In 1956, E. Michael proved his famous convex-valued selection theorems for l.s.c. mappings de ned on spaces with higher separation axioms (paracompact, collectionwise normal, normal and countably paracompact, normal, and perfectly normal), [39]. In 1959, he generalized the convex-valued selection theorem for mappings de ned on paracompact spaces by replacing \convexity" with \ -paraconvexity", for some xed constant 0 < 1 (see, [42]). In 1993, P.V. Semenov generalized this result by replacing with some continuous function f : (0;1) ! [0; 1) (functional paraconvexity) satisfying a certain property called (PS), [63]. In this thesis, we demonstrate that the classical Michael selection theorem for l.s.c. mappings with a collectionwise normal domain can be reduced only to compact-valued mappings modulo Dowker's extension theorem for such spaces. The idea used to achieve this reduction is also applied to get a simple direct proof of that selection theorem of Michael's. Some other possible applications are demonstrated as well. We also demonstrate that the -paraconvex-valued and the functionally-paraconvex valued selection theorems remain true for C 0 (Y )-valued mappings de ned on -collectionwise normal spaces, where is an in nite cardinal number. Finally, we prove that these theorems remain true for C (Y )-valued mappings de ned on -PF-normal spaces; and we provide a general approach to such selection theorems. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2012.

vi, 99 leaves : ill. ; 26 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1974

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references and index.

Thesis (M.Sc.). / Written for the Dept. of Mathematics and Statistics. Title from title page of PDF (viewed 2009/06/25). Includes bibliographical references.

Thesis (M.A.)--Boston University

The subject of this thesis is mathematical proof involving the use of computers. The proof in 1976 of the Four-Colour Theorem, in which an essential lemma is proved using a computer program which took over 1200 hours of computer time to complete, raises philosophical questions concerning the epistemological status of the proof and the extent to which its acceptance as a proof effects an alteration in the traditional concept of mathematical proof. Section I provides an exposition of this proof and a discussion of the Kantian conception of a priori knowledge to provide a background for the following analysis of the philosophical controversy which immediately developed after the publication of the proof. The unsurveyable length of the proof gave rise to the view that its structure was fundamentally empirical and closer to a scientific experiment than a traditional a priori proof. Objectors to this view claimed that the proof differed from most others only in that its empirical content was greater. No essential qualitative difference was involved. These views are examined, and an analysis of those of Frege and J.S. Mill are used to support the opinion that a detailed reassessment of the a priori/a posteriori distinction is necessary to clarify the issues raised by this type of proof. Section II provides an account of recent developments in epistemology with particular reference to the a priori/ a posteriori distinction and favours an analysis of this distinction based on differences in types of psychological process required to generate knowledge. It is maintained that this type of "psychologistic" analysis provides a clarification rather than a rejection of the Kantian conception of the distinction and shows clearly that the Four-Colour Theorem does significantly differ from previous purely formal proofs. _ The conclusion is that acceptance of unsurveyably long computer proofs by the mathematical community involves relinquishing a characteristic of proof formerly held to be essential.

"There is no more fundamental theorem in calculus than the mean-value theorem. Much of the theory of calculus depends, either directly or indirectly, on this theorem. As a consequence of its importance, the theorem has been investigated by a number of mathematicians with the result that various modifications and extensions of the basic theorem have been made"--Introduction. / "May 1956." / Typescript. / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Arts." / Includes bibliographical references (leaf 17).

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