Table of contents

Some concepts of differential geometry and topology which have applications in the theory of classical gauge fields are explained by means of simple examples. A guide to the mathematical literature is given.

The review is devoted to a discussion of the definition and properties of energy in Einstein's theory of gravitation. Asymptotically flat space-time is defined in terms of admissible asymptotically Cartesian coordinates and a corresponding group of coordinate transformations. A Lagrange function is introduced on such a space-time, and a generalized Hamiltonian formulation of the theory of gravitation is constructed in accordance with Dirac's method. The energy is defined as the generator of displacement with respect to the asymptotic time. It is shown that the total energy of the gravitational field and the matter fields with normal energy-momentum tensor is positive and vanishes only in the absence of matter fields and gravitational waves. The proof follows Witten's proof but contains a number of corrections and improvements. Various standard criticisms of the energy concept in general relativity are discussed and shown to be without substance.

Experimental and theoretical research on the recombination photoluminescence of free hot electrons in semiconductors (primarily GaAs) is reviewed. The polarization characteristics are discussed. These characteristics reflect, in particular, a momentum alignment of the electrons by linearly polarized light and an effect of a ripple in the constant-energy surfaces in the valence band. The dependence of the linear polarization on the spectrum is discussed in connection with various mechanisms for the energy relaxation of the hot electrons. The depolarization of the hot-electron photoluminescence in a magnetic field is discussed. A procedure is discussed for determining the energy relaxation times and the scale times for intervalley transitions through an analysis of depolarization curves. The energy distribution of the hot electrons is found from the hot-electron photoluminescence spectrum. The recombination luminescence of hot holes is discussed. These holes appear when the semiconductor is illuminated in the spin-split-off subband.

Characteristic properties of electromagnetic radiation are investigated for radiating systems in which variation of parameters is described by model smooth functions permitting derivation of exact solutions. Picked as examples of radiating systems are a moving charge with continuously variable velocity (with respect to magnitude and direction), a dipole with continuously variable dipole moment, and a charge uniformly moving in a medium with permittivity continuously varying in space or time. Exact expressions are derived for the electromagnetic field and spectral energy density of radiation in all problems under consideration. It is shown that the common property of electromagnetic radiation in such systems is exponential decline of the spectrum in the high frequency region.

This paper reviews the current literature on the formation and interaction of solitons in optical fibers. A number of original results of numerical modeling of the phenomena indicated are presented.