The 3-metric gij and extrinsic curvature (second fundamental form) Kij are the fundamental variables describing the geometry in any space+time decomposition of the Einstein equations. (g,K) describe the local geometry of a single space-like hypersurface M , and it is then natural to describe the evolution of the space-time geometry by a 1-parameter family (g(t), K(t)), describing the local geom...

We attempt a unification of the Quantum and the Statistical Mechanics of a single spinless particle. We consider a five dimensional (5D) space-time with no matter. We postulate that all the physical information on a 5D massless particle propagation is provided by the statistics over null paths in this 5D space-time. If the 5D metric does not depend on the fifth space-like coordinate, then the p...

We give a new algorithm for enumerating all possible embeddings of a metric space (i.e., the distances between every pair within a set of n points) into R Cartesian space preserving their l∞ (or l1) metric distances. Its expected time is O(n 2 log n) (i.e. within a poly-log of the size of the input) beating the previous O(n) algorithm. In contrast, we prove that detecting l ∞ embeddings is NP-c...