# Volume 47, № 8, 1995

### On optimization of weight quadrature formulas

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1011–1021

We obtain asymptotically optimal quadrature formulas on the class*H* ^{ ω }[-1, 1] for an arbitrary continuous weight function which is positive on [-1, 1] almost everywhere and for a wide class of moduli of continuity ω(*t*).

### Bifurcation of an equilibrium state of a singularly perturbed system with lag

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1022–1028

We consider a system of singularly perturbed differential-difference equations with periodic right-hand sides. A representation of the integral manifold of this system is obtained. The bifurcation of an invariant torus from an equilibrium state and subfurcation of periodic solutions are studied.

### Exponents of elements of a normal basis of the ideal of algebraic functions on a three-sheeted Riemannian surface

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1029–1037

On a three-sheeted Riemannian surface*R* of genus ρ given by an irreducible algebraic equation, we construct normal bases of the ideal of algebraic functions that are multiples of some integer divisors. A method for constructing such normal bases was given in [V. E. Kruglov,*Dokl. Akad. Nauk SSSR*,**321**, No. 1, 11–13 (1991)]. The relations obtained for the exponents of the constructed elements enable one to determine the number of solutions of the Riemann problem for any integer divisor and to find partial indices in the problems of factorization of matrices of permutation type.

### Interpolation Whitney constants

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1038–1043

We obtain new estimates for interpolation Whitney constants.

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

Lopatin A. K., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1044–1068

In this paper, we apply the theory developed in parts I-III [*Ukr. Math. Zh.*,**46**, No. 9, 1171–1188; No. 11, 1509–1526; No. 12, 1627–1646 (1994)] to some classes of problems. We consider linear systems in zero approximation and investigate the problem of invariance of integral manifolds under perturbations. Unlike nonlinear systems, linear ones have centralized systems, which are always decomposable. Moreover, restrictions connected with the impossibility of diagonalization of the coefficient matrix in zero approximation are removed. In conclusion, we apply the method of local asymptotic decomposition to some mechanical problems.

### Approximation of classes of continuous functions by generalized de la Vallée-Poussin sums

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1069–1079

We introduce generalized de la Vallée-Poussin sums and study their approximation properties for the classes of continuous periodic functions*C* _{β,} ^{ψ∞} .

### Characterization of locally connected continua in Euclidean spaces

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1080–1088

We prove that, for any locally connected bounded continuum in the Euclidean space E^{ n },*n*≥2, there exists a sequence of imbeddings of the segment [0, 1] into E^{ n } uniformly convergent to a continuous mapping of [0, 1] onto this continuum.

### On the sum of two Lie algebras with finite-dimensional commutants

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1089–1096

We prove that an infinite-dimensional Lie algebra over an arbitrary field which is decomposable into the sum of two of its subalgebras with finite-dimensional commutants is almost solvable.

### Best trigonometric and bilinear approximations for the Besov classes of functions of many variables

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1097–1111

We obtain order estimates for the best trigonometric and bilinear approximations for the classes*B* _{ p,θ } ^{ r } of functions of many variables.

### Lower bounds for widths of classes of convolutions of periodic functions in the metrics of $C$ and $L$

Serdyuk A. S., Stepanets O. I.

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1112-1121

We give new sufficient conditions for kernels to belong to the set $C_{y, 2n}$ introduced by Kushpel'. These conditions give the possibility of extending the set of kernels belonging to $C_{y, 2n}$. On the basis of these results, we obtain lower bounds for the Kolmogorov widths of classes of convolutions with these kernels. We show that these estimates are exact in certain important cases.

### Nonlocal two-point boundary-value problems in a layer with differential operators in the boundary condition

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1122–1128

We obtain criteria of well-posedness and strong well-posedness (smoothing of solutions as compared with given functions) of boundary-value problems for linear partial differential evolution equations in an infinite layer. The boundary condition is nonlocal and gives a relation between the values of the unknown function and its derivatives with respect to spatial coordinates on shifts of connected components of the boundary of the layer inside the layer.

### Construction of a solution of a quasilinear partial differential equation of parabolic type with oscillating and slowly varying coefficients

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1129–1135

We study a boundary-value problem for a partial differential equation of parabolic type with coefficients in the form of Fourier series with coefficients and frequency slowly varying in time.

### Ivan Aleksandrovich Lukovskii (on his 60th birthday)

Korenevsky D. G., Koshlyakov V. N., Mitropolskiy Yu. A., Samoilenko A. M.

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1136-1137

### On stability of an*n*th-order equation in a critical case

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1138–1143

We obtain sufficient conditions for the Lyapunov stability of the trivial solution of a nonautonomous*n*th-order equation in the case where the root of the boundary characteristic equation is equal to zero and has multiplicity greater than one.

### On finite groups all irreducible characters of which take at most two nonzero values

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1144–1148

We give a description of finite groups all irreducible characters of which take at most two nonzero values.

### Topology of nonstationary attractors in spaces of control processes and synergetic model in flight dynamics

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1149–1152

We give a definition of nonstationary attractors that can originally exist in spaces of control processes and formulate topological conditions for an arbitrary set to belong to the class of nonstationary attractors. We also present a synergetic model for the ascent of an airplane.