# Volume 51, № 11, 1999

### Parametrix method for a parabolic equation on a Riemannian manifold

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1443–1448

We present a scheme of construction of a fundamental solution of a parabolic equation on a Riemannian manifold with nonpositive curvature.

### On the boundedness of the total variation of the logarithm of a Blaschke product

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1449–1455

We establish that, for a Blaschke product B(*z*) convergent in the unit disk, the condition - ∞ < \(\smallint _0^1 \log (1 - t)n(t,B)dt\) is sufficient for the total variation of log*B* to be bounded on a circle of radius*r*, 0 <*r* < 1. For products B(*z*) with zeros concentrated on a single ray, this condition is also necessary. Here, n(*t, B*) denotes the number of zeros of the function*B* (*z*) in a disk of radius*t*.

### On the best approximation of classes $W^r H^{ω}$ by algebraic polynomials in the space $L_1$

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1456-1466

We obtain an asymptotic equality for the best approximation of the classes $W^r H^{ω}$ by algebraic polynomials in the space $L_1$ for convex upwards, regularly varying moduli of continuity.

### A singularly perturbed spectral problem for a biharmonic operator with Neumann conditions

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1467–1475

We study a mathematical model of a composite plate that consists of two components with similar elastic properties but different distributions of density. The area of the domain occupied by one of the components is infinitely small as $ε → 0$. We investigate the asymptotic behavior of the eigenvalues and eigenfunctions of the boundary-value problem for a biharmonic operator with Neumann conditions as $ε → 0$. We describe four different cases of the limiting behavior of the spectrum, depending on the ratio of densities of the medium components. In particular, we describe the so-called Sanches-Palensia effect of local vibrations: A vibrating system has a countable series of proper frequencies infinitely small as $ε → 0$ and associated with natural forms of vibrations localized in the domain of perturbation of density.

### Stability analysis with respect to two measures of impulsive systems under structural perturbations

Martynyuk A. A., Stavroulakis I. P.

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1476–1484

The asymptotic stability with respect to two measures of impulsive systems under structural perturbations is investigated. Conditions of asymptotic (ρ_{0}, ρ)-stability of the system in terms of the fixed signs of some special matrices are established.

### On convergence classes of Dirichlet series

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1485–1494

We establish conditions for the coefficients of a Dirichlet series under which this series belongs to a certain class of convergence.

### Approximation of one class of differentiable functions by piecewise-Hermitian*L*-splines

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1495–1504

The exact values of the upper bounds of deviations of piecewise-Hermitian*L*-splines are found for certain classes of functions determined by systems of linear differential operators with continuous coefficients.

### Approximation of continuous functions defined on the real axis by generalized Zygmund operators

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1505–1512

We establish estimates for upper bounds of deviations of generalized Zygmund operators on the classes of continuous $(ψ, β)$-differentiable functions defined on the real axis.

### Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable Hamiltonian systems. II

Prykarpatsky Ya. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1513–1528

By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville—Arnold integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytical 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. We also consider the problem of existence of adiabatic invariants associated with a slowly perturbed Hamiltonian system.

### Almost layer finiteness of a periodic group without involutions

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1529–1533

We prove a theorem that characterizes the class of almost layer finite groups in the class of periodic groups without involutions: If the normalizer of any nontrivial finite subgroup of a periodic conjugate biprimitive finite group without involutions is almost layer finite, then the group itself is almost layer finite.

### A priori estimates of solutions of linear parabolic problems with coefficients from Sobolev spaces

Romanenko I. B., Skrypnik I. V.

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1534–1548

We consider the general initial-boundary-value problem for a linear parabolic equation of arbitrary even order in anisotropic Sobolev spaces. We prove the existence and uniqueness of a solution and establish an*a priori* estimate for it.

### Approximation of locally integrable functions on the real line

Stepanets O. I., Wang Kunyang, Zhang Xirong

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1549–1561

We introduce the notion of generalized \(\bar \psi \) -derivatives for functions locally integrable on the real axis and investigate problems of approximation of the classes of functions determined by these derivatives with the use of entire functions of exponential type.

### On the existence of solutions for a differential inclusion of fractional order with upper-semicontinuous right-hand side

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1562–1565

We prove a theorem on the existence of solutions of the differential inclusion \(D_0^\alpha u(x) \in F(x,u(x)), u_{1 - \alpha } (0) = \gamma , \left( {u_{1 - \alpha } (x) = 1_0^{1 - \alpha } u(x)} \right),\) where \(\alpha \in (0,1), D_0^\alpha u(x) \left( {1_0^{1 - \alpha } u(x)} \right)\) is the Riemann-Liouville derivative (integral) of order α, and the multivalued mapping*F(x, u)* is upper semicontinuous in*u*.

### Quasiconformal mappings and radii of normal systems of neighborhoods

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1566–1568

We establish a new geometric criterion for plane homeomorphisms to belong to the class of*q*-quasiconforrnal mappings.

### On synchronization of symmetrically perturbed systems with quadratic nonlinearity

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1569–1573

We establish several synchronization conditions for the dynamics of symmetrically perturbed systems with quadratic nonlinearity.

### On a smooth solution of a nonlinear periodic boundary-value problem

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1574–1576

We establish conditions for the existence of a smooth solution of a quasilinear hyperbolic equation*u* _{ tt } - u_{xx} = ƒ*(x, t, u, u, u* _{x}),*u* (0,*t) = u* (π,*t*) = 0,*u (x, t+ T) = u (x, t), (x, t*) ∈ [0, π] ×*R*, and prove a theorem on the existence and uniqueness of a solution.

### On the extension of even-positive-definite functions of one and two variables

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1577–1581

We prove theorems on the extension of even-positive-definite functions from a finite interval to the entire axis and from a strip onto the entire plane.

### Differentiability of integrals of real functions with respect to $L_0$-valued measures

Ukr. Mat. Zh. - 1999. - 51, № 11. - pp. 1582-1585

We obtain conditions for the convergence of expressions $(\mu (A))^{ - 1} \smallint _A fd\mu$ in $L_0$ as the set $A$ decreases.