# Volume 65, № 9, 2013

### Transitive Maps on Topological Spaces

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1163–1185

In the present paper, we consider the problem of existence of nonequivalent definitions of topological transitivity, which is a classical problem of the topological dynamics. In particular, we use the fact that all available definitions of this kind imply a condition imposed on the dynamical system. The main result of our investigations is the complete classification of these dynamical systems.

### Order Estimates for the Best Approximations and Approximations by Fourier Sums of the Classes of (ψ, β)-Differential Functions

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1186–1197

We establish exact-order estimates for the best uniform approximations by trigonometric polynomials on the classes *C* ^{ψ} _{β, p } of 2π-periodic continuous functions f defined by the convolutions of functions that belong to the unit balls in the spaces *L* _{ p }, 1 ≤ *p* < ∞, with generating fixed kernels Ψ_{β} ⊂ *L* _{ p′}, \( \frac{1}{p}+\frac{1}{{p^{\prime}}}=1 \) , whose Fourier coefficients decrease to zero approximately as power functions. Exactorder estimates are also established in the *L* _{ p } -metric, 1 < *p* ≤ ∞, for the classes *L* ^{ψ} _{β,1} of 2π -periodic functions f equivalent in terms of the Lebesgue measure to the convolutions of kernels Ψ_{β} ⊂ *L* _{ p } with functions from the unit ball in the space *L* _{1}. It is shown that, in the investigated cases, the orders of the best approximations are realized by Fourier sums.

### Base Changeable Sets and Mathematical Simulation of the Evolution of Systems

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1198–1218

We introduce the notion of base changeable sets and study the principal properties of these sets. Base changeable sets are required for the construction of the general theory of changeable sets. The problem studied in our paper is closely connected with the famous sixth Hilbert problem.

### An Improved Jackson Inequality for the Best Trigonometric Approximation

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1219–1226

The paper presents an improved Jackson inequality and the corresponding inverse inequality for the best trigonometric approximation in terms of the moduli of smoothness equivalent to zero on the trigonometric polynomials whose degree does not exceed a certain number. The deduced inequalities are analogous to Timan’s inequalities. The relations between the moduli of different orders are also considered.

### Finite Groups with System of *SE*-Supplemented Subgroups

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1227–1235

We study the influence of the properties of supplemented subgroups on the structure of finite groups. The conditions under which a normal subgroup of a finite group possesses cyclic chief *p*-factors are obtained.

### Numerical Method for the Solution of a Hypersingular Integral Equation of the Second Kind

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1236–1244

А numerical method for the solution of a hypersingular integral equation of the second kind obtained as a generalization of the well-known method is proposed. The existence and uniqueness theorem is proved under additional assumptions. The rate of convergence of an approximate solution to the exact solution is obtained.

### Thin Subsets of Groups

Protasov I. V., Slobodianiuk S. V.

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1245–1253

For a group *G* and a natural number m; a subset *A* of *G* is called m-thin if, for each finite subset *F* of *G*; there exists a finite subset *K* of *G* such that |*F* _{ g } ∩ *A*| ≤ *m* for all *g* ∈ *G* \ *K*: We show that each *m*-thin subset of an Abelian group *G* of cardinality ℵ_{ n }; *n* = 0, 1,… can be split into ≤ *m* ^{ n+1} 1-thin subsets. On the other hand, we construct a group G of cardinality ℵ_{ ω } and select a 2-thin subset of *G* which cannot be split into finitely many 1-thin subsets.

### On the Orlicz–Sobolev Classes and Mappings with Bounded Dirichlet Integral

Ryazanov V. I., Salimov R. R., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1254–1265

It is shown that homeomorphisms *f* in \( {{\mathbb{R}}^n} \) , *n* ≥ 2, with finite Iwaniec distortion of the Orlicz–Sobolev classes *W* ^{1,φ} _{loc} under the Calderon condition on the function φ and, in particular, the Sobolev classes *W* ^{1,φ} _{loc}, *p* > *n* - 1, are differentiable almost everywhere and have the Luzin (*N*) -property on almost all hyperplanes. This enables us to prove that the corresponding inverse homeomorphisms belong to the class of mappings with bounded Dirichlet integral and establish the equicontinuity and normality of the families of inverse mappings.

### Differential Equations with Absolutely Unstable Trivial Solutions

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1266–1275

We obtain conditions for the absolute instability of trivial solutions of the nonlinear differential equations.

### Mean-Value Theorem

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1276–1282

We propose a new approach to the classical mean-value theorem in which two mean values are used instead of one. This approach is of especial importance for complex functions because there are no available theorems of this kind for these functions.

### Parallel Affine Immersions ${M^n}\to {{\mathbb{R}}^{n+2 }}$ with Flat Connection

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1283–1300

We present a classification of parallel affine immersions $f : M^n→{M^n}\to {{\mathbb{R}}^{n+2 }}$ with flat connection according to the rank of the Weingarten mapping.