# Volume 45, № 1, 1993

### On connectedness of derivative sets

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 3–9

We give conditions under which the graph of certain multivalued mappings which arise in various differential processes is connected.

### Bernstein-type inequalities for $\mathcal{L}$-splines

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 10-20

New Berhstein-type inequalities are obtain for 2π-periodic $\mathcal{L}$ -splines associated with a differential operator $\mathcal{L}_\tau (D)$ of degree $r$ with fixed real coefficients.

### On the commutant of a multiplication operator and conditions of invertibility of its elements

Berezovsky N. I., Linchuk S. S.

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 21–25

We obtain a description of isomorphisms acting in the space of square summable functions and commuting with the operator of multiplication by a continuous piecewise monotone function. Preliminary corrections are made of some mistakes found in the statements of the paper [3].

### On approximation of convolution classes

Bushev D. M., Koval'chuk I. R.

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 26–31

Asymptotic equalities are found for the least upper bounds of the best approximations of some convolution classes with even kernel in the metric of a space $L_p$.

### Reconstruction of linear functionals on classes of analytic functions of two variables on the basis of generalized information

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 32–37

For classes of analytic functions defined in terms of the two-dimensional Hadamard composition, we propose a renewal method for linear functionals based on blending constructions. The best renewal methods are indicated, and the exact estimates of errors are given.

### Uniform estimates for monotonic polynomial approximation

Dzyubenko H. A., Listopad V. V., Shevchuk I. A.

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 38–43

The uniform estimate is established for a monotone polynomial approximation of functions whose smoothness decreases at the ends of a segment.

### On asymptotic normality of the least square estimators of an infinite-dimensional parameter

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 44–53

Asymptotic properties of the estimators of an infinite-dimensional parameter are studied.

### Estimates of the Bernstein widths and their analogs

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 54–59

The estimates of exact order are obtained for Bernstein widths. A new scale of widths intermediate between Kolmogorov and Bernstein ones is introduced.

### Transformations and inertia of solutions to linear matrix equations

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 60–68

Linear equations and operators in a space of matrices are investigated. The transformations of matrix equations which allow one to find the conditions of solvability and the inertial properties of Hermite solutions are determined. New families of matrices (collectives) are used in the theory of inertia and positive invertibility of linear operators and, in particular, in the problems of localization of matrix spectra and matrix beams.

### Construction of unital matrix polynomials with mutually distinct characteristic roots

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 69–77

We investigate the construction of unital matrix polynomials with mutually distinct characteristic roots, namely, their similarity and reducibility by the similarity transformation to block-triangular, block-diagonal, and, in particular, to triangular and diagonal forms. We also study the problem of extracting linear factors.

### Variables of the action-angle type on symplectic manifolds stratified by coisotropic tori

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 77–85

A symplectic manifold is considered under the assumption that a smooth symplectic action of a commutative Lie group with compact coisotropic orbits is defined on it. The problem of existence of variables of the action-angle type is investigated with a view to giving a detailed description of flows in Hamiltonian systems with invariant Hamiltonians. We introduce the notion of a nonresonance symplectic structure for which the problem of recognition of resonance and nonresonance tori is solved.

### Complete integrability of a hydrodynamic Navier-Stokes model of the flow in a two-dimensional incompressible ideal liquid with a free surface

Samoilenko V. G., Suyarov U. S.

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 86–90

We establish the complete integrability of a nonlinear dynamical system associated with the hydrodynamic Navier-Stokes equations for the flow of an ideal two-dimensional liquid with a free surface over the horizontal bottom. We show that this dynamical system is naturally connected with the nonlinear kinetic Boltzmann-Vlasov equation for a one-dimensional flow of particles with a point potential of interaction between particles.

### KP hierarchy and (1+1)-dimensional multicomponent integrable systems

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 91–104

New types of reduction of the Kadomtsev-Petviashvili (KP) hierarchy are considered on the basis of Sato's approach. As a result, we obtain a new multicomponent nonlinear integrable system. Bi-Hamiltonian structures for the new equations are presented.

### On linear almost periodic pulse systems

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 105–113

For a linear almost periodic system under pulse influence, the conditions are established under which this system is reducible (by a linear change of variables with a discontinuous almost periodic matrix) to a system without pulses but with a Bohr almost periodic right-hand side. The set of linear almost periodic pulse systems possessing only bounded solutions is studied.

### A Kolmogorov type criterion for the best approximating operator

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 114–119

For the best approximating operator, a criterion is established, equivalent to the well known Kolmogorov theorem, which characterizes the best approximation element. The practical use of this criterion is illustrated by examples.

### Some inverse problems for parabolic and elliptic differential-operator equations

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 120–127

Necessary and sufficient conditions are established for the unique solvability of problems of determining an unknown right-hand side of a differential equation with an unbounded operator coefficient under an additional boundary condition.

### Invariance principle for the least squares estimates

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 128–131

The weak convergence of random fields, constructed in terms of the least squares estimator of the regression coefficient of a random field (which is a two-parametric martingale difference), is established.

### Fluctuations of forced vibrations in a medium with resistance

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 132–135

For a differential oscillation equation with coefficients perturbed by Gaussian delta-correlated random processes with a random external force, we obtain closed moment equations. In a special case, the mathematical expectation and the covariance function are found.

### Theorem on integral inequalities with functional argument

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 136–139

A theorem on the integral Volterra-Fredholm inequalities with a functional limit is proved. By using this theorem, the theorem on integral inequalities with retarded argument is established.

### Method of partial averaging in the systems of standard form with discontinuous right-hand sides

Plotnikov V. A., Zverkova T. S.

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 140–142

The scheme of partial averaging for systems of standard form with discontinuous right-hand sides is presented. The solutions are justified and estimated without assuming periodicity of the right-hand sides of the systems.

### Minimal Morse functions on a pair of manifolds

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 143-144

The existence theorem for a minimal Morse function on a pair of manifolds $(M_n,N_k)$, where $n - k ≥ 3,\; k ≥ 6$, is proved.

### Lower bound for the area of an analytic curve inside a cube in the space $ℂ^n$

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 145-147

The exact lower bound of the area of an analytic curve which lies inside a cube in the space $ℂ$ and passes through its center is obtained.

### Estimates of stability of desynchronized systems

Khusainov D. Ya., Stadnik O. I.

Ukr. Mat. Zh. - 1993. - 45, № 1. - pp. 148–152

Linear difference systems with lag are considered. The sufficient conditions of stability are established and exponential decay coefficients for solutions are derived. The second Lyapunov method with Razumikhin's condition is used.