Browsing by Subject "mathematics and applied mathematics"

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    A contribution to the foundations of the theory of Quasifibration
    (1995) Witbooi, Peter Joseph; Hardie, K. A.
    The concept of quasifibration was invented by Dold and Thom (DT]. May (M2] approached quasifibrations from a new angle, making use of n-equivalences. This dissertation presents a study of the notion of n-equivalences and related types of maps. The first of our two main goals is to prove a result, Theorem 5.1, which generalizes the fundamental theorem (DT; Satz 2.2] by Dold and Thom on globalization of quasifibrations. Secondly we show that by means of adjunction or clutching constructions, this theorem enables us to retrieve the famous results of James (J2; Theorem 1.2 and Theorem 1.3] in his work on suspension of spheres. The results of James appear in the thesis as Theorem 13.8. For some of the applications we need a generalized version of n-equivalence. This generalization entails replacing, in the definition of n-equivalence, the isomorphisms by isomorphisms modulo a suitable Serre class [Se] of abelian groups. For the sake of having the thesis self-contained, we include a formal discussion of localization of 1-connected spaces and Serre classes of abelian groups. This summarizes the scope of the thesis. More detail on the content of the thesis will be given after we have sketched a historical perspective on quasifibrations.
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    Local connectedness of frames
    (2004) Mushaandja, Zechariah; Schauerte, Anneliese
    In this thesis, we undertake a systematic study of local connectedness of frames. Among other central ideas in this study is that of a connected congruence on a frame. We show that the two definitions of a connected congruence in literature (section 2.2) are not equivalent, and hence introduce a new term for one of them. We also prove that, using Baboolal's methods, if the Stone-Cech compactification βL is locally connected then L need not be locally connected for completely regular frame L. This happens in chapter 5.
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    Unbound linear operators in operator ranges
    (1986) Labuschagne, L. E.; Cross, R. W.
    Many results in operator theory for example some perturbation results, are at present known only in the Banach space case. The aim of this work is to provide a natural generalisation of such results by considering operator ranges (the image of a bounded operator defined everywhere on a Banach space) as well as investigating and characterizing some of the properties of operator ranges. For the sake of generality we will for the most part be considering unbounded or closed linear operators instead of continuous everywhere defined linear operators. We will not be attempting to give exhaustive coverage of unbounded linear operators but will try to give some insight into the use of operator range techniques in the theory of unbounded linear operators. The first chapter will be aimed mainly at defining and introducing concepts used in later chapters. In the second chapter we turn our attention to the conjugate of a linear operator whilst also briefly looking at projections in an operator range. Chapter three is concerned mainly with investigating and characterizing the closed range property of linear operators whereas in the first part of chapter four we will be proving some fairly well known results on compact, precompact and strictly singular operators to be used in chapter five. In the second half of chapter four we will investigate the relationship between weakly compact operators and pre-reflexive spaces. Chapter five will be dealing with perturbation of semi-Fredholm operators by first of all continuous and then by strictly singular operators. We close with a discussion of the instability of non-semi-Fredholm operators under compact and a-compact perturbations.