# Volume 67, № 6, 2015

### Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. I

Bodenchuk V. V., Serdyuk A. S.

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 719-738

We prove that the kernels of analytic functions of the form $${H}_{h,\beta }(t)={\displaystyle \sum_{k=1}^{\infty}\frac{1}{ \cosh kh} \cos \left(kt-\frac{\beta \pi }{2}\right),}h>0,\beta \in \mathbb{R},$$ satisfy Kushpel’s condition $C_{y,2n}$ starting from a certain number $n_h$ explicitly expressed via the parameter $h$ of smoothness of the kernel. As a result, for all $n ≥ n_h$ , we establish lower bounds for the Kolmogorov widths $d_{2n}$ in the space $C$ of functional classes that can be represented in the form of convolutions of the kernel $H_{h,β}$ with functions $φ⊥1$ from the unit ball in the space $L_{∞}$.

### Paley Effect for Entire Dirichlet Series

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 739–751

For the entire Dirichlet series $f(z) = ∑_{n = 0}${∞$ a_n e^{zλn}$, we establish necessary and sufficient conditions on the coefficients $a_n$ and exponents $λ_n$ under which the function $f$ has the Paley effect, i.e., the condition $$\underset{r\to +\infty }{ \lim \sup}\frac{ \ln {M}_f(r)}{T_f(r)}=+\infty$$ is satisfied, where $M_f (r)$ and $T_f (r)$ are the maximum modulus and the Nevanlinna characteristic of the function $f$, respectively.

### A Globally and $R$-Linearly Convergent Hybrid HS and PRP Method and its Inexact Version with Applications

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 752–762

We present a hybrid HS- and PRP-type conjugate gradient method for smooth optimization that converges globally and $R$-linearly for general functions. We also introduce its inexact version for problems of this kind in which gradients or values of the functions are unknown or difficult to compute. Moreover, we apply the inexact method to solve a nonsmooth convex optimization problem by converting it into a one-time continuously differentiable function by the method of Moreau–Yosida regularization.

### Development of Complex Analysis and Potential Theory at the Institute of Mathematics of the Ukrainian National Academy of Sciences in 1991–2013

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 763–779

### Properties of the Ceder Product

Maslyuchenko O. V., Maslyuchenko V. K., Myronyk O. D.

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 780-787

We study properties of the Ceder product $X ×_b Y$ of topological spaces $X$ and $Y$, where $b ∈ Y$, recently introduced by the authors. Important examples of the Ceder product are the Ceder plane and the Alexandroff double circle. In particular, for $i = 0, 1, 2, 3$ we establish necessary and sufficient conditions for the Ceder product to be a $T_i$ -space. We prove that the Ceder product $X ×_b Y$ is metrizable if and only if the spaces $X$ and $\overset{.}{Y}=Y\backslash \left\{b\right\}$ are metrizable, $X$ is $σ$-discrete, and the set $\{b\}$ is closed in $Y$. If $X$ is not discrete, then the point $b$ has a countable base of closed neighborhoods in $Y$.

### Solutions of the Quasilinear Elliptic Systems with Combined Critical Sobolev–Hardy Terms

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 788-808

We study the existence of multiple solutions for a quasilinear elliptic system. Based on the Ambrosetti–Rabinowitz mountain-pass theorem and the Rabinowitz symmetric mountain-pass theorem, we establish several existence and multiplicity results for the solutions and *G*-symmetric solutions under certain suitable conditions.

### On the Estimation of Strong Means of Fourier Series

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 809–819

We study problem of $(λ, φ)$ -strong summation of number series by the regular method $λ$ with power summation of the function $φ$. The accumulated results are extended to the case of Fourier expansions in trigonometric functions $f ϵ L_p, p > 1$, where $C$ is the set of $2π$-periodic continuous functions. Some results are also obtained for the estimation of strong means of the method $λ$ in $L_p, p > 1$, at the Lebesgue point $x$ of the function $f$ under certain additional conditions in the case where the function $φ$ tends to infinity as $u → ∞$ faster than the exponential function $\exp (βu) − 1, β > 0$.

### Leiko Network on the Surfaces in the Euclidean Space $E_3$

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 820–828

We introduce the notion of Leiko network as a generalization of the geodetic network on the surfaces of nonzero Gaussian curvature in the Euclidian space $E_3$ and study its characteristics. The conditions of preservation of the Leiko network under infinitesimal deformations of the surfaces are also obtained.

### Analog of the Montel Theorem for Mappings of the Sobolev Class with Finite Distortion

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 829-837

We study the classes of mappings with unbounded characteristic of quasiconformality and obtain a result on the normal families of open discrete mappings $f : D → ℂ \backslash \{a, b\}$ from the class $W\{\text{loc}^{1,1}$ with finite distortion that do not take at least two fixed values $a 6 ≠ b$ in $ℂ$ whose maximal dilatation has a majorant of finite mean oscillation at every point. This result is an analog of the well-known Montel theorem for analytic functions and is true, in particular, for the so-called $Q$-mappings.

### A Criterion for the Existence of Almost Periodic Solutions of Nonlinear Differential Equations with Impulsive Perturbation

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 838–848

We establish conditions for the existence of almost periodic solutions of nonlinear almost periodic differential equations with impulsive perturbation in Banach spaces without using the \( \mathcal{H} \)-classes of these equations.

### Bezout Rings of Stable Range 1.5

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 849–860

A ring $R$ has a stable range 1.5 if, for every triple of left relatively prime nonzero elements $a, b$ and $c$ in $R$, there exists $r$ such that the elements $a+br$ and $c$ are left relatively prime. Let $R$ be a commutative Bezout domain. We prove that the matrix ring $M_2 (R)$ has the stable range 1.5 if and only if the ring $R$ has the same stable range.

### $G$-Supplemented Modules

Koşar B., Nebiyev C., Sökmez N.

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 861–864

Following the concept of generalized small submodule, we define $g$ -supplemented modules and characterize some fundamental properties of these modules. Moreover, the generalized radical of a module is defined and the relationship between the generalized radical and the radical of a module is investigated. Finally, the definition of amply $g$ -supplemented modules is given with some basic properties of these modules.