No abstract

The theorems of Hawking and Penrose show that space-times are likely to contain incomplete geodesics. Such geodesics are said to end at a singularity if it is impossible to continue the space-time and geodesic without violating the usual topological and smoothness conditions on the space-time. In this book the different possible singularities are defined, and the mathematical methods needed to extend the space-time are described in detail. The results obtained (many appearing here for the first time) show that singularities are associated with a lack of smoothness in the Riemann tensor. While the Friedmann singularity is analysed as an example, the emphasis is on general theorems and techniques rather than on the classification of particular exact solutions.

We derive an expression for the singular part of the stress-energy tensor on a hypersurface in spacetime in terms of the discontinuities in fundamental forms associated with the surface for both the non-null case, where our results are standard, and the null case. We then derive the minimum conditions which must be satisfied in order to glue two spacetimes together along such a hypersurface. In both cases, the essential requirement is only that the naturally induced (possibly degenerate) 3-merrics on the hypersurface must agree.

A general method for obtaining continuous solutions to the biomagnetic inverse problem is outlined and illustrated with a wide range of test cases, in a variety of experimental geometries. Magnetic sources are discussed briefly, but the main emphasis is on ionic flows, both in free space and in a homogeneous conducting sphere. We describe a way of obtaining depth information from measurements taken in a single plane and show how instrumental noise affects the quality of our reconstructions. An iterative scheme is introduced, capable of pinpointing a number of localised sources with a minimum of prior assumptions. However, our method is most naturally adapted to distributed sources. A number of inversions of distributed sources demonstrate that the method is poweiful, accurate and convenient.

We have studied the dynamics of dewetting at the interface between polystyrene (PS) and poly(methyl methacrylate) (PMMA) as a function of the molecular weights of the two materials. Optical microscopy and atomic force microscopy have been used to follow the dewetting process and to image both the surface and interfacial structure in the vicinity of the growing holes. We determined the scaling behavior of the dewetting velocity as a function of the molecular weight of the dewetting species (PS). Different regimes are found depending on the PMMA molecular weight. For large PMMA molecular weights the dewetting velocity scales inversely with the PS viscosity, while for low PMMA molecular weights, the dewetting velocity is almost independent of the PS viscosity. The experimental data are in quantitative agreement with recent theoretical predictions.