Electron as the eigenstate of curvilinear space with torsion

Ideas previously enunciated by the author about the physical interpretation of curvilinear space with torsion are developed. With the new equations describing the eigenstates of such a space, the static centrosymmetric solution for gravitational and electric fields can be made consistent with the Newton and Coulomb laws. The asymptotic behavior of the axisymmetric solution at infinity is studied. The gyromagnetic ratio obtained is characteristic of the electron. Ural State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, p. 56–60, May, 1998.     

Electrostatic field of a thin, unclosed spherical shell and a torus

The electrostatic problem for a thin, unclosed spherical shell and a torus is reduced to paired summation equations in the Legendre polynomials by means of formulas relating the spherical and toroidal harmonic functions. The paired equations are transformed to a Fredholm integral equation of the second kind. Formulas are obtained for computing the charges of the conductors in the form of a series in a small parameter. The capacitance is computed for certain geometrical parameters of the conductors. Zh. Tekh. Fiz. 68, 1–6 (July 1998)     

Orthogonal modules and nonlinear cohomologies

Ordinary cohomologies in an algebra most often exhibit themselves either as objects that describe extensions (of algebras, groups, or modules) or as subjects hindering the existence of algebraic constructions. We introduce the notion of nonlinear cohomologies at a level of cochain complexes, furnish particular examples for Lie algebras, and give an interpretation of first nonlinear cohomologies in terms of “quasiextensions” of orthogonal modules. Translated fromAlgebra i Logika, Vol. 37, No. 5, pp. 522–541, September–October, 1998.     

Smooth Solution of the Compressible Navier–Stokes Equations in an Unbounded Domain with Inflow Boundary Condition

The barotropic compressible Navier–Stokes equations in an unbounded domain are studied. We prove the unique existence of the solution (u, p) of the system (1.1) in the Sobolev spaceH^{k + 3} × H^{k + 2}provided that the derivatives of the data of the problem are sufficiently small, wherek ≥ 0 is any integer. The proof follows from an analysis of the linearized problem, the solvability of the continuity equation, and the Schauder fixed point theory. Similar smoothness results are obtained for a linearized form of (1.1).     

Thermal gradient induced sidebranching in directional solidification

The polycrystalline perovskitelike manganese oxides La_1-xA_xMnO (A = Na, and K, ) have been fabricated by sol-gel technique. For all the compositions explored in this work, the average manganese oxidation state is practically constant, at for A = Na, and for A = K, respectively. A close relationship is confirmed to hold between the Curie temperature (T_c) and the bond distance of Mn-O. Results of magnetic measurements show that these materials can be utilized as suitable candidates for magnetic refrigerants with wide applied temperature span, for their significant entropy change and the easily tuned Curie temperature. Received: 12 September 1997 / Revised: 18 December 1997 / Accepted: 21 January 1998     

Loss and non-linearity saturation effects on the bistability of non-linear distributed feedback structures

We investigate theoretically the effects of optical absorption and saturation of the non-linear index of refraction on the bistability of non-linear distributed feedback structures (NLDFBs). By assuming that the Kerr non-linearity saturates in an exponential fashion, we obtain for the first time closed-form expressions for the so-called self-phase and cross-phase modulation terms. Our investigation shows that both absorption and, in particular, saturation significantly affect the bistable properties of this structure and in many cases eliminate this response completely. In some cases, however, saturation alters the NLDFB's transfer characteristics in a potentially useful manner. We find that weak levels of saturation may increase the contrast ratio between the intensities of the high and low bistable states. At increasing levels of saturation, where bistability is no longer observed, we find regimes where the NLDFB structure could possibly be used for optical amplification. This revised version was published online in November 2006 with corrections to the Cover Date.     

Singularities in cascade models of the Euler equation

The formation of singularities in the three-dimensional Euler equation is investigated. This is done by restricting the number of Fourier modes to a set which allows only for local interactions in wave number space. Starting from an initial large-scale energy distribution, the energy rushes towards smaller scales, forming a universal front independent of initial conditions. The front results in a singularity of the vorticity in finite time, and has scaling form as function of the time difference from the singularity. Using a simplified model, we compute the values of the exponents and the shape of the front analytically. The results are in good agreement with numerical simulations.