Four-Dimensional Lie Group Integration of the Klein–Fock Equation

A method of four-dimensional Lie group integration of the Fock–Klein equation is described in the present paper. The method is based on the formalism of co-adjoint representation orbits and on the non-commutative integration method. Right-invariant metrics being solutions of the Einstein equation are also classified for manifolds of four-dimensional Lie groups.     

Molecular flow and wall collision age distributions

The use of storage cells has become a standard technique for internal gas targets in conjunction with high energy storage rings. In case of spin-polarized hydrogen and deuterium gas targets the interaction of the injected atoms with the walls of the storage cell can lead to depolarization and recombination. Thus the number of wall collisions of the atoms in the target gas is important for modeling the processes of spin relaxation and recombination. It is shown in this article that the diffusion process of rarefied gases in long tubes or storage cells can be described with the help of the one-dimensional diffusion equation. Mathematical methods are presented that allow one to calculate collision age distributions (CAD) and their moments analytically. These methods provide a better understanding of the different aspects of diffusion than Monte Carlo calculations. Additionally it is shown that measurements of the atomic density or polarization of a gas sample taken from the center of the tube allow one to determine the possible range of the corresponding density weighted average values along the tube. The calculations are applied to the storage cell geometry of the HERMES internal polarized hydrogen and deuterium gas target. Received 9 July 2001 and Received in final form 18 September 2001     

Superstable semigroups of operators

In [NR] the authors introduced the notion of superstable operators on a Banach space E using ultrapowers E_u of E. In [HR] this notion was extended to strongly continuous one-parameter semigroups again by means of ultrapowers.It is the aim of the present paper to give an equivalent intrinsic definition of superstability (without the reference to ultrapowers). This definition allows us to improve the results of [NR] as well as of [HR]. We apply our results to semigroups of positive linear operators on Banach lattices and C^*-algebras, respectively.     

Relations for extrema of the work of aggregation in micellar solutions

Exact relations for the dependence of the positions of extrema of the aggregation work on the aggregation number axis on the surfactant monomer concentration are found for a dilute micellar solution. Relations for the half-widths of the vicinities of the extrema of the aggregation work are determined also. These half-widths are determined by the condition that, within the boundaries of their values, the aggregation work deviates from its extreme values by a thermal unit. The relations derived are illustrated by calculations based on the droplet and quasi-droplet models of spherical aggregates of nonionic surfactant. Some important applications of the obtained results to the theory of relaxation in micellar solutions are considered.     

A p-version finite element method for nonlinear elliptic variational inequalities in 2D

This article introduces and analyzes a p-version FEM for variational inequalities resulting from obstacle problems for some quasi-linear elliptic partial differential operators. We approximate the solution by controlling the obstacle condition in images of the Gauss–Lobatto points. We show existence and uniqueness for the discrete solution u _p from the p-version for the obstacle problem. We prove the convergence of u _p towards the solution with respect to the energy norm, and assuming some additional regularity for the solution we derive an a priori error estimate. In numerical experiments the p-version turns out to be superior to the h-version concerning the convergence rate and the number of unknowns needed to achieve a certain exactness of the approximation.     

Co-cliques and star complements in extremal strongly regular graphs

Suppose that the positive integer μ is the eigenvalue of largest multiplicity in an extremal strongly regular graph G. By interlacing, the independence number of G is at most 4μ² + 4μ − 2. Star complements are used to show that if this bound is attained then either (a) μ = 1 and G is the Schläfli graph or (b) μ = 2 and G is the McLaughlin graph.