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Vector interpolation functions which approximate electromagnetic vector fields are constructed in this thesis. These vector functions are to be used when the solution of Maxwell's equations involves an irrotational or solenoidal vector field. In addition the functions are chosen so that they can easily be used in the implementation of a finite element method. / Four bases are constructed. The first two span the spaces of solenoidal or irrotational two component vector polynomials of order one in two variables whereas the other two span the spaces of solenoidal or irrotational three component vector polynomials of order one in three variables. The vector polynomials are then used within the finite element method to approximate the two component current density J and electric field E over a conducting plate and the three component current density in a three dimensional wire.

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Thesis (M.Comp.Sc.-Res.)--University of Wollongong, 2007. / Typescript. Includes bibliographical references: leaf 57-62.

Thesis (M.S.)--Oregon State University, 2007. / Printout. Includes bibliographical references (leaves 76-78). Also available on the World Wide Web.

In 1980 Feigin and Fuchs classified the length 2 bounded representations of Vec(R), the Lie algebra of polynomial vector fields on the line, as a result of their work on the cohomology of Vec(R). This dissertation is concerned mainly with the uniserial (completely indecomposable) representations of Vec(R) with a single Casimir eigenvalue and weights bounded below. Such representations are composed of irreducible representations with semisimple Euler operator action, bounded weight space dimensions, and weights bounded below. These are known to be the tensor density modules with lowest weight λ, for any non-zero complex number λ, and the trivial module C, with Vec(R) actions π_λ and π_C, respectively. Our proofs are cohomology arguments involving the first cohomology groups of Vec(R) with values in the space of homomorphisms between two irreducible representations. These results classify the finite length uniserial extensions, with a single Casimir eigenvalue, of admissible irreducible Vec(R) representations with weights bounded below. In almost every case there is at most one uniserial representation with a given composition series. However, in the case of an odd length extension with composition series {π_1,π_C,π_1,…,π_C,π_1}, there is a one-parameter family of extensions. We also give preliminary results on uniserial representations of the Virasoro Lie algebra.

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On the tangent bundle of a Riemannian manifold the most natural choice of metric tensor field is the Sasaki metric. This immediately brings up the question of infinitesimal symmetries associated with the inherent geometry of the tangent bundle arising from the Sasaki metric. The elucidation of the form and the classification of the Killing vector fields have already been effected by the Japanese school of Riemannian geometry in the sixties. In this thesis we shall take up the conformal vector fields of the Sasaki metric with the help of relatively advanced techniques.

Made available in DSpace on 2016-06-02T20:28:23Z (GMT). No. of bitstreams: 1 DissRCC.pdf: 783903 bytes, checksum: 2bdd4b30ef7629658fe82c00625b3209 (MD5) Previous issue date: 2007-09-25 / Financiadora de Estudos e Projetos / In this work, we describe the Poincaré Compactification of polinomial vector fields and present two characterization of it. / Nese trabalho, descrevemos a Compactificação de Poincaré de campos vetorias polinomiais e apresentamos duas caracterizações dela.