Algebras with general commutation relations and their applications. II. Unitary-nonlinear operator equations

Various aspects of the calculus of functions of ordered self-adjoint operators are considered. Passage to the commutative limit in the case of general nonlinear commutation relations is studied. An asymptotic solution of the Cauchy problem and asymptotically self-similar solutions are constructed for unitary-nonlinear operator equations. Asymptotic solutions are found for the Hartree equation with Coulomb interaction.Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 13, pp. 145–267, 1979.     

Crystal structure and triboluminescence of the [Tb(BTFA)2(NO3)(TPPO)2] complex

The crystal structure of the [Tb(BTFA)₂(NO₃)(TPPO)₂] complex (TPPO is triphenylphosphine oxide, BTFA is benzoyltrifluoracetone), which exhibits strong triboluminescence, has been established by X-ray crystallography. The crystals are triclinic: a = 11.668(3) Å, b = 11.700(3) Å, c = 12.512(3) Å, α = 65.161(4°), β = 79.120(4)°, γ = 61.860(4)°, space group P1, Z = 1. The central terbium(III) atom coordinates two oxygen atoms from two triphenylphosphine oxide molecules (Tb-O, 2.264(3) and 2.273(3) Å), two oxygen atoms from the nitrate group (Tb-O, 2.460(3) and 2.476(3) Å), and four oxygen atoms from two benzoyltrifluoroacetonate groups (Tb-O, 2.329(3), 2.399(3), 2.351(3), and 2.367(3) Å). The coordination polyhedron of the Tb(III) atom is a distorted dodecahedron. The photoluminescence and triboluminescence spectra of the [Tb(BTFA)₂(NO₃)(TPPO)₂] complex are identical and caused by the f-f luminescence of Tb³⁺.     

On the theory of n-nilpotent groups

The setv (G) of all integers n fbr which the group G is n-nilpotent is studied. A number of theorems generalizing fundamental results of F. Levi is proved for .Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 653–664, June, 1969.     

On the hypothesis of gas rotation in the magnetomechanical effect

This study continues a cycle of works published in Optika i Spektroskopiya (Optics and Spectroscopy) on the magnetomechanical effect in a gas discharge. It is devoted to the hypothesis of the appearance of a torque acting on an object placed in a gas discharge in a magnetic field and arising due to the momentum transfer from the rotating neutral gas. The velocities of gas rotation required for the formation of the observed moment of forces are estimated. Measurements of the velocities performed using laser Doppler anemometry are analyzed. The method of observing dust structures in a magnetic field to reveal gas rotation is justified. Corresponding experiments with the use of the method of optical visualization for the observation of plasma-dust structures are conducted. The experimental dependence ω(r) obtained for a liquidlike structure is not consistent with the hypothesis of the transfer of rotation from the gas. Additional observations of dust plasma upon tilting of the discharge tube in the gravitational field are conducted. It is found that, in this case, the structure shifts from the central region of the stratum to the periphery and the rotation ceases. Neither the analysis nor the experiments performed in this study are consistent with the hypothesis of gas rotation in the magnetomechanical effect. The results of the experiments with dust structures are of independent interest for the physics of dust plasma.     

Real rank and squaring mappings for unital -algebras

It is proved that if is a compact Hausdorff space of Lebesgue dimension , then the squaring mapping , defined by , is open if and only if . Hence the Lebesgue dimension of can be detected from openness of the squaring maps . In the case it is proved that the map , from the selfadjoint elements of a unital -algebra into its positive elements, is open if and only if is isomorphic to for some compact Hausdorff space with .

     

Geometric dynamics on quantum Nano-surfaces and low-energy spectrum in a homogeneous magnetic field

Low-energy dynamics and spectrum of the Schrödinger operator on curved surfaces in a magnetic field are considered. Nano-and micro-algebraic structures generate a quantization of the surface. On this quantum space, the dynamics has a “surfing” behavior described via curvature invariants. Two types of surface deformations correspond to trapping and transistor types of dynamics.We observe the “negative mass” effect in the low-energy spectrum under trapping deformations. We also compute, in geometrical terms, the energy levels near the spectral bottom (these levels are below the minimal Landau level). The surfing dynamics and the spectrum are independent of the magnitude of the magnetic field.     

Methodology of using expert systems for structure elucidation of organic molecules by their spectra

Major axioms underlying structure elucidation of molecules by their characteristic spectral data are considered. The amount of information obtained by solving the problem on structure elucidation of molecules serves as a basis to evaluate the quality of solution. Conditions for computer-aided solution of problems using expert systems (ES) are analyzed. Reasons for incorrect or zero solutions and for inconsistencies leading to negative results are examined. The role of additional information as a necessary condition for correct solution is discussed. On the basis of this analysis, a general strategy for structure elucidation of molecules by their spectra using ES is suggested. The principal idea is to use the iteration method, gradually increasing the rigidity of constraints. Strategies named ARCHEOLOGIST and SCULPTOR are discussed. Examples are given. Russian Scientific Research Institute of Organic Synthesis. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 3, pp. 548–558, May–June, 1995. Translated by L. Smolina     

Algebra with polynomial commutation relations for the Zeeman effect in the Coulomb-Dirac field

We consider a model of particle motion in the field of an electromagnetic monopole (in the Coulomb-Dirac field) perturbed by homogeneous and inhomogeneous electric fields. After quantum averaging, we obtain an integrable system whose Hamiltonian can be expressed in terms of the generators of an algebra with polynomial commutation relations. We construct the irreducible representations of this algebra and its hypergeometric coherent states. We use these states to represent the eigenfunctions of the original problem in terms of the solutions of the model ordinary differential equation. We also present the asymptotic approximations of the eigenvalues in the leading term of the perturbation theory, where the degeneration of the spectrum is removed completely.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 127–147, January, 2005.