# Volume 63, № 12, 2011

### On the exponential decay of vibrations of damped elastic media

Vakarchuk M. B., Vakarchuk S. B.

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1579-1601

Exact inequalities of the Kolmogorov type are obtained in Hardy Banach spaces for functions of one complex variable analytic in the unit disk and functions of two complex variables analytic in the unit bidisk. We also present applications of these inequalities to problems of the theory of approximation of analytic functions of one and two complex variables.

### Approximation of (ψ, β)-differentiable functions of low smoothness by biharmonic Poisson integrals

Kharkevych Yu. I., Zhyhallo K. M.

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1602-1622

We solve the Kolmogorov – Nikol’skii problem for biharmonic Poisson integrals on the classes of (ψ, β)- differentiable periodic functions of low smoothness in the uniform metric.

### On *ss*-quasinormal and weakly *s*-supplemented subgroups of finite groups

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1623-1631

Suppose that $G$ is a finite group and $H$ is a subgroup of $G$. $H$ is called $ss$-quasinormal in $G$ if there is a subgroup $B$ of $G$ such that $G = HB$ and $H$ permutes with every Sylow subgroup of $B$; $H$ is called weakly $s$-supplemented in G if there is a subgroup T of G such that $G = HT$ and $H \bigcap T \leq H_{sG}$, where $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $s$-quasinormal in $G$. In this paper we investigate the influence of $ss$-quasinormal and weakly $s$-supplemented subgroups on the structure of finite groups. Some recent results are generalized and unified.

### Homogenization of a quasilinear parabolic problem with different alternating nonlinear Fourier boundary conditions in a two-level thick junction of the type 3:2:2

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1632-1656

We investigate the asymptotic behavior of a solution of a quasilinear parabolic boundary-value problem in a two-level thick junction of the type 3:2:2. This junction consists of a cylinder on which thin disks of variable thickness are $\varepsilon$-periodically threaded. The thin disks are divided into two levels, depending on their geometric structure and the conditions imposed on their boundaries. In this problem, we consider different alternating inhomogeneous nonlinear Fourier conditions. Moreover, the Fourier conditions depend on additional perturbation parameters. We prove theorems on the convergence of a solution of this problem as $\varepsilon \rightarrow 0$ for different values of these parameters.

### Inequalities for trigonometric polynomials in spaces with integral metric

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1657-1671

In the spaces $L_{\psi}(T)$ of periodic functions with metric $\rho( f , 0)_{\psi} = \int_T \psi (| f (x) |) dx $, where $\psi$ is a function of the modulus-of-continuity type, we investigate analogs of the classic Bernstein inequalities for the norms of derivatives and increments of trigonometric polynomials.

### Existence and exponential stability of periodic solution for fuzzy BAM neural networks with periodic coefficient

Dai-xi Liao, Li-hui Yang, Zhang Qian-hong

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1672-1684

A class of fuzzy bidirectional associated memory (BAM) networks with periodic coefficients is studied. Some sufficient conditions are established for the existence and global exponential stability of a periodic solution of such fuzzy BAM neural networks by using a continuation theorem based on the coincidence degree and the Lyapunov-function method. The sufficient conditions are easy to verify in pattern recognition and automatic control. Finally, an example is given to show the feasibility and efficiency of our results.

### Existence and exponential stability of periodic solution for fuzzy BAM neural networks with periodic coefficient

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1685-1698

We obtain conditions for the existence of solutions of nonlinear differential equations in the space of functions bounded on the axis by using a local linear approximation of these equations.

### Approximation of analytic functions by Bessel functions of fractional order

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1699-1709

We solve the inhomogeneous Bessel differential equation $$x^2y''(x) + xy'(x) + (x^2 - \nu^2)y(x) = \sum^{\infty}_{m=0} a_mx^m$$, where $\nu$ is a positive nonintegral number, and use this result for the approximation of analytic functions of a special type by the Bessel functions of fractional order.

### Generalization of Rubel's result for operators

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1710-1716

We describe all pairs of linear continuous operators that act in spaces of functions analytic in domains and satisfy a relation that is an operator analog of the Rubel equation.

### Regularity of growth of fourier coefficients of entire functions of improved regular growth

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1717-1721

We establish a criterion for the improved regular growth of entire functions of positive order with zeros on a finite system of half-lines in terms of their Fourier coefficients.

### Index of volume 63 of „Ukrainian Mathematical Journal”

Ukr. Mat. Zh. - 2011. - 63, № 12. - pp. 1724- 1729