3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item Radial dynamics of the large N limit of multimatrix models(2016-01-22) Masuku, MthokozisiMatrix models, and their associated integrals, are encoded with a rich structure, especially when studied in the large N limit. In our project we study the dynamics of a Gaussian ensemble of m complex matrices or 2m hermitian matrices for d = 0 and d = 1 systems. We rst investigate the two hermitian matrix model parameterized in \matrix valued polar coordinates", and study the integral and the quantum mechanics of this system. In the Hamiltonian picture, the full Laplacian is derived, and in the process, the radial part of the Jacobian is identi ed. Loop variables which depend only on the eigenvalues of the radial matrix turn out to form a closed subsector of the theory. Using collective eld theory methods and a density description, this Jacobian is independently veri ed. For potentials that depend only on the eigenvalues of the radial matrix, the system is shown to be equivalent to a system of non-interacting (2+1)-dimensional \radial fermions" in a harmonic potential. The matrix integral of the single complex matrix system, (d = 0 system), is studied in the large N semi-classical approximation. The solutions of the stationary condition are investigated on the complex plane, and the eigenvalue density function is obtained for both the single and symmetrically extended intervals of the complex plane. The single complex matrix model is then generalized to a Gaussian ensemble of m complex matrices or 2m hermitian matrices. Similarly, for this generalized ensemble of matrices, we study both the integral of the system and the Hamiltonian of the system. A closed sector of the system is again identi ed consisting of loop variables that only depend on the eigenvalues of a matrix that has a natural interpretation as that of a radial matrix. This closed subsector possess an enhanced U(N)m+1 symmetry. Using the Schwinger-Dyson equations which close on this radial sector we derive the Jacobian of the change of variables to this radial sector. The integral of the system of m complex matrices is evaluated in the large N semi-classical approximation in a density description, where we observe the emergence of a new logarithmic term when m 2. The solutions of the stationary condition of the system are investigated on the complex plane, and the eigenvalue density functions for m 2 are obtained in the large N limit. The \fermionic description" of the Gaussian ensemble of m complex matrices in radially invariant potentials is developed resulting in a sum of non-interacting Hamiltonians in (2m + 1)-dimensions with an induced singular term, that acts on radially anti-symmetric wavefunctions. In the last chapter of our work, the Hamiltonian of the system of m complex matrices is formulated in the collective eld theory formalism. In this density description we will study the large N background and obtain the eigenvalue density function.Item Gaussian processes for temporal and spatial pattern analysis in the MISR satellite land-surface data(2014-07-31) Cuthbertson, Adrian JohnThe Multi-Angle Imaging SpectroRadiometer (MISR) is an Earth observation instrument operated by NASA on its Terra satellite. The instrument is unique in imaging the Earth’s surface from nine cameras at different angles. An extended system MISR-HR, has been developed by the Joint Research Centre of the European Commission (JRC) and NASA, which derives many values describing the interaction between solar energy, the atmosphere and different surface characteristics. It also generates estimates of data at the native resolution of the instrument for 24 of the 36 camera bands for which on-board averaging has taken place prior to downloading of the data. MISR-HR data potentially yields high value information in agriculture, forestry, environmental studies, land management and other fields. The MISR-HR system and the data for the African continent have also been provided by NASA and the JRC to the South African National Space Agency (SANSA). Generally, satellite remote-sensing of the Earth’s surface is characterised by irregularity in the time-series of data due to atmospheric, environmental and other effects. Time-series methods, in particular for vegetation phenology applications, exist for estimating missing data values, filling gaps and discerning periodic structure in the data. Recent evaluations of the methods established a sound set of requirements that such methods should satisfy. Existing methods mostly meet the requirements, but choice of method would largely depend on the analysis goals and on the nature of the underlying processes. An alternative method for time-series exists in Gaussian Processes, a long established statistical method, but not previously a common method for satellite remote-sensing time-series. This dissertation asserts that Gaussian Process regression could also meet the aforementioned set of time-series requirements, and further provide benefits of a consistent framework rooted in Bayesian statistical methods. To assess this assertion, a data case study has been conducted for data provided by SANSA for the Kruger National Park in South Africa. The requirements have been posed as research questions and answered in the affirmative by analysing twelve years of historical data for seven sites differing in vegetation types, in and bordering the Park. A further contribution is made in that the data study was conducted using Gaussian Process software which was developed specifically for this project in the modern open language Julia. This software will be released in due course as open source.