# Volume 61, № 11, 2009

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

Babenko V. F., Parfinovych N. V.

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.

### Equivalence of closed 1-forms on surfaces with edge

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

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.

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

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 $L_q$.

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

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.

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

Kharkevych Yu. I., Zhyhallo T. V.

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 $C$.

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

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.

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

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 \( \mathcal{M} \).

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

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.

### On one atypical scheme of application of the second Lyapunov method

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.

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

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.

### On the stable range of matrix rings

Petrichkovich V. M., Zabavskii B. V.

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.

### Growth of generalized Temperley–Lieb algebras connected with simple graphs

Samoilenko Yu. S., Zavodovskii M. V.

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 \( {\tilde E_7} \).