Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2001. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Oregon, 2003. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 69-71). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2004. / Includes bibliographical references (leaves 51-52). Also available in electronic version. Access restricted to campus users.

Thesis (M.S.)--Tennessee Technological University, 2009. / Title from title page screen (viewed on Aug. 26, 2009). Bibliography: leaves 50-51.

Approaches to nding solutions to di erential equations are usually ad hoc. One of the more successful methods is that of group theory, due to Sophus Lie. In the case of ordinary di erential equations, the subsequent symmetries obtained allow one to reduce the order of the equation. In the case of partial di erential equations, the symmetries are used to nd (particular) group invariant solutions by reducing the number of variables in the original equation. In the latter case, these solutions are particularly popular in applications as they are often the only physically signi cant ones obtainable. As a result, it is now becoming traditional to apply this symmetry method to nd solutions to di erential equations in a systematic manner. Based upon the Lie algebra of symmetries of the equation, we expect a certain number of symmetries after the reductions. However, it has become increasingly observed that, after reduction, more symmetries than expected are often obtained. These are called Hidden Symmetries and they provide new routes for further reduction. The idea of our research is to give an overview of this phenomenon. In particular, we investigate the possible origins of these symmetries. We show that they manifest themselves as nonlocal symmetries (or potential symmetries), contact symmetries or nonlocal contact symmetries of the original equation as well as point symmetries of another equation of same order. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2012.

No description available.

In this thesis we study left-invariant control offine systems on the symmetry group of a. model of hyperbolic plane geometry, the matrix Lie group SO(1, 2)₀. We determine that there are 10 distinct classes of such control systems and for typical elements of two of these classes we provide solutions of the left-invariant optimal wntrol problem with quauratic costs. Under the identification of the Lie allgebra .so(l, 2) with Minkowski spacetime R¹̕'², we construct a controllabilility criterion for all left-invariant control affine systems on 50(1. 2)₀ which in the inhomogeneous case depends only on the presence or absence of an element in the image of the system's trace in R¹̕ ²which is identifiable using the inner product. For the solutions of both the optimal control problems, we provide explicit expressions in terms of Jacobi elliptic functions for the solutions of the reduced extremal equations and determine the nonlinear stability of the equilibrium points.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Materials Science and Engineering, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Includes bibliographical references. / by Gretchen Lynn Kalonji. / Ph.D.

The concept of complete symmetry groups has been known for some time in applications to ordinary differential equations. In this Thesis we apply this concept to partial differential equations. For any 1+1 linear evolution equation of Lie’s type (Lie S (1881) Uber die Integration durch bestimmte Integrale von einer Klasse linear partieller Differentialgleichung Archiv fur Mathematik og Naturvidenskab 6 328-368 (translation into English by Ibragimov NH in CRC Handbook of Lie Group Analysis of Differential Equations 2 473-508) containing three and five exceptional point symmetries and a nonlinear equation admitting a finite number of Lie point symmetries, the representation of the complete symmetry group has been found to be a six-dimensional algebra isomorphic to sl(2,R) s A3,1, where the second subalgebra is commonly known as the Heisenberg-Weyl algebra. More generally the number of symmetries required to specify any partial differential equations has been found to equal the number of independent variables of a general function on which symmetries are to be acted. In the absence of a sufficient number of point symmetries which are not solution symmetries one must look to generalized or nonlocal symmetries to remove the deficiency. This is true whether the evolution equation be linear or not. We report Ans¨ atze which provide a route to the determination of the required nonlocal symmetry or symmetries necessary to supplement the point symmetries for the complete specification of the equations. Furthermore we examine the connection of ordinary differential equations to partial differential equations through a common realisation of complete symmetry group. Lastly we revisit the notion of complete symmetry groups and further extend it so that it refers to those groups that uniquely specify classes of equations or systems. This is based on some recent developments pertaining to the properties and the behaviour of such groups in differential equations under the current definition, particularly their representations and realisations for Lie remarkable equations. The results seem to be quite astonishing. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2008.

Classical gauge theories are studied for spherically symmetric monopole solutions. The Higgs field is taken in the adjoint representation and in the limit of vanishing self-interaction. The equations of motion can be represented by a Lax pair. Using techniques from group representation theory, explicit solutions are obtained in the case of the principal embedding of the symmetry group SU(2) in the bigger gauge group. The parameters of the solutions can be chosen to give finite fields everywhere, and the large r behaviour of the Higgs field, which determines the symmetry breaking, is discussed. Some low rank groups are studied as examples. Next, some properties of the non-principal SU(2) embeddings and their classification are discussed. These are then used to obtain more solutions, but the problem has not been solved for the most general case.