# Volume 67, № 2, 2015

### Integrability Analysis of a Two-Component Burgers-Type Hierarchy

Blackmore D., Özçağ E., Prykarpatsky A. K., Soltanov K. N.

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 147-162

The Lax integrability of a two-component polynomial Burgers-type dynamical system is analyzed by using a differential-algebraic approach. Its linear adjoint matrix Lax representation is constructed. A related recursive operator and an infinite hierarchy of nonlinear Lax integrable dynamical systems of the Burgers–Korteweg–de-Vries type are obtained by the gradient-holonomic technique. The corresponding Lax representations are presented.

### Convergence of Multiple Fourier Series of Functions of Bounded Generalized Variation

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 163-173

The paper introduces a new concept of Λ-variation of multivariable functions and studies its relationship with the convergence of multidimensional Fourier series.

### On the Radius of Injectivity for Generalized Quasiisometries in the Spaces of Dimension Higher Than Two

Gol'berg A. L., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 174-184

We consider a class of local homeomorphisms more general than the mappings with bounded distortion. Under these homeomorphisms, the growth of the *p*-module (*n-*1 *< p ≤ n*) of the families of curves is controlled by an integral containing an admissible metric and a measurable function *Q.* It is shown that, under generic conditions imposed on the majorant *Q,* this class has a positive radius of injectivity (and, hence, a ball in which every mapping is homeomorphic). Moreover, one of the conditions imposed on *Q* is not only sufficient but also necessary for existence of a radius of injectivity.

### Problem of Optimal Control for a Semilinear Hyperbolic System of Equations of the First Order with Infinite Horizon Planning

Derev’yanko N. V., Kirilich V. M.

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 185–201

We establish necessary conditions for the optimality of smooth boundary and initial controls in a semilinear hyperbolic system of the first order. The problem adjoint to the original problem is a semilinear hyperbolic system without initial conditions. The suggested approach is based on the use of special variations of continuously differentiable controls. The existence of global generalized solutions for a semilinear first-order hyperbolic system in a domain unbounded in time is proved. The proof is based on the use of the Banach fixed-point theorem and a space metric with weight functions.

### Inequalities of Different Metrics for Differentiable Periodic Functions

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 202–212

We prove the following sharp inequality of different metrics: $$\begin{array}{cc}\hfill {\left\Vert x\right\Vert}_q\le {\left\Vert {\varphi}_r\right\Vert}_q{\left(\frac{{\left\Vert x\right\Vert}_p}{{\left\Vert {\varphi}_r\right\Vert}_p}\right)}^{\frac{r+1/q}{r+1/p}}{\left\Vert {x}^{(r)}\right\Vert}_{\infty}^{\frac{1/p-1/q}{r+1/p}},\hfill & \hfill q>p>0,\hfill \end{array}$$ for 2π -periodic functions $x ∈ L_{∞}^r$ satisfying the condition $$L{(x)}_p\le {2}^{1/p}{\left\Vert x\right\Vert}_p,$$ where $$L{(x)}_p:= \sup \left\{{\left\Vert x\right\Vert}_{L_p\left[a,b\right]}:a,b\in \left[0,2\pi \right],\kern0.5em \left|x(t)\right|>0,\kern0.5em t\in \left(a,b\right)\right\},$$ and $φ_r$ is the Euler spline of order $r$. As a special case, we establish the Nikol’skii-type sharp inequalities for polynomials and polynomial splines satisfying the condition (A).

### Arithmetic of Semigroup Semirings

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 213-229

We define semigroup semirings by analogy with group rings and semigroup rings. We study the arithmetic properties and determine sufficient conditions under which a semigroup semiring is atomic, has finite factorization, or has bounded factorization. We also present a semigroup-semiring analog (although not a generalization) of the Gauss lemma on primitive polynomials.

### Almost Periodic Solutions of Nonlinear Equations that are not Necessarily Almost Periodic in Bochner’s Sense

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 230-244

We introduce a new class of almost periodic operators and establish conditions for the existence of almost periodic solutions of nonlinear equations that are not necessarily almost periodic in Bochner’s sense.

### Positive Solutions of a Class of Operator Equations

Cvetković A. S., Milovanović G. V., Stanić M. P.

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 245-260

Positive solutions of a class of matrix equations were studied by Bhatia, et al., *Bull. London Math. Soc.*, 32, 214 (2000), *SIAM J. Matrix Anal. Appl.*, 14, 132 (1993) and 27, 103–114 (2005), by Kwong, *Linear Algebra Appl.*, 108, 177–197 (1988), and by Cvetkovi? and Milovanovi?, [*Linear Algebra Appl.*, 429, 2401–2414 (2008)]. Following the idea used in the last paper, we study a class of operator equations in infinite-dimensional spaces and prove that the positivity of solutions can be established for this class of equations under the condition that a certain rational function is positive semidefinite.

### Levy Downcrossing Theorem for the Arratia Flow

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 261-271

We study the total number of downcrossings of a fixed strip by the trajectories of a continuum system of particles from the Arratia flow. We prove the convergence of the product of the strip width by the total number of downcrossings of the strip to the total local time for the Arratia flow. This statement is an analog of the well-known Levy downcrossing theorem for a Wiener process.

### Smoothing of the Singularities of Functions Whose Integrals over the Balls on a Sphere are Zero

Savost’yanova I. M., Volchkov V. V.

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 272-278

We study functions defined on a sphere with prickled point whose integrals over all admissible “hemispheres” are equal to zero. A condition is established under which the point is a removable set for this class of functions. It is shown that this condition cannot be omitted or noticeably weakened.

### On Centralizing and Strong Commutativity Preserving Maps of Semiprime Rings

Gölbaşı Ö., Huang Shuliang, Koç E.

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 279-285

We study some properties of centralizing and strong commutativity preserving maps of semiprime rings.

### Volodymyr Vasyl'ovych Sharko

Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 286-288