# Volume 58, № 10, 2006

### $X$-permutable maximal subgroups of Sylow subgroups of finite groups

Guo W., Shum K. P., Skiba A. N.

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1299–1309

We study finite groups whose maximal subgroups of Sylow subgroups are permutable with maximal subgroups.

### Finitary groups and Krull dimension over the integers

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1310–1325

Let $M$ be any Abelian group. We make a detailed study for reasons explained in the Introduction of the normal subgroup $$F_\infty Aut M = \{ g \in Aut M: M(g - 1) is\;a \;minimax\; group\}$$ of the automorphism group $Aut M$ of $M$. The conclusions, although slightly weaker than one would hope, in that they do not fully explain the common behavior of the finitary and the Artinian-finitary subgroups of $Aut M$, are certainly stronger than one might reasonably expect. Our main focus is on residual properties and unipotence.

### On small motions of a “liquid-gas” system in a bounded domain

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1326–1334

We study small motions and free oscillations of a compressible stratified liquid, the structure of the spectrum, and the basis property of a system of eigenvectors and obtain asymptotic relations for eigenvalues.

### On estimate for numerical radius of some contractions

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1335–1339

For the numerical radius of an arbitrary nilpotent operator *T* on a Hilbert space *H*, Haagerup and de la Harpe proved the inequality \(w(T) \leqslant \left\| T \right\|cos\frac{\pi }{{n + 1}}\), where $n \geq 2$ is the nilpotency order of the operator *T*. In the present paper, we prove a Haagerup-de la Harpe-type inequality for the numerical radius of contractions from more general classes.

### Random processes in Sobolev-Orlicz spaces

Kozachenko Yu. V., Yakovenko T. O.

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1340–1356

We establish conditions under which the trajectories of random processes from Orlicz spaces of random variables belong with probability one to Sobolev-Orlicz functional spaces, in particular to the classical Sobolev spaces defined on the entire real axis. This enables us to estimate the rate of convergence of wavelet expansions of random processes from the spaces $L_P({\Omega})$ and $L_2({\Omega})$ in the norm of the space $L_q(\mathbb{R})$.

### On exact Bernstein-type inequalities for splines

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1357–1367

We establish new exact Bernstein-type and Kolmogorov-type inequalities. The main result of this work is the following exact inequality for periodic splines $s$ of order $r$ and defect 1 with nodes at the points $iπ/n, i ∈ Z, n ∈ N:$ $$\left\| {s^{(k)} } \right\|_q \leqslant n^{k + 1/p - 1/q} \frac{{\left\| {\varphi _{r - k} } \right\|_q }}{{\left\| {\varphi _r } \right\|_p }}\left\| s \right\|_p ,$$ where $k, r ∈ N, k < r, p = 1$ or $p = 2, q > p$, and $ϕr$ is the perfect Euler spline of order $r$.

### Asymptotic equivalence of solutions of linear Itô stochastic systems

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1368–1384

We investigate the problem of the asymptotic equivalence of stochastic systems of linear ordinary equations and stochastic equations in the sense of mean square and with probability one.

### On minimization of one integral functional by the Ritz method

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1385–1394

Using the variational method, we investigate a nonlinear problem with a Bernoulli condition in the form of an inequality on a free boundary. We prove a solvability theorem and establish the convergence of an approximate solution obtained by the Ritz method to the exact solution in certain metrics.

### Best approximations of the classes $B_{p,\,\theta}^{r}$ of periodic functions of many variables in uniform metric

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1395–1406

We obtain estimates exact in order for the best approximations of the classes $B_{\infty,\,\theta}^{r}$ of periodic functions of two variables in the metric of $L_{\infty}$ by trigonometric polynomials whose spectrum belongs to a hyperbolic cross. We also investigate the best approximations of the classes $B_{p,\,\theta}^{r},\quad 1 \leq p < \infty$, of periodic functions of many variables in the metric of $L_{\infty}$ by trigonometric polynomials whose spectrum belongs to a graded hyperbolic cross.

### Long-range order in quantum lattice systems of linear oscillators

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1407–1424

The existence of the ferromagnetic long-range order is proved for equilibrium quantum lattice systems of linear oscillators whose potential energy contains a strong ferromagnetic nearest-neighbor (nn) pair interaction term and a weak nonferromagnetic term under a special condition on a superstability bound. It is shown that the long-range order is possible if the mass of a quantum oscillator and the strength of the ferromagnetic nn interaction exceed special values. A generalized Peierls argument and a contour bound, proved with the help of a new superstability bound for correlation functions, are our main tools.

### Asymptotic normality of a discrete procedure of stochastic approximation in a semi-Markov medium

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1425–1433

We obtain sufficient conditions for the asymptotic normality of a jump procedure of stochastic approximation in a semi-Markov medium using a compensating operator of an extended Markov renewal process. The asymptotic representation of the compensating operator guarantees the construction of the generator of a limit diffusion process of the Ornstein-Uhlenbeck type.

### Necessary condition for the stabilizability of nonlinear systems with respect to a part of variables in the class of discontinuous controls

Kovalev A. M., Kravchenko N. V., Nespirnyi V. N.

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1434–1440

We investigate the problem of the existence of a discontinuous feedback that guarantees the stabilization of a nonlinear control system with respect to a part of variables. A solution of the system is defined in the Filippov sense. We establish a necessary condition for stabilization with respect to a part of variables in the class of discontinuous controls, which generalizes the Ryan condition to the case of stabilization with respect to a part of variables. An example of a mechanical system that cannot be stabilized with respect to a part of variables is considered.