Department of Mathematical Scienceshttp://hdl.handle.net/10500/30162016-06-28T22:27:02Z2016-06-28T22:27:02ZAlgebraic and multilinear-algebraic techniques for fast matrix multiplicationGouaya, Guy Mathiashttp://hdl.handle.net/10500/201802016-06-03T08:10:17Z2015-01-01T00:00:00ZAlgebraic and multilinear-algebraic techniques for fast matrix multiplication
Gouaya, Guy Mathias
This dissertation reviews the theory of fast matrix multiplication from a multilinear-algebraic point of view, as
well as recent fast matrix multiplication algorithms based on discrete Fourier transforms over nite groups.
To this end, the algebraic approach is described in terms of group algebras over groups satisfying the triple
product Property, and the construction of such groups via uniquely solvable puzzles.
The higher order singular value decomposition is an important decomposition of tensors that retains some of
the properties of the singular value decomposition of matrices. However, we have proven a novel negative result
which demonstrates that the higher order singular value decomposition yields a matrix multiplication algorithm
that is no better than the standard algorithm.
2015-01-01T00:00:00ZCohomologies on sympletic quotients of locally Euclidean Frolicher spacesTshilombo, Mukinayi Hermenegildehttp://hdl.handle.net/10500/199422016-02-18T11:09:13Z2015-08-01T00:00:00ZCohomologies on sympletic quotients of locally Euclidean Frolicher spaces
Tshilombo, Mukinayi Hermenegilde
This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the
Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we
study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will
give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies.
Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures.
2015-08-01T00:00:00ZForecasting annual tax revenue of the South African taxes using time series Holt-Winters and ARIMA/SARIMA ModelsMakananisa, Mangalani P.http://hdl.handle.net/10500/199032016-02-26T12:44:36Z2015-10-01T00:00:00ZForecasting annual tax revenue of the South African taxes using time series Holt-Winters and ARIMA/SARIMA Models
Makananisa, Mangalani P.
This study uses aspects of time series methodology to model and forecast major taxes such as Personal Income Tax (PIT), Corporate Income Tax (CIT), Value Added Tax (VAT) and Total Tax Revenue(TTAXR) in the South African Revenue Service (SARS).
The monthly data used for modeling tax revenues of the major taxes was drawn from January 1995 to March 2010 (in sample data) for PIT, VAT and TTAXR. Due to higher volatility and emerging negative values, the CIT monthly data was converted to quarterly data from the rst quarter of 1995 to the rst quarter of 2010. The competing ARIMA/SARIMA and Holt-Winters models were derived, and the resulting model of this study was used to forecast PIT, CIT, VAT and TTAXR for SARS fiscal years 2010/11, 2011/12 and 2012/13. The results show that both the SARIMA and Holt-Winters models perform well in modeling and forecasting PIT and VAT, however the Holt-Winters model outperformed the SARIMA model in modeling and forecasting the more volatile CIT and TTAXR. It is recommended that these methods are used in forecasting future payments, as they are precise about forecasting tax revenues, with minimal errors and fewer model revisions being necessary.
2015-10-01T00:00:00ZMinimum entropy techniques for determining the period of W UMA starsMcArthur, Ian Alberthttp://hdl.handle.net/10500/198982016-04-05T05:46:40Z2014-08-01T00:00:00ZMinimum entropy techniques for determining the period of W UMA stars
McArthur, Ian Albert
This MSc report discusses the attributes of W Ursae Majoris (W UMa) stars and an investigation into the Minimum Entropy (ME) method, a digital technique applied to the determination of their periods of variability. A Python code programme was written to apply the ME method to photometric data collected on W UMa stars by the All Sky
Automated Survey (ASAS). Starting with the orbital period of the binaries estimated by ASAS, this programme systematically searches around this period for the period which corresponds to the lowest value of entropy. Low entropy here means low scatter (or spread) of data across the phase-magnitude plane. The ME method divides the light curve plot area into a number of elements of the investigators choosing. When a particular orbital period is applied to this photometric data, the resulting distribution of this data in the light curve plane corresponds to a speci c number of data points in each element into which this plane has been divided. This data spread is measured and calculated in terms of entropy and the lowest value of entropy corresponds to the lowest spread of data across the light curve plane. This should correspond to the best light curve shape available from the data and therefore the most accurate orbital period available. Subsequent to the testing of this Python code on perfect sine waves, it was applied, and its results compared, to the 62 ASAS eclipsing binary stars which were investigated by Deb and Singh (2011). The method was then applied to selected stars from the ASAS data base.
2014-08-01T00:00:00Z