In this thesis we study classes of static spherically symmetric solutions to the Einstein and Einstein–Maxwell equations that may be used to model the interior of compact stars. We also study the spherical accretion of fluids on to bodies in both general relativity and the Newtonian theory of gravity. The condition for pressure isotropy is obtained upon specifying one of the gravitational potentials and the electric field intensity. A series solution was found after specifying a cubic form for the potential. The pressure and energy density appear to be non–singular and continuous inside the star. This solution admits an explicit equation of state that, in regions close to the stellar centre, may be approximated by a polytrope. Another class of exact solutions to the Einstein–Maxwell solutions was found with charge. These solutions are in the form of hypergeometric functions with two free parameters. For particular parameter values we recovered two previously known exact solutions that are reasonable models for the interior of compact stars. We demonstrated two new solutions for other choices of the parameters. One of these has well behaved pressure, energy density and electric field intensity variables within the star. The other was rejected as unphysical on the grounds that it has a negative energy density. This violates the energy conditions. We obtained the mass accretion rate and critical radius of a polytrope accreting onto a D– dimensional Schwarzschild black hole. The accretion rate, ˙M , is an explicit function of the black hole mass, M, as well as the gas boundary conditions and the dimensionality, D, of the spacetime. We also found the asymptotic compression ratios and temperature profiles below the accretion radius and at the event horizon. This generalises the Newtonian expressions of Giddings and Mangano (2008) which examined the accretion of TeV black holes. We obtained the critical radius and accretion rates of a generalised Chaplygin gas accreting on to body under a Newtonian potential. The accretion rate is about 2 - 4 times greater than that for neutral hydrogen. The Rankine–Hugoniot relations for shocked GCG flow were also found. We found general expressions for the pressure and density compression ratios. Some post shock states imply negative volumes. We suspect that these may be thermodynamically forbidden. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2012.

We formulate a deterministic mathematical model for the co-infection of HPV with HIV without treatment. Mathematical techniques were used to analyze the stability of the models in terms of basic reproduction numbers for disease-free equilibrium point and fixed point theory used for analysis of the endemic equilibrium point. The model incorporating HIV and HPV co-infection sought to investigate the impact of HIV infection in the natural history of HPV infection, and the impact of HPV infection in the natural history of HIV infection, over a period of time. Numerical simulations were carried out to illustrate the trends of progression of HIV and HPV in the case of co-infection. The results from our study showed that when both HIV and HPV infected individuals are active in the system then co-infection grows faster compared to one infection which is active in the system. Our study also showed that when we started with HPV infection in the community and introduces HIV infection after sometime has more impact in the growth of co-infection population compared to start with HIV infection and introduces HPV infection after sometime in the community. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2014.

A well known fact of classical analysis is that the Stone- Cech compacti cation of a Tychono space is achieved as the structure space of the commutative unitary ring of real valued continuous functions de ned on it. An extension of this result is obtained [1] using semiring of non-negative real valued continuous functions on the Tychono space. The present work makes a survey of this paper and attempts to extend this to the point free domain, in which it is shown that the collection of frame homomorphisms from the frame of non-negative reals to a frame is a semiring again. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2012.

Thesis (MSc (Mathematics))--Stellenbosch University, 2008. / See full text for abstract.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2007. / An exposition is given of [12], a paper by N. Krupnik, which is a discussion of the minimum number of idempotent generators of a complete matrix algebra Mn(F) over a field F, as well as direct sums of complete matrix algebras over F. It will, for example, be proved that, if n ≥ 2, then the minimum number of idempotent generators of a n × n matrix algebra is equal to 2 or 3. Krupnik made an incorrect statement in ([12], Theorem 5), namely that the minimum number of idempotent generators of m copies of an infinite field F, as an algebra over F, is m−1. This error was identified and corrected by A.V. Kelarev, A.B. van der Merwe and L. van Wyk in [11]. The thesis also includes an exposition of this correction. Furthermore an exposition will be given of the main result of [5], where E. Formanek showed that, if n ≥ 2, then there is a non-vanishing central polynomial for Mn(F), with F any field. The last mentioned result will be used in the exposition of [12].

Thesis (MSc (Mathematical Sciences. Computer Science))--University of Stellenbosch, 2011. / ENGLISH ABSTRACT: Ontologies are conceptual models of a domain of discourse and are used in a number of applications to model a field of knowledge. For example, SNOMED, an ontology of medical terminology, is widely used among medical professionals. Commercial ontologies, such as SNOMED, can have hundreds of thousands of concepts. People who want to use these ontologies need an understanding thereof, but the sheer magnitude of these ontologies hampers comprehension. It was within this context that the need arose for software tools that facilitate the understanding of ontologies. Given this background, our aim is to investigate a new area within the field of ontologies, namely, ontology comprehension. We make a contribution to it by developing an ontology comprehension framework and writing a software tool of our own. This software tool, PathViz, helps users to understand how different concepts in an ontology are related to each other and what effect entailments have on the way concepts in an ontology relate to each other. The ontology comprehension framework, PathViz and the reasoning measurement instruments were found useful for ontology comprehension by participants at an ontology workshop. / AFRIKAANSE OPSOMMING: Ontologieë is konseptuele modelle van ’n domein en word in verskeie toepassings gebruik om ’n kennisveld te modelleer. SNOMED is ’n voorbeeld van ’n ontologie van mediese terme wat baie gebruik word deur die mediese beroepslui. Kommersiële ontologieë, soos SNOMED, kan bestaan uit duisende konsepte. Dit is belangrik om hierdie ontologieë wat gebruik word te verstaan, maar die enorme omvang van hierdie ontologieë belemmer die verstaanproses. In hierdie konteks het die behoefte ontstaan vir programmatuur wat die verstaanproses van ontologieë vergemaklik. Met hierdie agtergrond inaggenome, is dit ons doel om ’n nuwe area in die veld van ontologieë te ondersoek, naamlik, Ontologie-begrip. Ons maak ’n bydra tot hierdie veld deur ’n raamwerk vir ontologie-begrip te ontwikkel en programmatuur van ons eie te skryf. Hierdie programmatuur, PathViz, help gebruikers om te verstaan hoe verskillende konsepte in ’n ontologie aan mekaar verwant is. Verder help dit gebruikers om te verstaan watter invloed afleidings uit die ontologieë het op konsepverwantskappe. Deelnemers aan ’n ontologie-werkswinkel het gevind dat die raamwerk vir ontologie-begrip, PathViz en die instrumente wat die invloed van die ontologie-redeneerder meet, ontologie-begrip bevorder.

The collocation method for solving ordinary differential equations is examined. A detailed comparison with other weighted residual methods is made. The orthogonal collocation method is compared to the collocation method and the advantage of the former is illustrated. The sensitivity of the orthogonal collocation method to different parameters is studied. Orthogonal collocation on finite elements is used to solve an ordinary differential equation and its superiority over the orthogonal collocation method is shown. The orthogonal collocation on finite elements is also used to solve a partial differential equation from chemical kinetics. The results agree remarkably with those from the literature. / Thesis (M.Sc.)-University of KwaZulu-Natal, 2008.

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The high sickle cell gene frequency has been hypothesised to be related to the protective advantage against malaria disease among heterozygous individuals. In this thesis, we study the interaction between the dynamics of malaria and sickle cell gene. The main aim is to investigate the impact of malaria treatment on the frequency of sickle cell gene. For this, we develop a mathematical model that describes the interactions between malaria and sickle cell gene under malaria treatment. The model includes both homozygous for the normal gene (AA) and heterozygous for sickle cell gene (AS) and assumes that AS individuals are not treated since they do not show clinical symptoms. We first analyse the model without malaria treatment, using singular perturbation techniques, basing on the fact that epidemiological and demographical dynamics occur on two different time scales (fast and slow dynamics). Our analysis on the fast time scale shows that high sickle cell gene frequency leads to high endemic levels for longer duration of parasitemia among heterozygous individuals. However, if the duration of parasitemia is reduced then high sickle cell gene frequency is associated with low endemic levels. We also note that on the slow time scale, the invasion ability of sickle cell gene is dependent on the malaria epidemiological parameters. The invasion coefficient given as the difference in the weighted death rates of AA and AS individuals is used as a measure to determine the establishment of sickle cell gene in the population. Results show that, the gene may establish itself if the weighted death rate of AA individuals is greater than that of AS individuals otherwise it fails. We note that, high mortality of AA individuals leads to establishment of sickle cell gene in the population. Then we analysed the model with treatment, our results indicate that the frequency of sickle cell gene decreases with an increase in the recovery rate of AA individuals. We thus conclude that eradication of malaria disease will lead to a reduction in sickle cell gene frequency. / AFRIKAANSE OPSOMMING: Daar word veronderstel dat die hoë sekelsel geenfrekwensie onder heterosigotiese individue verwant is aan die beskermende voordeel teen malaria siekte. In hierdie verhandeling ondersoek ons die wisselwerking tussen die dinamika van malaria en die sekelsel geen. Die hoofdoel is om die invloed van malaria behandeling op die frekwensie van die sekelsel geen te ondersoek. Hiervoor het ons ‘n wiskundige model ontwikkel, wat die wisselwerking tussen die dinamika van malaria en die sekelsel geen met malaria behandeling, beskryf. Die model sluit beide homosigotiese vir die normale geen (AA) en heterosigotiese vir die sekelsel geen (AS) in, en neem aan dat AS individue nie behandel is nie omdat hulle nie die eerste kliniese simptome getoon het nie. Ons ontleed eers die model sonder malaria behandeling, deur gebruik te maak van enkelvoudige pertubasie tegnieke, wat gegrond is op die feit dat epidemiologiese en demografiese dinamika plaasvind op twee verskillende tydskale (vinnige en stadige dinamika). Ons ontleding op die vinnige tydskaal dui dat hoë sekelsel geenfrekwensie onder heterosigotiese individue lei tot hoë endemiese vlakke vir ‘n langer duur van parasitemie. Nietemin, as die duur van parasitemie afneem, dan word hoë sekelsel geenfrekwensie verbind met lae endemiese vlakke. Ons neem ook waar dat op die stadige skaal die indringingsvermoë van die sekelsel afhanklik is van malaria se epidemiologiese parameters. Die indringingskoëffisiënt wat bereken word as die verskil van die geweegde sterftekoerse van AA en AS individue, word gebruik as ‘n maatstaf om die vestiging van die sekelsel geen in die bevolking te bepaal. Resultate toon dat die geen homself kan vestig as die geweegde sterftekoers van AA individue groter is as di e van die AS individue, andersins misluk dit. Ons let op dat hoë mortaliteit van AA individue lei tot die vestiging van die sekelsel geen in die bevolking. Daarna het ons die model wat behandeling insluit ge-analiseer en ons resultate toon dat die frekwensie van die sekelsel geen afneem met ‘n toename in die herstelkoers van AA individue. Ons kom dus tot die gevolgtrekking dat die uitwissing van malaria siekte sal lei tot die afname in sekelsel geenfrekwensie.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Some of the diagram lemmas of Homological Algebra, classically known for abelian categories, are not characteristic of the abelian context; this naturally leads to investigations of those non-abelian categories in which these diagram lemmas may hold. In this Thesis we attempt to bring together two different directions of such investigations; in particular, we unify the five lemma from the context of homological categories due to F. Borceux and D. Bourn, and the five lemma from the context of modular semi-exact categories in the sense of M. Grandis. / AFRIKAANSE OPSOMMING: Verskeie diagram lemmata van Homologiese Algebra is aanvanklik ontwikkel in die konteks van abelse kategorieë, maar geld meer algemeen as dit behoorlik geformuleer word. Dit lei op ’n natuurlike wyse na ’n ondersoek van ander kategorieë waar hierdie lemmas ook geld. In hierdie tesis bring ons twee moontlike rigtings van ondersoek saam. Dit maak dit vir ons moontlik om die vyf-lemma in die konteks van homologiese kategoieë, deur F. Borceux en D. Bourn, en vyflemma in die konteks van semi-eksakte kategorieë, in die sin van M. Grandis, te verenig.

Thesis (MSc)--University of Stellenbosch, 2007. / ENGLISH ABSTRACT: Buying and selling in financial markets are driven by demand. The demand can be quantified by the imbalance in the number of shares QB and QS transacted by buyers and sellers respectively over a given time interval t. The demand in an interval t is given by (t) = QB − QS. The local noise intensity is given by = h|aiqi − haiqii|i where i = 1, . . . ,N labels the transactions in t, qi is the number of shares traded in transaction i, ai = ±1 denotes buyer- initiated and seller- initiated trades respectively and h· · · i is the local expectation value computed from all the transactions during the interval t. In a paper [1] based on data from the New York Stock Exchange Trade and Quote database during the period 1995-1996, Plerou, Gopikrishnan and Stanley [1] reported that the analysis of the probability distribution P( | ) of demand conditioned on the local noise intensity revealed the surprising existence of a critical threshold c. For < c, the most probable value of demand is roughly zero; they interpreted this as an equilibrium phase in which neither buying nor selling predominates. For > c two most probable values emerge that are symmetrical around zero demand, corresponding to excess demand and excess supply; they interpreted this as an out-of-equilibrium phase in which the market behaviour is buying for half of the time, and selling for the other half. It was suggested [1] that the two-phase behaviour indicates a link between the dynamics of a financial market with many interacting participants and the phenomenon of phase transitions that occurs in physical systems with many interacting units. This thesis reproduces the two-phase behaviour by means of experiments using sequences of random variables. We reproduce the two-phase behaviour based on correlated and uncorrelatd data. We use a Markov modulated Bernoulli process to model the transactions and investigate a simple interpretation of the two-phase behaviour. We sample data from heavy-tailed distributions and reproduce the two-phase behaviour. Our experiments show that the results presented in [1] do not provide evidence for the presence of complex phenomena in a trading market; the results are a consequence of the sampling method employed. / AFRIKAANSE OPSOMMING: Aankope en verkope in finansi¨ele markte word deur aanvraag gedryf. Aanvraag kan gekwantifiseer word in terme van die ongebalanseerdheid in die getal aandele QB en QB soos onderskeidelik verhandel deur kopers en verkopers in ’n gegewe tyd-interval t. Die aanvraag in ’n interval t word gegee deur (t) = QB −QS. Die lokale geraasintensiteit word gegee deur = h|aiqi − haiqii|i waar i = 1, . . . ,N die transaksies in t benoem, qi die getal aandele verhandel in transaksies verwys, en h· · · i op die lokale verwagte waarde dui, bereken van al die tansaksies tydens die interval t. In ’n referaat [1] wat op data van die New York Effektebeurs se Trade and Quote databasis in die periode tussen 1995 en 1996 geskoei was, het Plerou, Gopikrishnan en Stanley [1] gerapporteer dat ’n analise van die waarskynlikheidsverspreiding P( | ) van aanvraag gekondisioneer op die lokale geraasintensiteit , die verrassende bestaan van ’n kritieke drempelwaarde c na vore bring. Vir < c is die mees waarskynlike aanvraagwaarde nagenoeg nul; hulle het dit ge¨ınterpreteer as ’n ekwilibriumfase waartydens n`og aankope n`og verkope die oormag het. Vir > c is die twee mees waarskynlike aanvraagwaardes wat te voorskyn kom simmetries rondom nul aanvraag, wat oorenstem met ’n oormaat aanvraag en ’n oormaat aanbod; hulle het dit geinterpreteer as ’n buite-ewewigfase waartydens die markgedrag die helfte van die tyd koop en die anderhelfte verkoop. Daar is voorgestel [1] dat die tweefase gedrag op ’n verband tussen die dinamiek van ’n finansiele mark met baie deelnemende partye, en die verskynsel van fase-oorgange wat in fisieke sisteme met baie wisselwerkende eenhede voorkom, dui. Hierdie tesis reproduseer die tweefase gedrag deur middel van eksperimente wat gebruik maak van reekse van lukrake veranderlikes. Ons reproduseer die tweefase gedrag gebaseer op gekorreleerde en ongekorreleerde data. Ons gebruik ’n Markov-gemoduleerde Bernoulli proses om die transaksies te moduleer en ondersoek ’n eenvoudige interpretasie van die tweefase gedrag. Ons seem steekproefdata van “heavy-tailed” verspreidings en reproduseer die tweefase gedrag. Ons ekperimente wys dat die resultate in [1] voorgested is nie bewys lewer vir die teenwoordigheid van komplekse verskynsel in’n handelsmark nie; die resultate is as gevolg van die metode wat gebruik is vir die generering van die steekproefdata.