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The propagation of electromagnetic fields is described by the vector Helmholtz equation. Finite element analysis of the vector Helmholtz equation involves complications such as spurious modes that do not arise in the scalar Helmholtz equation. Both driven vector field problems as well as vector eigenvalue problems may be corrupted by these divergent, nonphysical fields. Additionally, boundary and interface conditions in vector field problems are more complex than in scalar problems. In this thesis, the cause of spurious modes is analyzed and a condition called the inclusion condition is shown to eliminate the spurious modes. Mixed order covariant projection finite elements are shown to avoid spurious corruptions in both driven and eigenvalue problems. The proposed elements do not involve globally imposed constraints or penalty functions, and boundary and interface conditions are easily imposed.

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This thesis considers a number of problems which are related to the study of random fractals. We define a class of iterations (which we call random hierarchical systems) of probability distributions, which are defined by applying a random map to a set of k independent and identically distributed random variables. Classical examples of this sort of iteration include the Strong Law of Large Numbers, Galton-Watson branching processes, and the construction of random self-similar sets. In Chapter 2, we consider random hierarchical systems on ℝ, under the condition that the random map is bounded above by a random weighted mean, and that the initial distribution is bounded below. Under moment conditions on the initial distribution we show that there exists almost sure convergence to a constant. In Chapters 3 to 5 we consider the asymptotics of some examples of random hierarchical systems, some of which arise when considering certain properties of random fractal graphs. In one example, which is related to first-passage percolation on a random hierarchical lattice, we show the existence of a family of non-degenerate fixed points and show that the sequence of distributions will converge to one of these. The results of some simulations are reported in Chapter 7. Part III investigates the spectral properties of random fractal graphs. In Chapter 8 we look at one example in detail, showing that there exist localised eigenfunctions which lead to certain eigenvalues having very high multiplicity. We also investigate the behaviour of the Cheeger constants of this example. We then consider eigenvalues of homogeneous random fractal graphs, which preserve some of the symmetry of deterministic fractals. We then use relationships between homogeneous graphs and more general random fractal graphs to obtain results on the eigenvalues of the latter. Finally, we consider a few further examples.

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Thesis (M. Phil.)--University of Hong Kong, 1998. / Includes bibliographical references.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 81-84). Also available on the Internet.