# Volume 53, № 10, 2001

### Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1299-1308

We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for *r* = 2, *k* = 1 or *r* = 3, *k* = 1, 2 and arbitrary *q, p* ∈ [1, ∞], the following unimprovable inequality holds for functions \(x \in L_\infty ^r \) : $$\left\| {x^{\left( k \right)} } \right\|_q \leqslant \frac{{\left\| {{\phi }_{r - k} } \right\|_q }}{{\left\| {{\phi }_r } \right\|_p^\alpha }}\left\| x \right\|_p^\alpha \left\| {x^{\left( k \right)} } \right\|_\infty ^{1 - \alpha } $$ where \(\alpha = \min \left\{ {1 - \frac{k}{r},\frac{{r - k + {1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-0em} q}}}{{r + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-0em} p}}}} \right\}\) and ϕ_{ r } is the perfect Euler spline of order *r*.

### Existence of a Global Classical Solution of One Problem Arising in Combustion Theory

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1309-1319

We consider a multidimensional free-boundary problem for a parabolic equation that arises in combustion theory. We prove the existence of a global classical solution. The idea of the method is as follows: first, we perform the differential–difference approximation of the problem and establish its solvability; then we prove uniform estimates and perform a limit transition.

### A Method for the Regularization of One Class of Dual Series Equations

Brovenko A. V., Melezhik P. N., Poedinchuk A. К.

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1320-1327

We propose a method for the regularization of one class of systems of dual series equations to which numerous problems in theoretical and mathematical physics are reduced.

### A Problem with Nonlocal Conditions for Partial Differential Equations with Variable Coefficients

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1328-1336

We establish conditions for the unique solvability of a problem for partial differential equations with coefficients dependent on variables *t* and *x* in a rectangular domain with nonlocal two-point conditions with respect to *t* and local boundary conditions with respect to *x*. We prove metric statements related to lower bounds of small denominators appearing in the course of solution of the problem.

### An Analog of the Poincaré Model for a Quaternion Hyperbolic Space

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1337-1342

We construct an analog of the Poincaré model for a quaternion hyperbolic space.

### Main Probability Characteristics of the Queuing System $G^k|G|1$

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1343-1357

For the queuing system *G* ^{κ}|*G*|1 with batch arrivals of calls, we present the distributions of the following characteristics: the length of a busy period, queue length in transient and stationary modes of the queuing system, total idle time of the queuing system, virtual waiting time to the beginning of the service, input stream of calls, output stream of served calls, etc.

### Application of the Averaging Method to the Investigation of Nonlinear Wave Processes in Elastic Systems with Circular Symmetry

Koval’chuk P. S., Kubenko V. D.

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1358-1367

We apply asymptotic methods of nonlinear mechanics (the Bogolyubov–Mitropol'skii averaging method) to the construction of approximate solutions of a system of nonlinear equations describing wave processes in elastic systems with circular symmetry. As an example, we study the dynamics of interaction of two flexural waves that propagate in a cylindrical shell under the conditions of free oscillations and periodic excitation.

### On One Regularity Condition for Quantum Quadratic Stochastic Processes

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1367-1379

We present necessary and sufficient conditions for the validity of a regularity condition for homogeneous quantum quadratic stochastic processes defined on von Neumann algebras.

### On Identities in Algebras $Q_{n,λ}$ Generated by Idempotents

Rabanovych V. I., Samoilenko Yu. S., Strilets O. V.

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1380-1390

We investigate the presence of polynomial identities in the algebras $Q_{n,λ}$ generated by $n$ idempotents with the sum $λe$ ($λ ∈ C$ and $e$ is the identity of an algebra). We prove that $Q_{4,2}$ is an algebra with the standard polynomial identity $F_4$, whereas the algebras $Q_{4,2},\; λ ≠ 2$, and $Q_{n,λ},\; n ≥ 5$, do not have polynomial identities.

### Random Walks in Random Media on a Cayley Tree

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1391-1401

We present sufficient conditions for the transience of random walks with bounded jumps in random media on a Cayley tree.

### New Exact Solutions of One Nonlinear Equation in Mathematical Biology and Their Properties

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1409-1421

The classical Lie approach and the method of additional generating conditions are applied to constructing multiparameter families of exact solutions of the generalized Fisher equation, which is a simplification of the known coupled reaction–diffusion system describing spatial segregation of interacting species. The exact solutions are applied to solving nonlinear boundary-value problems with zero Neumann conditions. A comparison of the analytic results and the corresponding numerical calculations shows the importance of the exact solutions obtained for the solution of the generalized Fisher equation.

### Periodic Atomic Quasiinterpolation

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1422-1426

We consider the approximation of periodic functions by using the atomic quasiinterpolation of the second and the first order. We obtain expressions for the coefficients of quasiinterpolants and present estimates for errors in the uniform metric.

### Upper Subrings of a Ring

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1427-1428

We describe maximal ideals of rings that are contained in the adjoint groups of their upper subrings.

### On Stabilization of Energy of a Conservative System Perturbed by a Random Process of “White-Noise” Type in the Itô Form

Bernatskaya J. N., Kulinich G. L.

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1429-1435

We investigate the problem of deterministic control over the behavior of the total energy of the simplest conservative nonlinear system with one degree of freedom without friction in the case of random perturbations by a process of the “white-noise” type in the Itô form. These perturbations act under a fixed angle to the vector of phase velocity of the conservative system.

### On One Class of Matrix Topological *-Algebras

Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1436-1439

We study a matrix algebra *M* _{n}(*U*), where *U* is a commutative topological nuclear entire (bounded, analytic) *-algebra. We prove that *M* _{n}(*U*) is also a topological nuclear entire (bounded, analytic) *-algebra.