# Volume 62, № 1, 2010

### Solvability of the boundary-value problem for the second-order elliptic differential-operator equation with spectral parameter in the equation and boundary conditions

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 3 - 14

We investigate the solvability of a boundary-value problem for second-order elliptic operator differential equation with a spectral parameter in the equation and boundary conditions. We also study the asymptotic behavior of eigenvalues corresponding to a homogeneous boundary-value problem.

### Periodic solutions of “predator–prey” systems with continuous delay and periodic coefficients

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 15 - 28

We prove the existence of positive ω-periodic solutions for some “predator–prey” systems with continuous delay of the argument for the case where the parameters of these systems are specified by ω-periodic continuous positive functions.

### Structure of finite inverse semigroup with zero, in which every stable order is fundamental or antifundamental

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 29 - 39

We find necessary and sufficient conditions for any stable order on a finite inverse semigroup with zéro to be fondamental or antifundamental.

### On the theory of the third-order equation with multiple characteristics containing the second time derivative

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 40 - 51

We construct a fundamental solution of the third-order equation with multiple characteristics containing the second time derivative, establish the estimates valid for large values of the argument, and study some properties of fundamental solutions necessary for the solution of boundary-value problems.

### Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point

Evtukhov V. M., Samoilenko A. M.

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 52 - 80

We establish conditions for the existence of solutions vanishing at a singular point for various classes of systems of quasilinear differential equations appearing in the investigation of the asymptotic behavior of solutions of essentially nonlinear nonautonomous differential equations of higher orders.

### Behavior of an almost semicontinuous Poisson process on a Markov chain upon attainment of a level

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 81–89

We consider almost semicontinuous processes defined on a Markov chain and obtain representations for the generatrices of the absolute maximum upon attainment of a positive level and the recovery time. Modified processes with two-step intensities of negative jumps are investigated.

### System of sticking diffusion particles of variable mass

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 90 - 103

We construct a mathematical model of an infinite system of diffusion particles with interaction whose masses affect the diffusion coefficient. The particles begin to move from a certain stationary distribution of masses. Their motion is independent up to their meeting. Then the particles become stuck and their masses are added. As a result, the diffusion coefficient varies as a function inversely proportional to the square root of the mass. It is shown that the mass transported by particles is also characterized by a stationary distribution.

### Best $m$-term trigonometric approximation for the classes $B^r_{p,θ}$ of functions of low smoothness

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 104–111

We obtain an exact-order estimate for the best $m$-term trigonometric approximation of the Besov classes $B^r_{p,θ}$ of periodic functions of many variables of low smoothness in the space $L_q, \; 1 < p ≤ 2 < q < ∞$.

### Inverse problem of spectral analysis of conflict dynamical systems

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 112–122

For conflict dynamical systems, we study the problem of the existence and description of initial measures that converge to measures with given spectral distributions.

### Approximations of classes $B^{Ω}_{p,θ}$ of functions of many variables by entire functions in the space $L_q (R^d)$

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 123–135

Exact-order estimates are obtained for the best approximations of the classes $B^{Ω}_{p,θ}$ of functions of many variables by entire functions of the exponential type in the space $L_q (R^d)$.

### Right Bézout ring with waist is a right Hermite ring

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 136–138

We study noncommutative rings in which the Jacobson radical contains a completely prime ideal. It is proved that a right Bézout ring in which the Jacobson radical contains a completely prime ideal is a right Hermite ring. We describe a new class of Bézout rings that are not elementary divisor rings.

### On the theory of hyper-$Q$-homeomorphisms

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 139–144

We show that if a homeomorphism $f$ of a domain $D ⊂ R^n,\; n ≥ 2$, is a hyper-$Q$-homeomorphism with $Q ∈ L_{\text{loc}^1$ , then $f ∈ ACL$. As a consequence, this homeomorphism has partial derivatives and an approximation differential almost everywhere.