Thesis (Ph. D. in Physics)--Vanderbilt University, Aug. 2007. / Title from title screen. Includes bibliographical references.

Made available in DSpace on 2016-05-17T16:50:53Z (GMT). No. of bitstreams: 0 Previous issue date: 2007-03-16. Added 1 bitstream(s) on 2016-05-17T16:54:31Z : No. of bitstreams: 1 000855888.pdf: 673100 bytes, checksum: f6c50190e13c6931b1a9076b72d32787 (MD5) / Apresentamos nesta dissertação uma revisão dos conceitos de geometria diferencial, onde estamos interessados em definir campos vetoriais que geram transformações de um parâmetro, formas diferenciais, variedades simpléticas e fibrados. Além disso, detalhamos o conceito de cohomologia de De Rham, o qual nos fornece uma ferramenta algébrica fundamental para analisar propriedades topológicas das variedades. A combinação desses conceitos, os quais suportam o nosso trabalho, permite-nos desenvolver teorias de localização equivariante de integrais definidas sobre espaços de fase clássicos, os quais também podem ser uma órbita co-adjunta. A localização é possível devido ao teorema de Duistermaat-Heckman, o qual nos permite escrever integrais como uma soma, ou integral, sobre o conjunto dos pontos críticos do espaço. Em seguida fazemos uma extensão para teorias de localização de integrais funcionais, onde é preciso definir o espaço dos loops. Nesse contexto aplicamos a formulação de localização equivariante tendo como base a conjectura de Atiyah-Witten para teorias supersimétricas, onde derivamos o teorema de índice de Atiyah-Singer para um operador de Dirac. O teorema de índice é aplicado no cálculo da anomalia quiral / We present in this dissertation a conceptual review of differential geometry, where we are interested in defining vector fields which are one-parameter transformation generators, differential forms, symplectic manifolds, and fiber bundles. In addition, we detail the concept about De Rham's cohomology, which provides us a fundamental algebraic tool to analyze topological properties of manifolds. The combination of these concepts, which are the background material of our work, allows us to develop equivariant localization theories of integrals defined on classical phase spaces, which can also be a co-adjoint orbit. The localization is possible because of the Duistermaat-Heckman theorem, which allows us to write integrals on the whole space just as a sum, or integral, on a critical points set. Further more, we do an extension to functional integrals localization theories, where it is needed to define loop spaces. In this context we apply equivariant localization formulation having the bases of Atiyah-Witten conjecture to supersymmetric theories, where we derive the Atiyah-Singer index theorem for a Dirac operator. The index theorem is applied to chiral anomaly calculation

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(Page numbers are given in brackets after sub-headings.) Chapter 1 :-Application of operational methods of solution to ordinary differential equations with constant coefficients. Motion of helical springs --- Free vibration ( p.2 ) --- Forced vibration ( p.7 ) --- Resonance ( p.10 ) ----- Linked springs ( p.12 ) --- Analogous electrical problems ( P. 15 ). Appendix A :-Laplace Transformations. ( p.18 ) Appendix B :- Brief explanation of the operational methods of solution used in the chapter. ( p.19 ) Appendix C :- Example of partial fraction working. ( p.21a) Chapter 2 :- Application of Fourier Series and of operational methods of solution to partial differential equations. Small transverse vibrations of a light elastic string ( p.22 ) Vibration of a string initially plucked at its middle point ( p.28 ) --- Alternative method of solution of the preceeding problem by operational methods ( p.31 ) --- Discussion of the shape assumed by the string at any time ( p.36 ) Chapter 3 :- Application of Bessel Functions to the solution of partial differential equations of the second order. Small vibrations of a light circular membrane ( p.39 ) Appendix A :- Transformation of the differential equation of motion from Cartesian to cylindrical coordinates. ( p.46 ). Appendix B :- Note on roots of Bessel Functions. ( p.47 ) Appendix C :- Bessel Functions - Some important results applied in the chapter. p.48 ). Chapter 4 :- Application of Legendre Polynomials to the solution of partial differential equations of the second order. Conduction of heat in a solid body ( p.50 ) ---------- Flow of heat in a solid sphere ( p.53 ). Appendix :-Statement of the Divergence Theorem used in the chapter. ( p.61 ). Chapter 5 :- Example of Simultaneous Partial Differential Equations. The flow of electricity in a long imperfectly insulated cable ( p.62 ) ---- Analogous problem of the flow of heat in rod of small uniform cross section ( p.69 ). / Thesis / Master of Arts (MA)

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Dans cette thèse, nous nous intéressons aux solutions de trou noir dans des théories de gravitation modifiées, inspirées par la Cosmologie ou la Théorie des Cordes. Les théories Einstein-Maxwell-Dilaton (EMD) comprennent des champs scalaires et de jauge additionnels, ainsi qu'un potentiel scalaire de Liouville en exponentielle. Dans les théories Einstein-Gauss-Bonnet, l'espace-temps est étendu à plus de quatre dimensions. Nous commençons par passer en revue et classer les solutions des théories EMD. Dans le cas où l'horizon du trou noir est planaire, l'obtention d'une équation maîtresse constitue l'un des principaux résultats. Elle alors permet d'intégrer totalement le système d'équations, au prix d'une contrainte sur les constantes de couplage, et sinon d'obtenir des solutions particulières. Dans le cas des théories Einstein-Gauss-Bonnet à six dimensions, le tenseur de Weyl de l'horizon intervient dans les équations par l'action de la constante de couplage du terme de Gauss-Bonnet : cela a pour effet de restreindre grandement la topologie de l'horizon, ce qui lève en partie la dégénérescence qui existait en Relativité Générale. Par la suite, nous étudions les propriétés thermodynamiques des trous noirs en Relativité Générale et en théories EMD. Pour ces dernières, nous montrons que des transitions de phases similaires à celles en Relativité Générale existent dans l'ensemble canonique. Plus généralement, ces propriétés dépendent de manière cruciale des constantes de couplage de la théorie. Pour finir, nous donnons une interprétation holographique aux trous noirs des théories EMD, en tant que bonnes approximations à l'Infra-Rouge. Nous calculons également les conductivités en courant continu et alternatif dans le cadre d'une application à la Matière Condensée, et trouvons certaines caractéristiques typiques des métaux étranges.

Les progrès des techniques de nanofabrication des dix dernières années ont permis la mise en place de protocoles visant à manipuler les charges uniques dans les nanostructures. Ces nouvelles techniques permettent d'envisager la réalisation d'expériences d'optique quantique avec des électrons. Cette thèse s'inscrit dans ce contexte. Le but de ce travail a été la construction d'un formalisme adapté à la description de telles expériences. Ce formalisme, construit en analogie avec la théorie de la cohérence quantique du champ électromagnétique de Glauber, souligne les similitudes et différences entre les photons se propageant dans le vide, et le transport électronique dans des conducteurs balistiques unidimensionnels. En particulier, il rend compte de la décohérence et de la relaxation en énergie des excitations électroniques en présence d'interactions. Un autre aspect de cette thèse a été la proposition de protocoles permettant de mesurer des quantités directement reliées aux propriétés de cohérence décrites par le formalisme de l'optique quantique électronique. En particulier, un protocole de tomographie quantique reposant sur l'effet Hanbury Brown et Twiss a été proposé pour reconstruire la cohérence à un corps d'une source monoélectronique. Ce protocole peut aussi être envisagé pour obtenir des informations sur les mécanismes de décohérence.

L'objet de cette thèse est l'étude de problèmes faisant intervenir les extrema du mouvement brownien en dimension 1 et 2. En dimension 1, y sont obtenues, en particulier, les distributions jointes du maximum et du temps d'atteinte de ce maximum pour n mouvements browniens indépendants sur un intervalle de temps fixé. En dimension 2, à l'aide des résultats en dimension 1, sont obtenues les valeurs moyennes du périmètre et de l'aire de l'enveloppe convexe de n chemins browniens indépendants, ouverts ou fermés. Quelques applications de ces résultats théoriques sont également présentées.

Accurate electronic structure calculations are becoming more and more important because of the increasing need for information about systems which are hard to perform experiments on. Databases compiled from theoretical results are also being used more than ever for applications, and the reliability of the theoretical methods are of utmost importance. In this thesis, the present limits on theoretical alloy calculations are investigated and improvements on the methods are presented. A short introduction to electronic structure theory is included as well as a chapter on Density Functional Theory, which is the underlying method behind all calculations presented in the accompanying papers. Multiple Scattering Theory is also discussed, both in more general terms as well as how it is used in the methods employed to solve the electronic structure problem. One of the methods, the Exact Muffin-Tin Orbital method, is described extensively, with special emphasis on the slope matrix, which energy dependence is investigated together with possible ways to parameterize this dependence. Furthermore, a chapter which discusses different ways to perform calculations for disordered systems is presented, including a description of the Coherent Potential Approximation and the Screened Generalized Perturbation Method. A comparison between the Exact Muffin-Tin Orbital method and the Projector Augmented-Wave method in the case of systems exhibiting both compositional and magnetic disordered is included as well as a case study of the MoRu alloy, where the theoretical and experimental discrepancies are discussed. The thesis is concluded with a short discussion on magnetism, with emphasis on its computational aspects. I further discuss a generalized Heisenberg model and its applications, especially to fcc Fe, and also present an investigation of the competing magnetic structures of FeNi alloys at different concentrations, where both collinear and non-collinear magnetic structures are included. For Invar-concentrations, a spin-flip transition is found and discussed. Lastly, I discuss so-called quantum corrals and possible ways of calculating properties, especially non-collinear magnetism, of such systems within perturbation theory using the force theorem and the Lloyd’s formula.

This thesis concerns the ternary Al-Cr-O system. (Al1-xCrx)2O3 solid solution thin films with 0.6<x<0.7 were deposited on Si(001) substrates at temperatures of 400-500 °C by reactive radio frequency magnetron sputtering from metallic targets of Al and Cr in a flow controlled Ar / O2 gas mixture. As-deposited and annealed (Al1-xCrx)2O3 thin films were analyzed by x-ray diffraction, elastic recoil detection analysis, scanning electron microscopy, transmission electron microscopy, and nanoindentation. (Al1-xCrx)2O3 showed to have face centered cubic structure with lattice parameter of 4.04 Å, which is in contrast to the typical corundum structure reported for these films. The as-deposited films exhibited hardness of ~ 26 GPa and elastic modulus of 220-235 GPa. Phase transformation from cubic to corundum (Al0.32Cr0.68)2O3 starts at 925 °C. Annealing at 1000 °C resulted in complete phase transformation, while no precipitates of alumina and chromia were observed. Studies on kinetics of phase transformation showed a two-step thermally activated process; phase transformation and grain growth with the apparent activation energies 213±162 and 945±27 kJ/mol, respectively.