Vol 38, No 3 (2017)

In the paper the boundary plane elasticity problem for the ring is solved usingMuskhelishvili complex potentialsmethod. The ring rotates uniformly under the influence of tangential forces, and the centrifugal forces of inertia are taken into account. Themutual rotation angle of the boundary points is obtained by analyzing the displacement field.

Strict superharmonicity of generalized reduced module as a function of a point (we call it Mityuk’s function) is established for the subclass of countably connected domains with unique limit point boundary component. The function just mentioned was first studied in detail by I.P. Mityuk and plays now an important role in the research of the exterior inverse boundary value problems of the theory of analytic functions in the multiply connected domains. At the heart of such a research one can see the fact that the critical points of Mityuk’s function are only maxima, saddles or semisaddles of corresponding surface. This fact is followed from the above strict superharmonicity.

Given a group G we study right and left zeros, idempotents, the minimal ideal, left cancellable and right cancellable elements of the semigroup N_<ω(G) of centered upfamilies and characterize groups G whose extensions N_<ω(G) are commutative. We finish the paper with the complete description of the structure of the semigroups N_<ω(G) for all groups G of cardinality |G| ≤ 4.

A commutative algebra B over the field of complex numbers with the bases {e₁, e₂} satisfying the conditions (e₁² + e₂²)² = 0, e₁² + e₂²)² ≠ 0, is considered. The algebra B is associated with the biharmonic equation. Consider a Schwartz-type boundary value problem on finding a monogenic function of the type Φ(xe₁ + ye₂) = U₁(x, y)e₁ + U₂(x, y)ie₁ + U₃(x, y)e₂ + U₄(x, y)ie₂, (x, y) ∈ D, when values of two components U₁, U₄ are given on the boundary of a domain D lying in the Cartesian plane xOy. We develop a method of its solving which is based on expressions of monogenic functions via corresponding analytic functions of the complex variable. For a half-plane and for a disk, solutions are obtained in explicit forms by means of Schwartz-type integrals.

We solve a boundary value problem on non-rectifiable contours for differentiable functions, and show that it gives us a tool for solving a number of problems for analytical and generalized analytical functions.

H. Behnke’s and E. Peschl’s definition of plänarkonvexitat leads to the Epstein-type inequalities when applies to the Hartogs domains in C². One-parameter series of such inequalities reveals the following rigidity phenomenon: the set of the parameters with contensive inequalities is exactly the segment which center corresponds to the well-known Nehari ball. The latter plays the crucial role in the forming the Gakhov class of all holomorphic and locally univalent functions in the unit disk with no more than one-pointed null-sets of the gradients of their conformal radii. The sufficient condition for the piercing of the Nehari sphere out of the Gakhov class is found. We deduce such a condition along the lines of the subordination approach to the proof of Haegi’s theorem about the inclusion of any convex holomorphic function into the Gakhov class.

In the present paper the smoothness loss of a continuation of solutions to convolution equations is studied. Also examples for some kinds of convolvers are given.

For upper bounds of the deviations of repeated Fejer sums taken over classes of periodic functions that admit analytic extensions to a fixed strip of the complex plane, we obtain asymptotic equalities. In certain cases, these equalities give a solution of the corresponding Kolmogorov–Nikolsky problem.

New sufficient conditions for injectivity in a finite-dimensional space were developed. They take the form of limits on the derivatives up to and including the second order, and ensure the existence of a homeomorphic continuity of the examined function across the whole plane. We have also analyzed the locally Lipschitz continuity. Conditions for injectivity in this case take the form of limits on the generalized Jacobian. The range of definition of the analyzed functions is typical for the issue of price equation with factors of production.

We study the spectral properties of four-electron systems in the Hubbard model in the quintet and singlet states. We proved the essential spectra of the four-electron systems in the quintet state is a single segment, and four-electron anti-bound states or four-electron bound states is absent. In the system exists two four-electron singlet states, and they are different origins. In the singlet states the essential spectra of four-electron systems is consists of the union no more three segments. Furthermore, in the system exists no more one four-electron anti-bound states or no more one bound states.

This article examines questions of unique solvability for an inverse boundary value problem to recover the coefficient and boundary regime of a nonlinear integro-differential equation with degenerate kernel. We propose a novel method of degenerate kernel for the case of inverse boundary value problem for the considered ordinary integro-differential equation of second order. By the aid of denotation, the integro-differential equation is reduced to a system of algebraic equations. Solving this system and using additional conditions, we obtained a system of two nonlinear equations with respect to the first two unknown quantities and a formula for determining the third unknown quantity. We proved the single-value solvability of this system using the method of successive approximations.

The direct Lyapunov method is used to prove the absolute linear instability of steadystate plane-parallel shear flows of an inviscid stratified incompressible fluid in the gravity field with respect to plane perturbations both in the Boussinesq and non-Boussinesq approximations. A strict description is given for the applicability of the known necessary condition for linear instability of steady-state plane-parallel shear flows of an ideal nonuniform (by density) incompressible fluid in the gravity field both in the Boussinesq and non-Boussinesq approximations (the Miles theorem). Analytical examples of illustrative character are constructed.

Conditions for the existence and uniqueness of a solution to a problem for a functional differential equation are presented. A special case of this equation is a functional differential equation derived previously by the authors for the distribution density of the brightness of light in interstellar space in the case of several clouds uniformly distributed in the equatorial plane of the Galaxy and having different optical densities.