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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.

More than 150 years after their invention by Hamilton, quaternions are now widely used in the aerospace and computer animation industries to track the paths of moving objects undergoing three-axis rotations. It is shown here that they provide a natural way of selecting an appropriate ortho-normal frame -- designated the quaternion-frame -- for a particle in a Lagrangian flow, and of obtaining the equations for its dynamics. How these ideas can be applied to the three-dimensional Euler fluid equations is then considered. This work has a bearing on the issue of whether the Euler equations develop a singularity in a finite time. Some of the literature on this topic is reviewed, which includes both the Beale-Kato-Majda theorem and associated work on the direction of vorticity by both Constantin, Fefferman & Majda and Deng, Hou and Yu. It is then shown how the quaternion formulation provides a further direction of vorticity result using the Hessian of the pressure., 22 pages. This review is based on an invited lecture given at the meeting Mathematical Hydrodynamics held at the Steklov Institute Moscow, in June 2006

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 ).