### Finite-dimensional approximations of diffusion measures in a Hilbert space

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1587–1592

For transition probabilities of diffusion processes in a Hilbert space, we construct finite-dimensional approximations and establish sufficient conditions for the equivalence of such measures under perturbation of the diffusion operator.

### A critical case of stability of one quasilinear difference equation of the second order

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1593–1603

We obtain sufficient conditions for the Perron stability of the trivial solution of a real difference equation of the form $$y_{n + 1} - 2\lambda _n y_n + y_{n - 1} = F(n,y_n ,\Delta y_{n - 1} ), n \in N$$ where \(y_n \in \left] { - 1,1} \right[,\left| {F(n,y_n ,\Delta y_{n - 1} )} \right| \le L_n \left( {\left| {y_n \left| + \right|\Delta y_{n - 1} } \right|} \right)^{1 + \alpha } ,L_n \ge 0\) and \(\alpha \in \left] {0, + \infty } \right[\) . The resuits obtained are valid for the case where \(\left| {\lambda _n } \right| = 1 + o(1), n \to + \infty \) .

### A multipoint problem for partial differential equations unresolved with respect to the higher time derivative

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1604–1613

We investigate the well-posedness of problems for partial differential equations unresolved with respect to the higher time derivative with multipoint conditions with respect to time. By using the metric approach, we determine lower bounds for small denominators appearing in the course of the solution of the problems.

### Structure of certain classes of groups with locally cyclic Abelian subgroups

Kuzennyi N. F., Maznichenko S. V.

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1614–1627

We investigate groups with locally cyclic Abelian subgroups. We give a constructive description of locally solvable groups of this type that contain a nonidentity periodic part; we also describe locally solvable groups all Abelian subgroups of which are cyclic.

### On lower bounds for the approximation of individual functions by local splines with nonfixed nodes

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1628–1637

For functions with the integrable βth power, where β = (*r* + 1 + 1/p)^{−1}, we obtain asymptotically exact lower bounds for the approximation by local splines of degree*r* and defect*k* ≥*r*/2 in the metric of*L* _{p}.

### Direct methods for the approximate solution of systems of singular integral equations in the case of nonnegative partial indices

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1638–1644

We justify direct methods for the approximate solution of systems of singular integral equations with Cauchy kernel on a unit circle in the case of nonnegative partial indices.

### Technical stability of autonomous control systems with variable structure

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1645–1658

We obtain conditions for the technical stability of autonomous dynamical systems with discontinuous control with respect to a given measure.

### Essentially unstable solutions of difference equations

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1659–1672

We study the essential instability of solutions of linear and nonlinear difference equations.

### Rate of convergence of a group of deviations on sets of $\bar{\psi}$−integrals

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1673-1693

We study functionals that characterize the strong summation of Fourier series on sets of $\bar{\psi}$−integrals in the uniform and integral metrics. As a result, we obtain estimates exact in order for the best approximations of functions from these sets by trigonometric polynomials.

### Systems of singularly perturbed degenerate integro-differential equations

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1694–1703

We construct asymptotic solutions of singularly perturbed homogeneous and inhomogeneous systems of integro-differential Fredholm-type equations with a degenerate matrix as the coefficient of the derivative.

### On the structure of the set of nonwandering points of a pair of coupled quadratic maps

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1704–1709

In the plane of parameters, we indicate values for which plane endomorphisms constructed by coupling two identical one-dimensional unimodal quadratic maps have an absorbing domain that contains an attractor and a nontrivial invariant subset of the set of nonwandering points.

### Investigation of smoothness conditions on the boundary for the strong linear convexity of a domain

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1710–1713

We construct a counterexample for the hypothesis that the strong linear convexity of a domain follows from the linear convexity if the set of singularities does not split the boundary.

### Co-adjoint orbits of compact Lie groups and generalized stereographic projection

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1714–1718

We generalize the notion of stereographic projection to the case of an arbitrary compact Lie group and find the explicit form of the local complex parametrization of an orbit of the corresponding group.

### On the best $m$-term trigonometric and orthogonal trigonometric approximations of functions from the classes $L^{Ψ}_{β,ρ}$

↓ Abstract

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1719-1721

We obtain estimates exact in order for the best trigonometric and orthogonal trigonometric approximations of the classes $L^{Ψ}_{β,ρ}$ of functions of one variable in the space $L_q$ in the case $2 < p < q < ∞$.

### Index of volume 51 of „Ukrainian Mathematical Journal"

Ukr. Mat. Zh. - 1999νmber=5. - 51, № 12. - pp. 1722-1728