M.Com. (Business Management) / Please refer to full text to view abstract

Ph.D. (Marketing Management) / The study was conducted for the purpose of providing a scientific research quality framework to provide guidelines and new knowledge for focus group research practice in South Africa. The need for a guiding focus group framework was borne out of the critical need for scientific and quality marketing information generated through focus group research. The growing need for quality focus group research information across industries in South Africa was evidently not met, due to the various challenges inherent within the local industry practice of focus group research. The various challenges impacting on the quality of focus group research were identified through an extensive literature investigation, and later confirmed through the empirical study which was preceded by an exploratory qualitative study. The major sources of research quality challenges identified and investigated for purposes of the study relate to non-systematic approaches to focus group research, malpractices and misuse of the focus group method leading to quality compromises. The unstreamlined approach to focus group research practice was found to be due to the different versions of the focus group process followed by many research practitioners, including their own self-guessed approaches. A further complication arises from lack of adherence to scientific procedures, either deliberately in the rush for profit or out of ignorance of research science with respect to the marketing research process. For these reasons, the quality of focus group research produced within the local marketing research industry may not me deemed scientific. Evidence abound, a desirable focus group research practice environment, guided through scientific application of the research method and conducive to the supply of quality focus group research needed to be created, in order to mitigate the quality challenges. The implications for questionable research quality were confirmed through the study to be dire for research user organisations across industries. Unsuspecting research users paid for and applied the defective research to inform their decision-making processes thereby risking failure in the face of market uncertainty.

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The broad goal is to classify all fuzzy subgroups of a given type of finite group. P.S. Das introduced the ntion of level subgroups to characterize fuzzy subgroups of finite grouops. The notion of equivalence of fuzzy subgroups which is used in this thesis was first introduced by Murali and Makamba. We use this equivalence to charterise fuzzy subgroups of inite Abelian groups (p-groups in particular) for a specified prime p. We characterize some crisp subgroups of p-groups and investigate some cases on equi valent fuzzy subgroups.

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In this thesis we consider the group of fractional linear transformations of a variable x over an algebraically closed field k. The purpose of the thesis is to determine all finite subgroups of this group whose orders are not divisible by the characteristic of k. / Science, Faculty of / Mathematics, Department of / Graduate

In this paper we obtain a generalization of the well known Riesz Representation Theorem to the case where the underlying space X is an infinite dimensional product of locally compact, regular and σ-compact topological spaces. In the process we prove that our measures on X correspond to projective limit measures of projective systems of regular Borel measures on the coordinate spaces. An example is given to show that σ-compactness of the coordinate spaces is necessary. / Science, Faculty of / Mathematics, Department of / Graduate

We are concerned with constructing examples of maximal abelian von Neumann subalgebras (MA subalgebras) in hyperfinite factors of type III. Our results will show that certain phenomena known to hold for the hyperfinite factor of type 11₁ also hold for type III factors. Let M and N be subalgebras of the factor α . We call M and N equivalent if M is the image of N by some automorphism of α . Let N(M) denote the subalgebra of α generated by all those unitary operators in α which induce automorphisms of M, and let N²(M), N³(M),... be defined in the obvious inductive fashion. Following J. Dixmier and S. Anastasio, we call a MA subalgebra M of α singular if N(M) = M, regular if N(M) = α, semi-regular if N(M) is a factor distinct from α, and m-semi-regular (m ≥ 2) if N(M),. . .N(m-1)(M) are not factors but N(m)(M) is a factor. The MA subalgebras of the hyperfinite 11₁ factor β have received much attention in the literature, in the papers of J. Dixmier, L. Pukanszky, Sister R. J. Tauer, and S. Anastasio. It is known that β contains a MA subalgebra of each type. Further, β contains pairwise inequivalent sequences of singular, semi-regular, 2-semi-regular, and 3-semi-regular MA subalgebras. The only hitherto known example of a MA subalgebra in a type III factor is regular. In 1956 Pukanszky gave a general method for constructing MA subalgebras in a class of (probably non-hyperfinite) type III factors. Because of an error in a calculation, the types of these subalgebras is not known. The main result of this thesis is the construction, in each of the uncountably many mutually non-isomorphic hyperfinite type III factors of R. Powers, of: (i) a semi-regular MA subalgebra (ii) two sequences of mutually inequivalent 2-semi-regular MA subalgebras 1 (iii) two sequences of mutually inequivalent 3-semi-regular MA subalgebras. Let α denote one of these type III factors and let β denote the hyperfinite 11₁ factor. Roughly speaking, whenever a non-singular MA subalgebra of β is constructed by means of group operator algebras, our method will produce a MA subalgebra of α of the same type. H. Araki and J. Woods have shown that α ⊗ β ≅ α, and it is therefore only necessary to construct MA subalgebras of α ⊗ β of the desired type. We obtain MA subalgebras of α ⊗ β by tensoring a MA subalgebra in α with one in β. In order to determine the type of such a MA subalgebra, we realize β as a constructible algebra and then regard α ⊗ β as a constructible algebra; this allows us to consider operators in α ⊗ β as functions from a group into an abelian von Neumann algebra. As a corollary to our calculations, we are able to construct mutually inequivalent sequences of 2-semi-regular and 3-semi-regular MA subalgebras of the hyperfinite 11₁ factor which differ from those of Anastasio. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis carries out some of classical integration theory in the context of an operator algebra. The starting point is measure on the projections of an abelian von Neumann algebra. This yields an integral on the self-adjoint operators whose spectral projections lie in the algebra. For this integral a Radon-Nikodym theorem, as well as the usual convergence theorems is proved. The methods and results of this thesis generalize, to non-commutative von Neumann Algebras [2, 3, 5]. (1) J. Dixmier Les Algèbres d'Opérateurs dans l'Espace Hilbertien. Paris, 1957. (2) H.A. Dye The Radon-Nikodym theorem for finite rings of operators, Trans. Amer. Math. Soc, 72, 1952, 243-230. (3) F.J. Murray and J. von Neumann, On Rings of Operators, Ann. Math. 37, 1936, 116-229. (4) F. RIesz and B. v. Sz.-Nagy, Functional Analysis, New York, 1955. (5) I.E. Segal A non-commutative extension of abstract integration, Ann. of Math. (2) 57, 1953, 401-457. / Science, Faculty of / Mathematics, Department of / Graduate