# Volume 58, № 9, 2006

### Invariance principle for one class of Markov chains with fast Poisson time. Estimate for the rate of convergence

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1155–1174

We obtain an estimate for the rate of convergence of normalized Poisson sums of random variables determined by the first-order autoregression procedure to a family of Wiener processes.

### Initial-value problem for the Bogolyubov hierarchy for quantum systems of particles

Gerasimenko V. I., Shtyk V. O.

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1175–1191

We construct cumulant (semi-invariant) representations for a solution of the initial-value problem for the Bogolyubov hierarchy for quantum systems of particles. In the space of sequences of trace-class operators, we prove a theorem on the existence and uniqueness of a solution. We study the equivalence problem for various representations of a solution in the case of the Maxwell-Boltzmann statistics.

### Mixed problem for a nonlinear ultraparabolic equation that generalizes the diffusion equation with inertia

Lavrenyuk S. P., Protsakh N. P.

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1192–1210

We consider a mixed problem for a nonlinear ultraparabolic equation that is a nonlinear generalization of the diffusion equation with inertia and the special cases of which are the Fokker-Planck equation and the Kolmogorov equation. Conditions for the existence and uniqueness of a solution of this problem are established.

### Cauchy problem for one class of pseudodifferential systems with entire analytic symbols

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1211–1233

Using functions convex downward, we describe a class of pseudodifferential systems with entire analytic symbols that contains Éidel’man parabolic systems of partial differential equations with continuous time-dependent coefficients. We prove a theorem on the correct solvability of the Cauchy problem for these systems in the case where initial data are generalized functions. We also establish the principle of localization of a solution of this problem.

### Asymptotic expansion of a semi-Markov random evolution

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1234–1248

We determine the regular and singular components of the asymptotic expansion of a semi-Markov random evolution and show the regularity of boundary conditions. In addition, we propose an algorithm for finding initial conditions for *t* = 0 in explicit form using the boundary conditions for the singular component of the expansion.

### Dissipativity of differential equations and the corresponding difference equations

Stanzhitskii A. N., Tkachuk A. M.

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1249–1256

We establish conditions under which the existence of a bounded solution of a difference equation yields the existence of a bounded solution of the corresponding differential equation. We investigate the relationship between the dissipativities of differential and difference equations in terms of Lyapunov functions.

### On the equivalence of some conditions for weighted Hardy spaces

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1257–1263

Let $G ∈ H_{σ}^p (ℂ+)$, where $H_{σ}^p (ℂ+)$ is the class of functions analytic in the half plane ℂ+ = {z: Re z > 0} and such that $$\mathop {\sup }\limits_{\left| \varphi \right| < \tfrac{\pi }{2}} \left\{ {\int\limits_0^{ + \infty } {\left| {G(re^{i\varphi } )} \right|^p e^{ - p\sigma r\left| {sin\varphi } \right|} dr} } \right\} < + \infty .$$ In the case where a singular boundary function $G$ is identically constant and $G(z) ≠ 0$ for all $z ∈ ℂ_{+}$, we establish conditions equivalent to the condition $G(z)\exp \left\{ {\frac{{2\sigma }}{\pi }zlnz - cz} \right\} \notin H^p (\mathbb{C}_+ )$, where $H^p (ℂ_{+})$ is the Hardy space, in terms of the behavior of $G$ on the real semiaxis and on the imaginary axis.

### On Artinian rings satisfying the Engel condition

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1264–1270

Let $R$ be an Artinian ring, not necessarily with a unit element, and let $R^{\circ}$ be the group of all invertible elements of $R$ under the operation $a \circ b = a + b + ab.$
We prove that $R^{\circ}$ is a nilpotent group if and only if it is an Engel group and the ring $R$ modulo its Jacobson radical is commutative. In particular,
the group $R^{\circ}$ is nilpotent if it is weakly nilpotent or $n$-Engel for some positive integer $n$. We also establish that $R$ is a strictly Lie-nilpotent ring if and only if R is an
Engel ring and $R$ modulo its Jacobson radical is commutative.

Нехай $R$ — артінове кільце, необов'язково з одиницею, i $R^{\circ}$ — група оборотних елементів кільця $R$ відносно операції $a \circ b = a + b + ab.$

### Integral analog of one generalization of the Hardy inequality and its applications

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1271–1275

Under certain conditions on continuous functions $μ, λ, a$, and $f$, we prove the inequality $$\int\limits_0^y {\mu (x)\lambda (x)f\left( {\frac{{\int_0^x {\lambda (t)a(t)dt} }}{{\int_0^x {\lambda (t)dt} }}} \right)dx \leqslant K\int\limits_0^y {\mu (x)\lambda (x)f(a(x))} dx,} y \leqslant \infty ,$$ and describe its application to the investigation of the problem of finding conditions under which Laplace integrals belong to a class of convergence.

### Global analyticity of solutions of nonlinear functional differential equations representable by Dirichlet series

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1276–1284

We show that, under certain additional assumptions, analytic solutions of sufficiently general nonlinear functional differential equations are representable by Dirichlet series of unique structure on the entire real axis $\mathbb{R}$ and, in some cases, on the entire complex plane $\mathbb{C}$. We investigate the dependence of these solutions on the coefficients of the basic exponents of the expansion into a Dirichlet series. We obtain sufficient conditions for the representability of solutions of the main initial-value problem by series of exponents.

### Volterra functional integral equation of the first kind with nonlinear right-hand side and variable limits of integration

Artykova J. A., Yuldashev T. K.

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1285–1288

We prove a theorem on the existence and uniqueness of a solution of a Volterra functional integral equation of the first kind with nonlinear right-hand side and nonlinear deviation. We use the method of successive approximations combined with the method of contracting mappings.

### On quadruples of projectors connected by a linear relation

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1289–1295

We describe the set of γ ∈ ℝ for which there exist quadruples of projectors P i for a fixed collection of numbers $\alpha_i \in \mathbb{R}_+, \quad i = \overline{1,4} $, такі, що $\alpha_1 P_1 + \alpha_2 P_2 + \alpha_3 P_3 + \alpha_4 P_4 = \gamma I$.

### Dmitry Petrina

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1296