Thesis (Ph. D.)--University of North Texas, May, 2009. / Title from title page display. Includes bibliographical references.

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In this thesis we first extend the notion of fuzzy normality to the notion of normality of a fuzzy subgroup in another fuzzy group. This leads to the study of normal series of fuzzy subgroups, and this study includes solvable and nilpotent fuzzy groups, and the fuzzy version of the Jordan-Hõlder Theorem. Furthermore we use the notion of normality to study products and direct products of fuzzy subgroups. We present a notion of fuzzy isomorphism which enables us to state and prove the three well-known isomorphism theorems and the fact that the internal direct product of two normal fuzzy subgroups is isomorphic to the external direct product of the same fuzzy subgroups. A brief discussion on fuzzy subgroups generated by fuzzy subsets is also presented, and this leads to the fuzzy version of the Basis Theorem. Finally, the notion of direct product enables us to study decomposable and indecomposable fuzzy subgroups, and this study includes the fuzzy version of the Remak-Krull-Schmidt Theorem.

The purpose of this paper is to present two proofs of Helly's Theorem and to use it in the proofs of several theorems classified in a group called Helly-type theorems.

Philosophiae Doctor - PhD / This study reports the development and use of an original methodology to measure inflation credibility, as well as the first results of such measurement in terms of an inflation credibility barometer. The barometer is an instrument measuring the degree of acceptance of the accuracy of historic inflation figures. Despite the lack of knowledge about inflation and the low inflation credibility recorded by this first calculation of an inflation credibility barometer for South Africa, valuable information about inflation is unveiled to the authorities. The research results serve as a benchmark, but cannot be compared to earlier research, as this study represents the first systematic measurement of inflation credibility in South Africa. The barometer yields better results than the limited current international measurement of perceptions of the accuracy of historic inflation figures. The barometer (i) reports the credibility of inflation figures as a figure between zero and 100; (ii) will highlight changes in credibility over time with repeated use; (iii) can be explained easily to the general public; (iv) provides for international comparison between countries; and (v) can be used by all countries. The use of inflation credibility barometers and changes in barometer readings over time can also serve as an early warning system for changes in inflation perceptions that might feed through to inflation expectations.

Development of a software system from existing components can surely have various benefits, but can also entail a series of problems. One type of problems is caused by a limited exchange of information between the developer and user of a component, i.e. the developer of a componentbased system. A limited exchange of information cannot only require the testing by the user but it can also complicate this tasks, since vital artifacts, source code in particular, might not be available. Self-testing components can be one response in such situation. This paper describes an enhancement of the Self-Testing COTS Components (STECC) Method so that an appropriately enabled component is not only capable of white-box testing its methods but also capable of black-box testing.

<p> P. Gabriel constructed rings of quotients by inverting elements of multiplicative sets which satisfy the Ore and the reversibility conditions. We employ this technique in our study of localizations of non-noetherian rings at Goldie semiprime ideals. The three types of clans developed in this thesis enable us to decompose in a unique fashion (weakly) classical sets of prime ideals into (weak) clans which, in essence, are minimal localizable sets of prime ideals, satisfying certain properties. We further show that these (weak) clans are mutually disjoint sets. The different types of rings, brought into consideration, exhibit many interesting properties in the context of our localization theory.</p> <p> We characterize the AR-property for the Jacobson radical of a semilocal ring by considering finitely generated modules. In the study of rings which are module-finite over their centres, we describe expressly the injective hull of the semilocal ring modulo its Jacobson radical. These two facts enable us to establish an interrelationship between the (strongly) classical semiprime ideals of the ring and those of its central subring. Furthermore, we show that under certain conditions the Q-sets are precisely all the minimal localizable sets of prime ideals of the ring. In the case of group rings, the flatness condition can be lifted without jeopardizing the validity of the assertion.</p> <p> Lastly, we apply localization technique to characterize the group theoretic notion of q-nilpotency.</p> / Thesis / Doctor of Philosophy (PhD)

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