It has been a successful practice to define a canonical pre-ordering on a normed space using the inclusion of faces of its closed dual unit ball. This pre-ordering reflects some geometric property in a natural way. In this article, we will give an algebraic description of this pre-ordering in the case of complex C*-algebras as well as that of their self-adjoint parts. In developing our theory we introduce the essential support of an element, which is closely related to the notion of peak projections studied recently by Blecher and Hay. As applications, we give some interesting facts about weak*-closed faces, and will identify the quasi-maximal elements and the quasi-minimal elements with respects to this pre-ordering. They are closely related to the extreme points and the smooth points of the unit sphere of the C*-algebra.

Motivated by the important work of Brown and York on quasilocal energy, we propose definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with a positive intrinsic curvature in a spacetime. We show that the quasilocal energy of the boundary of a compact spacelike hypersurface which satisfies the local energy condition is strictly positive unless the spacetime is flat along the spacelike hypersurface.

In this paper, self-similarity is illustrated and compared in deterministic and stochastic dissipative systems. Examples are (1) deterministic self-similarity in reaction-diffusion system and Navier-Stokes equations, where solutions eventually decay to zero due to balance of diffusion (viscosity) and nonlinearity;(2) statistical self-similarity in randomly advected passive scalar model of Kraichnan where solutions undergo turbulent decay due to roughness of advection;(3) selfsimilarity in blowup of solutions of fourth order nonlinear parabolic equations of the Cahn-Hillard type. Problems for future research are mentioned, especially those where self-similarity is conjectured based on numerical evidence or physical grounds but mathematically open.

The present paper concerns with the global structure and asymptotic behavior of the discontinuous solutions to flood wave equations. By solving a free boundary problem, we first obtain the global structure and large time behavior of the weak solutions containing two shock waves. For the Cauchy problem with a class of initial data, we use Glimm scheme to obtain a uniform BV estimate both with respect to time and the relaxation parameter. This yields the global existence of BV solution and convergence to the equilibrium equation as the relaxation parameter tends to 0.

Let <i>H</i> denote a spherical subgroup within a semisimple algebraic group <i>G</i>. In this paper we study the closures of the finitely many <i>H</i>-orbits in the flag variety of <i>G</i>. Using the language of Frobenius splitting we provide a criterion for these closures to have nice geometric and cohomological properties. We then show how the criterion applies to the spherical subgroups of minimal rank studied by N. Ressayre. Finally, we also provide applications of the criterion to orbit closures which are not multiplicity-free in the sense defined by M. Brion.