This paper concerns itself mainly with those functions from one topological or metric space to another that have closed graphs in the product space. Their relationship to closed, locally closed, compact, continuous and subcontinuous functions is studied in order to determine the relative strength of the closed graph condition. The paper collects and in some cases extends results found in papers by R. V. Fuller [2], P. E. Long [7] P. Kostyrko and T. Shalat [4], [5] and [6]. The main theorems deal with; 1) the characterization of continuous functions in terms of subcontinuity and the closed graph property; 2) a proof that if f has a closed graph then f is the limit of a sequence of continuous functions; and 3) a study of the operations under which the class of functions with closed graphs is closed. / Science, Faculty of / Mathematics, Department of / Graduate

A standard way of finding the unique zero on (0, 1) of a continuous decreasing functions with f(0) f(1)< 0, is to test the sign of f(1/2), then the sign of f(1/4) (if f(1/2)< 0) or the sign of f(3/4) (if f(1/2) >0), etc. In this way, the zero of f is localized in n steps to an interval of length 2⁻ⁿ. The unique maximum of a unimodal function on [0,1] can be similarly localized, but the unique maximum of a unimodal function on the unit square cannot. We start by generalizing these problems: let A be a compact subset of Eⁿ, let F be a set of real-valued functions on A, and for each f in F let S(f) be a point in the Cartesian product Aᵏ; S(f) is called a functional on F. Examples of such functionals are zeros, extrema, inflexion points, saddle points, etc., as well as sets of these. A test function T is a function of m real variables, which takes up only a finite number (≥2) of distinct values. An abscissa set Xᵢ is an ordered m-tuple (xlᵢ,...xmᵢ) ,, with each Xji in A. A sequential strategy is a way of selecting abscissa sets X₁, X₂,...,where the knowledge of T(f (xlᵢ, . ., f (xmᵢ)) is used to determine Xi+i. The N-th set of indeterminacy for S(f) is the largest subset of in which S(f) can lies consistent with the results of the first N-1 tests. A functional S(f) is sequentially localizable if a test function T and a sequential strategy exist, such that for every f in F the sets of indeterminacy shrink to a point (which must then be S(f) itself). First, several conditions are given to ensure the sequential localizability of a functional, these are presented in terms of certain topologies induced on Aᵏ and in terms of contraction maps. It is then shown that if a functional is localizable, there exists an optimal strategy under which the sets of indeterminacy converge fastest; further, the speed of localization is always exponential. Next, the concept of a random strategy and of random localizability is introduced, and it is shown that in many cases random localizability and sequential localizability are equivalent. Also, the speed of the former is not too much worse than the speed of the latter. Finally, optimal and near-optimal strategies are worked out for some functionals of interest. / Science, Faculty of / Mathematics, Department of / Graduate

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Thesis (D.A.)--Illinois State University, 1995. / Title from title page screen, viewed May 2, 2006. Dissertation Committee: John Dossey (chair), Ronald Halinski, Lotus Hershberger, Thomas Shilgalis. Includes bibliographical references (leaves 70-74) and abstract. Also available in print.

Thesis--Göttingen.

A research report submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2013 / Equations involving the gamma and hypergeometric functions are of great interest to mathematicians and scientists, and newly proven identities for these functions assist in finding solutions to differential and integral equations. In this work we trace a brief history of the development of the gamma and hypergeometric functions, illustrate the close relationship between them and present a range of their most useful properties and identities, from the earliest ones to those developed in more recent years. Our literature review will show that while continued research into hypergeometric identities has generated many new results, some of these can be shown to be variations of known identities. Hence, we will also discuss computer based methods that have been developed for creating and analysing such identities, in order to check for originality and for numerical validity.

Thesis--K.B. Ludwig-Maximilians-Universität zu München, 1909. / Vita.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references (leaves 92-95). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 97-105). Also available in electronic version. Access restricted to campus users.

Thesis (B.S.)--University of Missouri, School of Mines and Metallurgy, 1882. / The entire thesis text is included in file. Holograph [Handwritten and illustrated in entirety by author]. Title from title screen of thesis/dissertation PDF file (viewed January 6, 2009)