2017
Том 69
№ 2

All Issues

Volume 61, № 11, 2009

Article (Russian)

Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities

Babenko V. F., Parfinovych N. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1443-1454

We determine the exact values of the best (α, β)-approximations and the best one-sided approximations of classes of differentiable periodic functions by splines of defect 2. We obtain new sharp Jackson-type inequalities for the best approximations and the best one-sided approximations by splines of defect 2.

Article (Ukrainian)

Equivalence of closed 1-forms on surfaces with edge

Budnyts'ka T. V., Prishlyak A. O.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1455-1473

We investigate closed 1-forms with isolated zeros on surfaces with edge. A criterion for the topological equivalence of closed 1-forms is proved.

Article (Ukrainian)

Best orthogonal trigonometric approximations of the classes $B^{Ω}_{p,θ}$ of periodic functions of many variables

Voitenko S. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1473-1484

We obtain exact-order estimates for the best orthogonal trigonometric approximations of the classes $B^{Ω}_{p,θ}$ of periodic functions of many variables in the space $L_q$.

Article (English)

On removable sets of solutions of second-order elliptic and parabolic equations in nondivergent form

Gadjiev T. S., Mamedova V. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1485-1496

We consider nondivergent elliptic and parabolic equations of the second order whose leading coefficients satisfy the uniform Lipschitz condition. We find a sufficient condition for the removability of a compact set with respect to these equations in the space of Hölder functions.

Article (Ukrainian)

Approximation of (ψ, β)-differentiable functions by Poisson integrals in the uniform metric

Kharkevych Yu. I., Zhyhallo T. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1497-1515

We obtain asymptotic equalities for upper bounds of approximations of functions from the class $C_{β,∞} ψ$ by Poisson integrals in the metric of the space $C$.

Article (Ukrainian)

Asymptotic solutions of a system of differential equations with multiple turning point

Klyuchnyk I. H.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1516-1530

Using a transformation matrix, we asymptotically reduce a system of differential equations with a small parameter in the coefficients of a part of derivatives and with multiple turning point to an integrable system of equations.

Article (Russian)

$(o)$-Topology in *-algebras of locally measurable operators

Chilin V. I., Muratov M. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1531-1540

We consider the topology \( t\left( \mathcal{M} \right) \) of convergence locally in measure in the *-algebra \( LS\left( \mathcal{M} \right) \) of all locally measurable operators affiliated to the von Neumann algebra \( \mathcal{M} \). We prove that \( t\left( \mathcal{M} \right) \) coincides with the (o)-topology in \( L{S_h}\left( \mathcal{M} \right) = \left\{ {T \in LS\left( \mathcal{M} \right):T* = T} \right\} \) if and only if the algebra \( \mathcal{M} \) is σ-finite and is of finite type. We also establish relations between \( t\left( \mathcal{M} \right) \) and various topologies generated by a faithful normal semifinite trace on \( \mathcal{M} \).

Article (Ukrainian)

Method of local linear approximation in the theory of bounded solutions of nonlinear differential equations

Slyusarchuk V. Yu.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1541-1556

The conditions for the existence of solutions of nonlinear differential equations in a space of functions bounded on the axis are established by using local linear approximations of these equations.

Article (Ukrainian)

On one atypical scheme of application of the second Lyapunov method

Sosnitskii S. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1557-1563

The second Lyapunov method is applied to the analysis of stability of triangular libration points in a three-dimensional restricted circular three-body problem. It is shown that the triangular libration points are unstable.

Article (Russian)

Boundary-value problems for the wave equation with Lévy Laplacian in the Gâteaux class

Feller M. N.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1564-1574

We present the solutions of the initial-value problem in the entire space and the solutions of the boundary-value and initial-boundary-value problems for the wave equation $$\frac{∂^2U(t,x)}{∂x^2} = Δ_LU(t,x)$$ with infinite-dimensional Lévy Laplacian $Δ_L$ in the class of Gâteaux functions.

Brief Communications (Ukrainian)

On the stable range of matrix rings

Petrichkovich V. M., Zabavskii B. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1575-1578

It is shown that an adequate ring with nonzero Jacobson radical has a stable range 1. A class of matrices over an adequate ring with stable range 1 is indicated.

Brief Communications (Russian)

Growth of generalized Temperley–Lieb algebras connected with simple graphs

Samoilenko Yu. S., Zavodovskii M. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1579-1585

We prove that the generalized Temperley–Lieb algebras associated with simple graphs Γ have linear growth if and only if the graph Γ coincides with one of the extended Dynkin graphs \( {\tilde A_n} \), \( {\tilde D_n} \), \( {\tilde E_6} \), or \( {\tilde E_7} \). An algebra \( T{L_{\Gamma, \tau }} \) has exponential growth if and only if the graph Γ coincides with none of the graphs \( {A_n} \), \( {D_n} \), \( {E_n} \), \( {\tilde A_n} \), \( {\tilde D_n} \), \( {\tilde E_6} \), and \( {\tilde E_7} \).