# Volume 45, № 5, 1993

### On the reducibility of linear differential operators with unbounded operator coefficients

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 587–595

The general solutions of linear boundary-value problems for systems of ordinary differential equations under pulse influence are constructed by using semireciprocal matrices and the generalized Green matrix.

### Estimates of the supremum distribution for a certain class of random processes

Buldygin V. V., Kozachenko Yu. V.

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 596–608

Exponential estimates of the “tails” of supremum distributions are obtained for a certain class of pre-Gaussian random processes. The results obtained are applied to the quadratic forms of Gaussian processes and to processes representable as stochastic integrals of processes with independent increments.

### Stationary and periodic solutions of the operator Riccati equation under a random perturbation

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 609–615

Sufficient conditions are presented for the existence of stationary and periodic solutions of the operator Riccati equation under a random perturbation.

### Structure of the general solutions to boundary-value problems for ordinary differential equations under pulse influence studied by using semireciprocal matrices

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 616–625

The general solutions of linear boundary-value problems for systems of ordinary differential equations under pulse influence are constructed by using semireciprocal matrices and the generalized Green matrix.

### Chebyshev's recursion: Analytic principles and applications

Korzh S. A., Ovcharenko I. E., Ugrinovskii R. A.

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 626–646

We study different algebraic and algorithmic constructions related to the scalar product on the space of polynomials defined on the real axis and on the unit circle and to the Chebyshev procedure. A modern version of the Chebyshev recursion ($(m) - T$-recursion) is applied to check whether the Hankel and Toeplitz quadratic forms are positive definite, to determine the number of real (complex conjugate) roots of the polynomials, to localize the ordering of these roots, and to find bounds for the values of a function on a given set. We also consider the relation between the $(m) - T$-recursion and the method of moments in the study of Schrödinger operators for special classes of potentials.

### Potential theory for problems of diffraction on a layer between two parallel planes

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 647–662

We investigate the boundary-value problems that appear when studying the diffraction of acoustic waves on obstacles in a layer between two parallel planes. By using potential theory, these boundary-value problems are reduced to the Fredholm integral equations given on the boundary of the obstacles. The theorems on existence and uniqueness are proved for the Fredholm equations obtained and, hence, for the boundary-value problem.

### The best trigonometric approximations and the Kolmogorov diameters of the Besov classes of functions of many variables

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 663–675

The order estimates for the best trigonometric approximations and the Kolmogorov diameters of the classes $B^r_{p, \theta}$ of functions of many variables in the space $L_q$ are obtained for certain values of the parameters $p, q$.

### A method of summation of the Fourier-Jacobi series

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 676–680

A summation method is constructed for the Fourier-Jacobi series, which has properties similar to the properties of the de la Vallée-Poussin methods of summation of the Fourier series by the trigonometric system.

### On the growth of analytic functions represented by the Dirichlet series on semistrips

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 681–693

The behavior of the Dirichlet series with null abscissa of absolute convergence is studied on semistrips.

### Periodic solutions of monotone differential inclusions with fast and slow variables

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 694–703

We study the solvability of a periodic problem for monotone differential inclusions and the behavior of its solutions as the parameter changes.

### Correctness of the Cauchy problem for trinomial higher-order operator differential equations

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 704–714

Criteria are established for the correctness of the Cauchy problem for the equations $y^{(2n)} + Ay^{(n)} + By = 0, \quad t \in [0, \infty)$ where $n > 1, А, В$ are arbitrary commuting self-adjoint operators in a Hilbert space. For $n = 2$, the criterion is illustrated by the example of the equation describing the dynamics of an exponentially stratified rotating compressible fluid.

### On the expansion of solutions to differential equations with discontinuous right-hand side in a series in initial data and parameters

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 715–717

The conditions under which the solutions of equations with discontinuous right-hand sides depend on the initial data and parameters analytically are investigated. A definition is introduced, which specifies this dependence in the case where a surface of discontinuity exists.

### Normal congruences with focal bicylindrical surfaces

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 718–720

A system of equations determining normal con~uences with focal surfaces (bicylinders) is found in a biaxial space of hyperbolic type by using a moving frame of the space and the method of Cartan exterior forms. The geometric construction of these congruences is presented.

### Plane modules and distributive rings

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 721–724

Let $A$ be a semiprime ring entire over its center. We prove that the following conditions are equivalent:

(a) A is a ring distributive from the right (left);

(b) w.gl. $\dim (A) ≤ 1$; moreover, if $M$ is an arbitrary prime ideal of the ring $A$, then $A/M$ is a right Ore set.

### An estimate of the remainder term in the central limit theorem for $r$-independent random variables

Ukr. Mat. Zh. - 1993. - 45, № 5. - pp. 725–727

We suggest a method for the investigation of $r$-independent random variables by using multiplicative systems. An estimate of the remainder term in the central limit theorem for $r$-independent random variables is obtained.