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In this thesis, we study the Ricci flow and Ricci soliton equations on Riemannian manifolds which admit a certain degree of symmetry. More precisely, we investigate the Ricci soliton equation on connected Riemannian manifolds, which carry a cohomogeneity one action by a compact Lie group of isometries, and the Ricci flow equation for invariant metrics on a certain class of compact and connected homogeneous spaces. In the first case, we prove that the initial value problem for a cohomogeneity one gradient Ricci soliton around a singular orbit of the group action always has a solution, under a technical assumption. However, this solution is in general not unique. This is a generalisation of the analogous result for the Einstein equation, which was proved by Eschenburg and Wang in their paper "Initial value problem for cohomogeneity one Einstein metrics". In the second case, by studying the corresponding system of nonlinear ODEs, we identify a class of singular behaviours for the homogeneous Ricci flow on these spaces. The singular behaviours that we find all correspond to type I singularities, which are modelled on rigid shrinking solitons. In the case where the isotropy representation decomposes into two invariant irreducible inequivalent summands, we also investigate the existence of ancient solutions and relate this to the existence and non existence of invariant Einstein metrics. Furthermore, in this special case, we also allow the initial metric to be pseudo- Riemannian and we investigate the existence of immortal solutions. Finally, we study the behaviour of the scalar curvature for this more general situation and show that in the Riemannian case it always has to turn positive in finite time, if it was negative initially. By contrast, in the pseudo-Riemannian case, there are certain initial conditions which preserve negativity of the scalar curvature.

Achiral wide bite angle ligands have been shown to be highly active and to induce excellent chemo- and regioselectivities in many homogeneously catalyzed reactions. However, only a few examples of chiral wide bite angle ligands are known so far. A diphenyl ether backbone was selected to allow maximum synthetic versatility and potential for a modular approach to design and synthesize such chiral diphosphorus ligands. Three synthetic strategies have been explored in this thesis: i) introduction of chiral substituents in the ligand backbone, ii) the use of P-stereogenic donor atoms and iii) the synthesis of chiral mixed-donor ligands bearing chiral auxiliary groups on the phosphorus atoms. Functionalization of the 3,3'-positions of 2,2'-bis(diphenylphosphino)diphenyl ether by carboxylic acid or ether auxiliaries was achieved via straightforward four-step routes to generate a library of ligands that were tested in various catalytic reactions. In the Pd-catalyzed asymmetric allylic alkylation of l,3-diphenyl-2-propenyl acetate and cyclohexyl-2-enyl acetate with dimethyl malonate the enantioselectivity was found to depend on the size of the chiral auxiliary introduced within the diphenyl ether backbone and its proximity to the phosphorus donor groups and hence to the active metal centre. Two types of mixed donor bidentate diphosphorus ligands based on the diphenylether backbone have been established, i.e. phosphine-phosphite and phosphine-phosphonite derivatives. A small ligand library bearing different chiral auxiliaries was accomplished via straightforward syntheses that enable derivatization of the respective phosphite and phosphonite moieties in the final step. In the Rh-catalysed hydrogenation of several benchmark substrates high conversion and moderate to high enantioselectivities (up to 97% for dimethyl itaconate) were obtained. The enantioselectivity was influenced by the size of the ortho-substituent on the chiral auxiliary group of the phosphite or phosphonite fragment. Two modular synthetic approaches for the preparation of novel wide bite angle diphosphine ligands containing stereogenic P-atoms have been developed. Both protocols involved diphenylether as backbone and the chiral ephedrine based precursor (2R[subscript(P)],4S[subscript(C)],5R[subscript(C)])-oxazaphospholidine borane as initial auxiliary to induce chirality at phosphorus. Various novel diphosphines were isolated as highly enantioenriched compounds with dr-ratios up to 95:5.

Thesis (PhD)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: see full text for abstract. / AFRIKAANSE OPSOMMING: Sien volteks vir die opsomming.

In this thesis, the effect of fluorinating reagents on a selection of transition metal-mediated organic transformations was investigated. The first four chapters are focused on gold-catalysed nucleophilic addition processes performed in the presence of “F⁺” sources. Chapter 1 provides a general introduction to homogeneous gold catalysis and summarises the aims and objectives of the project. The effect of the electrophilic fluorinating reagent Selectfluor (82) on the gold-catalysed rearrangement of propargyl acetates 85 is discussed in Chapter 2. α-Fluoroenones 92 resulting from fluorodeacetylation of an allenyl acetate intermediate were delivered as the major products of these reactions (Scheme i). [Scheme i Gold-Catalysed Rearrangement-Fluorodeacetylation of Propargyl Acetates 85.] By contrast, performing the gold(I)-catalysed cyclisation of allenoates 102 in the presence of Selectfluor (82) led to products of oxidative coupling. The “F⁺” source in these processes most likely acts as an external oxidant in an Au<sup>I</sup</Au^III redox cycle. In Chapter 3, the cascade cyclisation-intramolecular arylation of benzyl-substituted substrates is discussed whilst the extension of the methodology towards intermolecular homocoupling and intermolecular alkynylation is presented in Chapter 4 (Scheme ii). [Scheme ii Gold-Catalysed Cyclisation-Oxidative Coupling of tert-Butyl Allenoates 102.] In Chapter 5, the feasibility of palladium-catalysed allylic [¹⁸F]radiofluorination was investigated using high-specific-activity [¹⁸F]fluoride. This study led to the development of the first transition metal-mediated C-¹⁸F bond-forming process of relevance for the preparation of radiotracers for PET imaging (Scheme iii). [Scheme iii Palladium-Catalysed Allylic [18F]Radiofluorination of Allylic Methyl Carbonate 227b.] Chapter 6 gives full experimental procedures and characterisation data for all compounds.

This thesis is dedicated to the study of the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for the existence of Einstein metrics with totally geodesic fibers in terms of Casimir operators. Some particular cases are studied, for instance, for normal base or fiber, symmetric fiber, Einstein base or fiber, for which the Einstein equations are manageable. We investigate the existence of such Einstein metrics for invariant bisymmetric fibrations of maximal rank, i.e., when both the base and the fiber are symmetric spaces and the base is an isotropy irreducible space of maximal rank. We find this way new Einstein metrics. For such spaces we describe explicitly the isotropy representation in terms subsets of roots and compute the eigenvalues of the Casimir operators of the fiber along the horizontal direction. Results for compact simply connected 4-symmetric spaces of maximal rank follow from this. Also, new invariant Einstein metrics are found on Kowalski n-symmetric spaces.

To categorise people is by society viewed as a basic cognitive process. The process is described to occur frequently. The purpose of this study was to investigate teachers experience and views concerning the categorization "we and them", and its relationship to the education of religion. The study aims to explore how the categorisation is expressed and valued. The study also aims to investigate the relationship between the teaching and the student groups heterogeneity/homogeneity, but also the relationship between the categorisation and the teaching. The data was conducted true semi-structured interviews with experienced teachers. The results of the study showed that "we and them" exist in the education of religion. The results of the study did not show an unambiguous perception of how the categorisation is expressed and viewed. The result also showed that the heterogeneity, alternatively homogeneity, of the student groups affects the education. The result also showed that the categorisation is affected by the heterogeneity or homogeneity, of the student groups. / Att kategorisera människor betraktas i den samtida kontexten som en av människans grundläggande kognitiva processer. Kategoriseringen och därmed indelning av människor är en ständigt pågående process. Syftet med denna studie är där av att undersöka i vilken utsträckning kategoriseringen av ett "vi" tillika ett "dom" är påtaglig i dagens religionsundervisning. Studien ämnar att undersöka religionslärares erfarenheter och åsikter kring en eventuell kategorisering av "vi och dom " i religionsundervisningen. Studien syftar till att undersöka hur en eventuell kategorisering av "vi och dom" yttrar sig och hur kategoriseringen värderas. Tillika syftar studien till att undersöka hur religionsundervisningen påverkas om elevgruppen är heterogen alternativt homogen, samt vilken betydelse heterogena-/homogena elevgrupper spelar för den eventuella kategoriseringen. Empirin erhölls genom semistrukturerade intervjuer med erfarna religionslärare. Studiens resultat visade att kategoriseringen "vi och dom" existerar i religionsundervisningen. Resultatet visade inte på någon entydig uppfattning rörande hur kategoriseringen yttrar sig tillika värderas. Vidare visade resultatet att religionsundervisningen påverkas utifrån elevgruppens heterogenitet alternativt homogenitet. Resultatet visade tillika att kategoriseringen "vi och dom" yttrar sig på olika vis, beroende på om religionsundervisningen bedrivs i en heterogen- eller en homogen elevgrupp.

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This thesis discusses the intermolecular hydroacylation reaction using cationic rhodium bis- phosphine complexes as catalysts. A series of small bite-angle rhodium bis-phosphine complexes have been prepared and characterised. The reactivity of these complexes has been investigated in order to gather information about the effect of subtle changes in the ligand design and they are compared to the previously reported catalysts. Chapter 2 presents the challenges involved in the synthesis of small bite-angle isopropyl and cyclohexyl PNP and PCP bis-phosphine ligand containing rhodium complexes. These complexes have been fully characterised and screened in intermolecular hydroacylation reaction using 2- (methylthio)benzaldehyde (<strong>E</strong>) and 1-octene or 1-octyne as substrates. The formed complexes were shown to be very efficient and regioselective alkyne hydroacylation catalysts. The mechanism of the hydroacylation reaction was investigated using the isopropyl PNP complex [Rh(ⁱPrPNMePⁱPr₂)(C₆H₅F)][BAr^F₄] (<strong>11b</strong>). Chapter 3 concentrates on developing new rhodium bis-phosphine complexes that involve a ligand incorporating the small bite-angle motif with the one of hemilability. The PNP complex [Rh((2-OMe-C₆H₄)₂PNMeP(2-OMe-C₆H₄)2)(C₆H₅F)][BAr^F₄] (<strong>41</strong>) was synthesised and analytically characterised. <strong>41</strong> was shown to be an active alkyne hydroacylation catalyst with more stability towards the catalyst deactivation pathway, reductive decarbonylation, compared to the previously investigated <strong>11b</strong>. Additionally mechanistic studies using <strong>41</strong> were carried out. The final chapter moves on to study the C-S activation ability of small bite-angle rhodium bis- phosphine complexes to remove the sulfur tether from the hydroacylation products at the end of the hydroacylation reaction. A screening is conducted to compare the reactivity of different small bite-angle ligands. Additionally, a detailed investigation is carried out to see the effect the C-S activation has on the hydroacylation reaction.

La synthèse d'amines chirales par catalyse asymétrique à l'aide de complexes de métaux de transition suscite un grand intérêt, puisque cela conduit à la production de molécules à haute valeur ajoutée. La voie catalytique offre une alternative avantageuse par rapport aux voies de synthèses conventionnelles, dont les inconvénients sont d'une part des produits de départ coûteux et d'autre part un grand nombre de produits secondaires, sans parler des étapes de synthèse souvent nombreuses. Dans le cadre d'un projet industriel visant à développer des outils catalytiques performants pour la synthèse d'amines chirales, nous nous sommes orientés vers le développement d'un système qui permettrait de réaliser l'hydroaminométhylation d'alcènes de façon énantiosélective. L'hydroaminométhylation est une réaction tandem combinant deux réactions catalytiques, l'hydroformylation et l'hydrogénation. A partir de molécules modèles, nous avons développé une approche permettant de rationnaliser la création de centres asymétriques en fonction du type de substrat utilisé, soit au cours de l'hydroformylation, soit au cours de l'hydrogénation de l'énamine intermédiaire. La réaction a été étudiée en version intermoléculaire et intramoléculaire. Le cycle catalytique de l'hydroformylation et celui de l'hydrogénation possèdent chacun des exigences différentes au niveau de la sphère de coordination du métal. De plus, les conditions de la réaction tandem ont une influence sur les espèces catalytiques formées et les sélectivités de la réaction. Grâce à des études de RMN sous pression, confirmées par des calculs théoriques, nous nous sommes attachés à étudier le comportement des complexes du rhodium mis en jeu, sous pression et dans les conditions d'hydroaminométhylation. Ces études nous ont permis d'approfondir la connaissance des espèces catalytiques impliquées dans cette réaction tandem. L'ensemble du travail de recherche a été mené en combinant les approches fondamentale et appliquée, nous permettant ainsi de proposer un outil catalytique adapté au substrat de départ considéré, dans la perspective d'une application industrielle de la réaction. / Amines are of great importance as building blocks or reactants in the chemical industry. The development of catalytic processes for their synthesis is thus of particular interest from an industrial point of viewsince they can afford an alternative to conventional synthetic pathways. In the context of an industrial project aiming to synthesize chiral amines, we focused on the development of a catalytic system adapted to the asymmetric hydroaminomethylation of alkenes. This tandem reaction includes two transition metal catalyzed reactions under CO and H2 pressure: the hydroformylation and the hydrogenation reactions. Starting from model molecules, we proposed an approach to carefully study the creation of asymmetric centers during the reaction sequence, either in the hydroformylation step or in the enamine intermediary hydrogenation step. The reaction was studied in its inter- and intramolecular version.Rhodium is often used for the hydroaminomethylation reaction, since it is able to complete both catalytic cycles of hydroformylation and hydrogenation. However, requirements in the coordination sphere of the metal center and species involved in each catalytic cycle are presumed to be different. Thanks to high pressure NMR experiments, combined with DFT calculations, the behaviour of rhodium complexes involved in the catalysis was investigated under conditions close to those of hydroaminomethylation. The knowledge of the rhodium species involved in the reaction was also improved thanks to these spectroscopic and catalytic experiments. The fundamental and applied approaches result in a deeper understanding of the tandem reaction sequence and allow us to design a catalytic system adapted to the starting substrate, in the purpose of an industrial application of the reaction.