# Volume 63, № 1, 2011

### Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 3-23

We investigate the approximation properties of the trigonometric system in $L_{2\pi}^{p(\cdot)}$. We consider the fractional order moduli of smoothness and obtain direct, converse approximation theorems together with a constructive characterization of a Lipschitz-type class.

### Spaces of generalized operators with bounded projection trace

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 24-39

We construct a theory of Banach spaces of "generalized" operators with bounded projection trace over the given Hilbert space. This theory can be efficient in investigating evolution problems for quantum systems with infinite number of particles.

### Rate of convergence in the Euler scheme for stochastic differential equations with non-Lipschitz diffusion and Poisson measure

Mishura Yu. S., Zubchenko V. P.

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 40-60

We study the rate of convergence and some other properties of the Euler scheme for stochastic differential equations with the non-Lipschitz diffusion and the Poisson measure.

### On some properties of Gelfond – Leontiev generalized integration operators

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 61-68

In a class of linear continuous operators acting in spaces of functions analytic in domains, we describe in various forms isomorphisms which commute with a degree of the Gelfond – Leontiev generalized integration. We also obtain images of all closed subspaces of a space of analytic functions which are invariant with respect to the degree of the Gelfond – Leontiev generalized integration.

### On branch points of three-dimensional mappings with unbounded characteristic of quasiconformality

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 69-79

For the open discrete mappings *f*: *D* \ {*b*} → **R**^{3} of the domain *D* ⊂ **R**^{3} satisfying relatively general geometric conditions in *D* \ {*b*} and having the essential singularity *b* ∈ **R**^{3}, we prove the following
statement.
Let *y*_{0} belong to **R**^{3} \ *f* (*D* \ {*b*}) and let the inner dilatation *K*_{I } (*x*, *f*) and the outer dilatation
K_{Ο }(*x*, *f*) of the mapping *f* at a point *x* satisfy certain conditions.
Denote by *B _{f }* the set of branch points of

*f*. Then for an arbitrary neighborhood

*V*of the point

*y*

_{0}, a set

*V*∩

*f*(

*B*) cannot be contained in the set

_{f }*A*such that

*g*(

*A*) =

*I*, where

*I*= {

*t*∈

**R**: |

*t*| < 1} and

*g*:

*U*→

**R**

^{n}is a quasiconformal mapping of the domain

*U*⊂

**R**

^{n}such that

*A*⊂

*U*.

### Existence and uniqueness of weighted pseudoinverse matrices and weighted normal pseudosolutions with singular weights

Deineka V. S., Galba E. F., Sergienko I. V.

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 80-101

For one of definitions of weighted pseudoinversion with singular weights, necessary and sufficient conditions for the existence and uniqueness are obtained. Expansions of weighted pseudoinverse matrices in matrix power series and matrix power products are obtained. Relationship is established between the weighted pseudoinverse matrices and the weighted normal pseudosolutions. Iterative methods for the calculation of both weighted pseudoinverse matrices and weighted normal pseudosolutions are constructed.

### Approximation of classes of analytic functions by a linear method of special form

Chaichenko S. O., Serdyuk A. S.

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 102-109

On classes of convolutions of analytic functions in uniform and integral metrics, we find asymptotic equations for the least upper bounds of deviations of trigonometric polynomials generated by certain linear approximation method of a special form.

### On the construction of a nonnegative solution for one class of Urysohn-type nonlinear integral equations on a semiaxis

Khachatryan A. Kh., Khachatryan Kh. A.

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 110-118

We investigate a class of Urysohn-type nonlinear integral equations with noncompact operator. We assume that Wiener-Hopf-Hankel-type linear integral operator is local minorant for initial Urysohn operator. We prove alternative theorem on the existence of positive solutions and investigate asymptotic behavior of obtained solutions at infinity.

### Morse Functions on Cobordisms

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 119-129

We study the homotopy invariants of crossed and Hilbert complexes. These invariants are applied to the calculation of exact values of Morse numbers of smooth cobordisms.

### On a new approach to the construction of hypercomplex number systems of rank two over the field of complex numbers

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 130-139

By using the introduced notion of a parameter of hypercomplex numerical system, we propose a new approach to the construction of a hypercomplex numerical systems of rank two over a field of complex numbers. We show that quadroplex (bicomplex) numbers and quaternions can be considered as a special cases of the universal system of hypercomplex numbers that correspond to some values of the mentioned parameter. We consider principal algebraic characteristics of the universal system of hypercomplex numbers. We also present examples of possible applications of numbers of universal hypercomplex system to certain values of the introduced parameter.

### Mechanical analogs of linear impulsive systems

Ukr. Mat. Zh. - 2011. - 63, № 1. - pp. 140-145

The linear system of differential equations with pulse influence is considered for which the condition of construction of its mechanical analogs is obtained.