We prove that the concentration of matter in a small volume is sufficient for the formation of trapped surfaces in a class of initial data for the Einstein equations. We consider momentarily static nonspherical initial data and formulate a condition for the existence of averaged trapped surfaces. The results are obtained in terms of the total rest mass \(M\) and the largest proper radius sup \(L\) of a smallest convex equipotential surface that encloses a nonspherical body.
It is shown that for every one-sided type-\(D\) Euclidean \(\cal {HH}\)-space with \({\mit \Lambda } \not = 0\) for which the local fundamental 2-form \({\mit \Phi }\) is nowhere vanishing, the Einstein equations can be locally reduced to a single, second-order nonlinear partial differential equation for one real function of three real variables.
Motion of a charged test particle in the field of an infinitely long massive charged filament is investigated. It is shown that the electric repulsion cannot prevent its fall on the filament. Hence, it follows that the Coulomb force cannot prevent the formation of naked linear singularity by collapse of charged matter.
The distribution of quantum states of a finite one-dimensional Heisenberg ferromagnet, consisting of \(N\) spins \(s\), over the discrete Brillouin zone has been analysed by means of stratification of the action of the translation group on the set of all \((2s + 1)^N\) magnetic configurations. It is shown that the rarefied bands are associated — through a secular eigenproblem — with irregular orbits, i.e. those on which the action is not free. An orthonormal complete basis, involving three exact quantum numbers: quasimomentum, the generalized star in the Brillouin zone, and the total magnetization, has been proposed.
The dynamical variational approach based on geometrical quantization is demonstrated to be capable in describing the most important quantum mechanical quantities. In particular, the consistent prescription for calculating the transition probabilities is presented. For several reasons, the method is expected to be better under control than the semiclassical methods in treating the systems whose classical counterparts are chaotic. The formal considerations are illustrated using an exactly solvable SU(3)-spin system.
The relationship between the “intermittent” increase of the scaled factorial moments and the singularities in the correlation functions is discussed. It is shown that for the existing data, far from the asymptotic region, the slopes measured from the increase of moments may be much smaller than the corresponding exponents in the correlation functions. Possible interpretations of some observed regularities of slope values for different processes are considered.
The experimental data for neutron interaction with \(^1\)H, \(^2\)H and \(^{12}\)C nuclei are parametrized in energy range 20 to 90 MeV. The energy dependence of the total and reaction cross section is represented in the form of a series of orthogonal polynomials. The logarithm of differential cross section for the angular distribution is expressed as a linear combination of Legendre polynomials with the energy dependence of their coefficients described by a series of orthogonal polynomials.
A nonlinear mapping which maps a variable having range from 0 to \(\infty \) onto another variable having range 0 to unity is used to find an approximant for the distribution of the spacing of levels of compound nucleus. It is shown to provide a very good approximation over the entire range of spacing.
