A dissertation presented to The Faculty of Science University of the Witwatersrand Johannesburg in ful lment of the requirements for the degree of Master of Science June 2015 / At nite N, the number of restricted Schur polynomials is greater than, or equal to the number of generalized restricted Schur polynomials. In this dissertation we study this dis- crepancy and explain its origin. We conclude that, for quiver gauge theories, in general, the generalized restricted Shur polynomials correctly account for the complete set of nite N constraints and they provide a basis, while the restricted Schur polynomials only account for a subset of the nite N constraints and are thus overcomplete. We identify several situations in which the restricted Schur polynomials do in fact account for the complete set of nite N constraints. In these situations the restricted Schur polynomials and the gen- eralized restricted Schur polynomials both provide good bases for the quiver gauge theory. Further, we demonstrate situations in which the generalized restricted Schur polynomials reduce to the restricted Schur polynomials and use these results to study the anomalous dimensions for scalar operators in ABJM theory in the SU(2) sector. The operators we consider have a classical dimension that grows as N in the large N limit. Consequently, the large N limit is not captured by summing planar diagrams { non-planar contributions have to be included. We nd that the mixing matrix at two-loop order is diagonalized using a double coset ansatz, reducing it to the Hamiltonian of a set of decoupled oscilla- tors. The spectrum of anomalous dimensions, when interpreted in the dual gravity theory, shows that the energy of the uctuations of the corresponding giant graviton is dependent on the size of the giant. The rst subleading corrections to the large N limit are also considered. These subleading corrections to the dilatation operator do not commute with the leading terms, indicating that integrability probably does not survive beyond the large N limit.

by Tsang Yue Kin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 120-122). / by Tsang Yue Kin. / Contents --- p.ii / List of Figures --- p.iv / List of Tables --- p.vi / Abstract --- p.vii / Acknowledgement --- p.viii / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1 --- Probability Density Function in Turbulence --- p.2 / Chapter 1.2 --- Scaling Laws in Turbulence --- p.6 / Chapter Chapter 2. --- The Problem --- p.12 / Chapter 2.1 --- Random Advection of a Passive Scalar --- p.12 / Chapter 2.2 --- Numerical Implementation --- p.14 / Chapter Chapter 3. --- Statistics of Scalar Fluctuation --- p.19 / Chapter 3.1 --- Results --- p.19 / Chapter 3.1.1 --- Probability Density Function of Scalar Fluctuation --- p.20 / Chapter 3.1.2 --- Exact Formula for PDF --- p.20 / Chapter 3.1.3 --- Other Velocity Statistics --- p.24 / Chapter 3.2 --- Conditional Averages q(x) and r(x) --- p.25 / Chapter 3.2.1 --- q(x) --- p.25 / Chapter 3.2.2 --- r(x) --- p.39 / Chapter 3.2.3 --- PDF calculated from various forms of q(x) and r(x) --- p.46 / Chapter 3.3 --- Other Velocity Statistics --- p.55 / Chapter Chapter 4. --- Statistics of Scalar Difference --- p.75 / Chapter 4.1 --- PDF of Scalar Difference --- p.75 / Chapter 4.1.1 --- Results --- p.76 / Chapter 4.1.2 --- Relation between PT{x) and 5S2n(T) --- p.80 / Chapter 4.2 --- Scaling of S2n(T) --- p.82 / Chapter 4.3 --- Other Velocity Statistics --- p.90 / Chapter 4.4 --- Discussions --- p.103 / Chapter Chapter 5. --- Conclusion --- p.105 / Appendix A. PDF of Stream Functions --- p.107 / Chapter A.1 --- General Formalism --- p.107 / Chapter A.2 --- Applications --- p.108 / Appendix B. Generation of Random Deviates --- p.112 / Appendix C. Check of Stationarity --- p.114 / Appendix D. Calculations of the coefficients of q(x) and r(x) --- p.117 / Bibliography --- p.120

Thesis (Ph. D.)--University of Canterbury, 2007. / Typescript (photocopy). Includes bibliographical references (p. 117-140). Also available via the World Wide Web.

Thesis (M.S.)--Rutgers University, 2008. / "Graduate Program in Electrical and Computer Engineering." Includes bibliographical references (p. 60-61).

by Chi-chung Lee. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1993. / Includes bibliographical references (leaves 45-47). / INTRODUCTION --- p.1 / Chapter CHAPTER 1 --- RADIAL SYMMETRY OF GROUND STATES --- p.7 / Chapter CHAPTER 2 --- EXISTENCE OF A GROUND STATE --- p.14 / Chapter CHAPTER 3 --- UNIQUENESS OF GROUND STATE I --- p.23 / Chapter CHAPTER 4 --- UNIQUENESS OF GROUND STATE II --- p.35 / REFERENCES --- p.45

Lo Ka Fai / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves [67]-68). / Text in English; abstracts in English and Chinese. / Lo Ka Fai / Abstract --- p.ii / Acknowledgements --- p.iii / List of Figures --- p.v / List of Tables --- p.ix / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- The First Problem: Heat Transfer and Large-scale Flow --- p.7 / Chapter 2.1 --- Circulating flow --- p.8 / Chapter 2.2 --- Shear flow --- p.10 / Chapter 3 --- Results and Discussions --- p.13 / Chapter 3.1 --- Circulating flow --- p.13 / Chapter 3.2 --- Shear flow --- p.17 / Chapter 4 --- The second problem: Intermittency --- p.22 / Chapter 5 --- Method of Analysis --- p.25 / Chapter 6 --- Results and discussions --- p.29 / Chapter 6.1 --- C=0.01 --- p.29 / Chapter 6.2 --- "C=0.05, 0.1 and 0.5" --- p.41 / Chapter 7 --- Conclusions --- p.60 / Chapter A --- Linear Scaling Exponents --- p.62 / Chapter B --- The Formula of and the Hierarchy --- p.64 / Bibliography --- p.66

by Pang Cheuk Sum. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Text in English; abstract also in Chinese. / by Pang Cheuk Sum. / Contents --- p.i / List of Figures --- p.iii / List of Tables --- p.v / Abstract --- p.vi / Acknowledgement --- p.viii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- The Problem --- p.7 / Chapter 2.1 --- Model of random advection --- p.7 / Chapter 2.2 --- Model without large scale flow --- p.8 / Chapter 2.3 --- "Model with, large-scale mean circulating flow" --- p.11 / Chapter 3 --- Reviews of the previous studies --- p.16 / Chapter 3.1 --- Review of previous studies on one-point scalar statistics --- p.16 / Chapter 3.2 --- Review of previous study on scalar difference --- p.18 / Chapter 4 --- Effects of a large-scale flow on one-point statistics --- p.20 / Chapter 4.1 --- Effects on mean scalar profile --- p.20 / Chapter 4.2 --- Effects on rms fluctuation --- p.23 / Chapter 4.3 --- Effects on probability density function --- p.26 / Chapter 5 --- Study of scalar difference --- p.34 / Chapter 5.1 --- Scalar difference at the center of lattice --- p.34 / Chapter 5.1.1 --- Probability density function --- p.34 / Chapter 5.1.2 --- Structure function --- p.43 / Chapter 5.1.3 --- Conditional averages --- p.52 / Chapter 5.1.4 --- Structure function at other position --- p.55 / Chapter 5.2 --- Effects of large-scale flow --- p.57 / Chapter 6 --- Summary --- p.65 / Chapter 6.1 --- Effects of a large-scale flow on passive scalar --- p.65 / Chapter 6.2 --- Statistics of scalar difference --- p.66 / Chapter A --- Simulation parameters --- p.67 / Chapter B --- Normalization of Probability Density Function --- p.68 / Bibliography --- p.69

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Zhou Quan = 关於热湍流中标量场混合和自组织结构的实验研究 / 周全. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 89-94). / Text in English; abstracts in English and Chinese. / Zhou Quan = Guan yu re tuan liu zhong biao liang chang hun he he zi zu zhi jie gou de shi yan yan jiu / Zhou Quan. / Abstract --- p.ii / Acknowledge --- p.iii / Table of Contents --- p.v / List of Figures --- p.xii / List of Tables --- p.xiii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Rayleigh-Benard System --- p.2 / Chapter 1.1.1 --- Physics Picture --- p.2 / Chapter 1.1.2 --- Basic Equations and Characteristic Parameters --- p.3 / Chapter 1.2 --- Passive and Active Scalars --- p.6 / Chapter 1.3 --- Large-Scale Circulation (LSC) --- p.7 / Chapter 2 --- Passive and Active Scalars in Convective Thermal Turbulence --- p.9 / Chapter 2.1 --- Experimental Setup and Measurement Techniques --- p.9 / Chapter 2.1.1 --- Rectangular Cell --- p.9 / Chapter 2.1.2 --- Temperature probe: thermistor --- p.12 / Chapter 2.1.3 --- Laser Induced Fluorescence (LIF) --- p.16 / Chapter 2.2 --- Passive Scalar Measurements --- p.24 / Chapter 2.2.1 --- Concentration Time Series --- p.24 / Chapter 2.2.2 --- Statistical Properties --- p.26 / Chapter 2.3 --- Active Scalar Measurements --- p.35 / Chapter 2.4 --- Relationship between passive scalar and active scalar --- p.41 / Chapter 2.4.1 --- Cross-correlation Functions --- p.41 / Chapter 2.4.2 --- Structure Functions --- p.48 / Chapter 2.5 --- Summary --- p.60 / Chapter 3 --- The Azimuthal Motion of the Mean Wind in Turbulent Rayleigh- Benard Convection --- p.63 / Chapter 3.1 --- Experimental Setup and Measurement Techniques --- p.64 / Chapter 3.1.1 --- Cylindrical Cell --- p.64 / Chapter 3.1.2 --- Particle Image Velocimetry (PIV) --- p.65 / Chapter 3.2 --- Azimuthal Motion of the Mean Wind in the Γ = 1.0 Cell --- p.68 / Chapter 3.2.1 --- Time-trace of the LSC orientation Φ --- p.69 / Chapter 3.2.2 --- Statistical properties of azimuthal motion of the mean wind --- p.73 / Chapter 3.3 --- Azimuthal Motion of the Mean Wind in theΓ = 0.5 Cell --- p.77 / Chapter 3.3.1 --- Time-trace of the LSC orientation Φ --- p.78 / Chapter 3.3.2 --- Statistical properties of azimuthal motion of the mean wind --- p.80 / Chapter 3.4 --- Summary --- p.85 / Chapter 4 --- Conclusion --- p.86 / Chapter 4.1 --- Scalar Mixing --- p.86 / Chapter 4.2 --- The Azimuthal Motion of the LSC --- p.87 / Chapter 4.3 --- Perspective for Further Investigation --- p.88 / Bibliography --- p.88

In the construction of Canonical Loop Quantum Gravity, General Relativity is rewritten in terms of the Ashtekar variables to simplify its quantisation. They consist of a densitised triad and a connection terms. The latter depends by definition and by construction on a free parameter β, called the Barbero–Immirzi parameter. This freedom is passed on to the quantum theory as it appears in the expressions of certain operators. Their discreet spectra depend on the arbitrary value of this parameter β, meaning that the scale of those spectra is not uniquely defined. To get around this ambiguity, we propose to consider a theory of Conformal Loop Quantum Gravity, by imposing a local conformal symmetry through the addition of a scalar field. We construct our theory starting from the usual Einstein–Hilbert action for General Relativity to which we add the action for the massless scalar field and rewrite it in terms of a new set of Ashtekar-like variables. They are constructed through a set of canonical transformations, which allow to move the Barbero–Immirzi parameter from the connection to the scalar variable. We then show that the theory can be quantised by fulfilling the conditions for a Dirac quantisation. Finally, we present some first elements of the quantum formalism. It is expected that with such a scalar-tensor theory, the quantum operators should not depend on the free parameter directly but rather on the dynamical scalar field, solving therefore the ambiguity.