Presowing Treatment of Spring Wheat Seeds by a Pulsed Electron Beam in the Atmosphere

High Energy Chemistry - The purpose of this work is to optimize the conditions of presowing treatment of spring wheat seeds with a low-energy pulsed electron beam. The absorbed dose in wheat has...     

Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method

For an arbitrary self-adjoint operator B in a Hilbert space , we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector with respect to the operator B, the rate of convergence to zero of its best approximation by exponential-type entire vectors of the operator B, and the k-modulus of continuity of the vector x with respect to the operator B. The results are used for finding a priori estimates for the Ritz approximate solutions of operator equations in a Hilbert space. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 633–643, May, 2005.     

Generalized wave polynomials and transmutations related to perturbed Bessel equations

The transmutation (transformation) operator associated with the perturbed Bessel equation is considered. It is shown that its integral kernel can be uniformly approximated by linear combinations of constructed here generalized wave polynomials, solutions of a singular hyperbolic partial differential equation arising in relation with the transmutation kernel. As a corollary of this result an approximation of the regular solution of the perturbed Bessel equation is proposed with corresponding estimates independent of the spectral parameter.     

On a Series Representation for Integral Kernels of Transmutation Operators for Perturbed Bessel Equations

A representation for the kernel of the transmutation operator relating a perturbed Bessel equation to the unperturbed one is obtained in the form of a functional series with coefficients calculated by a recurrent integration procedure. New properties of the transmutation kernel are established. A new representation of a regular solution of a perturbed Bessel equation is given, which admits a uniform error bound with respect to the spectral parameter for partial sums of the series. A numerical illustration of the application of the obtained result to solve Dirichlet spectral problems is presented.     

Parabolic Type Equations and Markov Stochastic Processes on Adeles

In this paper we study the Cauchy problem for new classes of parabolic type pseudodifferential equations over the rings of finite adeles and adeles. We show that the adelic topology is metrizable and give an explicit metric. We find explicit representations of the fundamental solutions (the heat kernels). These fundamental solutions are transition functions of Markov processes which are adelic analogues of the Archimedean Brownian motion. We show that the Cauchy problems for these equations are well-posed and find explicit representations of the evolution semigroup and formulas for the solutions of homogeneous and non-homogeneous equations.     

Characterization of the rate of convergence of one approximate method for the solution of an abstract Cauchy problem

We consider an approximate method for the solution of the Cauchy problem for an operator differential equation. This method is based on the expansion of an exponential in orthogonal Laguerre polynomials. We prove that the fact that an initial value belongs to a certain space of smooth elements of the operator A is equivalent to the convergence of a certain weighted sum of integral residuals. As a corollary, we obtain direct and inverse theorems of the theory of approximation in the mean. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 557–563, April, 2008.     

Approximation Theory in the Central Limit Theorem. Exact Results in Banach Spaces

Book Review

Approximation Theory in the Central Limit Theorem. Exact Results in Banach SpacesV. Paulauskas and A. Rackauskas: (MIA Series), Kluwer Academic Publishers, Dordrecht, Boston, London, 1989, 176 pp., ISBN 90-277-2875-9, Dfl154.00/US$79.00.     

A method for computation of scattering amplitudes and Green functions of whole axis problems

A method for the computation of scattering data and of the Green function for the one‐dimensional Schrödinger operator with a decaying potential is presented. It is based on representations for the Jost solutions in the case of a compactly supported potential obtained in terms of Neumann series of Bessel functions (NSBF). The representations are used for calculating a complete orthonormal system of generalized eigenfunctions of the operator H, which in turn allow one to compute the scattering amplitudes and the Green function of the operator H?λ with .     

Direct and inverse theorems of approximate methods for the solution of an abstract Cauchy problem

We consider an approximate method for the solution of the Cauchy problem for an operator differential equation based on the expansion of the exponential function in orthogonal Laguerre polynomials. For an initial value of finite smoothness with respect to the operator A, we prove direct and inverse theorems of the theory of approximation in the mean and give examples of the unimprovability of the corresponding estimates in these theorems. We establish that the rate of convergence is exponential for entire vectors of exponential type and subexponential for Gevrey classes and characterize the corresponding classes in terms of the rate of convergence of approximation in the mean. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 6, pp. 838–852, June, 2007.