In this thesis it will be shown that the hydrogen atom has a SU(2) × SU(2) symmetry generated by the quantum mechanical angular momentum and Runge-Lenz vector operators. Additionally, the hydrogenic atom will be studied with supersymmetric methods to identify a supersymmetry that relates different such systems. This thesis is intended to present the material in a manner accessible to people without background in Lie groups and supersymmetry, as well as fill in some calculations between steps that are not spelt out in the litterature.

A persistent challenge for behaviorally-based accounts of learning has been providing an account of learning that occurs in the absence of systematically programmed contingencies of reinforcement. Symmetry, one type of emergent behavior, has been repeatedly demonstrated with humans, but has been considerably more difficult to demonstrate with non-humans. In this study, pigeons were exposed to a go/no-go procedure in which hue stimuli were presented full screen on a touchscreen monitor. Pigeons learned 12 baseline relations in less than 30 days. Traditional measures used to evaluate symmetry indicated that, during tests, three of the four birds responded more to the reverse of relations that were reinforced in training than to the reverse of relations that were not reinforced in training. However, additional analyses of these data suggests that these differences were driven by one of two trial types and that symmetry was only observed for one of the two predicted relations. These data systematically replicate and extend work by Urcuioli and colleagues and point to areas where further research is needed.

>Magister Scientiae - MSc / The 9Be - 9B isospin doublet carries fundamental significance for both nuclear structure and nuclear astrophysics studies. The first excited 1/2+ state in 9Be is already well established. However, its isobaric analogue 1/2+ state in 9B has not been unambigously determined yet. Theoretically, two popular descriptions of the 9Bnucleus either use a cluster model with two unbound alpha particles held together by a covalent proton or using the shell model, as a 8Be core + proton in the sd shell. An experimental determination of the excitation energy of the first 1/2+ state in 9B will provide valuable information in validating the theoretical model that adequately describes such light unbound nuclei. Further, it will also provide a robust test of mirror (isospin) symmetry violations via measurements of mirror energy differences in the doublet. Although there have been several experimental attempts to characterize the first 1/2+ state in 9B several discrepancies still exist in reported values of the excitation energies. This thesis describes an experiment performed at iThemba LABS using the 9Be(3He,t)9B reaction to address the above issue. As a byproduct, the thesis also describes an additional determination of the excitation energy of the second J-pi = 1/2+, T = 3/2 state in 9B from the same experiment. This was performed in order to resolve a discrepancy related to the excitation energy of this state. The consequence of this measurement related to Isobaric Multiplet Mass Equation (IMME) for the excited T = 3/2, A = 9 quartet is discussed briefly.

One of the more recently established methods of analysis of di erentials involves the invariance properties of the equations and the relationship of this with the underlying conservation laws which may be physical. In a variational system, conservation laws are constructed using a well known formula via Noether's theorem. This has been extended to non variational systems too. This association between symmetries and conservation laws has initiated the double reduction of di erential equations, both ordinary and, more recently, partial. We apply these techniques to a number of well known equations like the damped driven Schr odinger equation and a transformed PT symmetric equation(with Schr odinger like properties), that arise in a number of physical phenomena with a special emphasis on Schr odinger type equations and equations that arise in Optics.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. September 26, 2014. / The aim of the dissertation is to extend on the work done by Hydon in [17]. We only consider second order ordinary difference equations and calculate their symmetry generators, first integrals and reduce their order, that is, find a general solution. We investigate the association between a symmetry generator and a first integral. Furthermore, we investigate when a reduced equation may be further reduced and lead to a double reduction. The examples considered are obtained from [17]. ii

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. July 2014. / This is the study of Lie algebraic properties of first integrals of scalar second-, third and higher-order ordinary differential equations (ODEs). The Lie algebraic classification of such differential equations is now well-known from the works of Lie [10] as well as recently Mahomed and Leach [19]. However, the algebraic properties of first integrals are not known except in the maximal cases for the basic first integrals and some of their quotients. Here our intention is to investigate the complete problem for scalar second-order and maximal symmetry classes of higher-order ODEs using Lie algebras and Lie symmetry methods. We invoke the realizations of low-dimensional Lie algebras. Symmetries of the fundamental first integrals for scalar second-order ODEs which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the point symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classi cation of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0, 1, 2 or 3 point symmetry cases. It is proved that the maximal algebra case is unique. By use of Lie symmetry group methods we further analyze the relationship between the first integrals of the simplest linear third-order ODEs and their point symmetries. It is well-known that there are three classes of linear third-order ODEs for maximal and submaximal cases of point symmetries which are 4, 5 and 7. The simplest scalar linear third-order equation has seven point symmetries. We obtain the classifying relation between the symmetry and the first integral for the simplest equation. It is shown that the maximal Lie algebra of a first integral for the simplest equation y000 = 0 is unique and four-dimensional. Moreover, we show that the Lie algebra of the simplest linear third-order equation is generated by the symmetries of the two basic integrals. We also obtain counting theorems of the symmetry properties of the first integrals for such linear third-order ODEs of maximal type. Furthermore, we provide insights into the manner in which one can generate the full Lie algebra of higher-order ODEs of maximal symmetry from two of their basic integrals. The relationship between rst integrals of sub-maximal linearizable third-order ODEs and their symmetries are investigated as well. All scalar linearizable third-order equations can be reduced to three classes by point transformations. We obtain the classifying relations between the symmetries and the first integral for sub-maximal cases of linear third-order ODEs. It is known, from the above, that the maximum Lie algebra of the first integral is achieved for the simplest equation. We show that for the other two classes they are not unique. We also obtain counting theorems of the symmetry properties of the rst integrals for these classes of linear third-order ODEs. For the 5 symmetry class of linear third-order ODEs, the first integrals can have 0, 1, 2 and 3 symmetries and for the 4 symmetry class of linear third-order ODEs they are 0, 1 and 2 symmetries respectively. In the case of sub-maximal linear higher-order ODEs, we show that their full Lie algebras can be generated by the subalgebras of certain basic integrals. For the n+2 symmetry class, the symmetries of the rst integral I2 and a two-dimensional subalgebra of I1 generate the symmetry algebra and for the n + 1 symmetry class, the full algebra is generated by the symmetries of I1 and a two-dimensional subalgebra of the quotient I3=I2. Finally, we completely classify the first integrals of scalar nonlinear second-order ODEs in terms of their Lie point symmetries. This is performed by first obtaining the classifying relations between point symmetries and first integrals of scalar nonlinear second order equations which admit 1, 2 and 3 point symmetries. We show that the maximum number of symmetries admitted by any first integral of a scalar second-order nonlinear (which is not linearizable by point transformation) ODE is one which in turn provides reduction to quadratures of the underlying dynamical equation. We provide physical examples of the generalized Emden-Fowler, Lane-Emden and modi ed Emden equations.

Obtivemos, a partir do reconhecimento de propriedades de simetria em uma classe de funções de onda tipo BCS, expressões analíticas para energias de BCS projetadas e taxas de transição de reações entre estados coletivos de emparelhamento isovetorial em núcleos. Como consequência, efetuamos um tratamento simplificado de coordenadas geradoras. Nesta tese, o objetivo é mais o de estabelecer um instrumento simples e eficiente para descrever estes estados coletivos O+, do que apresentar novos resultados. Testamos a validade, as limitações e vantagens desta nossa aproximação em modelos simples e núcleos. Entre os diversos resultados, nossas amplitudes espectroscópicas obtidas por GCM são comparáveis aos métodos de diagonallização exata no modelo de camada. De maneira geral, podemos afirmar: o que foi obtido encoraja-nos sobremaneira, especialmente se considerarmos que elas foram conseguidos por uma aproximação bastante simples. Também apresentamos alguns novos resultados de tal forma que podemos colocar em bases sólidas a nossa abordagem dos estados coletivos de emparelhamento isovetoriais. Em vista da simplicidade de nossa aproximação, o presente modelo pode ser considerado um bom candidato para um estudo sistemático dos estados coletivos de emparelhamento isovetorial, que em núcleos semi-pesados, são fortemente populados por reações de transferência de um par. / Analytical expressions for the projected-BCS energies and reaction transition rates among the isovector pairing collective states are obtained by the recognition of symmetry properties in a class of BCS wave functions. As a consequence it was possible to employ a simplified Generator Coordinate Method calculation. The main purpose of the present work is to establish and develop a simple yet powerful tool for the less complicated nuclear models and for some real nuclei. Among the various results, our GCM values of spectroscopic amplitude are shown to be comparable to those of the Shell Model calculations. This is indeed encouraging especially in view of the fact that they were reached using a simple approximation such as the present one. The results also clearly demonstrate that our model is on a rather sound basis and can be applied immediately to the study of isovector collective motions. The great simplicity of the present method, as compared with their earlier complicated versions, suggests that they might prove useful in the study of isovector pairing collective states which are strongly populated by pair transfer reactions in the medium weight nuclei.

In this thesis, we primarily focus on one common type of symmetry, the translational symmetry. We first review the current state-of-the-art methods for translational symmetry detection, and discuss their benefits and drawbacks. Towards an efficient, automatic and widely applicable translational symmetry detector, we develop a novel method for automatically detecting translational symmetry patterns, and extracting the corresponding lattices from images without pre-segmentation or reconstructing the underlying 3D geometry. In particular, we employ a region-based feature and fully utilize its regional properties (shape, orientation and well-defined boundary) to propose the repeated candidates. Compared with traditional treatments, which usually rely on point-based features and group them to propose repeated candidates, our treatment is more efficient and stable to perspective projection, distortion or noise. By clustering the candidate regions and indexing the major clusters using a GPU KD-tree, the parallel lattice formation processes turn out to be very efficient and achieve a real-time rate. By using a set of spatially varying vectors with a loose neighboring constraint to represent the underlying lattice, we successfully detect most of translational symmetry patterns over arbitrary surfaces, which can be planar or curve, without or with perspective projection, and even when suffered from global and local deformations. Moreover, the parallel searching and saving scheme enables us to simultaneously detect multiple disjoint symmetry patterns from an input images. / Symmetry has been an important concept in the nature, science and art. There is an abundant of biological, chemical, and artificial structures captured in many real-world images, exhibiting various forms of symmetry. The symmetry patterns and the repetitive elements reinforce the visual importance and usually make an image more attractive. Although our humans have an excellent innate ability in recognizing symmetry and perceiving its beauty, efficient and automatic symmetry detection from images remains a unsolved challenging problem in computer vision and graphics. Without understanding the high-level semantics of symmetry, editing such images while preserving the repetitions and their relations turns out to be difficult to perform, such as image resizing, image inpainting and image replacement. / The significant improvements of our method in both efficiency and accuracy make it a useful tool from which many applications can benefit. One of them is image resizing. We demonstrate that image resizing can be achieved more effectively if we have a better understanding of the image semantics. By analyzing the translational symmetry patterns, and detecting the underlying lattices in an image, we can summarize, instead of only distorting or cropping, the image content. This opens a new space for image resizing that allows us to manipulate, not only image pixels, but also the semantic cells in the lattice. As a general image contains both symmetry & non-symmetry regions and their natures are different, we propose to resize symmetry regions by summarization and non-symmetry region by optimized warping. In addition, by smoothing the intensity of cells across the lattice, we can further maintain the seamlessness of illumination during the summarization. As the difference in resizing strategy between symmetry regions and non-symmetry region leads to discontinuity at their shared boundary, we propose a framework to minimize the artifact. Experimental results show that, with the high-level knowledge of symmetry, our method outperforms the state-of-the-art resizing techniques. / Wu, Huisi. / Advisers: Tien-Tsin Wong; Pheng-Ann Heng. / Source: Dissertation Abstracts International, Volume: 73-06, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 92-100). / 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, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

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.

Este trabalho analisa de perto os efeitos da simetria quiral no fator de forma pion-nucleon onde o nucleon e estruturado como sendo um aglomerado de quarks constituintes confinados por meio de um potencial harmonico. O estudo e desenvolvido em etapas: primeiro estudamos a equivalencia entre a hipotese da supressao de pares e a simetria quiral num processo simples envolvendo pion, quark e uma particula escalar. Verificamos que, neste caso, nao ha equivalencia. A seguir construimos os vertices elementares quark-pion proprios, onde eliminamos a possibilidade de dupla contagem de particulas que existe quando empregamos os estados ligados de quarks para obter o potencial NN. Utilizamos estes vertices para construirmos os diagramas que descrevem a troca de um pion entre quarks de aglomerados distintos. Calculamos as amplitudes e realizamos a reducao para o limite nao-relativistico obtendo o potencial quark-quark. Aplicamos, entao, o potencial quark-quark na equacao de schrodinger que descreve o sistema de seis quarks na aproximacao de aglomerados para obtermos o potencial NN no espaco de configuracao. O fator de forma e obtido realizando-se a transformada de fourier do potencial NN e, assim, podemos verificar que a simetria quiral gera correcoes importantes. Alem desta analise, estudamos o metodo fock-tani aplicado ao caso onde os quarks trocam um pion entre si. / This work analyses very closely the chiral symmetry effects in the pion-nucleon form factor where the nucleon is constructed as a cluster, of three quarks confined by an harmonic potential. Chiral symmetry shows that the microscopic interactions between quarks and diquarks must be considered. The study is peformed step by step. First, we study the equivalence between the \"pair supression\" hypothesis and chiral symmetry in an elementary process with pion, quark and a scalar particle. In this case, our results show that the equivalence does not hold. Second, we obtain the form factor from the nucleon-nucleon potential expression applying a Fourier transformation on the potentia1 in coordinate space. The nucleon potential is obtained from the diagonalisation of the microscopic quark-quark potential using quark cluster bound state. The quark-quark potential is composed by one pion exchange between two quarks, quarks-diquarks and diquarks-diquarks. The processes are described by diagrams that are used to write relativistics amplitudes, then we reduce these amplitudes to the non-relativistic limit to find the potencial. In order to avoid double counting, we extract the particle part from diquark vertices. We also study the Fock-Tani method that is applied to the case with quark exchanges between clusters.