# Volume 58, № 12, 2006

### Order reduction for a system of stochastic differential equations with a small parameter in the coefficient of the leading derivative. Estimate for the rate of convergence

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1587–1601

In the metric $\rho(X, Y) = (\sup\limits_{0 \leq t \leq T} M|X(t) - Y(t)|^2)^{1/2} $ for an ordinary stochastic differential equation of order $p \geq 2$ with small parameter of the higher derivative, we establish an estimate of the rate of convergence of its solution to a solution of stochastic equation of order $p - 1$.

### Jacobi fields on a Riemann manifold

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1602–1613

Some properties of Jacobi fields on a manifold of nonpositive curvature are considered. As a result, we obtain relations for derivatives of one class of functions on the manifold.

### On small oscillations of a compressible stratified liquid

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1614–1623

We study the structure of the spectrum and the completeness and basis property of a system of eigenvectors.

### Problems for partial differential equations with nonlocal conditions. Metric approach to the problem of small denominators

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1624–1650

A survey of works of the authors and their disciples devoted to the investigation of problems with nonlocal conditions with respect to a selected variable in cylindrical domains is presented. These problems are considered for linear equations and systems of partial differential equations that, in general, are ill posed in the Hadamard sense and whose solvability in certain scales of functional spaces is established for almost all (with respect to Lebesgue measure) vectors composed of the coefficients of the problem and the parameters of the domain.

### Jacobi matrices associated with the inverse eigenvalue problem in the theory of singular perturbations of self-adjoint operators

Koshmanenko V. D., Tuhai H. V.

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1651–1662

We establish the relationship between the inverse eigenvalue problem and Jacobi matrices within the framework of the theory of singular perturbations of unbounded self-adjoint operators.

### Transfer of absolute continuity by a flow generated by a stochastic equation with reflection

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1663–1673

Let $\varphi_t(x),\quad x \in \mathbb{R}_+ $, be a value taken at time $t \geq 0$ by a solution of stochastic equation with normal reflection from the hyperplane starting at initial time from $x$. We characterize an absolutely continuous (with respect to the Lebesgue measure) component and a singular component of the stochastic measure-valued process $µ_t = µ ○ ϕ_t^{−1}$, which is an image of some absolutely continuous measure $\mu$ for random mapping $\varphi_t(\cdot)$. We prove that the contraction of the Hausdorff measure $H^{d-1}$ onto a support of the singular component is $\sigma$-finite. We also present sufficient conditions which guarantee that the singular component is absolutely continuous with respect to $H^{d-1}$.

### Best linear methods for the approximation of functions of the Bergman class by algebraic polynomials

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1674–1685

On concentric circles $T_{ϱ} = {z ∈ ℂ: ∣z∣ = ϱ},\; 0 ≤ ϱ < 1$, we determine the exact values of the quantities of the best approximation of holomorphic functions of the Bergman class $A_p, 2 ≤ p ≤ ∞$, in the uniform metric by algebraic polynomials generated by linear methods of summation of Taylor series. For $1 ≤ p < 2$, we establish exact order estimates for these quantities.

### Asymptotic normality of fluctuations of the procedure of stochastic approximation with diffusive perturbation in a Markov medium

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1686–1692

We consider the asymptotic normality of a continuous procedure of stochastic approximation in the case where the regression function contains a singularly perturbed term depending on the external medium described by a uniformly ergodic Markov process. Within the framework of the scheme of diffusion approximation, we formulate sufficient conditions for asymptotic normality in terms of the existence of a Lyapunov function for the corresponding averaged equation.

### Properties of entire solutions of differential equations

Sheremeta M. M., Sheremeta Z. M.

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1693–1703

We investigate the close-to-convexity and l-index boundedness of entire solutions of the differential equations $z^2w'' + \beta zw' + (\gamma z^2 — \beta)w = 0$ і$ zw'' + \beta w' + \gamma zw = 0$.

### Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1704–1714

The purpose of this paper is to present new oscillation theorems and nonoscillation theorems for the nonlinear Euler differential equation $t^2 x″' + g (x) = 0$. Here we assume that $x g(x) > 0$ if $x \neq 0$, but we do not necessarily require that $g (x)$ be monotone increasing. The obtained results are best possible in a certain sense. To establish our results, we use Sturm’s comparison theorem for linear Euler differential equations and phase plane analysis for a nonlinear system of Liénard type.

### Some extremal problems in the theory of nonoverlapping domains with free poles on rays

Bakhtin A. K., Targonskii A. L.

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1715–1719

We obtain new results on the maximization of the product of powers of the interior radii of pairwise disjoint domains with respect to certain systems of points in the extended complex plane.

### Some comments on regular and normal bitopological spaces

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1720–1724

Some properties of regular and normal bitopological spaces are established. The classes of sets inheriting the bitopological properties of regularity and normality are found. A theorem on a finite covering of pairwise normal spaces is proved. We also study the behavior of individual multivalued mappings, taking the axioms of bitopological regularity and normality into account.

### Index of volume 58 of „Ukrainian Mathematical Journal”

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1725-1728