# Volume 65, № 4, 2013

### Locally soluble AFA-groups

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 459-469

Let $A$ be an $\mathbf{R}G$-module, where $\mathbf{R}$ is a ring, $G$ is a locally solvable group, $C_G (A) = 1$, and each proper subgroup $H$ of $G$ for which $A/C_A(H)$ is not an Artinian $\mathbf{R}$-module is finitely generated. It is proved that a locally solvable group $G$ that satisfies these conditions is hyperabelian if R is a Dedekind ring. We describe the structure of $G$ in the case where $G$ is a finitely generated solvable group, $A/C_A(H)$ is not an Artinian $\mathbf{R}$-module and $\mathbf{R}$ is a Dedekind ring.

### On the separation problem for a family of Borel and Baire *G* -powers of shift measures on *R*

Pantsulaia G., Saatashvili G., Zerakidze Z.

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 470-485

The separation problem for a family of Borel and Baire G-powers of shift measures on R is studied for an arbitrary infinite additive group G by using the technique developed in [L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York (1974)], [ A. N. Shiryaev, Probability [in Russian], Nauka, Moscow (1980)], and [G. R. Pantsulaia, Invariant and Quasiinvariant Measures in Infinite-Dimensional Topological Vector Spaces, Nova Sci., New York, 2007]. It is proved that $T_n: R^n → R,\;n∈N$, defined by $$T_n(x_1,…,x_n) = -F^{-1}\left(n^{-1 } \# (\{ x_1,…,x_n \} \bigcap (-\infty;0])\right)$$ for $(x_1,…, x_n) ∈ R^n$ is a consistent estimator of a useful signal $θ$ in the one-dimensional linear stochastic model $$ξ_k = θ + ∆_k,\; k ∈ N,$$ where $\#(·)$ is a counting measure, $∆_k,\; k ∈ N$, is a sequence of independent identically distributed random variables on $R$ with a strictly increasing continuous distribution function $F$, and the expectation of $∆_1$ does not exist.

### Periodic solutions of a system with impulsive action at nonfixed times

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 486-493

We obtain conditions for the existence of a periodic solution of a system with impulses at variable times.

### Evolution free-boundary problem for a stationary system of the theory of elasticity

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 494-511

We consider an evolution free-boundary problem for a stationary linear system of the theory of elasticity that arises in the investigation of solid thin films in microelectronic devices. We prove its solvability on an arbitrary time interval under the condition that the initial data are sufficiently close to the stationary solution.

### Analogs of *S*-type spaces of partially even functions

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 512-521

We construct analogs of $S$-type spaces whose elements are functions that are even in a part of components of their arguments. We obtain a formula that expresses a power of a Bessel operator via the corresponding powers of a differential operator. This formula enables us to establish a relation between these spaces in terms of the Fourier – Bessel transformation and to clarify some basic properties of typical operations on their elements.

### Lebesgue-type inequalities for the de la Valee-Poussin sums on sets of analytic functions

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 522-537

For functions from the sets $C^{ψ}_{β} C$ and $C^{ψ}_{β} L_s,\; 1 ≤ s ≤ ∞$ generated by sequences $ψ(k) > 0$ satisfying the d’Alembert condition $\lim_{k→∞}\frac{ψ(k + 1)}{ψ(k)} = q,\; q ∈ (0, 1)$, we obtain asymptotically unimprovable estimates for the deviations of de la Vallee Poussin sums in the uniform metric in terms of the best approximations of the $(ψ, β)$-derivatives of functions of this sort by trigonometric polynomials in the metrics of the spaces $L_s$. It is proved that the obtained estimates are unimprovable in some important functional subsets of $C^{ψ}_{β} C$ and $C^{ψ}_{β} L_s$.

### On *-representations of λ-deformations of canonical commutation relations

Proskurin D. P., Yakymiv R. Ya.

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 538-545

We study irreducible integrable *-representations of the algebra $\mathfrak{U}_{\lambda, 2}$ generated by the following relations: $$\mathfrak{U}_{\lambda, 2} = \mathbb{C} \langle a_j, a_j^{*} \,| \,a_j^{*} a_j = 1 + a_ja_j^{*},\; a_1^{*}a_2 = \lambda a_2a_1^{*},\; a_2a_1 = \lambda a_1 a_2,\; j = 1, 2 \rangle .$$ For this *-algebra, we prove an analog of the von Neumann theorem on the uniqueness of an irreducible integrable representation.

### On Holomorphic Solutions of the Darwin Equations of Motion of Point Charges

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 546-554

The existence of holomorphic (in time) solutions of the nonrelativistic Darwin equations of motion of point charges is proved with the help of the Cauchy theorem.

### Generalizations of $\oplus$-supplemented modules

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 555-564

We introduce $\oplus$-radical supplemented modules and strongly $\oplus$-radical supplemented modules (briefly, $srs^{\oplus}$-modules) as proper generalizations of $\oplus$-supplemented modules. We prove that (1) a semilocal ring $R$ is left perfect if and only if every left $R$-module is an $\oplus$-radical supplemented module; (2) a commutative ring $R$ is an Artinian principal ideal ring if and only if every left $R$-module is a $srs^{\oplus}$-module; (3) over a local Dedekind domain, every $\oplus$-radical supplemented module is a $srs^{\oplus}$-module. Moreover, we completely determine the structure of these modules over local Dedekind domains.

### Projective method for equation of risk theory in the arithmetic case

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 565-582

We consider a discrete model of operation of an insurance company whose initial capital can take any integer value. In this statement, the problem of nonruin probability is naturally solved by the Wiener-Hopf method. Passing to generating functions and reducing the fundamental equation of risk theory to a Riemann boundary-value problem on the unit circle, we establish that this equation is a special one-sided discrete Wiener-Hopf equation whose symbol has a unique zero, and, furthermore, this zero is simple. On the basis of the constructed solvability theory for this equation, we justify the applicability of the projective method to the approximation of ruin probabilities in the spaces $l^{+}_1$ and $\textbf{c}^{+}_0$. Conditions for the distributions of waiting times and claims under which the method converges are established. The delayed renewal process and stationary renewal process are considered, and approximations for the ruin probabilities in these processes are obtained.

### A characterization of totally umbilical hypersurfaces of a space form by geodesic mapping

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 583-587

The idea of considering the second fundamental form of a hypersurface as the first fundamental form of another hypersurface has found very useful applications in Riemannian and semi-Riemannian geometry, specially when trying to characterize extrinsic hyperspheres and ovaloids. Recently, T. Adachi and S. Maeda gave a characterization of totally umbilical hypersurfaces in a space form by circles. In this paper, we give a characterization of totally umbilical hypersurfaces of a space form by means of geodesic mapping.

### Strongly alternative Dunford - Pettis subspaces of operator ideals

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 588-593

Introducing the concept of strong alternative Dunford – Pettis property (strong DP1) for the subspace M of operator ideals $\mathcal{U}(X, Y )$ between Banach spaces $X$ and $Y$, we show that M is a strong DP1 subspace if and only if all evaluation operators $\phi_x : \mathcal{M} → Y$ та $ψy∗ : \mathcal{M} → X^{*}$ are DP1 operators, where $\phi_x(T) = T x$ та $ψ_{y^{∗}} (T) = T^{∗}y^{∗}$ for $x ∈ X, y^{∗} ∈ Y$ and $T ∈ M$. Some consequences related to the concept of alternative Dunford – Pettis property in subspaces of some operator ideals are obtained.

### One method for the investigation of linear functional-differential equations

Cherepennikov V. B., Vetrova E. V.

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 594-600

We consider the scalar linear retarded functional differential equation $$\dot{x}(t) = ax(t - 1)+ bx \left( \frac tq \right) + f(t), \quad q > 1.$$ The study of linear retarded functional differential equations deals mainly with two initial-value problems: an initial-value problem with initial function and an initial-value problem with initial point (when one seeks a classical solution whose substitution into the original equation reduces it to an identity). In the present paper, an initial-value problem with initial point is investigated by the method of polynomial quasisolutions. We prove theorems on the existence of polynomial quasisolutions and exact polynomial solutions of the considered linear retarded functional differential equation. The results of a numerical experiment are presented.