We investigate a four-dimensional world, embedded into a five-dimensional spacetime, and find the fivedimensional Riemann tensor via generalization of the Gauss͑-Codacci͒ equations. We then derive the generalized equations of the four-dimensional world and also show that the square of the dilaton field is equal to Newton's constant. We find plausible constant and nonconstant solutions for the dilaton.

We show that the governing equations for two-dimensional gravity water waves with constant non-zero vorticity have a nearly-Hamiltonian structure, which becomes Hamiltonian for steady waves.Comment: 12 pages, AMS-LaTeX, no figures. To appear in Journal of Mathematical Fluid Mechanic

Phase-plane stability analysis of a dynamical system describing the Universe as a two-fraction fluid containing baryonic dust and real virial gas quintessence is presented. Existence of a stable periodic solution experiencing inflationary periods is shown. A van der Waals quintessence model is revisited and cyclic Universe solution again found.

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