Faculty of Science (ETDs)
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Item Two-dimensional turbulent classical and momentumless thermal wakes(University of the Witwatersrand, Johannesburg, 2023-07) Mubai, Erick; Mason, David PaulThe two-dimensional classical turbulent thermal wake and the two-dimensional momentumless turbulent thermal wake are studied. The governing partial differential equations result from Reynolds averaging the Navier-Stokes, the continuity and energy balance equations. The averaged Navier-Stokes and energy balance equations are closed using the Boussinesq hypothesis and an analogy of Fourier’s law of heat conduction. They are further simplified using the boundary layer approximation. This leads to one momentum equation with the continuity equation for an incompressible fluid and one thermal energy equation. The partial differential equations are written in terms of a stream function for the mean velocity deficit that identically satisfies the continuity equation and the mean temperature difference which vanishes on the boundary of the wake. The mixing length model and a model that assumes that the eddy viscosity and eddy thermal conductivity depend on spatial variables only are analysed. We extend the von Kármán similarity hypothesis to thermal wakes and derive a new thermal mixing length. It is shown that the kinematic viscosity and thermal conductivity play an important role in the mathematical analysis of turbulent thermal wakes. We obtain and use conservation laws and associated Lie point symmetries to reduce the governing partial differential equations to ordinary differential equations. As a result we find new analytical solutions for the two-dimensional turbulent thermal classical wake and momentumless wake. When the ordinary differential equations cannot be solved analytically we use a numerical shooting method that uses the two conserved quantities as the targets.Item The Large N Limit of Heavy Operator Excitations(University of the Witwatersrand, Johannesburg, 2023-07) Carlson, Warren Anthony; De Mello Koch, RobertOperators with bare dimension of order N are studied. These are restricted Schur polynomials labeled by Young diagrams with two long rows or two long columns and are heavy operators in the large N limit. A dramatic simplification of the action of the dilatation operator on these states is found, where the diagonalization of the dilatation operator reduces to solving three-term recursion relations. The solutions to these recursion relations reduce the spectrum of the dilatation operator to that of decoupled harmonic oscillators, showing that these systems are integrable at large N. Then, generating functions for bound states of two giant gravitons are constructed and an extension to more than two giant gravitons is sketched. These generating functions are integrals over auxiliary variables that encode the symmetrization and anti-symmetrization of the fields in the restricted Schur polynomials and give a simple construction of eigenfunctions of the dilatation operator. These generating functions are shown to be eigenfunctions of the dilatation operator in the large N limit. As a byproduct, this construction gives a natural starting point for systematic 1/N expansions of these operators. This includes the prospect to generate asymptotic representations of the symmetric group and its characters via the restricted Schur polynomials. Finally, the asymptotic expansion of the three-point function is computed in three BMN limits by varying one parameter in the large N limit. It is argued that these asymptotic expansions encode non-perturbative effects and are related by a parametric Stokes phenomenon.Item A Continuous Reinforcement Learning Approach to Self-Adaptive Particle Swarm Optimisation(University of the Witwatersrand, Johannesburg, 2023-08) Tilley, Duncan; Cleghorn, ChristopherParticle Swarm Optimisation (PSO) is a popular black-box optimisation technique due to its simple implementation and surprising ability to perform well on various problems. Unfortunately, PSO is fairly sensitive to the choice of hyper-parameters. For this reason, many self-adaptive techniques have been proposed that attempt to both simplify hyper-parameter selection and improve the performance of PSO. Surveys however show that many self-adaptive techniques are still outperformed by time-varying techniques where the value of coefficients are simply increased or decreased over time. More recent works have shown the successful application of Reinforcement Learning (RL) to learn self-adaptive control policies for optimisers such as differential evolution, genetic algorithms, and PSO. However, many of these applications were limited to only discrete state and action spaces, which severely limits the choices available to a control policy, given that the PSO coefficients are continuous variables. This dissertation therefore investigates the application of continuous RL techniques to learn a self-adaptive control policy that can make full use of the continuous nature of the PSO coefficients. The dissertation first introduces the RL framework used to learn a continuous control policy by defining the environment, action-space, state-space, and a number of possible reward functions. An effective learning environment that is able to overcome the difficulties of continuous RL is then derived through a series of experiments, culminating in a successfully learned continuous control policy. The policy is then shown to perform well on the benchmark problems used during training when compared to other self-adaptive PSO algorithms. Further testing on benchmark problems not seen during training suggest that the learned policy may however not generalise well to other functions, but this is shown to also be a problem in other PSO algorithms. Finally, the dissertation performs a number of experiments to provide insights into the behaviours learned by the continuous control policy.Item Predicting Future Stock Price with Sentiment Analysis: Recurrent vs. Attention Based Learning for Regression Tasks(University of the Witwatersrand, Johannesburg, 2023-08) Mcdonald, Bernard; Nasejje, JustineStock price prediction is a lucrative challenge as successful prediction could yield significant profits for investors – attracting research utilising novel data sources and modelling techniques. This research aimed to accurately predict the future closing price of the top five stocks of the NASDAQ100 index by leveraging Twitter data and recent advancements in machine learning. Three representations of large-scale Twitter data were derived: company, stock market, and general public sentiment. Company sentiment and stock market sentiment were Granger-causal (p < 0.10) for the closing price of four and two of the five companies considered, respectively. Five stock price prediction models were built: ARIMA, RNN, LSTM, GRU, and a novel Transformer model. A hyperparameter grid search selected feature subsets containing sentiment data as optimal in sixteen of the twenty (80%) model-dataset combinations fitted. Assessed using the RMSE, all the machine learning models outperformed the ARIMA model. The attention-based Transformer model outperformed the recurrent models in both predictive performance and model computational training efficiency. The model produced test RMSEs of 1.22, 2.07, 35.54, 16.61, and 4.95 when predicting the closing price of Apple, Microsoft, Amazon, Alphabet, and Facebook respectively.Item Relaxed Inertial Algorithm for Solving Equilibrium Problems(University of the Witwatersrand, Johannesburg, 2024) Elijah, Nwakpa ChidiIn this dissertation, we propose and study two relaxed inertial methods for solving equilibrium problems. In our first proposed method, we establish that the generated sequence of our proposed method weakly converges to a solution of the equilibrium problems. We apply this proposed method to variational inequality and fixed point problems. Further- more, a modification of the first method leads us to our second iterative method. Again, we established that the sequence generated by this method converges strongly to a solution of the equilibrium problems. Our proposed methods involve self-adaptive stepsizes and hence, do not require the fore knowledge of the Lipschitz constants for implementation. In each of our proposed methods, the convergence is established when the associated cost bifunction is pseudomonotone and satisfies the Lipschitz-type conditionItem Convergence Results for Inertial Regularized Bilevel Variational Inequality Problems(University of the Witwatersrand, Johannesburg, 2024) Okorie, Kalu Okam; Okeke, Chibueze ChristianIn this dissertation, we introduce and study the inertial forward-reflected-backward method for approximating a solution of bilevel variational inequality problems. Our proposed method involves a single projection onto a feasible set, one functional evaluation and adopts the inertial extrapolation term. These features make our algorithm cost-effective and efficient, which is desirable when the cost operator and the feasible set have a complex structure. We incorporate the regularization technique in our method and establish that the sequences generated by our method converge strongly to a solution of the bilevel variational inequality problem studied in this work; furthermore, we modified our method by replacing the stepsizes and projection onto a feasible set with a self-adaptive non-monotonic stepsizes and projection onto a constructive halfspace, respectively. The non-monotonic stepsizes ensure that our method performs without the previous detail of the Lipschitz constant, and the projection onto a constructive halfspace is cheap since its computation is through an explicit formula. These adjustments in our method ensure an improved performance, cheap computation and easy implementation of our method. We show the strong convergence result of the iterative sequences. Lastly, we give numerical experiments comparing the performance of the proposed methods with existing methodsItem Lessons for South Africa’s proposed social security retirement reforms from the experience of other sub-Saharan African countries(University of the Witwatersrand, Johannesburg, 2024) Walker, StephenThe South African government intends reforming its current social security system, including retirement benefits. Views on how this should be done vary, even within government. Proposals often take the experience of other countries into consideration but there is limited literature on the experience of other sub-Saharan African countries. The region is experiencing demographic change, especially reduced infant mortality, reduced fertility and increasing old age longevity. Here South Africa is advanced relative to other countries in the region, despite the high unemployment levels. South Africa’s informal sector is large relative to developed countries, but smaller than elsewhere in Sub-Saharan Africa. Countries in the region have tried several approaches when introducing reform. Level A non- contributory pensions in South Africa are advanced, relative to most countries in the region. Most other countries have mandatory, contributory, government-run level B funds, the closest equivalent in South Africa is the Unemployment Insurance Fund. DB level B schemes are the norm. However, many countries are experiencing strain on the financial sustainability of these schemes and a number of countries have had to increase scheme contributions or reduce benefits. Occupational retirement funds in South Africa are well established and have experienced significant reforms recently. South Africa’s level C2 occupational retirement fund coverage is not mandated by government but is high relative to other countries in the region, even those with compulsory coverage under level C1. South Africa is still relatively new to introducing contributory pensions for informal sector workers. Other countries have tried various approaches under both levels D1 and D2 without finding a perfect solution. The research shows that maximising coverage requires all pension types. Pension reform is an iterative process, there is no perfect solution and phasing-in change is best. Government should make decisions on what incremental improvements can be made and start implementing these. The sequencing of reforms is important, what happens at each level of pension provision will influence what can and should be done at the next level. South Africa should move towards universalisation of non-contributory pensions but needs to do so in a cost-effective manner. The experience in other countries in the region should be considered when setting goals for coverage of informal sector workers by a level D1 or D2 contributory scheme. Compulsory contributory pensions should be introduced for formal sector workers, shifting from a level C2 to a level C1 approach. Expanding the Unemployment Insurance Fund to cater for retirement benefits as opposed to creating a new level B national fund should be explored.Item Evaluating Pre-training Mechanisms in Deep Learning Enabled Tuberculosis Diagnosis(University of the Witwatersrand, Johannesburg, 2024) Zaranyika, Zororo; Klein, RichardTuberculosis (TB) is an infectious disease caused by a bacteria called Mycobacterium Tuberculosis. In 2021, 10.6 million people fell ill because of TB and about 1.5 million lives are lost from TB each year even though TB is a preventable and curable disease. The latest global trends in TB death cases are shown in 1.1. To ensure a higher survival rate and prevent further transmissions, it is important to carry out early diagnosis. One of the critical methods of TB diagnosis and detection is the use of posterior-anterior chest radiographs (CXR). The diagnosis of Tuberculosis and other chest-affecting dis- eases like Pneumoconiosis is time-consuming, challenging and requires experts to read and interpret chest X-ray images, especially in under-resourced areas. Various attempts have been made to perform the diagnosis using deep learning methods such as Convolutional Neural Networks (CNN) using labelled CXR images. Due to the nature of CXR images in maintaining a consistent structure and overlapping visual appearances across different chest-affecting diseases, it is reasonable to believe that visual features learned in one disease or geographic location may transfer to a new TB classificationmodel. This would allow us to leverage large volumes of labelled CXR images available online hence decreasing the data required to build a local model. This work will explore to what extent such pre-training and transfer learning is useful and whether it may help decrease the data required for a locally trained classifier. In this research, we investigated various pre-training regimes using selected online datasets to under- stand whether the performance of such models can be generalised towards building a TB computer-aided diagnosis system and also inform us on the nature and size of CXR datasets we should be collecting. Our experiment results indicated that both supervised and self-supervised pre-training between the CXR datasets cannot significantly improve the overall performance metrics of a TB. We noted that pre-training on the ChestX-ray14, CheXpert, and MIMIC-CXR datasets resulted in recall values of over 70% and specificity scores of at least 90%. There was a general decline in performance in our experiments when we pre-trained on one dataset and fine-tuned on a different dataset, hence our results were lower than baseline experiment results. We noted that ImageNet weights initialisation yields superior results over random weights initialisation on all ex- periment configurations. In the case of self-supervised pre-training, the model reached acceptable metrics with a minimum number of labels as low as 5% when we fine-tuned on the TBX11k dataset, although slightly lower in performance compared to the super-vised pre-trained models and the baseline results. The best-performing self-supervised pre-trained model with the least number of training labels was the MoCo-ResNet-50 model pre-trained on the VinDr-CXR and PadChest datasets. These model configura- tions achieved recall scores of 81.90% and a specificity score of 81.99% on VinDr-CXR pre-trained weights while the PadChest weights scored a recall of 70.29% and a speci- ficity of 70.22%. The other self-supervised pre-trained models failed to reach scores of at least 50% on both recall or specificity with the same number of labelsItem An Essay on Branching Time Logics(University of the Witwatersrand, Johannesburg, 2024) Marais, ChantelIn this thesis we investigate the Priorian logics of a variety of classes of trees. These classes of trees are divided in to irreflexive and reflexive trees, and each of these has a number of subclasses, for example, dense irreflexive trees, discrete reflexive trees, irreflexive trees with branches isomorphic to the natural numbers, etc. We find finite axiomatisations for the logics of these different classes of trees and show that each logic is sound and strongly / weakly complete with respect to the respective class of trees. The methods use to show completeness vary from adapting some known constructions for specific purposes, including unravelling and bulldozing, building a network step-by-step, filtering through a finite set of formulas, as well as using some new processes, namely refining the filtration and unfolding. Once the logics have been shown to be sound and complete with respect to the different classes of trees, we also show that most of these logics are decidable, using methods that include the finite model property, mosaics and conservative extensions. Lastly, we give a glimpse into the available research on other languages used to study branching time structures, including the Peircean and Ockhamist languages, and languages that include additional modal operators like “since” and “until”Item The role of invariants in obtaining exact solutions of differential equations(University of the Witwatersrand, Johannesburg, 2024) Ahmed, Mogahid Mamoon Abkar; Kara, A.H.We show here that variational and gauge symmetries have additional appli- cations to the integrability of differential equations. We present a general method to construct first integrals for some classes. In particular, we present a broad class of diffusion type equations, viz., the Fisher Kolmorov and Fitzhugh Nagumo equations, which satisfy the Painlev´e properties of their respective travelling wave forms and solitons. It is then shown how a study of invari- ance properties and conservation laws is used to ‘twice’ reduce the equations to solutions. We further constructing the first integrals of a large class of the well-known second-order Painlev´e equations. In some cases, variational and gauge symmetries have additional applications following a known Lagrangian in which case the first integral is obtained by Noether’s theorem. Generally, it is more convenient to adopt the ‘multiplier’ approach to find the first integrals. The main chapters of this thesis have either been published or submitted for publication in accredited journals. The contents of Chapters 2, 3 and 5 has been published ([54], [55]). All computations were done either by hand or Maple
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