Ukr. Mat. Zh. - 1993νmber=5. - 45, № 5. - pp. 587–595
Results on the reducibility of linear differential operators with unbounded operator coefficients to differential operators with a simpler structure are obtained.
Ukr. Mat. Zh. - 1993νmber=5. - 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.
Ukr. Mat. Zh. - 1993νmber=5. - 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. - 1993νmber=5. - 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.
Ukr. Mat. Zh. - 1993νmber=5. - 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.
Ukr. Mat. Zh. - 1993νmber=5. - 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. - 1993νmber=5. - 45, № 5. - pp. 663–675
Ukr. Mat. Zh. - 1993νmber=5. - 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.
Ukr. Mat. Zh. - 1993νmber=5. - 45, № 5. - pp. 681–693
The behavior of the Dirichlet series with null abscissa of absolute convergence is studied on semistrips.
Ukr. Mat. Zh. - 1993νmber=5. - 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.
Ukr. Mat. Zh. - 1993νmber=5. - 45, № 5. - pp. 704–714
On the expansion of solutions to differential equations with discontinuous right-hand side in a series in initial data and parameters
Ukr. Mat. Zh. - 1993νmber=5. - 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.
Ukr. Mat. Zh. - 1993νmber=5. - 45, № 5. - pp. 721–724
LetA 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, ifM is an arbitrary prime ideal of the ringA, thenA/M is a right Ore set.
Ukr. Mat. Zh. - 1993νmber=5. - 45, № 5. - pp. 725–727
We suggest a method for the investigation ofr-independent random variables by using multiplicative systems. An estimate of the remainder term in the central limit theorem forr-independent random variables is obtained.