# Volume 48, № 2, 1996

### Turning point in a system of differential equations with analytic operator

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 147-160

We construct uniform asymptotics for a solution of a system of singularly perturbed differential equations with turning point. We consider the case where the boundary operator analytically depends on a small parameter.

### Global solutions of a two-dimensional initial boundary-value problem for a system of semilinear magnetoelasticity equations

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 161-167

We prove the theorem on the existence and uniqueness of global solutions of a system of semilinear magnetoelasticity equations in a two-dimensional space.

### Approximation properties of systems of exponentials in one space of analytic functions

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 168-183

We obtain a criterion of completeness of a system of exponentials in the Hardy-Smirnov spaces in unbounded convex polygons and study the properties of incomplete systems of exponentials.

### Representation and investigation of solutions of a nonlocal boundary-value problem for a system of partial differential equations

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 184-194

We study the boundary-value problem for a system of partial differential equations with constant coefficients with conditions nonlocal in time. By using a metric approach, we prove the well-posedness of the problem in the scale of Sobolev spaces of functions periodic in space variables. By using matrix calculus, we construct an explicit representation of a solution.

### On the maximum principle for ultraparabolic equations

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 195-201

We prove the maximum principle and various modifications of it for one class of degeneration of parabolic equations.

### Space-time localization in problems with free boundaries for a nonlinear second-order equation

Berezovsky A. A., Mitropolskiy Yu. A., Shkhanukov-Lafishev M. Kh.

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 202-211

For thermal and diffusion processes in active media described by nonlinear evolution equations, we study the phenomena of space localization and stabilization for finite time.

### On one approach to the discretization of the lavrent’ev method

Pereverzev S. V., Solodkii S. G.

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 212-219

We propose a new scheme of discretization of the Lavrent’ev method for operator equations of the first kind with self-adjoint nonnegative operators of certain “smoothness.” This scheme is more economical in the sense of the amount of used discrete information as compared with traditional approaches.

### Nonlinear integrable systems related to the elliptic lie—baxter algebra

Pritula N. N., Sidorenko Yu. M., Strampp W.

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 220-235

We construct a hierarchy of Poisson Hamiltonian structures related to an “elliptic” spectral problem and determine the generating operators for the equation of asymmetric chiral *0* (3) — field.

### Approximation of classes of analytic functions by algebraic polynomials and kolmogorov widths

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 236-250

We obtain estimates of the best polynomial approximations, uniform in the closure $B$ of Faber domains of the complex plane $ℂ$, for functions continuous in $B$ and defined by Cauchy-type integrals with densities possessing certain generalized differential properties. We establish estimates exact in order for the Kolmogorov widths of classes of such functions in relevant functional spaces.

### On Isotopes of Groups. III

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 251-259

We describe normal congruences of group isotopes, establish criteria of homomorphism and isomorphism, and select the methods for description of isotopes up to isomorphism. In addition, we establish a criterion for a subset to be a subquasigroup of a group isotope and describe subquasigroups of certain classes of group isotopes. The obtained results are applied to the investigation of left distributive quasi-groups.

### Strong summability of orthogonal expansions of summable functions. I

Lasuriya R. A., Stepanets O. I.

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 260-277

We study the problem of strong summability of Fourier series in orthonormal systems of polynomial-type functions and establish local characteristics of the points of strong summability of series of this sort for summable functions. It is shown that the set of these points is a set of full measure in the region of uniform boundedness of systems under consideration.

### On the erugin and floquet-lyapunov theorems for countable systems of difference equations

Teplinsky Yu. V., Teplins’kyi O. Yu.

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 278-284

For linear difference equations in the space of bounded number sequences, we prove an analog of the Erugin theorem on reducibility and present sufficient conditions for the reducibility of countable linear systems of difference equations with periodic coefficients.

### Evgenii Yakovlevich Remez

Dzyadyk V. K., Mitropolskiy Yu. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 285-286

### Application of ateb-functions to the construction of solutions of some nonlinear partial differential equations

Ukr. Mat. Zh. - 1996. - 48, № 2. - pp. 287-288

We construct asymptotic approximations of one-frequency solutions of some nonlinear partial differential equations by using periodic Ateb-functions.