This paper presents a 6d world-volume action that describes the dynamics of the M theory five-brane in a flat 11d space-time background. The world-volume action has global 11d super-Poincaré invariance, as well as 6d general coordinate invariance and kappa symmetry, which are realized as local symmetries. The paper mostly considers a formulation in which general coordinate invariance is not man...

Lyapunov exponents (LEs) are key indicators of chaos in dynamical systems. In general relativity the classical definition of LE meets difficulty because it is not coordinate invariant and spacetime coordinates lose their physical meaning as in Newtonian dynamics. We propose a new definition of relativistic LE and give its algorithm in any coordinate system, which represents the observed changin...

We consider the double field formulation of closed bosonic string theory, and calculate Poisson bracket algebra symmetry generators governing both general coordinate local gauge transformations. Parameters these symmetries depend on a coordinate, defined as direct sum initial T-dual coordinate. When no antisymmetric is present, $C$-bracket appears Lie generalization in theory. With introduction...

A method to initialize an ensemble, introduced by Evensen (Physica, D 77:108–129, 1994a; J Geophys Res 99(C5):10143–10162, 1994b; Ocean Dynamics 53:343–367, 2003), was applied to the Ocean General Circulation Model (OGCM) HYbrid Coordinate Ocean Model (HYCOM) for the Pacific Ocean. Taking advantage of the hybrid coordinates, an initial ensemble is created by first perturbing the layer interface...

The study of dynamics in general relativity has been hampered by a lack of coordinate independent measures of chaos. Here I review a variety of invariant measures for quantifying chaotic dynamics in relativity that exploit the coordinate independence of fractal dimensions and symbolic entropies. 1 Time and chaos Historically, chaos theory was developed for Newtonian dynamics where time and spac...