In the first chapter we consider the question of uniqueness of conformal Ricci flow. We use an energy functional associated with this flow along closed manifolds with a metric of constant negative scalar curvature. Given initial conditions we use this functional to demonstrate the uniqueness of the solution to both the metric and the pressure function along conformal Ricci flow. In the next chapter we study backward Ricci flow of locally homogeneous geometries of 4-manifolds which admit compact quotients. We describe the longterm behavior of each class and show that many of the classes exhibit the same behavior near the singular time. In most cases, these manifolds converge to a sub-Riemannian geometry after suitable rescaling.

Orientador: Helio Vasconcelos Fagundes / Banca: José Geraldo Pereira / Banca: Maria Emília Xavier Guimarães / Resumo: No final do século XIX, L. Bianchi [1] fez a classificação das geometrias riemanianas em espaços tridimensionais, segundo seus possíveis grupos de isometrias. Parte de seus resultados foi adaptada, em uma linguagem mais moderna, por C. G. Behr [17] e outros, para o estudo de modelos cosmológicos espacialmente homogêneos mas não necessariamente isotrópicos. Esta dissertação expõe as idéias e resultados de Bianchi, e também os formalismos mais recentes no estudo desse problema. Por completeza, o espaço tridimensional do modelo de Kantowski-Sachs também é aqui incluído / Abstract: At the end of the 19th century, L. Bianchi [1] found a classification of the Riemannian geometries oni three-dimensional spaces, according to their possible isometry groups. A part of his results has been adapted, in a more modern language, by C. G. Behr [17] and others, for the study of cosmological models with homogeneous but not necessarily isotropic spatial sections. This dissertation presents Bianchi's ideas and results, and also more recent formalisms in the study of this problem. For completeness, the three-space of Kantowski-Sachs cosmological model is also here included / Mestre

Most biodiesel plants operate batch-wise using homogeneous alkali catalysts. Recently, several heterogeneous catalysts have been suggested in literature, as they have shown potential for overcoming most of the challenges associated with the application of homogeneous catalysts. Previous published techno-economic comparisons of the two technologies on large-scale processes located in the developed world, have revealed the economic superiority of heterogeneously catalysed processes. Hence, prospect exists for current homogeneously catalysed process plants to be converted to heterogeneously catalysed ones. The objective of this research was to investigate the actual cost benefit of converting a small-scale batch biodiesel plant from homogeneous to heterogeneous catalysed process. For this purpose, a small-scale batch biodiesel plant located in South Africa was taken as the base case homogeneous process. Aspen Batch Process Developer® software was used to perform the process simulations. The homogeneous process was converted to the heterogeneous one and results from process simulation were used to evaluate the economics of both processes, which were compared in terms of fixed capital cost, total manufacturing cost and profitability indicators. During economic evaluation, two types of cost factors were used: one prevailing in developed world and the other one relevant to South Africa. The sensitivity analysis of both processes was further performed in order to investigate the impact of some uncertain parameters on their profitability. Finally, a debottlenecking study was carried out. Results obtained from this study showed an increase in the annual throughput of biodiesel as well as significant savings in the total capital cost for the heterogeneous catalysed process relative to the homogeneous one. As regards the estimation of the total unit manufacturing cost of biodiesel, significant differences arose when using the two types of cost factors. Results of economic analyses estimated using cost factors relative to South Africa suggest an increase in the unit manufacturing cost of biodiesel while using the developed world’s cost factors suggests the opposite. This is due to the higher raw material and energy requirement for the CaO process, while knowing that the direct costs are a bigger proportion of the manufacturing costs estimated using the South African cost factors. Profitability and sensitivity analyses only provided positive results when estimated using the South African cost factors. In all cases, the heterogeneous catalysed process was found to be more promising than the homogeneous one over the prescribed project life. The study showed the importance of using cost factors relevant to a particular economic environment during techno-economic assessment of a process. It was also shown that there are economic benefits when replacing settling with centrifugation in biodiesel production processes. In summary, this thesis makes some important contributions. It presents the first process simulation for biodiesel production using Aspen Batch Process Developer® software and thereby proposes a methodology that is currently scarce in the literature. It also reports the first techno-economic analysis applied to the biodiesel field in South Africa and provides a preliminary insight to owners of biodiesel plants as regards the decision to convert or not their homogeneous catalysed plant to heterogeneous one.

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Over the past 15 years, ruthenium-catalyzed olefin metathesis has emerged as a cornerstone synthetic methodology in academia. Applications in fine-chemicals and pharmaceutical manufacturing, however, are just beginning to come on stream. Industrial uptake has been impeded by economic constraints associated with catalyst costs. These are due both to direct costs (exacerbated by intellectual property issues), and to further pressure exerted by the low turnover numbers attainable, and the need for extensive purification to remove ruthenium residues. From another perspective, however, these difficulties can be seen as arising from our rudimentary understanding of the fundamental organometallic chemistry of the Ru=CHR bond. In particular, we know little about the nature and reaction pathways of the Ru-methylidene unit present in the active species that propagates metathesis, and in the catalyst resting state. We know slightly more about the ruthenacyclobutane species, but still too little to guide us as to their non-metathetical reaction pathways, their contribution to deactivation relative to the methylidene species, and potential work-arounds. This thesis work was aimed at improving our understanding of the reactivity, speciation, and decomposition of key ruthenium intermediates in olefin metathesis. A major focus was the behaviour and deactivation of species formed from the second-generation Grubbs catalyst RuCl2(H2IMes)(PCy3)(=CHPh) (S-GII), which dominates ring-closing metathesis. Also studied were derivatives of the corresponding IMes catalyst A-GIIm, containing an unsaturated Nheterocyclic carbene (NHC) ligand. The methylidene complexes RuCl2(NHC)(PCy3)(=CH2) (GIIm) represent the resting state of the catalyst during ring-closing and cross-metathesis reactions: that is, the majority Ru species present during catalysis. Mechanistic studies of these key intermediates have been restricted, however, by the low yields and purity with which they could be accessed. Initial work therefore focused on designing a clean, high-yield route to the second-generation Grubbs methylidene complexes S-GIIm and A-GIIm. These routes were subsequently expanded to develop access to isotopically-labelled derivatives. Locating a 13C-label at the key alkylidene site, in particular, offers a powerful means of tracking the fate of the methylidene moiety during catalyst deactivation. Access to GIIm enabled detailed studies of the behaviour and decomposition of the Grubbs catalysts. First, the long-standing question of the impact of saturation of the NHC backbone (i.e. IMes vs. H2IMes) was examined. Dramatic differences in the behaviour of the two complexes were traced to profound differences in PCy3 lability arising from the diminished π-acidity of the IMes ligand. Secondly, the vulnerability of GIIm to nucleophiles was examined. This is an important issue from the perspective of decomposition by adventitious nucleophiles in the reaction medium during catalysis, but also reflects on substrate scope. For amine additives, the dominant deactivation pathway was shown to typically involve attack on the resting-state methylidene complex, not the metallacyclobutane, which has often been regarded as the most vulnerable intermediate. In addition, the sigma-alkyl intermediate formed by nucleophilic attack of displaced phosphine at the methylidene carbon was trapped by moving to the first-generation complex, and using a nitrogen donor (pyridine) that cannot promote decomposition via N–H activation pathways. Interception of this long-suspected species led to the proposal of “donorinduced” deactivation as a general decomposition pathway for Grubbs-class catalysts. Finally, the capacity of phosphine-free catalysts to overcome the shortcomings of the secondgeneration Grubbs catalysts was demonstrated, in a case study involving application of crossmetathesis (CM) to the synthesis of a high-value antioxidant. An efficient CM methodology was developed for the reaction of renewable essential-oil phenylpropenoids with vinyl acrylates. This work illustrates a new paradigm in sustainable metathesis. Rather than degrading unsaturated feedstocks via metathesis (a process that can be termed “metathe[LY]sis”), it demonstrates how metathesis with directly-functionalized olefins can be used to augment structure and function. From the perspective of organometallic chemistry and catalyst design, key comparisons built into this thesis are the effect of the NHC ligand (IMes vs. H2IMes) and its trans ancillary ligand on the efficient entry into catalysis; the susceptibility to nucleophilic attack of the alkylidene ligand (benzylidene vs. methylidene) vs. the metallacyclobutane; and the effect of replacing a phosphine ancillary ligand with a non-nucleophilic donor. From a practical standpoint, Chapter 2 brings new life to metathesis with the high-yield synthesis of the resting state species, Chapters 3 and 4 examine the deactivation, or death, of the methylidene complexes, and Chapter 5 establishes a new paradigm for olefin metathesis within the context of sustainable synthesis.

Let G be a split semisimple algebraic group over a field k. Our main objects of interest are twisted forms of projective homogeneous G-varieties. These varieties have been important objects of research in algebraic geometry since the 1960's. The theory of Chow motives and their decompositions is a powerful tool for studying twisted forms of projective homogeneous varieties. Motivic decompositions were discussed in the works of Rost, Karpenko, Merkurjev, Chernousov, Calmes, Petrov, Semenov, Zainoulline, Gille and other researchers. The main goal of the present thesis is to connect motivic decompositions of twisted homogeneous varieties to decompositions of certain modules over Hecke-type algebras that allow purely combinatorial description. We work in a slightly more general situation than Chow motives, namely we consider the category of h-motives for an oriented cohomology theory h. Examples of h include Chow groups, Grothendieck K_0, algebraic cobordism of Levine-Morel, Morava K-theory and many other examples. For a group G there is the notion of a versal torsor such that any G-torsor over an infinite field can be obtained as a specialization of a versal torsor. We restrict our attention to the case of twisted homogeneous spaces of the form E/P where P is a special parabolic subgroup of G. The main result of this thesis states that there is a one-to-one correspondence between h-motivic decompositions of the variety E/P and direct sum decompositions of modules DFP* over the graded formal affine Demazure algebra DF. This algebra was defined by Hoffnung, Malagon-Lopez, Savage and Zainoulline combinatorially in terms of the character lattice, the Weyl group and the formal group law of the cohomology theory h. In the classical case h=CH the graded formal affine Demazure algebra DF coincides with the nil Hecke ring, introduced by Kostant and Kumar in 1986. So the Chow motivic decompositions of versal homogeneous spaces correspond to decompositions of certain modules over the nil Hecke ring. As an application, we give a purely combinatorial proof of the indecomposability of the Chow motive of generic Severi-Brauer varieties and the versal twisted form of HSpin8/P1.

The two atom switching reactions referred to in the title were originally studied at temperatures greater than 2000°K in shock tubes by other investigators. For each reaction they proposed a direct four-center exchange mechanism in which one of the reactant molecules must be vibrationally excited, (the vibrational excitation mechanism or VEM). One of the predictions of the VEM is that molecules which are vibrationally hot but translationally cold should react through the four center transition state that leads to exchange. Using a mercury photosensitization technique, it is shown in the present work that excitation of CO to high vibrational levels is not sufficient to cause the CO self-exchange reaction. Similar attempts were also made to verify the VEM for the N2 reaction, but no exchange was observed. Kinetic modeling studies show that an atomic chain mechanism triggered by traces of oxygen impurity is responsible for all or much of the CO exchange observed in the shock tubes. Modeling studies show that many of the observed features of the N2 reaction are also correctly predicted by an atomic chain mechanism; however, the critical step in the mechanism, the N + N2 exchange reaction, has never been observed. Potential surface calculations show that at the restricted Hartree-Fock level of approximation the N3 potential surface has an energy barrier for exchange of over 80 kcal/mole, which is much too high if the atomic mechanism is to operate in the shock tubes. By comparison with similar calculations on N2o+, it is argued that the RHF calculations probably overestimate the true barrier height by about 80 kcal/mole, so the barrier to exchange on the N3 potential surface is probably no more than a few kcal/mole, and the N + N2 reaction is probably fast at high temperatures. Potential surface calculations on N4 show that the barrier to exchange through the four-center transition state is almost certainly much too high to account for the exchange observed in the shock tubes. Certain limitations on the rate law and energy barrier to exchange are derived for the VEM. It is concluded that both exchange reactions can be explained by atomic chain mechanisms and there is no need to invoke the VEM for either reaction.

A. Z. Petrov gave a complete list of all local group actions on a four-dimensional space-time that admit an invariant Lorentz metric, up to an equivalence relation. His list was compiled by directly constructing all possible Lie algebras of infinitesimal generators of group actions that preserve a Lorentz metric. The goal of this paper was to verify that classification by algebraically constructing a list of all possible three-dimensional homogeneous spaces and calculating which among them have a non-degenerate invariant metric.

<p> It is shown that the experimentally determined deflectional behavior of certain elastomer-lined sleeve bearings under static radial loads can be modeled to some extent by the compressive behavior of flat elastomeric slabs.</p> <p> An equation for the thermal bearing-bore change is developed using the conventional theory for the thermoelasticity of homogeneous cylinders. Some experimental results agree fairly well with this equation. An equation for the bore contraction due to liquid swell is developed in terms of thermoelasticity.</p> <p> Minimizing the lining thickness is recommended for minimizing both the radial deflections and the bore changes.</p> <p> An analysis is made of the frictional forces involved with the interference fit between the lining and its housing.</p> <p> Areas for further investigation are suggested.</p> / Thesis / Master of Engineering (MEngr)

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