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Inverse length scales for the 2d and 3d thermal convection equations on periodic boundary conditions are estimated directly from the PDEs using a ladder theorem. To within logarithmic dependence on the Rayleigh number RT, the 2d estimate approximately agrees with the inverse length scale estimate found by Manley, Foias and Temam(1987) which was based on the attractor dimension. In keeping with numerical experiments, it shows that RT1/2*RT1/2 grid points should be roughly enough to resolve the smallest features in the flow. Similar methods applied to the 3d case produce a curious result where an inverse length scale for the flow can be estimated rigorously for weak solutions. This estimate is proportional to RT2 which has an associated length scale which is unfortunately much too small to be of any practical use. The same methods also apply to double diffusive convection.

A method is described which enables multiple three-dimensional soliton-like solutions to be found for non-linear Klein-Gordon equations of the type Square Operator phi =F( phi ). The method is extendable to coupled equations Square Operator phi =F( phi , psi ); Square Operator psi =G( phi , psi ).

A comprehensive perturbation theory of laser light modulating an electron beam within a crystal shows that the finite size of the crystal plays an essential role. In particular the theory of the central diffraction spot supports the conclusions of Van Zandt and Meyer (1970). Bremsstrahlung provides an explanation of the light emitted from a central spot if but only if the incident electron beam is sufficiently monochromatic. The intensity of the light emitted by other processes is too weak to be observed.

The equations of motion of a relativistic electron in a spatially modulated magnetic field are solved for the two limiting cases of a strong or a weak field. The microwave frequency spectrum emitted in the strong field limit depends on a critical field parameter Bcr, in qualitative agreement with some recent experimental results of Friedman and Herndon. (see abstr. A17480 of 1972).