2017
Том 69
№ 4

All Issues

Volume 58, № 9, 2006

Article (Russian)

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

Baev A. V., Bondarev B. V.

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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.

Article (Ukrainian)

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

Gerasimenko V. I., Shtyk V. O.

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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.

Article (Ukrainian)

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

Lavrenyuk S. P., Protsakh N. P.

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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.

Article (Ukrainian)

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

Litovchenko V. A.

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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.

Article (Ukrainian)

Asymptotic expansion of a semi-Markov random evolution

Samoilenko I. V.

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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.

Article (Ukrainian)

Dissipativity of differential equations and the corresponding difference equations

Stanzhitskii A. N., Tkachuk A. M.

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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.

Brief Communications (Ukrainian)

On the equivalence of some conditions for weighted Hardy spaces

Dilnyi V. M.

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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.

Brief Communications (Russian)

On Artinian rings satisfying the Engel condition

Evstaf’ev R. Yu.

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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 $a \circ b = a + b + ab.$

Brief Communications (Ukrainian)

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

Mulyava O. M.

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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.

Brief Communications (Russian)

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

Murovtsev A. N.

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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.

Brief Communications (English)

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

Artykova J. A., Yuldashev T. K.

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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.

Brief Communications (Ukrainian)

On quadruples of projectors connected by a linear relation

Yusenko K. A.

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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 $\alpha_1 P_1 + \alpha_2 P_2 + \alpha_3 P_3 + \alpha_4 P_4 = \gamma I$.

Obituaries (Ukrainian)

Dmitry Petrina

Editorial Board

Full text (.pdf)

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