.

Explicit expressions for the numbers of labeled geodetic bicyclic, tricyclic, and tetracyclic graphs with a given number of vertices are obtained.

We consider the Cauchy problem with spatially localized initial data for a two-dimensional wave equation with variable velocity in a domain Ω. The velocity is assumed to degenerate on the boundary ∂Ω of the domain as the square root of the distance to ∂Ω. In particular, this problems describes the run-up of tsunami waves on a shallow beach in the linear approximation. Further, the problem contains a natural small parameter (the typical source-to-basin size ratio) and hence admits analysis by asymptotic methods. It was shown in the paper “Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation” [1] that the boundary values of the asymptotic solution of this problem given by a modified Maslov canonical operator on the Lagrangian manifold formed by the nonstandard characteristics associatedwith the problemcan be expressed via the canonical operator on a Lagrangian submanifold of the cotangent bundle of the boundary. However, the problem as to how this restriction is related to the boundary values of the exact solution of the problem remained open. In the present paper, we show that if the initial perturbation is specified by a function rapidly decaying at infinity, then the restriction of such an asymptotic solution to the boundary gives the asymptotics of the boundary values of the exact solution in the uniform norm. To this end, we in particular prove a trace theorem for nonstandard Sobolev type spaces with degeneration at the boundary.

A method of constructing an asymptotic solution of a singularly perturbed Volterra integral equation in the case of a spectral singularity of first order is proposed.

Necessary and sufficient isomorphism conditions for the second cohomology group of an algebraic group with an irreducible root system over an algebraically closed field of characteristic p ≥ 3h − 3, where h stands for the Coxeter number, and the corresponding second cohomology group of its Lie algebra with coefficients in simple modules are obtained, and also some nontrivial examples of isomorphisms of the second cohomology groups of simple modules are found. In particular, it follows from the results obtained here that, among the simple algebraic groups SL₂(k), SL₃(k), SL₄(k), Sp₄(k), and G₂, nontrivial isomorphisms of this kind exist for SL₄(k) and G₂ only. For SL₄(k), there are two simple modules with nontrivial second cohomology and, for G₂, there is one module of this kind. All nontrivial examples of second cohomology obtained here are one-dimensional.

The operator inclusion 0 ∈ A(x) + N(x) is studied. Themain results are concerned with the case where A is a bounded monotone-type operator from a reflexive space to its dual and N is a cone-valued operator. A criterion for this inclusion to have no solutions is obtained. Additive and homotopy-invariant integer characteristics of set-valued maps are introduced. Applications to the theory of quasi-variational inequalities with set-valued operators are given.

An analog of Pólya’s theorem on the estimate of the transfinite diameter for a class of multivalued analytic functions with finitely many branch points and of the corresponding class of admissible compact sets located on the associated (with this function) two-sheeted Stahl–Riemann surface is obtained.

A finite group whose all irreducible characters are rational valued is called a rational group. Using the concept of transversal action, we get a sufficient condition on non Abelian rational groups that guarantees every Sylow 2-subgroup is also rational. This gives a partial answer to an old conjecture rationality of Sylow 2-subgroup of rational group.

A nonlinear Sobolev-type equation that can be used to describe nonstationary processes in the semiconductor medium is studied. A number of families of exact solutions of this equation that can be expressed in terms of elementary functions and quadratures is obtained; some of these families contain arbitrary sufficiently smooth functions of one argument. The qualitative behavior of the resulting solutions is analyzed.

In 1955, M. A. Krasnosel’skii proved a fixed-point theorem for a single-valued map which is a completely continuous contraction (a hybrid theorem). Subsequently, his work was continued in various directions. In particular, it has stimulated the development of the theory of condensing maps (both single-valued and set-valued); the images of such maps are always compact. Various versions of hybrid theorems for set-valued maps with noncompact images have also been proved. The set-valued contraction in these versions was assumed to have closed images and the completely continuous perturbation, to be lower semicontinuous (in a certain sense). In this paper, a new hybrid fixed-point theorem is proved for any set-valued map which is the sum of a set-valued contraction and a compact set-valued map in the case where the compact set-valued perturbation is upper semicontinuous and pseudoacyclic. In conclusion, this hybrid theorem is used to study the solvability of operator inclusions for a new class of operators containing all surjective operators. The obtained result is applied to solve the solvability problem for a certain class of control systems determined by a singular differential equation with feedback.

We call a finite computational process using only arithmetic operations a rational algorithm. A rational algorithm that is able to check the congruence between arbitrary complex matrices A and B is currently not known. The situation may be different if A and B belong to a certain class of specialmatrices. For instance, there exist rational algorithms for the case where both matrices are Hermitian or unitary. In this paper, rational algorithms for checking the congruence between accretive or dissipative A and B are proposed.

The asymptotic behavior of the Hermite–Padé approximants of the first type for two beta functions are studied. The results are expressed in terms of equilibrium problems of logarithmic potential theory and in terms of meromorphic functions on Riemann surfaces.

Certain sufficient criteria for the types of partial boundedness of solutions with partially controllable initial conditions are obtained in terms of higher-order derivatives of the Lyapunov functions.

Using the notion of an existentially closed structure, we obtain embedding theorems for groups and Lie algebras. We also prove the existence of certain groups and Lie algebras with prescribed properties.

The sets with continuous selection from near-best approximations and the monotone path-connected sets are studied; several examples of such sets are also considered.