# Volume 67, № 9, 2015

### A New Characterization of PSL($2, q$) for Some $q$

Amiri S. S. S., Asboei A. K., Iranmanesh A.

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1155–1162

Let $G$ be a finite group and let $π_e (G)$ be the set of orders of elements from $G$. Let $k ∈ π_e (G)$ and let $m_k$ be the number of elements of order $k$ in $G$. We set nse $(G) := \{m_k | k ∈ π_e (G)\}$. It is proved that PSL($2, q$) are uniquely determined by nse (PSL($2, q$)), where $q ∈ \{5, 7, 8, 9, 11, 13\}$. As the main result of the paper, we prove that if $G$ is a group such that nse $(G) = nse (PSL(2, q))$, where $q ∈ {16, 17, 19, 23}$, then $G ≅ PSL(2, q)$.

### On the Optimal Recovery of Integrals of Set-Valued Functions

Babenko V. F., Babenko V. V., Polishchuk M. V.

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1163-1171

We consider the problem of optimization of the approximate integration of set-valued functions from the class specified by a given majorant of their moduli of continuity performed by using the values of these functions at *n* fixed or free points of their domain.

### Laplacian Generated by the Gaussian Measure and Ergodic Theorem

Bogdanskii Yu. V., Sanzharevskii Ya. Yu.

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1172-1180

We consider the Laplacian generated by the Gaussian measure on a separable Hilbert space and prove the ergodic theorem for the corresponding one-parameter semigroup.

### Perturbation Theory of Operator Equations in the FréChet and Hilbert Spaces

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1181-1188

The perturbation theory is constructed in the Fréchet and Hilbert spaces. An iterative process is proposed for finding branching solutions.

### One Problem Connected with the Helgason Support Problem

Savost’yanova I. M., Volchkov V. V., Volchkov V. V.

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1189-1200

We solve the problem of description of the set of continuous functions in annular subdomains of the *n*-dimensional sphere with zero integrals over all (*n -* 1)-dimensional spheres covering the inner spherical cap. As an application, we establish a spherical analog of the Helgason support theorem and new uniqueness theorems for functions with zero spherical means.

### Homogenized Model of Diffusion in Porous Media with Nonlinear Absorption on the Boundary

Goncharenko M. V., Khil’kova L. A.

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1201-1216

We consider a boundary-value problem used to describe the process of stationary diffusion in a porous medium with nonlinear absorption on the boundary. We study the asymptotic behavior of the solution when the medium becomes more and more porous and denser located in a bounded domain $Q$. A homogenized equation for the description of the main term of the asymptotic expansion is constructed.

### Classes of Analytic Functions Defined by a Differential Operator Related to Conic Domains

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1217–1231

Let $A$ be the class of functions $f(z) = z + ∑_{k = 2}^{ ∞} a_k z^k$ analytic in an open unit disc $∆$. We use a generalized linear operator closely related to the multiplier transformation to study certain subclasses of $A$ mapping $∆$ onto conic domains. Using the principle of the differential subordination and the techniques of convolution, we investigate several properties of these classes, including some inclusion relations and convolution and coefficient bounds. In particular, we get many known and new results as special cases.

### Relative Extensions of Modules and Homology Groups

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1232–1243

We introduce the concepts of relative (co)extensions of modules and explore the relationship between the relative (co)extensions of modules and relative (co)homology groups. Some applications are given.

### On the $C^{*}$-Algebra Generated by the Bergman Operator, Carleman Second-Order Shift, and Piecewise Continuous Coefficients

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1244–1252

We study the $C^{*}$ -algebra generated by the Bergman operator with piecewise continuous coefficients in the Hilbert space $L_2$ and extended by the Carleman rotation by an angle $π$. As a result, we obtain an efficient criterion for the operators from the indicated $C^{*}$ -algebra to be Fredholm operators.

### Multiple Haar Basis and its Properties

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1253–1264

In the Lebesgue spaces $L_p ([0, 1]^d ), 1 ≤ p ≤ ∞$, for $d ≥ 2$, we define a multiple basis system of functions $H^d = (h_n )_{n = 1}^{∞}$. This system has the main properties of the well-known one-dimensional Haar basis $H$. In particular, it is shown that the system $H^d$ is a Schauder basis in the spaces $L_p ([0, 1]^d ),\; 1 ≤ p < ∞$.

### On the Best Linear Approximation Method for Hölder Classes

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1265-1284

We find the exact values of one-dimensional linear widths for the Hölder classes of functions in the space *C* and the value of the best approximation of the Hölder classes of functions by a wide class of linear positive methods.

### Asymptotic Representations for the Solutions of One Class of Nonlinear Differential Equations of the Second Order

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 1285-1293

We establish asymptotic representations for the solutions of one class of nonlinear differential equations of the second order with rapidly and regularly varying nonlinearities.

### Ivan Oleksandrovych Lukovs’kyi (on his 80th birthday)

Ukr. Mat. Zh. - 2015. - 67, № 9. - pp. 12941297