This thesis is concerned with the 2-dimensional Toda equations and their geometric interpretation in form of r-adapted maps into flag manifolds, r-adapted maps are not only of interest due to their relation with the Toda equations, but also for their adaption to the m-synametric space structure of flag manifolds. This thesis studies the congruence question for r-adapted maps in flag manifolds. The main theorem of this thesis is a congruence theorem for г-holomorphic maps Ψ : S(^2) → G/T of constant curvature, where G can be any compact simple Lie group. It is supplemented by a congruence theorem for general r-holomorphic maps Ψ : S(^2) → G/T if G has rank 2, and a number of congruence theorems for isometric r-primitive Ψ : S(^2) → G/T of constant Kahler angle. The second group of congruence theorems is proved for the rank 2 case, as well as a selection of Lie groups with higher rank: SU(4),SU(5),F(_4),E(_6),E(_6),E(_8),Sp(n).

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. March 2016. / We investigate the two-dimensional turbulent wake and derive the governing equations for the mean velocity components using both the eddy viscosity and the Prandtl mixing length closure models to complete the system of equations. Prandtl’s mixing length model is a special case of the eddy viscosity closure model. We consider an eddy viscosity as a function of the distance along the wake, the perpendicular distance from the axis of the wake and the mean velocity gradient perpendicular to the axis of thewake. We calculate the conservation laws for the system of equations using both closure models. Three main types of wakes arise from this study: the classical wake, the wake of a self-propelled body and a new wake is discovered which we call the combination wake. For the classical wake, we first consider the case where the eddy viscosity depends solely on the distance along the wake. We then relax this condition to include the dependence of the eddy viscosity on the perpendicular distance from the axis of the wake. The Lie point symmetry associated with the elementary conserved vector is used to generate the invariant solution. The profiles of the mean velocity show that the role of the eddy viscosity is to increase the effective width of the wake and decrease the magnitude of the maximum mean velocity deficit. An infinite wake boundary is predicted fromthis model. We then consider the application of Prandtl’s mixing length closure model to the classical wake. Previous applications of Prandtl’s mixing length model to turbulent wake flows, which neglected the kinematic viscosity of the fluid, have underestimated the width of the boundary layer. In this model, a finite wake boundary is predicted. We propose a revised Prandtl mixing length model by including the kinematic viscosity of the fluid. We show that this model predicts a boundary that lies outside the one predicted by Prandtl. We also prove that the results for the two models converge for very large Reynolds number wake flows. We also investigate the turbulentwake of a self-propelled body. The eddy viscosity closure model is used to complete the system of equations. The Lie point symmetry associated with the conserved vector is derived in order to generate the invariant solution. We consider the cases where the eddy viscosity depends only on the distance along the wake in the formof a power law and when a modified version of Prandtl’s hypothesis is satisfied. We examine the effect of neglecting the kinematic viscosity. We then discuss the issues that arisewhenwe consider the eddy viscosity to also depend on the perpendicular distance from the axis of the wake. Mean velocity profiles reveal that the eddy viscosity increases the boundary layer thickness of the wake and decreases the magnitude of the maximum mean velocity. An infinite wake boundary is predicted for this model. Lastly, we revisit the discovery of the combination wake. We show that for an eddy viscosity depending on only the distance along the axis of the wake, a mathematical relationship exists between the classical wake, the wake of a self-propelled body and the combination wake. We explain how the solutions for the combination wake and the wake of a self-propelled body can be generated directly from the solution to the classical wake. / GR 2016

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 79-81. / by Ranee Kathryn Gupta. / Ph.D.

It is well-known that del Pezzo surfaces of degree 9 -- n. are in one-to-one correspondence to flat En bundles over elliptic curves which are anti-canonical curves of such surfaces. In my thesis, we study a broader class of rational surfaces which are called ADE surfaces. We construct Lie algebra bundles of any type on these surfaces, and extend the above correspondence to flat G bundles over elliptic curves, where G is a simple, compact and simply-connected Lie group of any type. Concretely, we establish a natural identification between the following two very different moduli spaces for a Lie group G of any type: the moduli space of rational surfaces with G-configurations and the moduli space of flat G-bundles over a fixed elliptic curve. / Zhang, Jiajin. / "July 2007." / Adviser: Leung Nai Chung Conan. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0357. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 77-79). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Thesis. 1977. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Vita. / Bibliography : leaves 176-178. / Ph.D.

Dissertação apresentada para obtenção do grau de Doutor, na Faculdade de Ciências da Universidade Clássica de Lisboa

<p>We consider one transaction costs model which was suggested by Cetin, Jarrow and Protter (2004) for an illiquid market. In this case the hedging strategy of programming traders can affect the assets prises. We study the corresponding partial differential equation (PDE) which is a non-linear Black-Scholes equation for illiquid markets. We use the Lie group analysis to determine the symmetry group of this equations. We present the Lie algebra of the Lie point transformations, the complete symmetry group and invariants. For different subgroups of the main symmetry group we describe the corresponding invariants. We use these invariants to reduce the PDE under investigation to ordinary differential equations (ODE). Solutions of these ODE's are subgroup-invariant solutions of the non-linear Black-Scholes equation. For some classes of those ODE's we find exact solutions and studied their properties.</p><p>% reduce non-linear PDE to ODE's. To some ODE's we find exact solutions.</p><p>%In this work we are studying one model for pricing derivatives in illiquid market. We discuss it structure and properties. Make a symmetry reduction for the PDE corresponding our model.</p>

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We consider one transaction costs model which was suggested by Cetin, Jarrow and Protter (2004) for an illiquid market. In this case the hedging strategy of programming traders can affect the assets prises. We study the corresponding partial differential equation (PDE) which is a non-linear Black-Scholes equation for illiquid markets. We use the Lie group analysis to determine the symmetry group of this equations. We present the Lie algebra of the Lie point transformations, the complete symmetry group and invariants. For different subgroups of the main symmetry group we describe the corresponding invariants. We use these invariants to reduce the PDE under investigation to ordinary differential equations (ODE). Solutions of these ODE's are subgroup-invariant solutions of the non-linear Black-Scholes equation. For some classes of those ODE's we find exact solutions and studied their properties. % reduce non-linear PDE to ODE's. To some ODE's we find exact solutions. %In this work we are studying one model for pricing derivatives in illiquid market. We discuss it structure and properties. Make a symmetry reduction for the PDE corresponding our model.

The action of various operations on the cohomology of nilpotent Lie algebras is studied. In the cohomology of any Lie algebra, we show that the existence of certain nontrivial compositions of higher cohomology operations implies the existence of hypercube-like structures in cohomology, which in turn establishes the Toral Rank Conjecture for that Lie algebra. We provide examples in low dimensions and exhibit an infinite family of nilpotent Lie algebras of arbitrary dimension for which such structures exist. A new proof of the Toral Rank Conjecture is also given for free two-step nilpotent Lie algebras.