# Volume 51, № 10, 1999

### On the order of growth of rectangular partial sums of double orthogonal series

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1299–1310

We obtain estimates of the order of growth of rectangular partial sums of double orthogonal series and establish their unimprovability on the set of all double orthogonal systems.

### A multipoint problem with multiple nodes for linear hyperbolic equations

Beresnevich V. V., Bernik V. I., Ptashnik B. I., Vasylyshyn P. B.

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1311–1316

We establish conditions for the unique solvability of a multipoint (with respect to the time coordinate) problem with multiple nodes for linear hyperbolic equations with constant coefficients in the class of functions periodic in the space variable. We prove metric statements concerning lower bounds of small denominators that appear in the course of construction of a solution of the problem.

### Eigenvalue problems with discontinuous eigenfunctions and their numerical solutions

Deineka V. S., Sergienko I. V., Skopetskii V. V.

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1317–1323

We consider new eigenvalue problems with discontinuous eigenfunctions and construct computational algorithms whose accuracy is not worse than the accuracy of analogous known algorithms for problems with smooth eigenfunctions.

### Primary graded groups with complementable non-Frattini subgroups

Chernikov N. S., Dovzhenko S. A.

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1324–1333

We describe primary graded groups (in particular, locally graded, RN-groups) with complementable non-Frattini subgroups.

### Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1334–1341

Pseudodifferential equations of the form $v(D_{\chi})y = f$ (where $v$ is a function holomorphic at zero and $D_{\chi}$ is a pseudodifferential operator) are studied on spaces of test functions of non-Gaussian infinite-dimensional analysis. The results obtained are applied to construct a generalized translation operator $T^{\chi}_y = \chi(\langle y, D_{\chi}\rangle)$ the already mentioned spaces and to study its properties. In particular, the associativity, the commutativity, and another properties of $T^{\chi}_y$ which are analogs of the classical properties of a generalized translation operator.

### Bifurcation of the state of equilibrium in the system of nonlinear parabolic equations with transformed argument

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1342–1351

We consider a system of nonlinear parabolic equations with transformed argument and prove the existence of integral manifolds. We investigate the bifurcation of an invariant torus from the state of equilibrium.

### On the best approximation of functions of $n$ variables

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1352–1359

We propose a new approach to the solution of the problem of the best approximation, by a certain subspace for functions of*n* variables determined by restrictions imposed on the modulus of, continuity of certain partial derivatives. This approach is based on the duality theorem and on the representation of a function as a countable sum of simple functions.

### Large deviations for Bayes discrimination of a finite number of simple hypotheses

Gabriel' L. A., Lin'kov Yu. N.

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1360–1367

We consider the problem of discrimination of a finite number of simple hypotheses in the general scheme of statistical experiments. Under conditions of the validity of theorems on large deviations for the logarithm of likelihood ratio, we investigate the asymptotic behavior of probabilities of errors of the Bayes criterion. We obtain the asymptotics of the amount of Shannon information contained in an observation and in the Bayes criterion.

### Structure of a general solution of systems of nonlinear difference equations

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1368–1378

We investigate the structure of a general solution of systems of nonlinear difference equations with continuous argument in a neighborhood of the state of equilibrium.

### Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. I

Prikarpatskii Ya. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1379–1390

By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville-Arnol’d integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytic method for the investigation of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically the structure of quasiperiodic solutions of the Hamiltonian system under consideration.

### On quasiconformal mappings corresponding to the beltrami equation

Samkharadze I. G., Samsoniya Z. V.

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1391–1397

By using methods of integral equations, we investigate problems of conformal and quasiconformal mappings of close domains.

### Optimization of projection schemes of digitization of ill-posed problems

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1398–1410

We construct new projection schemes of digitization of ill-posed problems, which are optimal in the sense of the amount of discrete information used. We establish that the application of self-adjoint projection schemes to digitization of equations with self-adjoint operators is not optimal.

### On the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1411–1416

For nonholonomic systems, we introduce the notion of the function of Hamiltonian action, with the use of which we investigate the stability of nonholonomic systems in the case where the equilibrium state under consideration is a critical point of the corresponding Lagrangian (Whittaker system).

### Lyapunov transformation and stability of differential equations in banach spaces

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1417–1424

A sufficient condition of exponential stability of regular linear systems with bifurcation on a Banach space is proved.

### On locally graded groups with minimality condition for a certain system of nonhypercentral subgroups

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1425–1430

We characterize groups without nontrivial perfect sections (in particular, solvable groups) with the minimality condition for the subgroups without hypercentral subgroups of finite index.

### Bott functions and the euler characteristic

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1431–1432

In terms of the Euler characteristic, we obtain the condition of existence of Bott functions on differentiable manifolds that have a set of critical points formed by connected homeomorphic submanifolds.

### On the application of numerical methods to the solution of nonlinear second-order differential equations with random deviations of argument

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1433–1441

We consider the application of the Krylov-Bogolyubov-Mitropol’skii asymptotic method and Runge-Kutta methods to the investigation of oscillating solutions of quasilinear second-order differential equations with random deviations of argument. For specific equations, we obtain approximate numerical solutions and characteristics of random oscillations.