# Volume 44, № 8, 1992

### Equivalence of a part of derived chains of boundary-value problems for second-order ordinary differential equations

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1003–1011

A method is proposed for obtaining criteria for the equivalence of a part of derived chains of boundary-value problems for second-order differential equations with a spectral parameter in boundary conditions.

### Averaging in hyperbolic systems subject to weakly dependent random perturbations

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1011–1020

The first initial boundary value problem is considered for a hyperbolic equation with a small parameter for an external action described by some stochastic process satisfying some of the conditions of weak dependence. Averaging of the coefficients over the temporal variable is conducted. The existence is assumed of a unique generalized solution both for the initial stochastic problem and for the problem with an “averaged” equation, which turns out to be deterministic. For the probability of deviation of a solution of the initial equation from the solution of the “averaged” problem, exponential bounds are established of the type of S. N. Bernshtein inequalities for the sums of independent random variables.

### Points of strong summability of Fourier series

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1020–1031

The points of a function f ? L at which there are given estimates of the rate of convergence to zero of the strong arithmetic means of its Fourier series and the trigonometrically conjugate series are characterized.

### On a class of orthogonalizers for exponential systems with real frequencies

Gubreev G. M., Ignatenko T. R.

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1031–1044

We describe complete minimal families of exponential functions and Riesz bases of exponential functions which admit orthogonalizers of a special type. We obtain a complete description of all orthogonalizers of the class of complete minimal family of exponential functions under consideration and formulate a simple condition which guarantees the uniqueness of the orthogonalizer.

### On the oscillation and asymptotic behavior of the solutions of a certain system of differential-functional equations of neutral type

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1044–1049

### Behavior of integral curves in the neighborhood of optimal integral manifolds

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1049–1060

We describe an explicit construction of optimal integral manifolds [1] for a quasilinear system of differential equations that uses the method of successive approximations. We study the behavior of integral curves in the neighborhood of optimal integral manifolds. We cite a numerical method of synthesis of optimal control and prove its justification.

### Investigation of stability conditions of stochastical perturbed systems with lag

Khusainov D. Ya., Nechaeva I. G.

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1060–1064

Linear stochastic differential systems with a single lag are considered. Sufficient conditions for uniform (in lag) stability in the mean-square under constantly acting perturbations are obtained.

### An integral equation of convolution type with two kernels and its abstract analog

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1065–1078

We consider an integral equation of the convolution type with two kernels, generated by functions from some Banach algebras, and a linear equation with two coefficients in abstract rings with factorial pairs of sub-rings. Theorems and formulas have been proved, characterizing the general relation of the solvability problem of the equations with the factorization properties of elements constructed from the kernels and coefficients.

### Continuous relationship between a parameter and the solutions of impulse evolution systems

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1078–1083

A theorem on continuous dependence of the solutions of a nonlinear impulse evolution system on a parameter is proved; this theorem can be used as a basis for an averaging principle.

### Well-posed problems in a layer with differential operators in boundary conditions

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1083–1090

### Energy density and flux in nonrelativistic quantum mechanics

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1090–1095

A number of mathematical consequences of the Schroedinger equation\(i\hbar \dot \psi = {\rm H}_\psi \) are given and interpreted as local energy and momentum conservation laws. Several Hamiltonians are treated.

### On the Fourier transform of the Hamming norm

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1095–1102

### Construction of asymptotic solutions of linear singularly perturbed systems of second order with degeneracy

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1102–1112

### Classes of planar topological maps with first generalized derivatives

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1114–1116

We consider the class of planar topological maps with first generalized derivatives. A geometric method for the study of the properties of this class based on the use of regular systems of neighborhoods is given.

### Approximation of solutions of Goursat's problem for a nonlinear equation of hyperbolic type by Bernshtein type polynomials

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1116–1123

Using Bernshtein type polynomials for functions of two variables, a scheme is given for successive approximations, and its uniform convergence to the unique solution of the Goursat problem for a nonlinear hyperbolic equation is proved.

### A new approach to solving the stationary Fokker-Planck-Kolmogorov equation for a randomly oscillating nonlinear system

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1123–1129

It is shown that the Fokker-Planck-Kolmogorov equation in terms of amplitude and phase may, in the stationary case, be reduced to a first order partial differential equation which we call the stationary reduced Fokker-Planck-Kolmogorov. A method for approximate solution of the reduced equation is presented which does not need assumptions on the smallness of nonlinearity of a system and intensity of random influences.

### Automatic continuity, bases, and radicals in metrizable algegbras

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1129–1132

The automatic continuity of a linear multiplicative operator T: X?Y, where X and Y are real complete metrizable algebras and Y semi-simple, is proved. It is shown that a complex Frechét algebra with absolute orthogonal basis (x_{i}) (orthogonal in the sense that x_{i}X_{j}=0 if i ? j) is a commutative symmetric involution algebra. Hence, we are able to derive the well-known result that every multiplicative linear functional defined on such an algebra is continuous. The concept of an orthogonal Markushevich basis in a topological algebra is introduced and is applied to show that, given an arbitrary closed subspace Y of a separable Banach space X, a commutative multiplicative operation whose radical is Y may be introduced on X. A theorem demonstrating the automatic continuity of positive functionals is proved.

### Limits of analytic vector measures

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1133–1135

The article attempts to determine when a vector measure is the limit of a sequence of analytic vector measures in the sense of convergence in semivariation and when it is the limit of a sequence of such measures in variation.

### On the diameters of certain classes of analytic functions. II

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1135–1138

In the spaces E_{q}(?), q?1, we consider the classes W^{r}E_{p}(?), p?1, consisting of analytic functions f(z) ? E_{P}(?) the integral moduli of continuity of whose r-th derivatives are majorized by a given nonnegative nondecreasing function ?.

### An estimate of the convergence rate in the central limit theorem for two-parameter martingale differences

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1138–1141

### Continuity of the metric-projection operators

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1141–1144

### Stability and controllability of the motion of dynamical systems far off from equilibrium positions

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1145–1149

### Characterization of residual ?-algebras

Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1149–1152