Commencing with a brief survey of Lie-group theory and diierential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Lie-group structure: Runge{Kutta{Munthe-Kaas schemes, Fer and Magnus expansions. This is followed by complexity analysis of Fer and Magnus expansions, whose conclusion is that for order four, six and eight an appropriately discreti...

Suppose we take a set of greyscale images obtained from videophotography of a moving object in three dimensions, such as an aeroplane, and suppose we compute moments or fourier descriptors, or some other set of smooth features of the resulting image to get a point in a feature space Fn. Then provided we compute enough moments or other descriptors, i.e. provided n is big enough, and provided the...

Differential equations arising in fluid mechanics are usually derived from the intrinsic properties of mechanical systems, in the form of conservation laws, and bear symmetries, which are not generally preserved by a finite difference approximation, and leading to inaccurate numerical results. This paper proposes a method that enables us to build a scheme that preserves some of those symmetries.