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Borel determinacy states that if G(T;X) is a game and X is Borel, then G(T;X) is determined. Proved by Martin in 1975, Borel determinacy is a theorem of ZFC set theory, and is, in fact, the best determinacy result in ZFC. However, the proof uses sets of high set theoretic type (N1 many power sets of ω). Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of Borel Determinacy. To prove this, Friedman produces a model of ZC and a Borel set of Turing degrees that neither contains nor omits a cone; so by another theorem of Martin, Borel Determinacy is not a theorem of ZC. This paper contains three main sections: Martin's proof of Borel Determinacy; a simpler example of Friedman's result, namely, (in ZFC) a coanalytic set of Turing degrees that neither contains nor omits a cone; and finally, the Friedman result.

We aim to explain the intuition behind several large cardinal axioms, give characterization theorems for these axioms, and then discuss a few of their properties. As a capstone, we hope to introduce a new large cardinal notion and give a similar characterization theorem of this new notion. Our new notion of near strong compactness was inspired by the similar notion of near supercompactness, due to Jason Schanker.

Sidman and Tailby (1982) established procedures to analyze the nature of stimulus to stimulus relations established by conditional discriminations. Their research describes specific behavioral tests to determine the establishment of properties that define the relations of equivalence. An equivalence relation requires the demonstration of three conditional relations: reflexivity, symmetry, and transitivity. The equivalence stimulus paradigm provides a method to account for novel responding. The research suggests that equivalence relations provide a more efficient and effective approach to the assessment, analysis, and instruction of skills. The present research examined the effectiveness of the formation of an equivalence class in teaching young children coin value relations. The second aspect of the study was to determine if there was a relationship between equivalence class formation and generalization of the skills established to other settings. Five children, 4- and 5-years old, were selected to participate in the study based on their lack of skills in the area of coin values and purchasing an item with dimes or quarters equaling fifth cents. The experimental task was presented on a Macintosh computer with HyperCard programming. The experimental stimuli consisted of pictures of dimes, quarters, and Hershey candy bars presented in match-to-sample procedures. Two conditional discriminations were taught (if A then B and if B then C.). The formation of an equivalence class was evaluated by if C then A. Generalization across settings was tested after the formation of an equivalence class by having the children purchase a Hershey candy bar with dimes at a play store. A multiple baseline experimentaldesign was used to demonstrate a functional relationship between the formation of an equivalence class and generalization of skills across settings. The present research provides supportive evidence that coin value relations can be taught to young children using equivalence procedures. The study also demonstrated generalization of novel, untaught stimuli across settings, after the formation of an equivalence class. A posttest on generalization across settings was conducted 3 months after the study. Long-term stability of equivalence relations was demonstrated by three of the subjects. / Department of Special Education

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

Assuming the axiom of determinacy, we give a new proof of the strong partition relation on ω1. Further, we present a streamlined proof that J<λ+(a) (the ideal of sets which force cof Π α < λ) is generated from J<λ+(a) by adding a singleton. Combining these results with a polarized partition relation on ω1