# Volume 54, № 1, 2002

### Optimization of Nonlinear Systems of Stochastic Difference Equations

Dzhalladova I. A., Valeyev K. G.

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 3-14

We present new results concerning the synthesis of optimal control for systems of difference equations that depend on a semi-Markov or Markov stochastic process. We obtain necessary conditions for the optimality of solutions that generalize known conditions for the optimality of deterministic systems of control. These necessary optimality conditions are obtained in the form convenient for the synthesis of optimal control. On the basis of Lyapunov stochastic functions, we obtain matrix difference equations of the Riccati type, the integration of which enables one to synthesize an optimal control. The results obtained generalize results obtained earlier for deterministic systems of difference equations.

### Pade–Chebyshev Approximants for One Class of Functions

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 15-19

By using the method of generalized moment representations proposed by Dzyadyk in 1981, we construct the Pade–Chebyshev approximants for one class of functions that is an analog of the class of Markov functions.

### On Some Problems of the Asymptotic Theory of Linear Differential Equations of the nth Order

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 20-42

We investigate smoothness properties of the roots of algebraic equations with almost constant coefficients and construct a transformation, which may be efficiently used for the investigation of the asymptotic behavior of a fundamental family of solutions of a broad class of nonautonomous linear differential equations of the *n*th order.

### Complete Asymptotics of the Deviation of a Class of Differentiable Functions from the Set of Their Harmonic Poisson Integrals

Kharkevych Yu. I., Zhyhallo K. M.

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 43-52

On a class of differentiable functions *W* ^{r} and the class \(\overline W ^r \) of functions conjugate to them, we obtain a complete asymptotic expansion of the upper bounds \(\mathcal{E}(\mathfrak{N},A\rho )_C \) of deviations of the harmonic Poisson integrals of the functions considered.

### On the Limit Distribution of Integrals of Shot-Noise Processes

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 53-62

We establish limit theorems for integrals of shot-noise processes and study the asymptotic behavior of the moments of integrals of this type.

### Upper and Lower Bounds of a Solution of the Cauchy Problem for a Stochastic Differential Equation of Parabolic Type with Power Nonlinearities (Weak Source)

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 63-70

We study the time evolution of a solution of the Cauchy problem for a stochastic differential equation of the parabolic type with power nonlinearities. We construct upper and lower bounds for this solution.

### Asymptotic Behavior of Solutions of Systems of Functional Differential Equations with Nonlinear Deviation of Argument

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 71-77

For systems of functional differential equations with nonlinear deviation of argument, we obtain sufficient conditions for the existence of families of their solutions that are continuously differentiable for *t* ∈ *R* ^{+}. We also investigate the asymptotic behavior of these solutions.

### Spatially-Homogeneous Boltzmann Hierarchy as Averaged Spatially-Inhomogeneous Stochastic Boltzmann Hierarchy

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 78-93

We introduce the stochastic dynamics in the phase space that corresponds to the Boltzmann equation and hierarchy and is the Boltzmann–Grad limit of the Hamiltonian dynamics of systems of hard spheres. By the method of averaging over the space of positions, we derive from it the stochastic dynamics in the momentum space that corresponds to the space-homogeneous Boltzmann equation and hierarchy. Analogous dynamics in the mean-field approximation was postulated by Kac for the explanation of the phenomenon of propagation of chaos and derivation of the Boltzmann equation.

### Some Pseudoparabolic Variational Inequalities with Higher Derivatives

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 94-105

We consider a pseudoparabolic variational inequality with higher derivatives. We prove the existence and uniqueness of a solution of this inequality with a zero initial condition.

### Direct and Inverse Theorems in the Theory of Approximation of Functions in the Space $S^p$

Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 106-124

We continue the investigation of approximation properties of the space $S^p$. We introduce the notion of kth modulus of continuity and establish direct and inverse theorems on approximation in the space $S^p$ in terms of the best approximations and moduli of continuity. These theorems are analogous to the well-known theorems of Jackson and Bernshtein.

### Helly Theorem and Related Results

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 125-128

By using the classical Helly theorem, one cannot obtain information about a family of convex compact sets in the *n*-dimensional Euclidean space if it is known that only subfamilies consisting of *k* elements, 0 < *k* ≤ *n*, have nonempty intersections. We modify the Helly theorem to fix this issue and investigate the behavior of generalized convex families.

### Orders of Power Growth near the Critical Strip of the Riemann Zeta Function

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 129-132

We study the asymptotic behavior of the functions ζ(*z*) and ζ^{−1}(*z*) near the line *x* = 1.

### On the Existence of Periodic Solutions of a System of Two Differential Equations with Pulse Influence

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 133-137

We investigate the problem of the existence of periodic solutions of a system of two linear differential equations with pulse influence on a plane in the case where a stable knot or a stable focus is a singular point of this system.

### Asymptotic Behavior of Solutions of Nonlinear Difference Equations with Continuous Argument

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 138-141

We establish conditions for the existence and uniqueness of continuous asymptotically periodic solutions of nonlinear difference equations with continuous argument.

### On Representations of C*-Algebras O*n*, α of the Cuntz Type

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 142-145

We show that Cuntz-type algebras *O* _{n, k}, *n* ≥ *k* ≥ 2, and *O* _{ n, k + 1/2}, *n* ≥ 4, *k* ≥ 2, are *-wild (this implies that the description of all *-representations of these algebras is a very complicated problem).