# Masters Degrees (Applied Mathematics)

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Item Semi-tetrad decomposition of spacetime with conformal symmetry.(2018) Hansraj, Chevarra.; Maharaj, Sunil Dutt.; Goswami, Rituparno.Show more In this thesis, we study the kinematical and dynamical properties of a general spacetime that admits a conformal Killing vector. A 1+1+2 decomposition of the spacetime is performed using the fluid 4-velocity and a preferred spatial direction in the 3-space. The Lie derivatives of the 4-velocity vector and the preferred spatial direction vector are calculated and analyzed. We compare our results with the 1+3 decomposition of Maartens et al (1986), and find new results in the form of a scalar equation and constraint equation owing to the further decomposition. This provides new insights into the behaviour of the acceleration, expansion, shear and vorticity scalars which are not possible in the 1+3 decomposition. The general energy momentum tensor for an anisotropic fluid is considered and decomposed using the semi-tetrad covariant approach. We take the Lie derivative along the conformal Killing vector and apply to Einstein’s field equations. This makes it possible to generate a set of constraint equations in the new geometrical variables. All the geometrical and thermodynamical quantities are written in terms of the 1+1+2 decomposition variables. This is a new result. We also find a system of equations that must be satisfied by the thermodynamical variables when a conformal symmetry exists applied to the perfect fluid case. We show that the conformal factor satisfies a damped wave equation with a potential.Show more Item First integrals for spherically symmetric shear-free perfect fluid distributions.(2018) Gumede, Sfundo Cebolenkosi.; Maharaj, Sunil Dutt.; Goswami, Rituparno.Show more In this dissertation we study spherically symmetric shear-free spacetimes. In particular we analyse the integrability of and find exact solutions to the Emden- Fowler equation yxx = f(x)y2; which is the master equation governing the behaviour of shear-free neutral perfect fluid distributions. We first review the study of Maharaj et al (1996) by finding a first integral to this master equation. This first integral is subject to the integrability condition which we use to find restrictions on the function f(x): We show that this first integral is a generalisation of particular solutions obtained by Stephani (1983) and Srivastava (1987). Furthermore, we use a similar method to obtain a new first integral of the master equation. This is achieved by multiplying the Emden-Fowler equation by an integrating factor. We then study the integrability condition, which is an integral equation, related to the new first integral. We find that the integrability condition can be written as a third order differential equation whose solution can be expressed in terms of elementary functions and elliptic integrals. In general the solution of the integrability condition is given parametrically. We believe that this is a new result. A particular form of f(x) is identified which corresponds to repeated roots of a cubic equation giving an explicit solution.Show more Item Bounds on the extremal eigenvalues of positive definite matrices.(2018) Jele, Thokozani Cyprian Martin.; Singh, Virath Sewnath.; Singh, Pravin.Show more The minimum and maximum eigenvalues of a positive de nite matrix are crucial to determining the condition number of linear systems. These can be bounded below and above respectively using the Gershgorin circle theorem. Here we seek upper bounds for the minimum eigenvalue and lower bounds for the maximum eigenvalue. Intervals containing the extremal eigenvalues are obtained for the special case of Toeplitz matrices. The theory of quadratic forms is discussed in detail as it is fundamental in obtaining these bounds.Show more Item Solutions of the Volterra integro-differential equation.(2018) Ali, Yusuf.; Singh, Virath Sewnath.; Singh, Pravind Sewsanker.; Narain, Rivendra BasanthShow more Integro-di erential equations has found extensive applications in the eld of engineering, sciences and mathematical modelling of various physical and biological phenomena. In this thesis we focus on the Volterra type integro-di erential equation which has been used to model biological species co-existing, heat di usion, electromagnetic theory etc. In recent years much research has focused on nding approximate solutions of the integro-di erential equation by polynomial methods, speci cally focusing on the Lagrange collocation and piecewise cubic Hermite collocation methods. A further aspect to the thesis will be on analytical methods, mainly the applications of Lie group theory to the Volterra type equation. Lie group theory is one of the most powerful methods applied to obtain solutions of di erential equations. We will present the linear independent symmetries of the Volterra type equation of the rst and second kind. In addition, we shall apply the Laplace transform and it's inverse to determine general solutions for selected forms of kernel, speci cally those with convolution integrands.Show more Item Tracing star formation in groups and filaments around a young active galaxy cluster at z ∼ 1.46.(2022) Khumalo, Nondumiso.; Hilton, Matthew James.Show more Abstract available in PDF.Show more Item The evolution of galaxies in Sunyaev-Zel’dovich selected galaxy clusters from ACT DR5.(2023) Ragavan, Damien Cole.; Hilton, Matthew James.Show more Abstract available in PDF.Show more Item HI intensity mapping and cross-correlation science with HIRAX.(2020) Naidoo, Warren.; Moodley, Kavilan.Show more Abstract available in PDF.Show more Item Spectral methods for nonlocal wave problems.(2019) Pillay, Marc.; Shindin, Sergey Konstantinovich.; Parumasur, Nabendra.Show more Abstract available in PDF.Show more Item Algebraizing deductive systems.(1995) Van Alten, Clint Johann.; Raftery, James Gordon.; Sturm, Teo.Show more Abstract available in PDF.Show more Item Hyperspheres of static charged fluids in standard and modified gravity.(2019) Moodly, Lushen.; Hansraj, Sudan.Show more Abstract available in PDF.Show more Item New models in general relativity and Einstein-Gauss-Bonnet gravity.(2021) Naicker, Shavani.; Maharaj, Sunil Dutt.; Brassel, Byron Perry.Show more We generate the Einstein-Gauss-Bonnet field equations in five dimensions for a spherically symmetric static spacetime. The matter distributions considered are both neutral and charged. The introduction of a coordinate transformation brings the condition of isotropic pressure to a single master ordinary differential equation that is an Abel equation of the second kind. We demonstrate that the master equation can be reduced to a first order nonlinear canonical differential equation. Firstly, we consider uncharged gravitating matter. Several new classes of exact solutions are found in explicit and implicit forms. One of the potentials is determined completely. The second potential satisfies a constraint equation. Secondly, we study charged gravitating matter with Maxwell’s equations. We find new classes of exact charged solutions in explicit and implicit forms using two approaches. In the first approach, we can find new exact models without integration. In the second approach the Abelian pressure isotropy equation has to be integrated, which we demonstrate is possible in a number of cases. The inclusion of the electromagnetic field provides an extra degree of freedom that leads to viable exact solutions. An interesting feature characterising the new models is that a general relativity limit does not exist. Our new solutions exist only in Einstein-Gauss-Bonnet gravity. In addition, we have considered the dynamics of a shear-free fluid in Einstein gravity in higher dimensions with nonvanishing heat flux in a spherically symmetric manifold. This endeavour generates new exact models, being a generalisation of models developed in earlier treatments.Show more Item Using machine learning techniques to identify strong gravitational lensing systems in DES.(2021) Headley, Annarien Gertruida.; Hilton, Matthew James.; Sinayskiy, Ilya.; Pillay, Anban Woolaganathan.Show more Gravitational lensing systems enable astronomers to look into the distant universe by magnifying distant objects. Strong gravitational lensing systems are an incredibly rare phenomenon, with only a total about 1; 000 having been discovered. The use of machine learning (ML) has enabled the search for these systems, in the vast sky surveys that currently exist, to be narrowed down. This work investigates the use of ML techniques to identify strong gravitational lensing systems within the Dark Energy Survey (DES). We use the ML technique of convolutional neural networks (CNNs), a deep neural network architecture, as they are able to perform various image processing tasks efficiently. We generated a dataset of 96; 768 images to train and validate our CNN, half of which contains images from DES and the other half containing simulated lenses. The images from DES are scored with a 0, and the simulated lenses with a 1. Our CNN gained an accuracy of 99:73 0:07% and a mean loss of 0:81 after being evaluated against an unseen dataset similar to that of the training data. We also evaluated our CNN against 389 real lenses, and gained an accuracy of 11:92 2:75% and a mean loss of 10:49. Our CNN correctly predicted 57 389 (14:65%) lenses. In this thesis, we present our CNN and the lenses that were correctly identified. In previous studies using CNNs to identify gravitational lensing systems, accuracies between 20% to 40% were achieved, thus the accuracy of 15% achieved by our CNN is competitive. Our results could be improved by training our CNN on the all morphologies of known lensing systems, not only those containing bright arcs. Results could also be improved by ensuring that all the images are centred on the lens, and not the source images. These results and limitations are presented and discussed in the thesis. By looking at accurately simulated, as well as real lenses, one can train the CNN to be more precise.Show more Item Discontinuous Galerkin finite element method for the stokes problem using rectangular meshes.(2020) Ekpe, Wonder Kudzo.; Arunakirinathar, Kanagaratnam.Show more Abstract available in PDF.Show more Item Analysis of models arising from heat conduction through fins using Lie symmetries and Tanh method.(2021) Bulunga, Vusi Andile.; Mhlongo, Mfanafikile Don.Show more Abstract available in PDF.Show more Item Fischer matrices and character tables of group extensions.(1994) Whitley, Nicola Susan.; Moori, Jamshid.Show more Abstract available in PDF.Show more Item The efficiency of incomplete block designs in on-farm trials.(2002) Ndugwa, Robert Peter.; Njuho, Peter Mungai.Show more Abstract available in pdf.Show more Item Exact solutions in pure lovelock gravity.(2019) Gabuza, Nomfundo.; Hansraj, Sudan.Show more Item Some aspects of strong gravity effects on the electromagnetic field of a radio pulsar magnetosphere : solving the Maxwell’s equations.(2019) Sellick, Kathleen A.; Ray, Subharthi.Show more The general relativistic (GR) effects of a neutron star play a substantial role on the physics at the stellar surface. These neutron stars also host a very strong magnetic field and spin with periods of a few seconds to as high as milliseconds. In order to account for the motion of charged particles in the magnetosphere immediately outside the stellar surface, it is essential to include the GR effects in the Maxwell’s equations. To account for the frame dragging effects due to the stellar spin, we have, in this dissertation, considered a 3+1 decomposition of the spacetime and applied them to find the solutions to Maxwell’s equations of an isolated neutron star in a vacuum, for different cases. In order to derive our solutions we made use of the vector spherical harmonics in a curved spacetime. We first considered an aligned dipole magnetic field from which we formed a general formalism for the magnetic and electric fields for higher orders. We then considered an orthogonal dipole magnetic field for which we solved only for the non-rotating case. In a realistic scenario for a radio pulsar, the radio beams which originate from the pole caps of the magnetic field, have a finite angle with the spin axis and hence it is necessary to find a model for an oblique rotator. This study will be helpful in the future for the understanding of the charged particle interaction at the pulsar pole caps and hence for the emission mechanism of a radio pulsar.Show more Item Numerical solution of the Klein-Gordon equation in an unbounded domain.(2018) Lukumon, Gafari Abiodun.; Parumasur, Nabendra.; Shindin, Sergey Konstantinovich.Show more Abstract available in PDF file.Show more Item Giant radio halos and relics in ACTPol clusters.(2017) Sikhosana, Sinenhlanhla Precious.; Moodley, Kavilan.; Knowles, Kenda Leigh.Show more Galaxy clusters are the largest gravitationally-bound structures in the universe. They act as the largest astrophysical laboratories in the universe and are extremely interesting objects to study as they are at crossroads between astrophysics and cosmology. In previous decades the most prominent cluster studies were focused on thermal processes in the intracluster medium (ICM). However, recent studies have shown that non-thermal studies give a different perspective on ICM processes. Giant radio halos and radio relics are examples of this non-thermal diffuse radio emission. Giant radio halos are believed to originate from synchrotron radiation resulting from the re-acceleration of relativistic electrons in the cluster's magnetic field by the turbulent energy following merger activity. Radio relics, another form of non-thermal diffuse radio emission, have been identi ed as possible tracers of merger shock waves. The study of diffuse radio emission has a number of open questions such as; the observed bimodality in the radio power versus X-ray luminosity plot. The bimodality could partly be due to the identi cation of halos and relics in clusters without a well-de ned selection function. In this thesis, we studied giant radio halos and relics in a homogeneous, mass-selected sample of sixteen clusters selected via the Sunyaev- Zel'dovich (SZ) effect by the Atacama Cosmology Telescope (ACT) with polarization sensitive receivers (ACTPol). We carried out a radio wavelength study using data obtained from the Giant Metrewave Radio Telescope (GMRT) for four of these clusters. This subsample of four clusters will be added to the larger sample, eight of which have archival data, and four of which will be proposed for observations in the next GMRT observation cycle. We used the GMRT data at 610 MHz to search for diffuse radio emission in each cluster. We applied various uv-cuts and tapers to isolate the low-resolution emission in the target fi eld. For two of the four observed clusters, we tentatively discovered extended radio emission at a signifi cance level of at least 3o' We then measured radio fluxes for compact sources in the cluster region. We were able to calculate spectral indices for the compact sources that were cross-matched in FIRST.Show more