# Volume 63, № 11, 2011

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

Goncharenko M. V., Khruslov E. Ya.

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1443-1459

We consider a homogenized system of equations that is a macroscopic model of nonstationary vibrations of an elastic medium with a large number of small cavities filled with viscous incompressible liquid (wet elastic medium). It is proved that the solution of the initial boundary-value problem for this system in a bounded domain $\Omega$ tends to zero in the metric of $L_2(\Omega)$ exponentially with time.

### Two-dimensional pseudospherical surfaces with degenerate Bianchi transformation

Gor'kavyi V. A., Nevmerzhitskaya E. N.

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1460-1468

We classify two-dimensional pseudospherical surfaces with degenerate Bianchi transformation in a multidimensional Euclidean space.

### Fundamental solutions of the Cauchy problem for some degenerate parabolic equations of the Kolmogorov type

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1469-1500

Fundamental solutions of the Cauchy problem for three classes of degenerate parabolic equations are investigated. These equations are natural generalizations of the classical Kolmogorov equation of the diffusion with the inertia.

### On the representation of groups approximated by finite *p*-groups

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1501-1511

Faithful triangular representations of finite Kaloujnine $p$-groups are presented. This allows us to obtain a triangular representation of the projective limit $P_p$ of groups $P_{p,n}$. The obtained representation is studied by using a language of matrix templates specially developed for this purpose. As an example, we present a triangular representation of the well-known self-similar Gupta – Sidki 3-group.

### Skitovich-Darmois theorem for finite Abelian groups

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1512-1523

Let $X$ be a finite Abelian group, let $\xi_i,\; i = 1, 2, . . . , n,\; n ≥ 2$, be independent random variables with values in $X$ and distributions $\mu_i$, and let $\alpha_{ij},\; i, j = 1, 2, . . . , n$, be automorphisms of $X$. We prove that the independence of n linear forms $L_j = \sum_{i=1}^{n} \alpha_{ij} \xi_i$ implies that all $\mu_i$ are shifts of the Haar distributions on some subgroups of the group $X$. This theorem is an analog of the Skitovich – Darmois theorem for finite Abelian groups.

### On the Jackson theorem for periodic functions in metric spaces with integral metric. II

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1524-1533

In the spaces $L_{\psi}(T^m)$ of periodic functions with metric $\rho(f, 0)_{\psi} = \int_{T^m}\psi(|f(x)|)dx$ , where $\psi$ is a function of the type of modulus of continuity, we study the direct Jackson theorem in the case of approximation by trigonometric polynomials. It is proved that the direct Jackson theorem is true if and only if the lower dilation index of the function $\psi$ is not equal to zero.

*Q* -permutable subgroups of finite groups

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1534-1543

A subgroup $H$ of a group $G$ is called $Q$-permutable in $G$ if there exists a subgroup $B$ of $G$ such that (1) $G = HB$ and (2) if $H_1$ is a maximal subgroup of $H$ containing $H_{QG}$, then $H_1B = BH_1 < G$, where $H_{QG}$ is the largest permutable subgroup of $G$ contained in $H$. In this paper we prove that: Let $F$ be a saturated formation containing $U$ and $G$ be a group with a normal subgroup $H$ such that $G/H \in F$. If every maximal subgroup of every noncyclic Sylow subgroup of $F∗(H)$ having no supersolvable supplement in $G$ is $Q$-permutable in $G$, then $G \in F$.

### A common fixed point for generalized (ψ, φ)_{f,g} - weak contractions

Abbas M., Parvaneh V., Razani A.

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1544-1554

We extend the common fixed point theorem established by Zhang and Song in 2009 to generalized (ψ, φ)_{f,g} weak contractions. Moreover, we give an example that illustrates the main result. Finally, some common fixed
point results are obtained for mappings satisfying a contraction condition of the integral type in complete
metric spaces.

### On weakly *s* -normal subgroups of finite groups

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1555-1564

Assume that $G$ is a finite group and $H$ is a subgroup of $G$. We say that $H$ is $s$-permutably imbedded in $G$ if, for every prime number p that divides $|H|$, a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $s$-permutable subgroup of $G$; a subgroup $H$ is $s$-semipermutable in $G$ if $HG_p = G_pH$ for any Sylow $p$-subgroup $G_p$ of $G$ with $(p, |H|) = 1$; a subgroup $H$ is weakly $s$-normal in $G$ if there are a subnormal subgroup $T$ of $G$ and a subgroup $H_{*}$ of $H$ such that $G = HT$ and $H \bigcap T ≤ H_{*}$, where $H_{*}$ is a subgroup of $H$ that is either $s$-permutably imbedded or $s$-semipermutable in $G$. We investigate the influence of weakly $s$-normal subgroups on the structure of finite groups. Some recent results are generalized and unified.

### Optimization of interval formulas for approximate integration of set-valued functions monotone with respect to inclusion

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1565-1569

The best interval quadrature formula is obtained for the class of convex set-valued functions defined on the segment [0, 1] and monotone with respect to inclusion.

### Approximation for absolutely continuous functions by Stancu Beta operators

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1570-1576

In this paper, we obtain an exact estimate for the first-order absolute moment of Stancu Beta operators by means of the Stirling formula and integral operations. Then we use this estimate for establishing a theorem on approximation of absolutely continuous functions by Stancu Beta operators.