Symmetry operators and separation of variables in the (2+1)(2+1)-dimensional Dirac equation with external electromagnetic field

We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a $(2+1)$-dimensional Riemann (curved) spacetime. Nonequivalent complete sets of mutually commuting symmetry operators are classified in a $(2+1)$-dimensional Minkowski (flat) space. For each of the sets we carry out a complete separation of variables in the Dirac equation and find a corresponding electromagnetic potential permitting separation of variables.Comment: 24 pages, version accepted for publication in Int. J. Geom. Methods Mod. Phy

The Shapovalov determinant for the Poisson superalgebras

Among simple Z-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C_{2}: these are the Lie superalgebra k^L(1|6) of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series: the finite dimensional Lie superalgebra sh(0|2k) of special Hamiltonian fields in 2k odd indeterminates, and the Kac--Moody version of sh(0|2k). Using C_{2} we compute N. Shapovalov determinant for k^L(1|6) and sh(0|2k), and for the Poisson superalgebras po(0|2k) associated with sh(0|2k). A. Shapovalov described irreducible finite dimensional representations of po(0|n) and sh(0|n); we generalize his result for Verma modules: give criteria for irreducibility of the Verma modules over po(0|2k) and sh(0|2k)

Quantum integrable systems in three-dimensional magnetic fields: the Cartesian case

In this paper we construct integrable three-dimensional quantum-mechanical systems with magnetic fields, admitting pairs of commuting second-order integrals of motion. The case of Cartesian coordinates is considered. Most of the systems obtained are new and not related to the separation of variables in the corresponding Schr\"odinger equation.Comment: 8 page

Influence of annealing on nanocrystal formation in Ni amorphous alloy

It is established that nanocrystals in the alloy samples increase in size with increasing annealing temperature. The rise of resistivity upon annealing at temperatures below 300°C and its significant decrease at higher temperatures is consistent with the change in the amorphous alloy microstructureyesBelgorod State Universit

On the new approach to variable separation in the time-dependent Schr\"odinger equation with two space dimensions

We suggest an effective approach to separation of variables in the Schr\"odinger equation with two space variables. Using it we classify inequivalent potentials $V(x_1,x_2)$ such that the corresponding Schr\" odinger equations admit separation of variables. Besides that, we carry out separation of variables in the Schr\" odinger equation with the anisotropic harmonic oscillator potential $V=k_1x_1^2+k_2x_2^2$ and obtain a complete list of coordinate systems providing its separability. Most of these coordinate systems depend essentially on the form of the potential and do not provide separation of variables in the free Schr\" odinger equation ($V=0$).Comment: 21 pages, latex, to appear in the "Journal of Mathematical Physics" (1995

Representations of the Generalized Lie Algebra sl(2)_q

We construct finite-dimensional irreducible representations of two quantum algebras related to the generalized Lie algebra \ssll (2)_q introduced by Lyubashenko and the second named author. We consider separately the cases of $q$ generic and $q$ at roots of unity. Some of the representations have no classical analog even for generic $q$. Some of the representations have no analog to the finite-dimensional representations of the quantised enveloping algebra $U_q(sl(2))$, while in those that do there are different matrix elements.Comment: 14 pages, plain-TEX file using input files harvmac.tex, amssym.de

THE WORLD OF CHILDHOOD AND YOUTH IN RUSSIAN NOBLE FAMILIES OF THE FIRST HALF OF THE NINETEENTH CENTURY IN THE CONTEXT OF STRATIFICATION APPROACH

traditions. Hence, it is one of the criteria for evolutionary or revolutionary changes in a particular society, accentuating the culture of everyday life and the connection of generations. In a rigidly structured estate society, such as the population of the Russian Empire, the world of childhood and youth developed in the context of social traditionalism, adopted in one or another estate community. Methodology: The basis of the study is based on the principles of historicism, objectivity, and science. To prove these theses, comparative-historical, problem-chronological, logical-analytical methods were used, which made it possible to reliably reconstruct the world of childhood and youth in noble families of Russia of the first half of the nineteenth century. Result: The most detailed system of socialization of children and adolescents was developed in an environment of a noble family, distinguished by more strict and complex etiquette, where pride in belonging to a noble class was inculcated at an early age. But the feeling of belonging to the upper class among the young nobles was different and mediated by many factors, first of all, the ancient lineage, the level of the material wealth of the family, education, upbringing,  and culture. That is, the variability of the socialization of children and adolescents was extensive. Applications: This research can be used for universities, teachers, and students. Novelty/Originality: In this research, the model of The World of Childhood and Youth in Russian Noble Families of the First Half of the Nineteenth Century in the Context of Stratification Approach is presented in a comprehensive and complete manner