# Volume 50, № 5, 1998

### A limit theorem for mixing processes subject to rarefaction. II

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 603–612

The limit theorem proved in the first part of this paper is applied to the well-known schemes of processes subject to rarefaction arising in queuing theory, mathematical biology, and in problems for counters.

### Integral equations in the linear theory of elasticity in semiinfinite domains

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 613–622

We investigate linear integral equations on a semiaxis that appear in the course of construction of solutions of boundary-value problems in the theory of elasticity in such domains as a semiinfinite strip or a cylinder. By using the Mellin transformation and the theory of perturbations of linear operators, we establish general results concerning the solvability and asymptotic properties of solutions of the equations considered. We give examples of application of the general statements obtained to specific integral equations in the theory of elasticity.

### Invariant symmetric restrictions of a self-adjoint operator. I

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 623–631

We prove necessary and sufficient conditions of the *S*-invariance of a subset dense in a separable Hilbert space *H*.

### Estimates of eigenvalues of self-adjoint boundary-value problems with periodic coefficients

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 632–638

We consider self-adjoint boundary-value problems with discrete spectrum and coefficients periodic in a certain coordinate. We establish upper bounds for eigenvalues in terms of the eigenvalues of the corresponding problem with averaged coefficients. We illustrate the results obtained in the case of the Hill vector equation and for circular and rectangular plates with periodic coefficients.

### Diametral theory of algebraic surfaces and geometric theory of invariants of groups generated by reflections. I

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 639–653

A review of the current state of the diametral theory of algebraic hypersurfaces in the real Euclidean space is given.

### On solutions of a second-order quasilinear differential system representable by Fourier series with slowly varying parameters

Kostin A. V., Shchegolev S. A.

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 654–664

For a second-order quasilinear differential system whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, we prove that, under certain conditions, there exists a particular solution with a similar structure in the case of purely imaginary roots of the characteristic equation for the matrix of coefficients of the linear part.

### On the application of Ateb-functions to the construction of an asymptotic solution of the perturbed nonlinear Klein-Gordon equation

Mitropolskiy Yu. A., Sokil B. I.

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 665–670

For the perturbed nonlinear Klein-Gordon equation, we construct an asymptotic solution by using Ateb-functions. We consider autonomous and nonautonomous cases.

### On an estimate of the domain of attraction for systems with aftereffect

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 671–678

We consider the problem of construction of the domain of attraction for systems with aftereffect. The structure of solutions of the Cauchy problem in the vector space of states of a system is clarified with examples. We describe a constructive method for the estimation of the domain of attraction from the inside with the use of the Lyapunov functions. This method is used for the estimation of the effect of delay in a control device in the process of solution of the problem of uniaxial orientation of a spacecraft.

### Transformation of formal expansions of solutions of linear differential equations in a parameter into continued *RITZ*-fractions

Rozhankovskaya M. I., Syavavko M. S.

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 679–686

We use the apparatus of *RITZ*-fractions to improve the convergence of series that represent the formal solution of linear differential equations with parameter under boundary or initial conditions. We establish conditions for the existence of this solution in the case where the parameter of the equation tends to infinity. The case of a small parameter is also considered.

### Optimal control over evolution stochastic systems and its application to stochastic models of financial mathematics

Biirdeinyi A. G., Svishchuk A. V.

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 687–698

We consider problems of optimal stabilization of controlled evolution stochastic systems in semi-Markov media and their application to financial stochastic models.

### Information complexity of projection algorithms for the solution of Fredholm equations of the first kind. I

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 699–711

We construct a new system of discretization of the Fredholm integral equations of the first kind with linear compact operators *A* and free terms from the set Range (*A(A*A*)^{V}), v > 1/2. The approach proposed enables one to obtain the optimal order of error on such classes of equations by using a considerably smaller amount of discrete information as compared with standard schemes.

### Contour-solid properties of finely holomorphic functions

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 712–723

We prove contour-solid theorems for holomorphic functions defined in finely open sets of the complex plane.

### A theorem of the Phragmén-Lindelöf type for solutions of an evolution equation of the second order with respect to time variable

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 724–731

We consider a solution *u(x, t)* of the general linear evolution equation of the second order with respect to time variable given on the ball Π(*T*) = {(*x,t): xε R* ^{n}, *t* ε [0, *T*]} and study the dependence of the behavior of this solution on the behavior of the functions at infinity.

### On the Sendov problem on the Whitney interpolation constant

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 732–734

For a function *ƒ* continuous on [0, 1] and satisfying the equalities \(f(0) = f\left( {\frac{1}{3}} \right) = f\left( {\frac{2}{3}} \right) = f(1) = 0,\) we prove that \(|f(x)| \le 2\omega _4 \left( {\frac{1}{4},f} \right),{\rm{ }}x \in [0,1],\) where ω_{4}(*t,ƒ*) is the fourth modulus of smoothness of the function *ƒ*.

### Approximation of functions from Weyl-Nagy classes by Zygmund averages

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 735–738

For functions from the Weyl-Nagy classes, in the uniform metric, we calculate exact-order estimates of the deviations of the Zygmund sums.

### On finite groups with cyclic Abelian subgroups

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 739–741

Solvable and minimal unsolvable finite groups with cyclic Abelian subgroups are constructively described.

### On subgroups lifting modulo central commutant

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 742–745

We consider a finitely generated group *G* with the commutant of odd order \(p_1^{n_1 } \ldots p_s^{n_s } \) located at the center and prove that there exists a decomposition of *G/G′* into the direct product of indecomposable cyclic groups such that each factor except at most *n* _{l} + ... + *n* _{s} factors lifts modulo commutant.