# Volume 59, № 8, 2007

*c *^{*} -Supplemented subgroups and *p *-nilpotency of finite groups

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1011–1019

A subgroup $H$ of a finite group $G$ is said to be $c^{*}$-supplemented in $G$ if there exists a subgroup $K$ such that $G = HK$ and $H ⋂ K$ is permutable in $G$. It is proved that a finite group $G$ that is $S_4$-free is $p$-nilpotent if $N_G (P)$ is $p$-nilpotent and, for all $x ∈ G \backslash N_G (P)$, every minimal subgroup of $P ∩ P^x ∩ G^{N_p}$ is $c^{*}$-supplemented in $P$ and (if $p = 2$) one of the following conditions is satisfied:

(a) every cyclic subgroup of $P ∩ P^x ∩ G^{N_p}$ of order 4 is $c^{*}$-supplemented in $P$,

(b) $[Ω2(P ∩ P^x ∩ G^{N_p}),P] ⩽ Z(P ∩ G^{N_p})$,

(c) $P$ is quaternion-free, where $P$ a Sylow $p$-subgroup of $G$ and $G^{N_p}$ is the $p$-nilpotent residual of $G$.

This extends and improves some known results.

### On the invertibility of the operator *d*/*dt * + *A* in certain functional spaces

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1020–1025

We prove that the operator $\cfrac{d}{dt} + A$ constructed on the basis of a sectorial operator $A$ with spectrum in the right half-plane of $ℂ$ is continuously invertible in the Sobolev spaces $W_p^1 (ℝ, D_{α}),\; α ≥ 0$. Here, $D_{α}$ is the domain of definition of the operator $A^{α}$ and the norm in $D_{α}$ is the norm of the graph of $A^{α}$.

### Consistent estimator in multivariate errors-in-variables model in the case of unknown error covariance structure

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1026–1033

We consider a linear multivariate errors-in-variables model *AX* ? *B*, where the matrices *A* and *B* are observed with errors and the matrix parameter *X* is to be estimated. In the case of lack of information about the error covariance structure, we propose an estimator that converges in probability to *X* as the number of rows in *A* tends to infinity. Sufficient conditions for this convergence and for the asymptotic normality of the estimator are found.

### Constancy of upper-continuous two-valued mappings into the Sorgenfrey line

Fotii O. H., Maslyuchenko V. K.

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1034–1039

By using the Sierpiński continuum theorem, we prove that every upper-continuous two-valued mapping of a linearly connected space (or even a c-connected space, i.e., a space in which any two points can be connected by a continuum) into the Sorgenfrey line is necessarily constant.

### Weak convergence of integral functionals of random walks weakly convergent to fractional Brownian motion

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1040–1046

We consider a random walk that converges weakly to a fractional Brownian motion with Hurst index *H* > 1/2. We construct an integral-type functional of this random walk and prove that it converges weakly to an integral constructed on the basis of the fractional Brownian motion.

### Best approximation by holomorphic functions. Application to the best polynomial approximation of classes of holomorphic functions

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1047–1067

We find necessary and sufficient conditions under which a real function from $L_p(\mathbb{T}),\; 1 \leq p < \infty$, is badly approximable by the Hardy subspace $H_p^0: = \{f \in H_p:\; F(0) = 0\}$. In a number of cases, we obtain exact values for the best approximations in the mean of functions holomorphic in the unit disk by functions that are holomorphic outside the unit disk. We use obtained results in determining exact values of the best polynomial approximations and га-widths of some classes of holomorphic functions. We find necessary and sufficient conditions under which the generalized Bernstein inequality for algebraic polynomials on the unit circle is true.

### On the boundary behavior of imbeddings of metric spaces into a Euclidean space

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1068–1074

We investigate the boundary behavior of so-called *Q*-homeomorphisms with respect to a measure in some metric spaces. We formulate a series of conditions for the function *Q*(*x*) and the boundary of the domain under which any *Q*-homeomorphism with respect to a measure admits a continuous extension to a boundary point.

### Multiple Fourier sums and ψ-strong means of their deviations on the classes of ψ-differentiable functions of many variables

Lasuriya R. A., Stepanets O. I.

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1075–1093

We present results concerning the approximation of ψ-differentiable functions of many variables by rectangular Fourier sums in uniform and integral metrics and establish estimates for φ-strong means of their deviations in terms of the best approximations.

### Bounded approximate synthesis of the optimal control for the wave equation

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1094–1104

We consider the problem of optimal control for the wave equation. For the formulated problem, we find the optimal control in the form of a feedback in the case where the control reaches a restriction, construct an approximate control, and substantiate its correctness, i.e., prove that the proposed control realizes the minimum of the quality criterion.

### Asymptotics of the values of approximations in the mean for classes of differentiable functions by using biharmonic Poisson integrals

Kalchuk I. V., Kharkevych Yu. I.

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1105–1115

Complete asymptotic decompositions are obtained for values of exact upper bounds of approximations of functions from the classes $W^r_1,\quad r \in N,$ and WJr, $\overline{W}^r_1,\quad r \in N\backslash\{1\}$, by their biharmonic Poisson integrals.

### On the geometric results of A. V. Pogorelov

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1116–1130

We present a survey of the principal results of A. V. Pogorelov in the field of geometry.

### Isomonodromic deformations and the differential Galois theory

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1131-1134

We show how a solution of an inverse problem of the differential Galois theory can be used to construct isomonodromic deformations.

### A note on mixed summation-integral-type operators

Gupta M. K., Manoj Kumar, Rupen Pratap Singh

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1135–1139

Very recently Deo, in the paper “Simultaneous approximation by Lupas operators with weighted function of Szasz operators” [J. Inequal. Pure Appl. Math., **5**, No. 4 (2004)] claimed to introduce the integral modifications of Lupas operators. These operators were first introduced in 1993 by Gupta and Srivastava. They estimated the simultaneous approximation for these operators and called them Baskakov-Szasz operators. There are several misprints in the paper by Deo. This motivated us to perform subsequent investigations in this direction. We extend the study and estimate a saturation result in simultaneous approximation for the linear combinations of these summation-integral-type operators.

### Application of the FD-method to the solution of the Sturm-Liouville problem with coefficients of special form

Klymenko Ya. V., Makarov V. L.

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1140–1147

We use the functional-discrete method for the solution of the Strum-Liouville problem with coefficients of a special form and obtain the estimates of accuracy. The numerical experiment is performed by using the Maple-10 software package.

### Bifurcation of solutions of a linear Fredholm boundary-value problem

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1148–1152

We establish constructive conditions for the appearance of solutions of a linear Fredholm boundary-value problem for a system of ordinary differential equations in the critical case and propose an iterative procedure for finding these solutions. The range of values of a small parameter for which the indicated iterative procedure is convergent is estimated.