Thesis (Ph. D.)--University of Texas at Austin, 2005. / Vita. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Theses (M.A.)--Marshall University, 2006. / Title from document title page. Includes abstract. Document formatted into pages: contains iv, 48 including illustrations. Bibliography: p. 47-48.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2007. / Includes bibliographical references (leaves 81-82). Also available in electronic version.

Thesis (Ph.D.)--York University, 2004. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 148-156). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ99161

Magister Scientiae - MSc / The main purpose of this thesis is to study the interplay between relational structures and topology , and to portray pairings in terms of some finite poset models and order preserving maps. We show the interrelations between the categories of topological spaces, closure spaces and relational structures. We study the 4-point non-Hausdorff model S4 weakly homotopy equivalent to the circle S1. We study pairings of some objects in the category of relational structures similar to the multiplication S4 x S4- S4 S4 fails to be order preserving for posets. Nevertheless, applying a single barycentric subdivision on S4 to get S8, an 8-point model of the circle enables us to define an order preserving poset map S8 x S8- S4. Restricted to the axes, this map yields weak homotopy equivalences S8 x S8, we obtain a version of the Hopf map S8 x S8s - SS4. This model of the Hopf map is in fact a map of non-Hausdorff double map cylinders. / South Africa

The technique of recursive subdivision can be visualised, loosely, as successively chopping off the corners of a polyhedron to make it less pointed. If the polyhedron is represented as a mesh of points connected by edges, repeated application of the subdivision results in progressively finer meshes tending in the limit to a surface. The subdivision is determined by the weightings given to the respective points and their neighbours.

This thesis is concerned with a domain of linearizability, otherwise known as a Siegel disc, around an irrationally indi erent fixed point of a complex analytic map. In particular, we investigate the existence of Siegel discs and examine the properties of their boundary curves for golden mean rotation number. The key tool used is the idea of a renormalization operator acting on a space of functions. Firstly, a computer-assisted proof is discussed and verified, which establishes the existence of a fixed point of the relevant renormalization operator. In particular, the proof yields a ball of functions around an approximate fixed point that is guaranteed to contain the true fixed point. The rigorous computational techniques which allow computers to be used for this purpose are then discussed. Given the existence of the renormalization fixed point, we verify certain topological conditions, known as the necklace hypotheses, on the action of the maps making up the fixed point. This proves the existence of a Siegel disc having a Holder continuous (invariant) boundary curve for all maps attracted to the fixed point. Further, it is shown that the motion on the boundary is conjugate to a pure rotation, that the boundary curve passes through a critical point of the map, and that the conjugator is not differentiable on a dense set of points. Finally, by viewing the invariant curve as the limit set of an iterated function system (IFS), a further investigation is made to get rigorous bounds on the fractal dimension of the Siegel disc boundary. This involves calculating bounds on the contractivities and coercivities of the maps of the IFS and solving corresponding partition equations. In particular, a rigorous upper bound on the dimension of 1.08523 is obtained.

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Various notions of compactness in a fuzzy topological space have been introduced by different authors. The aim of this thesis is to compare them. We find that in a T₂ space (in the sense that no fuzzy net converges to two fuzzy points with different supports) all these notions are equivalent for the whole space. Furthermore, for N-compactness and f-compactness (being the only notions that are defined for an arbitrary fuzzy subset) we have equivalence under a stronger condition, namely, a T₂ space in the sense that every prime prefilter has an adherence that is non-zero in at most one point