Spectroscopy of nanoscale crystalline structural elements

The study of structural elements, nanoparticles, microblocks and other nanoscale objects was an important part of the study of crystals non-equilibrium properties. The behavior of nanoscale structures allows us to judge the dynamics of the crystal lattice during doping, deformation, and interactions with radiation. Along with x-ray and electron microscopic studies, optical methods for determining the size of nanoscale objects, the energy of their electrons, and the symmetry of electronic States are increasingly being used. Among nanoscale objects, proton-separated structural elements (PSE) attract special attention in connection with the development of crystal structure block-hierarchical (BH) model. In this paper, we consider the possibility of calculating the size of PSE crystals in a model of quantum-dimensional structures. According to this model except values of the crystal potential in PSE, you should consider the area of high electron density, the existence of which is beyond the scope of conventional theory. Experimental data allow us to determine the position of this zone as localized around the atomic core of the PSE. Note that the atomic backbone generally coincides with the unit cell, that is, it consists of the same number of atoms and has the same point symmetry group.

Some General New Einstein Walker Manifolds

Lie symmetry group method is applied to find the Lie point symmetry group of a system of partial differential equations that determines general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.

Symmetry Reduction, Exact Solutions, and Conservation Laws of the (2+1)-Dimensional Dispersive Long Wave Equations

By means of the generalized direct method, we investigate the (2+1)-dimensional dispersive long wave equations. A relationship is constructed between the new solutions and the old ones and we obtain the full symmetry group of the (2+1)-dimensional dispersive long wave equations, which includes the Lie point symmetry group S and the discrete groups D. Some new forms of solutions are obtained by selecting the form of the arbitrary functions, based on their relationship. We also find an infinite number of conservation laws of the (2+1)-dimensional dispersive long wave equations.

APPLYING SYMMETRY PROPERTIES OF FINITE SYSTEM TO ITS FIELD THEORY DESCRIPTION

The self-energy is received using the theory group and field techniques, based on the wave functions of the classes of point symmetry group and accounting for the two-particle system. It contains components responsible not only for relaxation processes but also for the oscillating ones, caused by a different degree of occupation of the class of point symmetry group by particles of the structure. Analyzing the character of energy oscillations depending on the degree of the class occupation, the mechanisms of diffusion, catalysis, chemical reactions and Le-Shatelje principle are described.