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<p>The main purpose of this thesis is to study the interplay between relational structures and topology , and to portray pairings in terms of some finite poset models and order preserving maps. We show the interrelations between the categories of topological spaces, closure spaces and relational structures. We study the 4-point non-Hausdorff model S4 weakly homotopy equivalent to the circle S1. We study pairings of some objects in the category of relational structures similar to the multiplication S4 x S4- S4 S4 fails to be order preserving for posets. Nevertheless, applying a single barycentric subdivision on S4 to get S8, an 8-point model of the circle enables us to define an order preserving poset map S8 x S8- S4. Restricted to the axes, this map yields weak homotopy equivalences S8 x S8, we obtain a version of the Hopf map S8 x S8s - SS4. This model of the Hopf map is in fact a map of non-Hausdorff double map cylinders.</p>

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We consider the problem of characterising Besov-Lipshitz and Triebel-Lizorkin spaces using kernels with limited smoothness and decay. This extends the work of H.-Q. Bui et al in [4] and [5] from kernels in S to more general kernels, including the Poisson kernel. We overcome the difficulty of defining the convolution of a general kernel with a distribution by using the concept of a bounded distribution introduced by E. Stein [12]. The characterisations we obtain are valid for the full range of indices.

The construction of a non-separable reflexive Banach space on which every operator is the sum of a scalar multiple of the identity operator and an operator of separable range is presented. Using a result of Rao, a sufficient condition is given for Banach spaces with smooth norms to be decomposable. It is shown that operators on Banach spaces of co-dimension one in their biduals are the sum of a scalar multiple of the identity operator and a weakly compact operator. The Banach spaces of bounded operators L(1¹, 1^p) (1<p<ꝏ) and L(1^p, 1^r), 1 < p ≤ r ≤ p¹ < ꝏ, where 1/p + 1/p¹ = 1, are shown to be primary. The spaces of bounded diagonal operators and compact diagonal operators on a seminormalized Schauder basis β, the multiplier algebras L<sup>d</sub>(X, β) and K_d(X, β), are introduced and studied. New examples of these multiplier algebras are presented and a theorem of Sersouri is extended. A necessary and sufficient condition is given for c_o to embed in K_d(X, β). A sufficient condition is given on a semi-normalized Schauder basis β of a reflexive hereditarily indecomposable Banach space Y to ensure that K_d(Y, β) has the RNP. It is shown that the algebra L_d(X, β) is semisimple and that on the algebra K_d(X, β) derivations are automatically continuous. By representing diagonal operators as stochastic processes a general method of constructing multiplier algebras is given. A non trivial multiplier invariance for the normalized Haar basis of L¹[0,1] is proved.

Types were introduced by Krivine and Maurey, in a refinement of a result by Aldous showing that infinite dimensional subspaces of L_r contain Ωp for some 1≤pꝏ) . A synthesis of these ideas was provided by Garling whose representation of types as random measures was the motivation for much of this work. This thesis aims to investigate the structure of the representation, and to provide concrete representations for differing Banach spaces. Chapter one contains the necessary preliminaries for the later chapters, and finishes by introducing the representation due to Garling of types on L_ϕ(X) as random measures on τ(X) The second chapter consists of two parts. In the first part we examine the structure of the map between types on L_p(X) and random measures on τ(X) . We show that convolution is preserved by the mapping, and give an explicit representation of the space of types on L₁(Ω_p). The second part is concerned with representations of τ(X) . We give conditions for the decomposition of τ(X) into X*S(X) , and derive representations for the space of types on L₁(L_2k). The third chapter studies differentiability of types. We extend differentiability from X to τ(X) , and develop ideas that will be used in the study of uniqueness. In chapter four we consider questions concerning the uniqueness of measures and random measures on X and τ(X) . We construct spaces where the representation of types as random measures is not in uniquely determined. We prove that if a certain uniqueness property for measures on X fails then Ωⁿ₁ embeds in X.

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Mode of access: World Wide Web. / Thesis (Ph. D.)--University of Hawaii at Manoa, 2005. / Includes bibliographical references (leaves 160-161). / Electronic reproduction. / Also available by subscription via World Wide Web / viii, 161 leaves, bound ill. 29 cm

This thesis is concerned with placing 'public art' within the broader modernist spatialisation of social relations. The research takes place around two related enquiries. The first emerges from questions raised by the art critic Rosalyn Deutsche in regard to the proposition that public art functions as both a profession and technology that attempts to pattern space 'so that docile and useful bodies are created by and deployed within it'. Following such questions, this thesis seeks to scrutinise the ways in which discourses on public art might also operate in enabling, maintaining or even disrupting everyday practices and socio-spatial relations. Secondly, as a foray into methodologies of public art research, the thesis considers Foucauldian 'governmentality' approaches in terms of what these might have to offer an investigation of public art. / Thesis (PhD )--University of South Australia, 2008.

Bibliography: leaves 122-124. / v, 129 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / The aim of the thesis is to extend the notion of Apq space from its historical context in the work of Herz and to recognise such spaces as preduals of spaces of intertwining operators of induced representations as suggested by the work of Rieffel (abstract). / Thesis (Ph.D.)--University of Adelaide, Dept. of Mathematics, 1996?