2017
Том 69
№ 9

All Issues

Volume 57, № 4, 2005

Article (Russian)

Estimates for the Rate of Convergence in Ordinary Differential Equations under the Action of Random Processes with Fast Time

Bondarev B. V., Kovtun E. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 435–457

We study the procedure of averaging in the Cauchy problem for an ordinary differential equation perturbed by a certain Markov ergodic process. We establish several estimates for the rate of convergence of solutions of the original problem to solutions of the averaged one.

Article (Russian)

On the Solvability of Impulsive Differential-Algebraic Equations

Perestyuk N. A., Vlasenko L. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 458–468

We establish theorems on the existence and uniqueness of a solution of the impulsive differential-algebraic equation $$\frac{d}{{dt}}[Au(t)] + Bu(t) = f(t,u(t)),$$ where the matrix A may be singular. The results are applied to the theory of electric circuits.

Article (Ukrainian)

Congruences of a Permutable Inverse Semigroup of Finite Rank

Derech V. D.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 469–473

We describe the structure of any congruence of a permutable inverse semigroup of finite rank.

Article (Ukrainian)

Properties of a Solution of an Inhomogeneous Hyperbolic Equation with Random Right-Hand Side

Dovhai B. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 474–482

We consider an inhomogeneous hyperbolic equation with zero initial and boundary conditions and a random centered sample-continuous Gaussian right-hand side. We establish conditions for the existence of a solution of the first boundary-value problem of mathematical physics in the form of a series uniformly convergent in probability in terms of a covariance function. An estimate for the distribution of the supremum of a solution of this problem is obtained.

Article (Ukrainian)

On Periodic Solutions of One Class of Systems of Differential Equations

Korol' I. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 483–495

We study the problem of the existence of periodic solutions of two-dimensional linear inhomogeneous periodic systems of differential equations for which the corresponding homogeneous system is Hamiltonian. We propose a new numerical-analytic algorithm for the investigation of the problem of the existence of periodic solutions of two-dimensional nonlinear differential systems with Hamiltonian linear part and their construction. The results obtained are generalized to systems of higher orders.

Article (Russian)

Summation of Fourier-Laplace Series in the Space $L(S^m)$

Lasuriya R. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 496–504

We establish estimates of the rate of convergence of a group of deviations on a sphere in the space $L(S^m),\quad m > 3$.

Article (Russian)

On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point

Mokhon'ko O. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 505–513

The statement of the Malmquist theorem (1913) about the growth of meromorphic solutions of the differential equation \(f' = \frac{{P(z,f)}}{{Q(z,f)}}\), where P(z, f) and Q(z, f) are polynomials in all variables, is proved in the case of solutions with isolated singular point at infinity.

Article (Ukrainian)

Groups with Almost Normal Subgroups of Infinite Rank

Kuchmenko S. N., Semko N. N.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 514–532

We study classes of groups whose subgroups of some infinite ranks are almost normal.

Article (Russian)

Best $n$-Term Approximations with Restrictions

Stepanets O. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 533–553

We determine exact values of the best $n$-term approximations with restrictions on polynomials used for the approximation of $\lambda, q$-ellipsoids in the spaces $S^{p,\, \mu}_{\varphi}$.

Brief Communications (Russian)

Multiplicity of Continuous Mappings of Domains

Zelinskii Yu. B.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 554–558

We prove that either the proper mapping of a domain of an n-dimensional manifold onto a domain of another n-dimensional manifold of degree k is an interior mapping or there exists a point in the image that has at least |k|+2 preimages. If the restriction of f to the interior of the domain is a zero-dimensional mapping, then, in the second case, the set of points of the image that have at least |k|+2 preimages contains a subset of total dimension n. In addition, we construct an example of a mapping of a two-dimensional domain that is homeomorphic at the boundary and zero-dimensional, has infinite multiplicity, and is such that its restriction to a sufficiently large part of the branch set is a homeomorphism.

Brief Communications (Russian)

On Initial Data of a Simple Conservative Scattering System That Can Be Transferred to Zero by a Sequence of Inputs from l ²

Nudel'man M. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 559–563

We describe the lineal of initial data of a simple conservative scattering system which can be transferred to zero by a sequence from l ². The proof is based on the known connection between the Lax - Phillips scattering theory and the theory of unitary operator nodes developed by B. Szokefalvi - Nagy, C. Foias, and M. S. Brodskii.

Brief Communications (Russian)

Approximation of Classes of ψ-Integrals of Periodic Functions of Many Variables by Rectangular Linear Means of Their Fourier Series

Bodraya V. I., Novikov O. A., Rukasov V. I.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 564–570

We obtain asymptotic equalities for deviations of rectangular linear means of Fourier series on classes of ψ-integrals of multivariable functions

Brief Communications (Ukrainian)

On the Stability of the Maximum Term of the Entire Dirichlet Series

Skaskiv O. B., Trakalo O. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 571–576

We establish necessary and sufficient conditions for logarithms of the maximal terms of the entire Dirichlet series $F(z) = \sum^{+\infty}_{n=0}a_n e^{z\lambda_n}$ and $A(z) = \sum^{+\infty}_{n=0}a_n b_n e^{z\lambda_n}$ to be asymptotically equivalent as ${\rm Re}\;z \rightarrow +\infty$ outside some set of finite measure.