# Volume 68, № 8, 2016

### On wave operators for the multidimensional electromagnetic Schrödinger operator in divergent form

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1011-1020

We prove the existence of wave operators for the multidimensional electromagnetic Schr¨odinger operator in divergent form by the Cook method. Moreover, under certain conditions on the coefficients of the given operator, we establish the isometry of its wave operators and determine the initial domains of these operators.

### Jackson-type inequalities with generalized modulus of continuity and exact values of the $n$-widths of the classes of $(ψ,β)$-differentiable functions in $L_2$. II

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1021-1036

In the space $L_2$, we determine the exact values of some $n$-widths for the classes of functions such that the generalized moduli of continuity of their $(\psi, \beta)$ - derivatives or their averages with weight do not exceed the values of the majorants $\Phi$ satisfying certain conditions. Specific examples of realization of the obtained results are also analyzed.

### Problem without initial conditions for a countable semilinear hyperbolic system of first-order equations

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1043-1055

We derive sufficient conditions for the solvability of the problem without initial conditions for a countable semilinear hyperbolic system of first-order equations and establish conditions for the classical solvability of the initial-boundary value problem for countable hyperbolic systems of semilinear equations of the first-order in a semistrip.

### Groups of deviations of Fourier series in generalized Hölder spaces

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1056-1067

We study the rate of convergence of the values of analogs of the functionals of strong approximation of Fourier series in generalized $L$-Hölder spaces.

### Exponential and infinitary divisors

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1068-1079

We study several problems in the field of modified divisors; more precisely, from the theory of exponential and infinitary divisors.We analyze the behavior of modified divisors, sum-of-divisors, and totient functions. Our main results are connected with the asymptotic behavior of mean values and explicit estimates of the extremal orders.

### Widths of the anisotropic Besov classes of periodic functions of several variables

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1080-1091

We establish the exact-order estimates of Kolmogorov and orthoprojective widths of anisotropic Besov classes of periodic functions of several variables in the spaces $L_q$.

### Asymptotics of normalized control with Markov switchings

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1092-1101

We study the process of transfer of Markov perturbations and control over this process under the condition of existence of the equilibrium point of the quality criterion. For this control, we construct a normalized process and establish its asymptotic normality in the form of the Ornstein – Uhlenbeck process in the case where the transfer process changes under the influence of Markov switchings along a new trajectory of evolution from the state in which it was at the time of switching.

### Spectral problem for Sturm – Liouville operator with retarded argument which contains a spectral parameter in boundary condition

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1102-1114

We prove the existence of wave operators for the multidimensional electromagnetic Schr¨odinger operator in divergent form by the Cook method. Moreover, under certain conditions on the coefficients of the given operator, we establish the isometry of its wave operators and determine the initial domains of these operators.

### Nonlocal mixed-value problem for a Boussinesq-type integrodifferential equation with degenerate kernel

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1115-1131

We consider the problem of one-valued solvability of the mixed-value problem for a nonlinear Boussinesq type fourth-order integrodifferential equation with degenerate kernel and integral conditions. The method of degenerate kernel is developed for the case of nonlinear Boussinesq type fourth-order partial integrodifferential equation. The Fourier method of separation of variables is employed. After redenoting, the integrodifferential equation is reduced to a system of countable system of algebraic equations with nonlinear and complex right-hand side. As a result of the solution of this system of countable systems of algebraic equations and substitution of the obtained solution in the previous formula, we get a countable system of nonlinear integral equations (CSNIE). To prove the theorem on one-valued solvability of the CSNIE, we use the method of successive approximations. Further, we establish the convergence of the Fourier series to the required function of the mixed-value problem. Our results can be regarded as a subsequent development of the theory of partial integrodifferential equations with degenerate kernel.

### Properties of the isolated solutions bounded on the entire axis for a system of nonlinear ordinary differential equations

Abildayeva A. D., Dzhumabaev D. S.

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1132-1138

We establish the conditions of continuous dependence on the right-hand side for the “isolated” solutions of a system of nonlinear ordinary differential equations bounded on the entire axis.

### A note on $SΦ$-supplemented subgroups

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1139-1141

We give new and brief proofs of the results obtained by X. Li and T. Zhao in [\mathrm{S}\Phi -supplemented subgroups of finite groups // Ukr. Math. J. – 2012. – 64, № 1. – P. 102–109].

### Characterization of the group $G_2(5)$ by the prime graph

Darafsheh M. R., Nosratpour P.

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1142-1146

Let $G$ be a finite group. The prime graph of $G$ is a graph $\Gamma (G)$ with vertex set $\pi (G)$ and the set of all prime divisors of $|G|$, where two distinct vertices $p$ and $q$ are adjacent by an edge if $G$ has an element of order $pq$. We prove that if $G\Gamma (G) = \Gamma (G_2(5))$, then $G$ has a normal subgroup $N$ such that $\pi (N) \subseteq \{ 2, 3, 5\}$ and $G/N \sim = G_2(5)$.

### Univalence criteria and quasiconformal extensions

Ukr. Mat. Zh. - 2016. - 68, № 8. - pp. 1147-1151

We establish more general conditions for the univalence of analytic functions in the open unit disk $U$. In addition, we obtain a refinement to the criterion of quasiconformal extension for the main result.