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In this thesis, we report a novel symmetry breaking system in single-beam optical trap. The breaking of symmetry is observed in Brownian dynamics of a linked pair of beads with substantially differing radii (500nm and 100nm). Such composite beads were originally conceived as a manipulation means to study of Brownian interactions between mesoscopic biological agents of the order of 100 – 200 nm (viruses or bacteria) with cell surfaces. During the initial testing of the composite bead system, we discovered that the system displayed thermally activated transitions and energetics of symmetry breaking. This thesis, while making a brief overview of the biological relevance of the composite bead system, focuses primarily on the analysis and experimentation that reveals the complex dynamics observed in the system. First, we theoretically analyze the origin of the observed symmetry breaking using electromagnetic theory under both Gaussian beam approximation and full Debye-type integral representation. The theory predicts that attachment of a small particle to a trapped microsphere results in creation of a bistable rotational potential with thermally activated transitions. The theoretical results are then verified using optical trapping experiments. We first quantify the top-down symmetry breaking based on measurement of the kinetic transition rates. The rotational potential is then explored using an experiment employing a novel algorithm to track rotational state of the composite bead. The results of the theory and experiments are compared with results of a Brownian dynamics simulation based on Smart Monte Carlo algorithm.

Nuclei near N=90 and Z=64 have recently been suggested to be `tetrahedral-magic'. One of the main signatures for tetrahedral symmetry is a vanishing quadrupole moment in low-lying negative-parity bands, resulting in very weak or even vanishing E2 matrix elements. With N=90 and Z=62, the transitional nucleus 152Sm is a potential candidate for relatively stable tetrahedral symmetry. Its structure was investigated using deuteron inelastic scattering with a 22 MeV polarized beam at the MP tandem Van de Graaff accelerator of the TU/LMU Munich. The scattered deuterons were momentum analyzed using the Q3D spectrometer. The experimental spectra obtained have allowed the extraction of high-quality differential cross-sections and analyzing powers for levels up to 1.8 MeV. The low-lying negative-parity bands are observed to be strongly populated and the angular distributions associated to their levels exhibit several structural features. The overall agreement is relatively good when considering strong intra-band E2 transitions, but further calculations must be performed to allow precise matrix element extraction. In particular, a simple population pathway test on the 1- state has demonstrated that calculations with vanishing E2 transitions in the negative-parity band are also capable of reproducing its experimental angular distributions. Therefore, the presence of tetrahedral symmetry signature in 152Sm is not excluded. / This work has been supported in part by the Natural Sciences and Engineering Research Council of Canada.

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New four dimensional N = 2 field theories can be engineered from compactifying six dimensional (2, 0) superconformal field theory on a punctured Riemann surface. Hitchin’s equation is defined on this Riemann surface and the fields in Hitchin’s equation are singular at the punctures. Four dimensional theory is entirely determined by the data at the punctures. Theory without lagrangian description can also be constructed in this way. We first construct new four dimensional generalized superconformal quiver gauge theory by putting regular singularity at the puncture. The algorithm of calculating weakly coupled gauge group in any duality frame is developed. The asymptotical free theory and Argyres-Douglas field theory can also be constructed using six dimensional method. This requires introducing irregular singularity of Hithcin’s equation. Compactify four dimensional theory down to three dimensions, the corresponding N = 4 theory has the interesting mirror symmetry. The mirror theory for the generalized superconformal quiver gauge theory can be derived using the data at the puncture too. Motivated by this construction, we study other three dimensional theories deformed from the above theory and find their mirrors. The surprising relation of above four dimensional gauge theory and two dimensional conformal field theory may have some deep implications. The S-duality of four dimensional theory and the crossing symmetry and modular invariance of two dimensional theory are naturally related.

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.

Chapter one is an overall introduction reviewing the applications of compounds possessing rotational symmetry, in particular their role as asymmetric reagents/catalysts in various reactions. The attempted syntheses of two different types of C₃-symmetric homochiral triols are described in chapter two. Unfunctionalised and functionalised triamides were synthesised from the corresponding amine and 1,3,5-benzenetricarbonyl trichloride. Unfunctionalised N-substituted 1,3,5-triazacychlohexanes (triazines) were synthesised from the corresponding amine and formaldehyde, however reactions between chiral amino alcohols and formaldehyde failed to give the desired trihydroxy triazines. Chapter three describes the syntheses of a range of unfunctionalised chiral C₃-symmetric compounds, as well as a review of C₃-symmetric systems as host molecules. The successful synthesis of a range of unfunctionalised triethrs was carried out by reacting 1,3,5-tris(bromomehyl)mesitylene with some selected terpenes and other alcohols. Their role as host molecules was investigated, however they failed to bind with any of the various metals that were used. Their asymmetric induction capabilities were also tested when one of the triethers was used as a catalyst in the Et₂Zn/aldehyde reaction. The desired alcohol was indeed isolated, however it was found by ¹H nmr of the Mosher's ester to be a reacemic mixture. In chapter four the synthesis of the unfunctionalised triethers that was described in chapter three was extended to the synthesis of C₃-symmetric triethers containing functional groups. A range of tridentate ligands were synthesised containing either amino, ester, keto or hydroxyl terminal groups. In addition to these, trihydroxy ligands containing amino linkages in place of ethereal linkages were also synthesised. From these compounds three of the trihydroxy ligands were used in complexation reactions with Sn, Ti, Si, Ni, and Cu. The reactions using Si, Ni and Cu failed to give the desired complexes. However, strong evidence was obtained for the existence of the desired Sn and Ti complexes. Once again using the Et₂Zn/aldehyde reaction preliminary asymmetric induction studies were carried out using one of the Ti complexes. The reaction showed some selectivity in favour of the R enantiomer. Chapter four also describes the syntheses of some hexadentate ligands, in particular the synthesis of a chiral C₃-symmetric hexol, a C₃-symmetric ligand containing three β keto-ester groups, and also a 1,3-diketo derivative.

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.

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The present thesis reports psychophysical and modeling studies on the spatial properties of visual mechanisms mediating the perception of mirror symmetry in human vision. In a first set of experiments, patterns were filtered for power spectra that decayed with spatial frequency according to variable slopes. Results revealed that symmetry detection is optimal if contrast energy is roughly equated across log-frequency bands (i.e. 1/f2) and that, under such conditions, spatial scales contribute equally and independently to symmetry perception. In a second study, random-noise patterns were filtered for various orientation bands. Results showed that symmetry perception is possible at all orientations, is mediated by oriented mechanisms, and is computed independently in different orientation channels. Data also revealed that the dimensions of the spatial integration region (IR) for symmetry vary with orientation in a way that approximately matches the spatial distribution of information in the stimulus. Finally, symmetry detection was measured for bandpass textures of variable spatial density and variable contrast polarity. For such patterns, it was found that symmetry is computed at a spatial scale proportional to stimulus density and that mechanisms insensitive to contrast polarity (i.e. second-order) are involved in the scale-selection process. / Overall, results from empirical and modeling work revealed an intimate link between symmetry perception and the properties of spatial filters. In particular, I argue that the size of the IR tends to vary such that a fixed amount of information is integrated irrespective of the spatial properties of the stimulus. Implications for the functional architecture of symmetry perception are discussed, and a paradigm for future research in symmetry perception is proposed in which spatial filtering is extended to higher orders of spatial complexity.