Spectral theory is the study of Mark Kac's famous question [K], "can one hear the shape of a drum?" That is, can we determine the geometrical or topological properties of a manifold by using its Laplace Spectrum? In recent years, the problem has been extended to include the study of Riemannian orbifolds within the same context. In this thesis, on the one hand, we answer Kac's question in the negative for orbifolds that are spherical space forms of dimension higher than eight. On the other hand, for the three-dimensional and four-dimensional cases, we answer Kac's question in the affirmative for orbifold lens spaces, which are spherical space forms with cyclic fundamental groups. We also show that the isotropy types and the topology of the singularities of Riemannian orbifolds are not determined by the Laplace spectrum. This is done in a joint work with E. Stanhope and D. Webb by using P. Berard's generalization of T. Sunada's theorem to obtain isospectral orbifolds. Finally, we construct a technique to get examples of orbifold lens spaces that are not isospectral, but have the same asymptotic expansion of the heat kernel. There are several examples of such pairs in the manifold setting, but to the author's knowledge, the examples developed in this thesis are among the first such examples in the orbifold setting.

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

In 1967, J. A. Erdős showed, using a matrix theory approach that the semigroup Sing[sub]n of singular endomorphisms of an n-dimensional vector space is generated by the set E of idempotent endomorphisms of rank n - 1. This thesis gives an alternative proof using a linear algebra and semigroup theory approach. It is also shown that not all the elements of E are needed to generate Sing[sub]n. Necessary conditions for a subset of E to generate found; these conditions are shown to be sufficient if the vector space is defined over a finite field. In this case, the minimum order of all subsets of E that generate Sing[sub]n is found. The problem of determining the number of subsets of E that generate Sing[sub]n and have this minimum order is considered; it is completely solved when the vector space is two-dimensional. From the proof given by Erdős, it could be deduced that any element of Sing[sub]n could be expressed as the product of, at most, 2n elements of E. It is shown here that this bound may be reduced to n, and that this is best possible. It is also shown that, if E+ is the set of all idempotent of Singn, then (E+)n−1 is strictly contained in Sing[sub]n. Finally, it is shown that Erdős's result cannot be extended to the semigroup Sing of continuous singular endomorphisms of a separable Hilbert space. The sub semigroup of Sing generated by the idempotent of Sing is determined and is, clearly, strictly contained in Sing.

University of Macau / Faculty of Science and Technology / Department of Mathematics

Aspects of fuzzy vector spaces and fuzzy groups are investigated, including linear independence, basis, dimension, group order, finitely generated groups and cyclic groups. It was necessary to consider cardinality of fuzzy sets and related issues, which included a question of ways in which to define functions between fuzzy sets. Among the results proved, are the additivity property of dimension for fuzzy vector spaces, Lagrange's Theorem for fuzzy groups ( the existing version of this theorem does not take fuzziness into account at all), a compactness property of finitely generated fuzzy groups and an extension of an earlier result on the order-homomorphisms. An open question is posed with regard to the existence of a basis for an arbitrary fuzzy vector space.

L will denote a completely distributive lattice with an order reversing involution. The concept of an L-uniform space is introduced. An extension theorem concerning L-uniformly continuous functions is proved. A characterisation of L-uniformizability, involving L-complete regularity is given. With respect to L--completely regular spaces it is shown that the topological modification of an L-completely regular space is completely regular. Furthermore it is shown that the topologically generated L-topology of a completely regular space is L-completely regular.

The notion of a filter F ∈ 2²x has been extended to that of a : prefilter: ƒ ∈ 1²x, generalised filter ƒ ∈ 2²x x and fuzzy filter ᵩ ∈ 1¹x. A uniformity is a filter with some other conditions and the notion of a uniformity D ∈ 2²xxx has been extended to that of a : fuzzy uniformity d ∈ 1²xxx , generalised uniformity ∈ 1²xxx and super uniformity b ∈ 1¹x. We establish categorical embeddings from the category of uniform spaces into the categories of fuzzy uniform spaces, generalised uniform spaces and super uniform spaces and also categorical embeddings into the category of super uniform spaces from the categories of fuzzy uniform spaces and generalised uniform spaces.

The objective of this thesis is to develop certain aspects of the theory of (L,M)-fuzzy topological spaces, where L and M are complete lattices (with additional conditions when necessary). We obtain results which are to a large extent analogous to results given in a series of papers of Šostak (where L = M = [0,1]) but not necessarily with analogous proofs. Often, our generalizations require a variety of techniques from lattice theory e.g. from continuity or complete distributive lattices.

The objective of this thesis is a survey of crisp and fuzzy sober topological spaces. We begin by examining sobriety of crisp topological spaces. We then extend this to the L- topological case and obtain analogous results and characterizations to those of the crisp case. We then brie y examine semi-sobriety of (L;M)-topological spaces.

Certain properties of the solution u of the equation Pu = v in a Banach space will be investigated. It will be assumed that v is a prescribed element of the space, P is a transformation defined on a closed subset in the space and consisting of the sum of a linear transformation and a contraction mapping, and that P and v depend on a real variable λ. which assumes values over the half-open positive interval 0 < λ ≤ λₒ. Then a theorem will be proved, establishing the existence and uniqueness of the solution u(λ) of P(λ)u(λ) = v(λ) . Under the hypothesis that P and v possess asymptotic expansions as λ→0, it will be shown that asymptotic solutions exist, that they are asymptotically unique, and that they possess asymptotic expansions which may be determined by a recursive process from those of P and v. The results obtained will be applied to particular types of Banach spaces, such as finite-dimensional Euclidean spaces, spaces of Lebesgue-square-summable functions and of continuous functions over a closed interval. / Science, Faculty of / Mathematics, Department of / Graduate