Multiple scattering corrections to the Beer-Lambert law are analyzed by means of a rigorous small-angle solution to the radiative transfer equation. Transmission functions for predicting the received radiant power-a directly measured quantity in contrast to the spectral radiance in the Beer-Lambert law-are derived. Numerical algorithms and results relating to the multiple scattering effects for laser propagation in fog, cloud, and rain are presented.

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An approach is outlined for computing the different orders of scattering in any medium that possesses a phase function with a strong forward peak. Computations are done for the case of a Gaussian laser beam incident on such a medium. The formulation adopted does reproduce the natural divergence of general Gaussian beams without the need to assume the presence of point sources or the need to assume perfectly collimated beams within the region of interest. Results are discussed for the case of water cloud particles with a strongly forward-peaked phase function for the incident laser radiation.

The chaotic behavior of solutions to Einstein's equations has recently been studied by Barrow within the framework of the dynamical systems theory. Barrow's program of gravitational turbulence is implemented in part by considering the solutions of type VIIo and IX as well as some intermediate types. Quantitative measures of chaos, such as the power spectrum and Lyapunov characteristic exponent, are computed. By converting the equations of motion for the cosmic scale factors to stochastic Langevin's equations, the Mixmaster cosmology in the presence of driving noise terms is investigated. Possible sources of noise can be attributed to an imperfect cancellation of the effective vacuum energy density and the energy density associated with the Higgs field. An ensemble average over random trajectories leads to the suppression of chaotic behavior for type-IX cosmology.