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Let<i>M</i> be a space-time whose local mass density is non-negative everywhere. Then we prove that the total mass of<i>M</i> as viewed from spatial infinity (the ADM mass) must be positive unless<i>M</i> is the flat Minkowski space-time. (So far we are making the reasonable assumption of the existence of a maximal spacelike hypersurface. We will treat this topic separately.) We can generalize our result to admit wormholes in the initial-data set. In fact, we show that the total mass associated with each asymptotic regime is non-negative with equality only if the space-time is flat.

Gaseous fluids may move slowly, as smoke does, or at high speed, such as occurs with explosions. High-speed gas flow is always accompanied by low-speed gas flow, which produces rich visual details in the fluid motion. Realistic visualization involves a complex dynamic flow field with both low and high speed fluid behavior. In computer graphics, algorithms to simulate gaseous fluids address either the low speed case or the high speed case, but no algorithm handles both efficiently. With the aim of providing visually pleasing results, we present a hybrid algorithm that efficiently captures the essential physics of both low- and high-speed gaseous fluids. We model the low speed gaseous fluids by a grid approach and use a particle approach for the high speed gaseous fluids. In addition, we propose a physically sound method to connect the particle model to the grid model. By exploiting complementary strengths and avoiding weaknesses of the grid and particle approaches, we produce some animation examples and analyze their computational performance to demonstrate the effectiveness of the new hybrid method.

We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles ≤ π (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles ≤ π, possibly with boundary consisting of totally geodesic hyperbolic turnovers.To that end we first generalize the local rigidity result contained in [Wei] to the setting of hyperbolic cone-3-manifolds of finite volume as above. We then use the techniques developed in [BLP] to deform the cone-manifold structure to a complete non-singular or a geometric orbifold structure, where global rigidity holds due to Mostow-Prasad rigidity, cf. [Mos], [Pra], in the hyperbolic case,resp. [deR], cf. also [Rot], in the spherical case. This strategy has already been implemented successfully by [Koj] in the compact hyperbolic case if the singular locus is a link using HodgsonKerckhoff local rigidity, cf. [HK].

In this paper we consider a discontinuous Galerkin discretization of the ideal magnetohydrodynamics (MHD) equations on unstructured meshes, and the divergence free constraint ($\nabla \cdot \B = 0$) of its magnetic field $\B$. We first present two approaches for maintaining the divergence free constraint, namely the approach of a locally divergence free projection inspired by locally divergence free elements \cite{Li2005}, and another approach of the divergence cleaning technique given by Dedner et al. \cite{Dedner2002}. By combining these two approaches we obtain a efficient method at the almost same numerical cost. Finally, numerical experiments are performed to show the capacity and efficiency of the scheme.