D.Phil.

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This thesis addresses the design of quantizers for two-dimensional vectors, where the scalar components are quantized sequentially. Specifically, design algorithms for unrestricted polar quantizers (UPQ) and successively refinable UPQs (SRUPQ) for vectors in polar coordinates are proposed. Additionally, algorithms for the design of sequential scalar quantizers (SSQ) for vectors with correlated components in Cartesian coordinates are devised. Both the entropy-constrained (EC) and fixed-rate (FR) cases are investigated. The proposed UPQ and SRUPQ design algorithms are developed for continuous bivariate sources with circularly symmetric densities. They are globally optimal for the class of UPQs/SRUPQs with magnitude thresholds confined to a finite set. The time complexity for the UPQ design is $O(K^2 + KP_{max})$ in the EC case, respectively $O(KN^2)$ in the FR case, where $K$ is the size of the set from which the magnitude thresholds are selected, $P_{max}$ is an upper bound for the number of phase levels corresponding to a magnitude bin, and $N$ is the total number of quantization bins. The time complexity of the SRUPQ design is $O(K^3P_{max})$ in the EC case, respectively $O(K^2N^{'2}P_{max})$ in the FR case, where $N'$ denotes the ratio between the number of bins of the fine UPQ and the coarse UPQ. The SSQ design is considered for finite-alphabet correlated sources. The proposed algorithms are globally optimal for the class of SSQs with convex cells, i.e, where each quantizer cell is the intersection of the source alphabet with an interval of the real line. The time complexity for both EC and FR cases amounts to $O(K_1^2K_2^2)$, where $K_1$ and $K_2$ are the respective sizes of the two source alphabets. It is also proved that, by applying the proposed SSQ algorithms to finite, uniform discretizations of correlated sources with continuous joint probability density function, the performance approaches that of the optimal SSQs with convex cells for the original sources as the accuracy of the discretization increases. The proposed algorithms generally rely on solving the minimum-weight path (MWP) problem in the EC case, respectively the length-constrained MWP problem or a related problem in the FR case, in a weighted directed acyclic graph (WDAG) specific to each problem. Additional computations are needed in order to evaluate the edge weights in this WDAG. In particular, in the EC-SRUPQ case, this additional work includes solving the MWP problem between multiple node pairs in some other WDAG. In the EC-SSQ (respectively, FR-SSQ) case, the additional computations consist of solving the MWP (respectively, length-constrained MWP) problem for a series of other WDAGs. / Dissertation / Doctor of Philosophy (PhD)

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Extensions to general relativity are often considered as possibilities in the quest for a quantum theory of gravity on one hand, or to resolve anomalies within cosmology on the other. Scalar fields, found in many areas of physics, are frequently studied in this context. This is partly due to their manifestation in the effective four dimensional theory of a number of underlying fundamental theories, most notably string theory. This thesis is concerned with the effects of scalar fields on cosmological and black hole solutions. By comparison, an analysis of an inhomogeneous cosmological model which requires no extensions to general relativity is also undertaken. In chapter three, examples of numerical solutions to black hole solutions, which have previously been shown to be linearly stable, are found. The model includes at least two scalar fields, non-minimally coupled to electromagnetism and hence possesses non-trivial contingent primary hair. We show that the extremal solutions have finite temperature for an arbitrary coupling constant. Chapter four investigates the effects of higher order curvature corrections and scalar fields on the late-time cosmological evolution. We find solutions which mimic many of the phenomenological features seen in the post-inflation Universe. The effects due to non-minimal scalar couplings to matter are also shown to be negligible in this context. Such solutions can be shown to be stable under homogeneous perturbations. Some restrictions on the value of the slope of the scalar coupling to the Gauss-Bonnet term are found to be necessary to avoid late-time superluminal behaviour and dominant energy condition violation. A number of observational tests are carried out in chapter five on a new approach to averaging the inhomogeneous Universe. In this "Fractal Bubble model" cosmic acceleration is realised as an apparent effect, due to quasilocal gravitational energy gradients. We show that a good fit can be found to three separate observations, the type Ia supernovae, the baryon acoustic oscillation scale and the angular scale of the sound horizon at last scattering. The best fit to the supernovae data is χ² ≃ 0:9 per degree of freedom, with a Hubble parameter at the present epoch of H0 = 61:7+1:4 -1:3 km sec⁻¹ Mpc⁻¹ , and a present epoch volume void fraction of 0:76 ± 0:05.

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We are interested in examining the long-time decay rate of a passive scalar in two-dimensional flows. The focus is on the effect of boundary conditions for kinematically prescribed velocity fields with random or periodic time dependence. Scalar evolution is followed numerically in a periodic geometry for families of flows that have either a slip or a no-slip boundary condition on a square or plane layer subdomain D. The boundary conditions on the passive scalar are imposed on the boundary C of the domain D by restricting to a subclass invariant under certain symmetry transformations. The scalar field obeys constant (Dirichlet) or no-flux (Neumann) conditions exactly for a flow with the slip boundary condition and approximately in the no-slip case. At late times the decay of a passive scalar, for example temperature, is exponential in time with a decay rate gamma(kappa), where kappa is the molecular diffusivity. Scaling laws of the form gamma(kappa) ~ C*kappa^alpha for small kappa are obtained numerically for a variety of boundary conditions on flow and scalar, and supporting theoretical arguments are presented. In particular when the scalar field satisfies a Neumann condition on all boundaries, alpha ~ 0 for a slip flow condition; for a no-slip condition we confirm results in the literature that alpha ~ 1/2 for a plane layer, but find alpha ~ 2/3 in a square subdomain D where the decay is controlled by stagnant flow in the corners. For cases where there is a Dirichlet boundary condition on one or more sides of the subdomain D, the exponent measuring the decay of the scalar field is alpha ~ 1/2 for a slip flow condition and alpha ~ 3/4 for a no-slip condition. The scaling law exponents alpha for chaotic time-periodic flows are compared with those for similarly constructed random flows. Motivated by the theory of passive scalar field, in Part II of this work we extend the investigation of the evolution of passive scalar for the flows addressed specifically in Part I. Based on an ensemble averaging over random velocity fields, the theoretical results obtained confirm the scaling laws computed numerically for a single, long realisation of random flows. In analogy with Lebedev and Turitsyn (2004) and Salman and Haynes (2007) our results show very good agreement between such an ensemble theory and applications. In part III of our study, we expand upon the work set out in the previous parts of this thesis in terms of the polar-co-ordinate system. We analyse the structures of flows driven near to a corner with a link to Moffatt corner eddies. A long-time exponential decay rate gamma(kappa)=C*kappa^alpha has been obtained confirming our numerical and theoretical results predicted in Part I and Part II in this work. The exponent alpha is determined in a structure of Moffatt corner eddies.

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.