The aim of this work is to study the Hamiltonian quaternions H and quaternion alge- bras. The first two chapters are based on the article Quaternion algebras by K. Conrad and the rest on the book Quaternion algebras by J. Voight. In the beginning, we mainly develop the theory about quaternions, quaternion algebras and study equivalent condi- tions for being a split or non-split quaternion algebra. After that, we also characterize, up to isomorphism, quaternion algebras over several fields such as R, C or Fp. In the third chapter, the thesis deals with orders in quaternion algebras, especially Lipschitz and Hurwitz order. The fourth chapter is dedicated to the relationship between unit quaternions and rotations in R3 , thanks to which we can characterize finite subgroups of H1 , or equivalently H× . This result will be used in the last chapter, where we are mainly focused on the problem of characterization of the group of units in orders in Hamiltonian quaternions. 1

Nous avons investigué les propriétés des champs de jauge de spin élevé libres à travers une étude de divers aspects de leur dynamique hamiltonienne. Pour des champs se propageant sur un espace-temps plat, les contraintes issues de l'analyse hamiltonienne de ces théories de jauge ont été identifiées et résolues par l'introduction de prépotentiels, dont l'invariance de jauge comprend, de façon intrigante, à la fois des difféomorphismes linéarisés généralisés et des transformations d'échelle de Weyl généralisées et linéarisées. Cela a motivé notre étude systématique des invariants conformes pour les spins élevés. Les invariants correspondants ont été construits à l'aide du tenseur de Cotton, dont nous avons établi les propriétés essentielles (symétrie, conservation, trace nulle; invariance, complétude). Avec ces outils géométriques, l'analyse hamiltonienne a pu être complétée et une action du premier ordre écrite en termes des prépotentiels. Nous avons constaté que cette action possédait une invariance manifeste par dualité électromagnétique; cette invariance, combinée à l'invariance de jauge des prépotentiels, fixe d'ailleurs uniquement l'action. En outre, de façon générale, cette action s'est révélée être exactement celle obtenue à travers une réécriture des équations du mouvement des spins élevés comme des conditions d'auto-dualité tordue (non manifestement covariantes).Avec un intérêt pour les extensions supersymétriques, nous avons amorcé la généralisation de cette étude aux champs fermioniques. Le champ de masse nulle libre de spin 5/2 a été soumis à la même analyse, et son prépotentiel s'est révélé partager l'invariance de jauge conforme déjà observée dans le cas bosonique général. Le supermultiplet incorporant les spins 2 et 5/2 a ensuite été considéré, et une symétrie rigide de son action, combinant une transformation de dualité électromagnétique du spin 2 avec une transformation de chiralité du spin 5/2 a été construite pour commuter avec la supersymétrie. Dans une autre direction, nous avons étudié les propriétés d'un champ tensoriel chiral de symétrie mixte dans un espace-temps plat à six dimensions: une (2,2)-forme. Son analyse hamiltonienne a été réalisée, des prépotentiels introduits et l'action de premier ordre obtenue s'est encore une fois révélée être la même que celle obtenue à travers une réécriture des équations du mouvement comme des conditions d'auto-chiralité (non manifestement covariante).Finalement, nous nous sommes penchés sur les charges de surface des champs fermioniques et bosoniques de spin élevé se propageant sur un espace-temps à courbure constante. Cela a été réalisé par une analyse hamiltonienne de ces systèmes, les contraintes étant identifiées aux générateurs des transformations de jauge. Injectant dans ces générateurs des valeurs des paramètres des transformations de jauge correspondant à des transformations impropres de jauge (imposant une réelle variation physique sur les champs) a ensuite permis d'évaluer la valeur de ces générateurs pour des champs résolvant les équations du mouvement: elle s'est bien révélée finie et non-nulle, constituant les charges de surface de ces théories. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

(Hamiltonian chaos and its application to anomalous dynamics in turbulent environment) RMP-induced ELM control has been tested on several tokamaks. It is believed that increase of electron transport across the magnetic field plays an important role. Edge plasma turbulence also affects dynamics in the edge region of tokamak. We study the simultaneous effect of plasma turbulence and RMP-induced stochastic magnetic field within the single-particle framework. We find out that the plasma turbulence is an important element of dynamics that should be taken into account in further (especially single-particle) studies.

This thesis focuses on developing new approaches to solving the ground state properties of quantum many-body Hamiltonians, and the goal is to develop a systematic approach which properly balances efficiency and accuracy. Two new formalisms are proposed in this thesis: the Variational Discrete Action Theory (VDAT) and the Off-Shell Effective Energy Theory (OET). The VDAT exploits the advantages of both variational wavefunctions and many-body Green's functions for solving quantum Hamiltonians. VDAT consists of two central components: the Sequential Product Density matrix (SPD) and the Discrete Action associated with the SPD. The SPD is a variational ansatz inspired by the Trotter decomposition and characterized by an integer N, and N controls the balance of accuracy and cost; monotonically converging to the exact solution for N → ∞. The Discrete Action emerges by treating the each projector in the SPD as an effective discrete time evolution. We generalize the path integral to our discrete formalism, which converts a dynamic correlation function to a static correlation function in a compound space. We also generalize the usual many-body Green's function formalism, which results in analogous but distinct mathematical structures due to the non-abelian nature of the SPD, yielding discrete versions of the generating functional, Dyson equation, and Bethe-Salpeter equation. We apply VDAT to two canonical models of interacting electrons: the Anderson impurity model (AIM) and the Hubbard model. We prove that the SPD can be exactly evaluated in the AIM, and demonstrate that N=3 provides a robust description of the exact results with a relatively negligible cost. For the Hubbard model, we introduce the local self-consistent approximation (LSA), which is the analogue of the dynamical mean-field theory, and prove that LSA exactly evaluates VDAT for d=∞. Furthermore, VDAT within the LSA at N=2 exactly recovers the Gutzwiller approximation (GA), and therefore N>2 provides a new class of theories which balance efficiency and accuracy. For the d=∞ Hubbard model, we evaluate N=2-4 and show that N=3 provides a truly minimal yet precise description of Mott physics with a cost similar to the GA. VDAT provides a flexible scheme for studying quantum Hamiltonians, competing both with state-of-the-art methods and simple, efficient approaches all within a single framework. VDAT will have broad applications in condensed matter and materials physics. In the second part of the thesis, we propose a different formalism, off-shell effective energy theory (OET), which combines the variational principle and effective energy theory, providing a ground state description of a quantum many-body Hamiltonian. The OET is based on a partitioning of the Hamiltonian and a corresponding density matrix ansatz constructed from an off-shell extension of the equilibrium density matrix; and there are dual realizations based on a given partitioning. To approximate OET, we introduce the central point expansion (CPE), which is an expansion of the density matrix ansatz, and we renormalize the CPE using a standard expansion of the ground state energy. We showcase the OET for the one band Hubbard model in d=1, 2, and ∞, using a partitioning between kinetic and potential energy, yielding two realizations denoted as K and X. OET shows favorable agreement with exact or state-of-the-art results over all parameter space, and has a negligible computational cost. Physically, K describes the Fermi liquid, while X gives an analogous description of both the Luttinger liquid and the Mott insulator. Our approach should find broad applicability in lattice model Hamiltonians, in addition to real materials systems. The VDAT can immediately be applied to generic quantum models, and in some cases will rival the best existing theories, allowing the discovery of new physics in strongly correlated electron scenarios. Alternatively, the OET provides a practical formalism for encapsulating the complex physics of some model and allowing extrapolation over all phase space. Both of the formalisms should find broad applications in both model Hamiltonians and real materials.

Indiana University-Purdue University Indianapolis (IUPUI) / The essence of Bayesian data analysis is to ascertain posterior distributions. Posteriors generally do not have closed-form expressions for direct computation in practical applications. Analysts, therefore, resort to Markov Chain Monte Carlo (MCMC) methods for the generation of sample observations that approximate the desired posterior distribution. Standard MCMC methods simulate sample values from the desired posterior distribution via random proposals. As a result, the mechanism used to generate the proposals inevitably determines the efficiency of the algorithm. One of the modern MCMC techniques designed to explore the high-dimensional space more efficiently is Hamiltonian Monte Carlo (HMC), based on the Hamiltonian differential equations. Inspired by classical mechanics, these equations incorporate a latent variable to generate MCMC proposals that are likely to be accepted. This dissertation discusses how such a powerful computational approach can be used for implementing statistical models. Along this line, I created a unified computational procedure for using HMC to fit various types of statistical models. The procedure that I proposed can be applied to a broad class of models, including linear models, generalized linear models, mixed-effects models, and various types of semiparametric regression models. To facilitate the fitting of a diverse set of models, I incorporated new parameterization and decomposition schemes to ensure the numerical performance of Bayesian model fitting without sacrificing the procedure’s general applicability. As a concrete application, I demonstrate how to use the proposed procedure to fit a multivariate generalized additive model (GAM), a nonstandard statistical model with a complex covariance structure and numerous parameters. Byproducts of the research include two software packages that all practical data analysts to use the proposed computational method to fit their own models. The research’s main methodological contribution is the unified computational approach that it presents for Bayesian model fitting that can be used for standard and nonstandard statistical models. Availability of such a procedure has greatly enhanced statistical modelers’ toolbox for implementing new and nonstandard statistical models.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第17924号 / 情博第506号 / 新制||情||89(附属図書館) / 30744 / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 梅野 健, 教授 中村 佳正, 教授 船越 満明 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第20632号 / 工博第4370号 / 新制||工||1679(附属図書館) / 京都大学大学院工学研究科分子工学専攻 / (主査)教授 佐藤 啓文, 教授 白川 昌宏, 教授 山本 量一 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DGAM

京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第21350号 / 工博第4509号 / 新制||工||1702(附属図書館) / 京都大学大学院工学研究科社会基盤工学専攻 / (主査)教授 後藤 仁志, 教授 KIM Chul-Woo, 准教授 KHAYYER,Abbas / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DGAM

Using the idea that a quantum mechanical system drops to its ground state as its temperature goes to absolute zero several operators are devised to enable the approximation of the lowest order energy eigenstate of a given symmetry; as well as an approximation to the energy eigenvalue of the same order.

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