Let R be any of the following rings: the smooth functions on R^2n with the Poisson bracket, the Hamiltonian vector fields on a symplectic manifold, the Lie algebra of smooth complex vector fields on C, or a variety of rings of functions (real or complex valued) over 2nd countable spaces. Then if H is any other Polish ring and φ:H →R is an algebraic isomorphism, then it is also a topological isomorphism (i.e. a homeomorphism). Moreover, many such isomorphisms between function rings induce a homeomorphism of the underlying spaces. It is also shown that there is no topology in which the ring of real analytic functions on R is a Polish ring.

Statistics / M. Sc. (Statistics)

Thesis (MMUS (Music))--Stellenbosch University, 2010. / ENGLISH ABSTRACT: The South African-born British composer Priaulx Rainier (1903-1986) wrote her Requiem (1955-1956) for solo tenor and choir to a text by surrealist poet David Gascoyne. The poem (completed in 1940) contradicts the commemorative genre of the requiem and instead anticipates the prospective victims of the war that was to come. Due to the disturbances caused by the war, Rainier only started working on the music fifteen years later in 1955. Existing discourse on Rainier has been shaped by the thoughtless regurgitation of opinions, reviews and clichéd biographical models. Since very few detailed analyses of Rainier’s works exist, this thesis attempts to address this gap in research on this composer. It therefore aims to contribute to a more balanced, evidence-based discourse. The significance of the Requiem is that it was said by commentators to indicate a period of increasing abstraction in Rainier’s oeuvre. The findings regarding tonality in this work may therefore serve as a point of reference in future analyses of works preceding and following the Requiem. In the literature review, recurring issues in the discourse on Rainier (such as the numerous references to her childhood in Natal as an influence on her works and the descriptions of her works as possessing a masculine gender identity) is highlighted. The contextualisation that follows includes a reception-based periodisation of Rainier’s works and a “biography” of the Requiem, with a special focus on the intersection between the symbolic world of David Gascoyne and the Requiem and the sculptor Barbara Hepworth’s influence on Rainier’s work. Underpinning these contextual considerations is a set theory analysis of the work that attempts to illustrate the composer’s aspiration towards musical abstraction as a creative force. The findings of the analysis are also contextualised with regard to existing notions of Rainier’s style and tonality in her music. / AFRIKAANSE OPSOMMING: Die Suid-Afrikaans gebore Britse komponis Priaulx Rainier (1903-1986) se Requiem (1955-1956) vir solo-tenoor en koor is ’n toonsetting van ’n gedig deur die surrealistiese digter David Gascoyne. Die gedig weerspreek die gedenkende karakter van die requiem genre deurdat dit vooruitwys na die slagoffers van die oorlog wat op hande was. As gevolg van die onderbrekings wat deur die oorlog veroorsaak was, het Rainier eers vyftien jaar later, in 1955, aan die musiek begin werk. Die ondeurdagte herhaling van opinies, resensies en biografiese clichés kenmerk huidige diskoers oor Rainier. Aangesien daar min gedetailleerde analises van Rainier se werke in die literatuur bestaan, poog hierdie tesis om bestaande leemtes in die navorsing oor Rainier te vul. Dit streef ook na die ontwikkeling van ’n meer gebalanseerde, bewysgebaseerde diskoers. Die belang van hierdie werk setel hoofsaaklik daarin dat dit volgens sommige bronne dui op ’n kenteringsmoment in Rainier se oeuvre, waarna haar werke meer abstrak geword het. Die bevindinge rakende tonaliteit in hierdie werk kan dus dien as ’n verwysingspunt vir toekomstige analises van Rainier se werke wat die Requiem voorafgaan en volg. Herhalende elemente in die diskoers oor Rainier (soos die invloed van haar kinderjare in Natal op haar werke en die beskrywings van ’n manlike gender-identeit in haar werke) word in die literatuurstudie uitgelig. Die konstekstualisering wat volg bestaan uit ’n literatuur-gebaseerde periodisering van haar werke en ’n “biografie” van die Requiem wat fokus op die verhouding tussen simboliek in die werke van David Gascoyne en die Requiem en die invloed van die beeldhouer Barbara Hepworth op Rainier se werke. Die kontekstualisering word gevolg deur ’n ‘set’-teorie-analise van die werk waarin gepoog word om die komponis se strewe na musikale abstraksie as kreatiewe inspirasie aan te toon. Die bevindinge van die analise word dan vergelyk met die bestaande opvattinge oor styl en tonaliteit in Rainier se werke. / COMPOSITIONS: In Paradisum for SATB choir 2 Droom for mezzo-soprano and piano: 11 • Kuspad 14 • Sy 17 • Glasblaser 20 • Glasprater 23 String trio arrangements from Droom: • Glasblaser 26 • Glasprater 30 Het Hom! for tenor saxophone 34 The beauty in sorrow for string quartet: 37 • First Movement 40 • Second Movement 49 • Third Movement 54 Symphonata for orchestra: 61 • First Movement 62 • Second Movement 66 • Third Movement 71 • Fourth Movement 77

In this thesis, a discrete Markov chain model for analyzing memory interference in multiprocessors, is presented.

The purpose of this thesis is to study the equivalence relation between sets A and B: A o B if and only if there exists a one to one function f from A onto B. In Chapter I, some of the fundamental properties of the equivalence relation are derived. Certain basic results on countable and uncountable sets are given. In Chapter II, a number of theorems on equivalent sets are proved and Dedekind's definitions of finite and infinite are compared with the ordinary concepts of finite and infinite. The Bernstein Theorem is studied and three different proofs of it are given. In Chapter III, the concept of cardinal number is introduced by means of two axioms of A. Tarski, and some fundamental theorems on cardinal arithmetic are proved.

The aim of this work is the study of infinite conformal iterated function systems. More specifically, we investigate some properties of a limit set J associated to such system, its Hausdorff and packing measure and Hausdorff dimension. We provide necessary and sufficient conditions for such systems to be bi-Lipschitz equivalent. We use the concept of scaling functions to obtain some result about 1-dimensional systems. We discuss particular examples of infinite iterated function systems derived from complex continued fraction expansions with restricted entries. Each system is obtained from an infinite number of contractions. We show that under certain conditions the limit sets of such systems possess zero Hausdorff measure and positive finite packing measure. We include an algorithm for an approximation of the Hausdorff dimension of limit sets. One numerical result is presented. In this thesis we also explore the concept of positively recurrent function. We use iterated function systems to construct a natural, wide class of such functions that have strong ergodic properties.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 100-102. / by Robert Alan Proctor. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: p. 135-138. / by James William Walker. / Ph.D.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Bibliography: leaves 106-109. / by Rae Michael Andrew Shortt. / Ph.D.

The manufacturer claims that there is only one solution to the puzzle Instant Insanity II. However, a recent paper shows that there are two solutions. Our goal is to find ways in which we only have one solution. We examine the permutation groups of the puzzle and use modern algebra to attempt to fix the puzzle. First, we find the permutation group for the case when there is only one empty slot at the top. We then examine the scenario when we add an extra column or an extra row to make the game a 4 × 5 puzzle or a 5 x 4 puzzle, respectively. We consider the possibilities when we delete a color to make the game a 3 × 3 puzzle and when we add a color, making the game a 5 × 5 puzzle. Finally, we determine if solution two is a permutation of solution one.