# Volume 46, № 12, 1994

### Large deviation probabilities for *UH*-statistics

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1611–1620

For*U*-statistics taking values in a Hilbert space, we obtain estimates of the rate of convergence in the central limit theorem.

### Nonlocal boundary-value problem for parabolic equations

Mel'nyk O. M., Ptashnik B. I., Zadorozhna N. M.

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1621–1626

We study the problem for Shilov parabolic equations of arbitrary order with constant coefficients with conditions nonlocal in time and periodic in space variables. We establish conditions for the existence and uniqueness of a classical solution of the problem and prove metric theorems on lower bounds of small denominators appearing in the construction of a solution of the problem.

### Bogolyubov averaging and normalization procedures in nonlinear mechanics. III

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1627–1646

We describe the technique of normalization based on the method of asymptotic decomposition in the space of representation of a finite-dimensional Lie group. The main topics of the theory necessary for understanding the method are outlined. Models based on the Van der Pol equation are investigated by the method of asymptotic decomposition in the space of homogeneous polynomials (the space of representation of a general linear group in a plane) and in the space of representation of a rotation group on a plane (ordinary Fourier series). The comparison made shows a dramatic decrease in the necessary algebraic manipulations in the second case. We also discuss other details of the technique of normalization based on the method of asymptotic decomposition.

### Elliptic boundary-value problems in complete scales of Nikol'skii-type spaces

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1647–1654

We consider an elliptic boundary-value problem on an infinitely smooth manifold with, generally speaking, disconnected boundary. It is established that the operator of this problem is a Fredholm operator when considered in complete scales of functional spaces that depend on the parameters *s* ε ℝ,*p*ε[1, ∞] and, for sufficiently large s≥0, coincide with the classical Nikol'skii spaces on a manifold.

### Points of strong summability of fourier series

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1655–1664

We study strong means of deviations of partial sums of expansions of functions *f* in systems of functions of polynomial type.

### On some problems in perturbation theory of smooth invariant tori of dynamical systems

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1665–1699

Problems related to perturbation theory of smooth invariant tori of dynamical systems in a *n*-dimensional Euclidean space*R* ^{n} are considered. The clarification of these problems plays an important role for perturbation theory suggested by the author in [1] and extends the scope of its application.

### On the universal ultrametric space

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1700–1706

An ultrametric space in which any separable ultrametric space can be isometrically imbedded is constructed. We describe the method for isometric imbedding of any separable ultrametric space into *l* _{1},*l* _{2} and*c* _{0} based on the application of this universal space.

### On invariance of some properties of solutions under perturbation of a pulse system of differential equations

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1707–1713

Sufficient conditions for the invariance of boundedness and stability properties of solutions under perturbation of a pulse system of differential equations are established.

### Minimal handle decomposition of smooth simply connected five-dimensional manifolds

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1714–1720

A theorem on the existence of the unique minimal topologic handle decomposition of differentiable simply connected five-dimensional manifolds is proved. For a decomposition of this sort, the number of handles of each index is given.

### Method for solution of convolution-type integral equations

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1721–1723

The spectral method for solution of convolution-type integral equations in the basis of Chebyshev-Laguerre polynomials is reduced to the representation in matrix form. This enables one to construct algorithms of reconstruction of input signals directly from the discrete values of the output signals and to estimate the influence of an input data error on the precision of reconstruction of a signal.

### Index of Volume 46 of Ukrainian Mathematical Journal

Ukr. Mat. Zh. - 1994. - 46, № 12. - pp. 1724-1729