2019
Том 71
№ 8

# Volume 49, № 9, 1997

Anniversaries (Ukrainian)

### Bohdan Iosypovych Ptashnyk

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1155–1156

Article (Russian)

### On one direct method for the approximate solution of a periodic boundary-value problem

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1157–1161

We propose a direct method for the approximate solution of integral equations that arise in the course of approximate solution of a periodic boundary-value problem for linear differential equations by the method of boundary conditions. We show that the proposed direct method is optimal in order.

Article (English)

### A unified approach for univalent functions with negative coefficients using the Hadamard product

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1162–1170

For given analytic functions ϕ(z) = z + Σ n=2 λ n z n , Ψ(z) = z + Σ n=2 μ with λ n ≥ 0, μ n ≥ 0, and λ n ≥ μ n and for α, β (0≤α<1, 0<β≤1), let E(φ,ψ; α, β) be of analytic functions ƒ(z) = z + Σ n=2 a n z n in U such that f(z)*ψ(z)≠0 and $$\left| {(f(z)*\varphi (z))/((f(z)*\psi (z)) - 1\left| { < \beta } \right|(f(z)*\varphi (z))/((f(z)*\psi (z)) + (1 - 2\alpha )} \right|$$ for z∈U; here, * denotes the Hadamard product. Let T be the class of functions ƒ(z) = z - Σ n=2 |a n | that are analytic and univalent in U, and let E T (φ,ψ;α,β)=E(φ,ψ;α,β)∩T. Coefficient estimates, extreme points, distortion properties, etc. are determined for the class E T (φ,ψ;α,β) in the case where the second coefficient is fixed. The results thus obtained, for particular choices of φ(z) and ψ(z), not only generalize various known results but also give rise to several new results.

Article (Ukrainian)

### Tangential limit values of a biharmonic poisson integral in a disk

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1171–1176

Let C 0 be a curve in a disk D={|z|<1} that is tangent to the circle at the point z=1, and let C θ be the result of rotation of this curve about the origin z=0 by an angle θ. We construct a bounded function biharmonic in D that has a zero normal derivative on the boundary and for which the limit along C θ does not exist for all θ, 0≤θ≤2π.

Article (Russian)

### Investigation of stabilization of a mathematical model of a dynamical system with random influence in the resonance case

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1177–1181

We construct and investigate a mathematical model of a dynamical system with random influence stabilized by increasing the frequency of random influence.

Article (Ukrainian)

### On the existence of a solution of a quasilinear mixed problem for a quasiwave equation

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1182–1185

We establish conditions for the existence and uniqueness of a solution of a quasilinear mixed problem for a quasiwave equation.

Article (Ukrainian)

### Singular perturbations of finite rank. Point spectrum

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1186–1194

We establish necessary and sufficient conditions for the appearance of an additional point spectrum under singular perturbations of finite rank.

Article (Ukrainian)

### Boltzmann-Enskog limit for equilibrium states of systems of hard spheres in the framework of a canonical ensemble

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1195–1205

We prove the existence of the Boltzmann-Enskog limit for an equilibrium system of hard spheres. On the basis of analysis of the Kirkwood-Salsburg equations, we show that the limit distribution functions are constants that can be represented as series in density.

Article (Russian)

### The problem of extension for two-parameter kernels

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1206–1212

We solve the problem of construction of a two-parame to either a multiplicative or a coordinatewise two-parameter semigroup. The construction is carried out on the basis of the “initial family of kernels.”

Article (Ukrainian)

### On the stability of an invariant torus

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1212-1222

We consider the problem of the asymptotic stability for trivial invariant torus of a linear extension of dynamical system on a torus. We formulate and prove sufficient criteria of asymptotic stability and study conditions for the existence and uniqueness of the Lyapunov functions of fixed sign.

Article (Ukrainian)

### On Navier-Stokes fields with linear vorticity

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1223–1229

We describe all Navier-Stokes fields with vorticity linear in space variables.

Article (Russian)

### New polynomials of knots

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1230–1235

For some knots and links with respect to regular isotopy, we introduce a new invariant, which is a Laurent polynomial in three variables. The properties of this invariant are studied.

Article (Ukrainian)

### Multipoint problem for typeless systems of differential equations with constant coefficients

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1236–1249

For typeless systems of differential equations with constant coefficients, we investigate the well-posedness of the problem with multipoint conditions for a selected variable and 2π-periodic conditions for the other coordinates. The conditions of univalent solvability are established and the metric theorems are proved for lower bounds of small denominators that appear in the construction of solutions of the problems.

Article (Russian)

### On estimates of approximation characteristics of the Besov classes of periodic functions of many variables

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1250–1261

We obtain order estimates for some approximate characteristics of the Besov classes B p,ϑ r of periodic functions of many variables.

Article (Ukrainian)

### Lie symmetries and exact solutions of nonlinear equations of heat conductivity with convection term

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1262–1270

The Lie symmetries of nonlinear diffusion equations with convection term are completely described. The Lie ansatzes and exact solutions of a certain nonlinear generalization of the Murray equation are constructed. An example of the family of non-Lie solutions of the Murray equation is given.

Article (Russian)

### On information complexity of certain classes of operator equations

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1271–1277

For some classes of operator equations of the second kind, we obtain an estimate of information complexity exact in order. We construct a new projection-type method that realizes the optimal estimate.

Article (Ukrainian)

### On some properties of degenerate linear systems

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1278–1296

Для лінійної системи звичайних диференціальних рівнянь з виродженою матрицею при похідних знайдено умови звідності до центральної канонічної форми, встановлено структуру загального розв'язку та умови розв'язності задачі Коші, досліджено питання про періодичні розв'язки.