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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.

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.

This thesis is concerned with the role of symmetry in low-level image segmentation. Early detection of local image properties that could indicate the presence of an object would be useful in segmentation, and it is proposed here that approximate bilateral symmetry, which is common to many natural and man made objects, is a candidate local property. To be useful in low-level image segmentation the representation of symmetry must be relatively robust to noise interference, and the symmetry must be detectable without prior knowledge of the location and orientation of the pattern axis. The experiments reported here investigated whether bilateral symmetry can be detected with and without knowledge of the axis of symmetry, in several different types of pattern. The pattern properties found to aid symmetry detection in random dot patterns were the presence of compound features, formed from locally dense clusters of dots, and contrast uniformity across the axis. In the second group of experiments, stimuli were designed to enhance the features found to be important for global symmetry detection. The pattern elements were enlarged, and grey level was varied between matched pairs, thereby making each pair distinctive. Symmetry detection was found to be robust to variation in the size of matched elements, but was disrupted by contrast variation within pairs. It was concluded that the global pattern structure is contained in the parallelism between extended, cross axis regions of uniform contrast. In the third group of experiments, detection performance was found to improve when the parallel structure was strengthened by the presence of matched strings, rather than pairs of elements. It is argued that elongation, parallelism, and approximate alignment between pattern constituents are visual properties that are both presegmentally detectable, and sufficient for the representation of global symmetric structure. A simple computational property of these patterns is described.

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