No 3 (2008)

Solvability of the generalized solution for the time-dependent modified transport equation with initial and boundary conditions problem is proved. Modified transport equation differs from the common equation by replacement integrated composed function, being the solution, by a constant function, from the same class of the solution. That modification is more convenient for some nonlinear inverse problems in comparison with the common transport equation.

A constructive method for finding some first integrals of equations of motion of dissipative systems with infinite number of degrees of freedom is suggested.

This paper provides a gentle introduction to the foundational ideas, concepts and results in the field of science dedicated to the theory of statistical function estimation and pattern recognition. The so-called VC Theory of Vapnik and Chervonenkis is introduced and explored gradually. The emphasis is placed on helping the reader appreciate the importance of the extension of the classical law of large numbers to function spaces, and the key role that "new" concepts such as Empirical Risk Minimization (ERM) principle, ERM consistency, VC dimension, and complexity control play in constructing algorithms that yield function estimators with optimal properties. As much as possible, each key concept is introduced via a tangible example, with the hope of helping the reader grasp the essential core of the foundational concept under exploration.

The symmetric implicit operator-difference multi-layer schemes for solving the timedependent Schrцdinger equation based on decomposition of the evolution operator via the explicit Magnus expansion up to the sixth order of accuracy with respect to the time step are presented. Reduced schemes for solving the Cauchy problem of a set of coupled timedependent Schrцdinger equations with respect to the hyperradial variable are devised by using the Kantorovich expansion of the wave packet over a set of appropriate parametric basis angular functions. The implicit algebraic schemes for numerical solving the problem with symmetric operators, using discretization of the component of the wave package by hyperradial variable by the high order finite-element method are formulated. The convergence and efficiency of the numerical schemes are demonstrated in numerical calculations of the exactly solvable models of one-dimensional oscillator with time-dependent frequency, two-dimensional oscillator in time-dependent external field by using the conventual angular basis.

We consider a nonlinear generalization of the Yukawa meson theory of nuclear forces that belongs to the class of theories with the Lagrangians proposed by L. Shiff. Relying on this basis, we study nonlinear equations for a nuclear potential and dynamic equations for nuclear particles which contain one unknown function of this potential. To found it, we suggest a principle which implies the proportionality of the nuclear field potential to the potential energy of nucleons in the field. The found equations are applied to study the dynamics of nuclear particles moving under the simultaneous action of nuclear and electromagnetic forces. It is shown that in the case of sufficiently intensive interaction of nucleons, the suggested equations give the value 14.9 for the dimensionless constant of strong interaction close to the experimental value.