# Volume 49, № 7, 1997

### Functional law of the iterated logarithm for fields and its applications

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 883–894

For a Wiener field with an arbitrary finite number of parameters, we construct the law of the iterated logarithm in the functional form. We consider the problem for random fields of a certain type to reside within curvilinear boundaries without assuming that the Cairoli—Walsh condition is satisfied.

### Degenerate orbits of adjoint representation of orthogonal and unitary groups regarded as algebraic submanifolds

Boyars'kyi O. M., Skrypnik T. V.

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 895–905

We suggest a method for describing some types of degenerate orbits of orthogonal and unitary groups in the corresponding Lie algebras as level surfaces of a special collection of polynomial functions. This method allows one to describe orbits of the types *SO*(2*n*)/*SO*(2*k*)×*SO*(2)^{ n−k }, *SO*(2*n*+1)/*SO*(2*k*+1)×*SO*(2)^{ n−k }, and (*S*)*U*(*n*)/(*S*)(*U*(2*k*)×*U*(2)^{ n−k }) in *so(2n)*, *so(2n+1)*, and *(s)u(n)*, respectively. In addition, we show that the orbits of minimal dimensions of the groups under consideration can be described in the corresponding algebras as intersections of quadries. In particular, this approach is used for describing the orbit *CP* ^{ n−1}⊂*u(n)*.

### On one generalization of the averaging method

Dzhalladova I. A., Valeyev K. G.

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 906–911

We consider one case where it is possible to establish sufficient conditions for the convergence and analyticity of matrix series used for the construction of a system of moment equations.

### Probability for a Wiener process to reside in tube domains for a long period of time

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 912–917

We give various representations of asymptotics for the probability for a Wiener process to reside within a curvilinear strip during extended time intervals.

### Finite groups with complementable maximal primary cyclic subgroups

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 918–926

We study finite nonprimary groups with complementable maximal primary cyclic subgroups and give a description of all supersolvable groups of this sort.

### On one problem of minimization of area

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 927–936

We consider the problem of asymptotically optimal location of disks with equal radii for the minimization of the are of the figure bounded by a given curve and a connected union of these disks.

### On axiomatizations of Boolean algebras

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 937–942

We construct some new axiomatic systems for the Boolean algebra. In particular, an axiomatic system for disjunction and logical negation consists of three axioms. We prove the independence of the axiomatic systems proposed.

### Existence and stability of periodic solutions for chains of connected oscillators

Maistrenko Yu. L., Popovich O. V.

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 943–950

We consider a nonlinear system of difference equations. This system corresponds to chains of *N* symmetrically connected oscillators with sufficiently general type of connection, which includes, among others, local and global connection. We prove a theorem on the existence and stability of space-time periodic solutions of such systems for sufficiently small values of the parameter of connection ɛ.

### Approximation of certain classes of differentiable functions by generalized splines

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 951–957

We find the exnet value of the best (α, β)-approximation by generalized Chebyshev splines for a class of functions differentiable with weight on [−1, 1].

### Solutions of the Maxwell equations describing the spectrum of hydrogen

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 958–969

We obtain a new class of solutions of the Maxwell equations describing the spectrum of hydrogen. We prove that, instead of the quantum-mechanical Dirac equation, the ordinary classical Maxwell equations can be applied to the solution of many problems in atomic and nuclear physics.

### Nonlinear difference equations with asymptotically stable solutions

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 970–980. July

We establish conditions of asymptotic stability for all solutions of the equation *X* _{ n+1}=*F*(*X* _{ n }), *n*≥0, in the Banach space *E* in the case where *r(F′(x))<1 ∀ x ∈ E, r′(x)* is the spectral radius of *F′(x)*. An example of an equation with an unstable solution is given.

### On uniformly stable linear quasiperiodic systems

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 981–987

In a finite-dimensional complex space, we consider a system of linear differential equations with quasiperiodic skew-Hermitian matrix. The space of solutions of this system is a sum of one-dimensional invariant subspaces. Over a torus defined by a quasiperiodic matrix of the system, we investigate the corresponding one-dimensional invariant bundles (nontrivial in the general case). We find conditions under which these bundles are trivial and the system can be reduced to diagonal form by means of the Lyapunov quasiperiodic transformation with a frequency module coinciding with the frequency module of the matrix of the system.

### On inequalities of Kolmogorov-Hörmander type for functions bounded on a discrete net

Babenko V. F., Vakarchuk M. B.

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 988–992

We obtain a strengthened version of the Hörmander inequality for functions ƒ: ℝ → ℝ, in which, instead of ‖ƒ‖_{∞}, we use the least upper bound of the values of *f* on a discrete set of points.

### One-dimensional problems with free boundaries in ecology

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 993–997

Statements of problems with free boundaries are given for nonlinear parabolic equations arising in ecology and medicine. Some constructive methods for their solution are considered.

### Exact solution of one boundary-value problem

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 998–1001

We study the boundary-value perlodic problem *u* _{ tt }−*u* _{ xx }=*F*(*x, t*), *u*(0, *t*)=*u*(π, *t*)=0, *u*(*x*, *t*+*T*)=*u*(*x, t*), (*x, t*) is **R** ^{2}. By using the Vejvoda-Shtedry operator, we determine a solution of this problem.

### Remark on the Lebesgue constant in the Rogosinski Kernel

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 1002–1004

For every *n*, we compute the Lebesgue constant of Rogosinski kernel with any preassigned accuracy.

### Limit theorem for the maximum of dependent Gaussian random elements in a Banach space

Ukr. Mat. Zh. - 1997. - 49, № 7. - pp. 1005–1008

The well-known Nisio result on the asymptotie equality for the maximum of real-valued Gaussian random variables is generalized to the case of Gaussian random variables taking values in a Banach space.