# Volume 67, № 3, 2015

### Well-Posed Solvability of a Nonlocal Boundary-Value Problem for the Systems of Hyperbolic Equations with Impulsive Effects

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 291-303

We consider a nonlocal boundary-value problem for a system of hyperbolic equations with impulsive effects. The relationship is established between the well-posed solvability of the nonlocal boundary-value problem for a system of hyperbolic equations with impulsive effects and the well-posed solvability of a family of two-point boundary-value problems for a system of ordinary differential equations with impulsive effects. Sufficient conditions for the existence of a unique solution of the family of two-point boundary-value problems for a system of ordinary differential equations with impulsive effects are obtained by method of introduction of functional parameters. The algorithms are proposed for finding the solutions. The necessary and sufficient conditions of the well-posed solvability of a nonlocal boundary-value problem for a system of hyperbolic equations with impulsive effects are established in the terms of the initial data.

### Scattered Subsets of Groups

Banakh T. O., Protasov I. V., Slobodianiuk S. V.

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 304-312

We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset *A* of a group *G* is scattered if and only if *A* does not contain any piecewise shifted *IP* -subsets. For an amenable group *G* and a scattered subspace *A* of *G,* we show that *μ*(*A*) = 0 for each left invariant Banach measure *μ* on *G.* It is also shown that every infinite group can be split into ℵ_{0} scattered subsets.

### Two-Weight Criteria for the Multidimensional Hardy-Type Operator in $p$-Convex Banach Function Spaces and Some Applications

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 313-325

The main aim of the paper is to prove a two-weight criterion for the multidimensional Hardy-type operator from weighted Lebesgue spaces into *p*-convex weighted Banach function spaces. The problem for the dual operator is also considered. As an application, we prove a two-weight criterion of boundedness of the multidimensional geometric mean operator from weighted Lebesgue spaces into weighted Musielak–Orlicz spaces.

### On One Uniqueness Theorem for a Weighted Hardy Space

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 326–332

A uniqueness theorem is proved for the space of functions analytic in the right half plane and satisfying the condition $$\underset{\left|\upvarphi \right|<\frac{\uppi}{2}}{ \sup}\left\{{\displaystyle \underset{0}{\overset{+\infty }{\int }}{\left|f\left(r{e}^{i\varphi}\right)\right|}^p{e}^{-p\sigma r\left| \sin \varphi \right|}dr}\right\}<+\infty .$$

### Generalized Elastic Line Deformed on a Nonnull Surface by an External Field in the 3-Dimensional Semi-Euclidean Space $\mathbb{E}_1^3$

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 333–340

We deduce intrinsic equations for a generalized elastic line deformed on the nonnull surface by an external field in the semi-Euclidean space $\mathbb{E}_1^3$ and give some applications.

### Investigation of the Functional Properties and Spaces of Multipliers for Group $L(p, q)(G)$-Algebras

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 341–354

Let $G$ be a locally compact Abelian group (noncompact and nondiscrete) with Haar measure. Suppose that $1 < p < ∞$ and $1 ≤ q ≤ ∞$. The purpose of the paper is to define temperate Lorentz spaces and study the spaces of multipliers on Lorentz spaces and characterize them as the spaces of multipliers of certain Banach algebras.

### Approximate Synthesis of Distributed Bounded Control for a Parabolic Problem with Rapidly Oscillating Coefficients

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 355-365

We study the problem of finding the optimal control in the form of feedback (synthesis) for a linear-quadratic problem in the form of a parabolic equation with rapidly oscillating coefficients and distributed control on the right-hand side (whose Fourier coefficients obey certain restrictions in the form of inequalities) and a quadratic quality criterion. We deduce the exact formula of synthesis and justify its approximate form corresponding to the replacement of rapidly oscillating coefficients by their averaged values.

### A Procedure of Complete Averaging for Fuzzy Differential Inclusions on a Finite Segment

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 366-374

We justify the applicability of the method of complete averaging on a finite segment for differential inclusions with fuzzy right-hand sides containing a small parameter.

### Kronrod–Reeb Graphs of Functions on Noncompact Two-Dimensional Surfaces. I

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 375-396

We consider continuous functions on two-dimensional surfaces satisfying the following conditions: they have a discrete set of local extrema; if a point is not a local extremum, then there exist its neighborhood and a number $n ∈ ℕ$ such that a function restricted to this neighborhood is topologically conjugate to Re $z^n$ in a certain neighborhood of zero. Given $f : M^2 → ℝ$, let $Γ_{K−R} (f)$ be a quotient space of $M^2$ with respect to its partition formed by the components of the level sets of $f$. It is known that, for compact $M^2$, the space $Γ_{K−R} (f)$ is a topological graph. We introduce the notion of graph with stalks, which generalizes the notion of topological graph. For noncompact $M^2$, we establish three conditions sufficient for $Γ_{K−R} (f)$ to be a graph with stalks.

### On Sylow Subgroups of Some Shunkov Groups

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 397-405

We study Shunkov groups with the following condition: the normalizer of any finite nontrivial subgroup has an almost layer-finite periodic part. Under this condition, we establish the structure of Sylow 2-subgroups in this group.

### Necessary and Sufficient Conditions for the Existence of Weighted Singular-Valued Decompositions of Matrices with Singular Weights

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

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 406–426

A weighted singular-valued decomposition of matrices with singular weights is obtained by using orthogonal matrices. The necessary and sufficient conditions for the existence of the constructed weighted singular-valued decomposition are established. The indicated singular-valued decomposition of matrices is used to obtain a decomposition of their weighted pseudoinverse matrices and decompose them into matrix power series and products. The applications of these decompositions are discussed.

### On the Holomorphy of Developable Vector Fields on Almost Hermitian Manifolds

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 427–430

We introduce the notion of absolutely developable and biholomorphic vector fields defined on almost Hermitian manifolds. It is shown that any developable vector field on a K¨ahlerian manifold is an absolutely developable vector field. It is also proved that, on a nearly Kählerian manifold, an absolutely developable vector field ξ preserves the almost complex structure if and only if ξ is a special concircular vector field. In addition, we conclude that, on a quasi-Kählerian or Hermitian manifold, a biholomorphic vector field ξ is a special concircular vector field.