Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let g denote the complexification of the Lie algebra of U, g=u^C. Each u-compatible triangular decomposition g= n_- + h + n_+ determines a Poisson Lie group structure pi_U on U. The Evens-Lu construction produces a (U, pi_U)-homogeneous Poisson structure on X. By choosing the basepoint in X appropriately, X is presented as U/K where K is the fixed point set of an involution which stabilizes the triangular decomposition of g. With this presentation, a connection is established between the symplectic foliation of the Evens-Lu Poisson structure and the Birkhoff decomposition of U/K. This is done through reinterpretation of results of Pickrell. Each symplectic leaf admits a natural torus action. It is shown that these actions are Hamiltonian and the momentum maps are computed using triangular factorization. Finally, local formulas for the Evens-Lu Poisson structure are displayed in several examples.

In this thesis we seek exact solutions to the isotropic Einstien-Maxwell system that model the interior of relativistic stars. The field equations are transformed to a simpler form using the transformation of Durgapal and Bannerji (1983); the integration of the system is reduced to solving the condition of pressure isotropy. This condition is a recurrence relation with variable rational coe±cients which can be solved in general. New classes of solutions of linearly independent functions are obtained in terms of special functions and elementary functions for different spatial geometries. Our results contain models found previously including the superdense Tikekar (1990) neutron star model, the uncharged isotropic Maharaj and Leach (1996) solutions, the Finch and Skea (1989) model and the Durgapal and Bannerji (1983) superdense neutron star. Our general class of solutions also contain charged relativistic spheres found previously, including the model of Hansraj and Maharaj (2006) and the model of Thirukkanesh and Maharaj (2006). In addition, two exact analytical solutions describing the interior of a charged strange quark star are obtained by applying the MIT bag equation of state. We regain the Mak and Harko (2004) solution for a charged quark star as a special case. / Thesis (Ph. D.) University of KwaZulu-Natal, Westville, 2007.

A number of symmetric porphyrins containing both zinc(II) and lead(II) central metal ions have been prepared. Several routes to these compounds have been investigated and their merits evaluated. The conjugation of the chromophores has been extended by the inclusion of different <I>trans</I>- substituted arylethynyl substituents in the <I>meso</I>-position. An improved multigram method for the preparation of [bis-trimethylsilyethynylporphinato]zinc(II), which does not necessitate chromatographic purification, is described. The use of [5-triisopropylsilylethynylporphinato]zinc(II) for the preparation of donor/acceptor prophyrins via Sonogashira coupling, several of which were prepared, is also described. Investigations of the binding of 4(3<I>H</I>)pyrimidone with [5,5'-bis-[10,20-bis(3,5-di-<I>tert</I>-butylphenyl)-15-tri-iso-propylsilylethynyl-porphinyl]-1,3-butyn-1,4-diyl]dizinc(II) were made. The association constant was found to be 2500 ± 140 mol^-1 (ΔG = -4.6 Kcal/mol), significantly greater than that for 4(3<I>H</I>)pyrimidone alone (444 ± 5 mol^-1), as determined by NMR dilution experiments. The crystal structures of [5,15-bis(3,5-bis-<I>tert</I>-butylphenyl)porphinato]zinc(II), [5,15-bis-(3,5-di-<I>tert</I>-butylphenyl)-10,20-bis-triisopropylsilylethynylporphinato]lead(II), [5,10,15,20-<I>tetrakis</I>-triisopropylsilylethynylporphinato]lead(II), 5,15-bis-(3,5-bis-<I>tert</I>-butylphenyl)-10,20-bis-phenylethynylporphyrin, [5,15-bis-(3,5-bis-<I>tert</I>-butylphenyl)-10,20-bis-(3-hydroxy-3-methyl-butynyl)porphinato]zinc(II),5,10,15,20-<I>tetrakis</I>-triisopropylsilylethynylporphyrin and [5,15-bis-(3,5-bis-<I>tert</I>-butylphenyl)-10-(4-N, N-dimethylaminophenylethynyl)-20-triisopropylsilylethynylporphinato]zinc(II) are also described.

The ring of symmetric functions Λ, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the symmetric group. One may define a coproduct on Λ by the plethystic addition on alphabets. In this way the ring of symmetric functions becomes a Hopf algebra. The Littlewood–Richardson numbers may be viewed as the structure constants for the co-product in the Schur basis. The first part of this thesis, inspired by the umbral calculus of Gian-Carlo Rota, is a study of the co-algebra maps of Λ. The Macdonald polynomials are a somewhat mysterious qt-deformation of the Schur functions. The second part of this thesis contains a proof a generating function identity for the Macdonald polynomials which was originally conjectured by Kawanaka.

Thesis (Ph. D.)--University of Washington, 1994. / Vita. Includes bibliographical references (leaves [139]-141).

Thesis (Ph.D.)--University of Hong Kong, 2000. / Includes bibliographical references (leaves 50-53) Also available in print.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 41-42).

Thesis (Ph. D.)--University of Wisconsin, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

First published in the Bulletin of the American Mathematical Society in Vol.71, 1965, published by the American Mathematical Society

First published in the Bulletin of the American Mathematical Society in Vol.74, 1968, published by the American Mathematical Society