Doctoral Degrees (Mathematics and Computer Science Education)

Permanent URI for this collectionhttps://hdl.handle.net/10413/7140

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A simulation modeling approach to aid research into the control of a stalk-borer in the South African Sugar Industry.
(2008) Horton, Petrovious Mitchell.; Sibanda, Precious.; Hearne, John W.; Conlong, Desmond Edward.; Apaloo, Joseph.
The control of the African stalk borer Eldana saccharina Walker (Lepidoptera: Pyralidae) in sugarcane fields of KwaZulu-Natal, South Africa has proved problematical. Researchers at the South African Sugarcane Research Institute (SASRI) have since 1974 been intensively investigating various means of controlling the pest. Among the methods of control currently being investigated are biological control, chemical control, production of more resistant varieties and crop management. These investigations, however, require many years of experimentation before any conclusions can be made. In order to aid the research currently being carried out in the Entomology Department at SASRI (to investigate biological control strategies, insecticide application strategies and the carry-over decision), a simulation model of E. saccharina growth in sugarcane has been formulated. The model is cohort-based and includes the effect of temperature on the physiological development of individuals in each life-stage of the insect. It also takes into account the effect of the condition of sugarcane on the rate of E. saccharina infestation, by making use of output from the sugarcane growth model CANEGRO. Further, a crop damage index is defined that gives an indication of the history of E. saccharina infestation levels during the sugarcane’s growth period. It is linked to the physiological activity of the borer during the period spent feeding on the stalk tissue. The damage index can further be translated into length of stalks bored and hence the percentage of the stalk length bored can be calculated at each point in the simulation using the total length of stalks calculated in the CANEGRO model. Using an industry accepted relationship between percent stalks damaged and reduction in sucrose content of the crop, reductions in losses in the relative value of the crop when the various control measures are implemented can be compared. Relationships between the reduction in percent stalk length bored (and hence gains in the relative value of the crop) and the various control strategies are obtained.
An algebraic study of residuated ordered monoids and logics without exchange and contraction.
(1998) Van Alten, Clint Johann.; Raftery, James Gordon.
Please refer to the thesis for the abstract.
Amplitude-shape method for the numerical solution of ordinary differential equations.
(1997) Parumasur, Nabendra.; Banasiak, Jacek.; Mika, Janusz R.
In this work, we present an amplitude-shape method for solving evolution problems described by partial differential equations. The method is capable of recognizing the special structure of many evolution problems. In particular, the stiff system of ordinary differential equations resulting from the semi-discretization of partial differential equations is considered. The method involves transforming the system so that only a few equations are stiff and the majority of the equations remain non-stiff. The system is treated with a mixed explicit-implicit scheme with a built-in error control mechanism. This approach proved to be very effective for the solution of stiff systems of equations describing spatially dependent chemical kinetics.
An exploration of preservice teachers’ use of educational technologies as visualization tools when teaching mathematics.
(2022) Zulu, Mzwandile Wiseman.; Mudaly, Vimolan.
This interpretive qualitative study explores the use of educational technologies by preservice teachers as visualization tools during mathematics teaching at secondary schools. Sfard’s commognitive framework and Koehler and Mishra’s technological pedagogical content knowledge theoretical frameworks undergird the study. Data were collected from ten preservice mathematics teachers at a university in the province of KwaZulu-Natal, South Africa. Performance tests, semi-structured interviews, focus group discussions, and observations were employed to collect data, which was analyzed using reflexive thematic analysis. The study found that preservice teachers employed two primary visualization strategies when they engaged in mathematical problem-solving: symbolic mental visualization, which they combined with their understanding of word usage, endorsed narratives and routines to arrive at a solution; and graphic visual mediators, such as diagrams, which they sketched to contextualize the problem statement and verify they solutions and use of mathematical word usage, routines, and endorsed narratives. Participants were found to be unable to solve a mathematics problem if they had not visualized it effectively; using a graphic visual mediator to understand the problem statement did not, however, guarantee success when solving a problem. A relationship was found between the visualization techniques that the participants used in their own attempts to solve mathematical problems and the visualization techniques they used in their lesson planning and teaching of mathematics in the same content area. Participants used innovative strategies to mediate learning, including educational technologies that facilitated visual mediators to enhance learners’ engagement with concepts. Synergies were found between the elements of the commognition and TPACK frameworks as these were used in tandem to analyze data. A model was developed (C+TPACK) that integrates the key elements of these frameworks. Further research is recommended to establish the viability, credibility and generalizability of the model.
Anisotropic stars in general relativity.
(2004) Chaisi, Mosa.; Maharaj, Sunil Dutt.
In this thesis we seek new solutions to the anisotropic Einstein field equations which are important in the study of highly dense stellar structures. We first adopt the approach used by Maharaj & Maartens (1989) to obtain an exact anisotropic solution in terms of elementary functions for a particular choice of the energy density. This class of solution contains the Maharaj & Maartens (1989) and Gokhroo & Mehra (1994) models as special cases. In addition, we obtain six other new solutions following the same approach for different choices of the energy density. All the solutions in this section reduce to one with the energy density profile f-L ex r-2 . Two new algorithms are generated, Algorithm A and Algorithm B, which produce a new anisotropic solution to the Einstein field equations from a given isotropic solution. For any new anisotropic solution generated with the help of these algorithms, the original isotropic seed solution is regained as a special case. Two examples of known isotropic solutions are used to demonstrate how Algorithm A and Algorithm B work, and to obtain new anisotropic solutions for the Einstein and de Sitter models. Anisotropic isot~ermal sphere models are generated given the corresponding isotropic (f-L ex r-2 ) solution of the Einstein field equations. Also, anisotropic interior Schwarzschild sphere models are found given the corresponding isotropic (f-L ex constant) solution of the field equations. The exact solutions and line elements are given in each case for both Algorithm A and Algorithm B. Note that the solutions have a simple form and are all expressible in terms of elementary functions. Plots for the anisotropic factor S = J3(Pr - pJJ/2 (where Pr and Pl. are radial and tangential pressure respectively) are generated and these point to physically viable models.
An APOS analysis of the teaching and learning of factorisation of quadratic expressions in grade 10 mathematics classrooms.
(2021) Vilakazi, Aubrey Sifiso.; Bansilal, Sarah.
The South African Curriculum and Assessment Policy Statements (CAPS) document, for the Further Education and Training Phase (FET) Mathematics Grades 10-12 (2011) shows that the factorisation of algebraic quadratic expressions or equations pervades the mathematics of the secondary school. As a result, for learners to be successful at mathematics in Grade 12, they need to know a great deal of algebra, particularly the factorisation of quadratics. It is therefore important for us as mathematics educators to identify areas in the factorization of quadratics that teachers and learners are struggling to learn and apply. With this in mind, the study sets to embark on an APOS analysis of the teaching and learning of factorisation of quadratic expressions in Grade 10 mathematics classrooms. Following on from the research questions, this study is located within the principles of the mixed methods case study approach. The combination of methodologies has allowed me to identify broad trends across the groups of learners and those of educators as a whole as well as differences within the participants of the groups themselves. The participants of the study were the groups of Grade 10 learners from the two participating schools, as well as the Grade 10 mathematics teachers from the two circuits of Ilembe District. Five sources of data were used. Firstly, data were generated from 25 teachers from the two circuits who participated in the teachers’ questionnaires. A second data collection instrument was the classroom lessons’ observations of the six teachers. A third data source was the learner group activity and learners’ interviews administered to 12 learners. A fourth data source was the unstructured interviews with six teachers. The final instrument was the analysis of the 205 Grade 10 mathematics 2019 March common paper learners’ scripts. This study was guided by the theory of constructivism and more specifically Action, Process, Object, Schema (APOS) theory which views learning as changes in conception. As an individual engages with a concept, the conception changes from an initial external view towards seeing the concept as a totality upon which other Actions and Processes can act. This study has found that, firstly, teachers and learners tend to rely too much on the use of rules in factoring certain quadratics. In so doing, a prototype of the quadratic expression concept is perceived which consists of isolated and disconnected concepts. As a result, most learners were not able to factor the trinomial quadratic of 𝑎≠1, since they perceived the factoring of 𝑎𝑥2+𝑏𝑥+𝑐 with 𝑎=1 and that of 𝑎≠1 as two different procedures. Secondly, there are also students whose mental constructions (conception) are limited to Action levels in terms of APOS theory. The findings of the study suggest that teachers and learners should be able to consider quadratic expressions as one big idea and follow the fundamental considerations when factoring the quadratic expressions. Furthermore the use of multi-methods in factoring quadratics is encouraged and needed for students to better understand the connections between different methodologies for conceptual development.
An APOS exploration of conceptual understanding of the chain rule in calculus by first year engineering students.
(2011) Jojo, Zingiswa Mybert Monica.; Brijlall, Deonarain.; Maharaj, Aneshkumar.
The main issue in this study is how students conceptualise mathematical learning in the context of calculus with specific reference to the chain rule. The study focuses on how students use the chain rule in finding derivatives of composite functions (including trigonometric ones). The study was based on the APOS (Action-Process-Objects-Schema) approach in exploring conceptual understanding displayed by first year University of Technology students in learning the chain rule in calculus. The study consisted of two phases, both using a qualitative approach. Phase 1 was the pilot study which involved collection of data via questionnaires which were administered to 23 previous semester students of known ability, willing to participate in the study. The questionnaire was then administered to 30 volunteering first year students in Phase 2. A structured way to describe an individual student's understanding of the chain rule was developed and applied to analyzing the evolution of that understanding for each of the 30 first year students. Various methods of data collection were used namely: (1) classroom observations, (2) open-ended questionnaire, (3) semi-structured and unstructured interviews, (4) video-recordings, and (5) written class work, tests and exercises. The research done indicates that it is essential for instructional design to accommodate multiple ways of function representation to enable students to make connections and have a deeper understanding of the concept of the chain rule. Learning activities should include tasks that demand all three techniques, Straight form technique, Link form technique and Leibniz form technique, to cater for the variation in learner preferences. It is believed that the APOS paradigm using selected activities brought the students to the point of being better able to understand the chain rule and informed the teaching strategies for this concept. In this way, it is believed that this conceptualization will enable the formulation of schema of the chain rule which can be applied to a wider range of contexts in calculus. There is a need to establish a conceptual basis that allows construction of a schema of the chain rule. The understanding of the concept with skills can then be augmented by instructional design based on the modified genetic decomposition. This will then subject students to a better understanding of the chain rule and hence more of calculus and its applications.
The application of Rasch measurement theory to improve the functioning of a mathematics assessment instrument.
(2021) Ngirishi, Harrison.; Bansilal, Sarah.
Assessment is an integral part of the teaching and learning process. Concerns about student performance in assessments often drive the teaching and learning. In South Africa there has been numerous concerns about poor learning outcomes in mathematics and this has led to calls for all stakeholders to work together to try and find solutions. This study focuses on the assessment of mathematics with particular interest in the KZN provincial Grade 12 mathematics trial examination paper 2. The study explored the use of Rasch analysis in improving the functioning of the mathematics assessment instrument. The aim of the study was to use the Rasch analysis to report on the functioning of the test instrument in measuring proficiency in mathematics, checking on the targeting and reliability of the test instrument, explain anomalies where data did not fit the Rasch model, investigate differential item functioning (DIF), response dependency and multidimensionality. The study also sought the teachers’ views about the findings of the Rasch analysis. A sequential explanatory design was used in this study, where the Rasch analysis provided the theoretical framework for the analysis of the quantitative data. The qualitative analysis of the teachers’ responses helped to get more understanding of the results of the quantitative analysis of the leaners’ responses. The study found that the assessment instrument was difficult for this particular cohort, some items displayed DIF for language and response dependency due to some teachers not applying continuous accuracy marking. The study revealed that some teachers were not applying the continuous accuracy marking process. Items which carried more than two accuracy marks, showed misfit to the Rasch model. Teachers cited not applying continuous accuracy marking due to time constraints and large number of learners in classes. Teachers blamed poor performance on learners’ lack of basic understanding, adequate preparation and motivation, societal influences, poor understanding of proof type questions, allocation of many accuracy marks on one item and the language barrier. The recommendations of this study if implemented may help teachers in the teaching process and examiners in producing fair assessment instruments. The recommendations may lead to improvement of mathematics results.
Aspects of distance and domination in graphs.
(1995) Smithdorf, Vivienne.; Swart, Hendrika Cornelia Scott.; Dankelmann, Peter A.
The first half of this thesis deals with an aspect of domination; more specifically, we investigate the vertex integrity of n-distance-domination in a graph, i.e., the extent to which n-distance-domination properties of a graph are preserved by the deletion of vertices, as well as the following: Let G be a connected graph of order p and let oi- S s;:; V(G). An S-n-distance-dominating set in G is a set D s;:; V(G) such that each vertex in S is n-distance-dominated by a vertex in D. The size of a smallest S-n-dominating set in G is denoted by I'n(S, G). If S satisfies I'n(S, G) = I'n(G), then S is called an n-distance-domination-forcing set of G, and the cardinality of a smallest n-distance-domination-forcing set of G is denoted by On(G). We investigate the value of On(G) for various graphs G, and we characterize graphs G for which On(G) achieves its lowest value, namely, I'n(G), and, for n = 1, its highest value, namely, p(G). A corresponding parameter, 1](G), defined by replacing the concept of n-distance-domination of vertices (above) by the concept of the covering of edges is also investigated. For k E {a, 1, ... ,rad(G)}, the set S is said to be a k-radius-forcing set if, for each v E V(G), there exists Vi E S with dG(v, Vi) ~ k. The cardinality of a smallest k-radius-forcing set of G is called the k-radius-forcing number of G and is denoted by Pk(G). We investigate the value of Prad(G) for various classes of graphs G, and we characterize graphs G for which Prad(G) and Pk(G) achieve specified values. We show that the problem of determining Pk(G) is NP-complete, study the sequences (Po(G),Pl(G),P2(G), ... ,Prad(G)(G)), and we investigate the relationship between Prad(G)(G) and Prad(G)(G + e), and between Prad(G)(G + e) and the connectivity of G, for an edge e of the complement of G. Finally, we characterize integral triples representing realizable values of the triples b,i,p), b,l't,i), b,l'c,p), b,l't,p) and b,l't,l'c) for a graph.
Aspects of functional variations of domination in graphs.
(2003) Harris, Laura Marie.; Henning, Michael Anthony.; Hattingh, Johannes H.
Let G = (V, E) be a graph. For any real valued function f : V >R and SCV, let f (s) = z ues f(u). The weight of f is defined as f(V). A signed k-subdominating function (signed kSF) of G is defined as a function f : V > {-I, I} such that f(N[v]) > 1 for at least k vertices of G, where N[v] denotes the closed neighborhood of v. The signed k-subdomination number of a graph G, denoted by yks-11(G), is equal to min{f(V) I f is a signed kSF of G}. If instead of the range {-I, I}, we require the range {-I, 0, I}, then we obtain the concept of a minus k-subdominating function. Its associated parameter, called the minus k-subdomination number of G, is denoted by ytks-101(G). In chapter 2 we survey recent results on signed and minus k-subdomination in graphs. In Chapter 3, we compute the signed and minus k-subdomination numbers for certain complete multipartite graphs and their complements, generalizing results due to Holm [30]. In Chapter 4, we give a lower bound on the total signed k-subdomination number in terms of the minimum degree, maximum degree and the order of the graph. A lower bound in terms of the degree sequence is also given. We then compute the total signed k-subdomination number of a cycle, and present a characterization of graphs G with equal total signed k-subdomination and total signed l-subdomination numbers. Finally, we establish a sharp upper bound on the total signed k-subdomination number of a tree in terms of its order n and k where 1 < k < n, and characterize trees attaining these bounds for certain values of k. For this purpose, we first establish the total signed k-subdomination number of simple structures, including paths and spiders. In Chapter 5, we show that the decision problem corresponding to the computation of the total minus domination number of a graph is NP-complete, even when restricted to bipartite graphs or chordal graphs. For a fixed k, we show that the decision problem corresponding to determining whether a graph has a total minus domination function of weight at most k may be NP-complete, even when restricted to bipartite or chordal graphs. Also in Chapter 5, linear time algorithms for computing Ytns-11(T) and Ytns-101(T) for an arbitrary tree T are presented, where n = n(T). In Chapter 6, we present cubic time algorithms to compute Ytks-11(T) and Ytks-101l(T) for a tree T. We show that the decision problem corresponding to the computation of Ytks-11(G) is NP-complete, and that the decision problem corresponding to the computation of Ytks-101 (T) is NP-complete, even for bipartite graphs. In addition, we present cubic time algorithms to computeYks-11(T) and Yks-101(T) for a tree T, solving problems appearing in [25].
Coagulation-fragmentation dynamics in size and position structured population models.
(2008) Noutchie, Suares Cloves Oukouomi.; Banasiak, Jacek.
One of the most interesting features of fragmentation models is a possibility to breach
Conformally invariant relativistic solutions.
(1993) Maharaj, M. S.; Maharaj, Sunil Dutt.; Maartens, Roy.
The study of exact solutions to the Einstein and Einstein-Maxwell field equations, by imposing a symmetry requirement on the manifold, has been the subject of much recent research. In this thesis we consider specifically conformal symmetries in static and nonstatic spherically symmetric spacetimes. We find conformally invariant solutions, for spherically symmetric vectors, to the Einstein-Maxwell field equations for static spacetimes. These solutions generalise results found previously and have the advantage of being regular in the interior of the sphere. The general solution to the conformal Killing vector equation for static spherically symmetric spacetimes is found. This solution is subject to integrability conditions that place restrictions on the metric functions. From the general solution we regain the special cases of Killing vectors, homothetic vectors and spherically symmetric vectors with a static conformal factor. Inheriting conformal vectors in static spacetimes are also identified. We find a new class of accelerating, expanding and shearing cosmological solutions in nonstatic spherically symmetric spacetimes. These solutions satisfy an equation of state which is a generalisation of the stiff equation of state. We also show that this solution admits a conformal Killing vector which is explicitly obtained.
Conservation laws models in networks and multiscale flow optimization.
(2011) Tchoukouegno Ngnotchouye, Jean Medard.; Banda, Mapundi K.; Sibanda, Precious.
The flow of fluids in a network is of practical importance in gas, oil and water transport for industrial and domestic use. When the flow dynamics are understood, one may be interested in the control of the flow formulated as follows: given some fluid properties at a final time, can one determine the initial flow properties that lead to the desired flow properties? In this thesis, we first consider the flow of a multiphase gas, described by the drift flux model, in a network of pipes and that of water, modeled by the shallow water equations, in a network of rivers. These two models are systems of partial differential equations of first order generally referred to as systems of conservation laws. In particular, our contribution in this regard can be summed up as follows: For the drift-flux model, we consider the flow in a network of pipes seen mathematically as an oriented graph. We solve the standard Riemann problem and prove a well posedness result for the Riemann problem at a junction. This result is obtained using coupling conditions that describe the dynamics at the intersection of the pipes. Moreover, we present numerical results for standard pipes junctions. The numerical results and the analytical results are in agreement. This is an extension for multiphase flows of some known results for single phase flows. Thereafter, the shallow water equations are considered as a model for the flow of water in a network of canals. We analyze coupling conditions at the confluence of rivers, precisely the conservation of mass and the equality of water height at the intersection, and implement these results for some classical river confluences. We also consider the case of pooled stepped chutes, a geometry frequently utilized by dams to spill floodwater. Here we consider an approach different from the engineering community in the sense that we resolve the dynamics by solving a Riemann problem at the dam for the shallow water equations with some suitable coupling conditions. Secondly, we consider an optimization problem constrained by the Euler equations with a flow-matching objective function. Differently from the existing approaches to this problem, we consider a linear approximation of the flow equation in the form of the microscopic Lattice Boltzmann Equations (LBE). We derive an adjoint calculus and the optimality conditions from the microscopic LBE. Using multiscale analysis, we obtain an equivalent macroscopic result at the hydrodynamic limit. Our numerical results demonstrate the ability of our method to solve challenging problems in fluid mechanics.
Constructions and justifications of a generalization of Viviani's theorem.
(2013) Govender, Rajendran.; De Villiers, Michael David.
This qualitative study actively engaged a group of eight pre-service mathematics teachers (PMTs) in an evolutionary process of generalizing and justifying. It was conducted in a developmental context and underpinned by a strong constructivist framework. Through using a set of task based activities embedded in a dynamic geometric context, this study firstly investigated how the PMTs experienced the reconstruction of Viviani’s theorem via the processes of experimentation, conjecturing, generalizing and justifying. Secondly, it was investigated how they generalized Viviani’s result for equilateral triangles, further across to a sequence of higher order equilateral (convex) polygons such as the rhombus, pentagon, and eventually to ‘any’ convex equi-sided polygon, with appropriate forms of justifications. This study also inquired how PMTs experienced counter-examples from a conceptual change perspective, and how they modified their conjecture generalizations and/or justifications, as a result of such experiences, particularly in instances where such modifications took place. Apart from constructivsm and conceptual change, the design of the activities and the analysis of students’ justifications was underpinned by the distinction of the so-called ‘explanatory’ and ‘discovery’ functions of proof. Analysis of data was grounded in an analytical–inductive method governed by an interpretive paradigm. Results of the study showed that in order for students to reconstruct Viviani’s generalization for equilateral triangles, the following was required for all students: *experimental exploration in a dynamic geometry context; *experiencing cognitive conflict to their initial conjecture; *further experimental exploration and a reformulation of their initial conjecture to finally achieve cognitive equilibrium. Although most students still required the aforementioned experiences again as they extended the Viviani generalization for equilateral triangles to equilateral convex polygons of 4 sides (rhombi) and five sides (pentagons), the need for experimental exploration gradually subsided. All PMTs expressed a need for an explanation as to why their equilateral triangle conjecture generalization was always true, and were only able to construct a logical explanation through scaffolded guidance with the means of a worksheet. The majority of the PMTs (i.e. six out of eight) extended the Viviani generalization to the rhombus on empirical grounds using Sketchpad while two did so on analogical grounds but superficially. However, as the PMTs progressed to the equilateral pentagon (convex) problem, the majority generalized on either inductive grounds or analogical grounds without the use of Sketchpad. Finally all of them generalized to any convex equi-sided polygon on logical grounds. In so doing it seems that all the PMTs finally cut off their ontological bonds with their earlier forms or processes of making generalizations. This conceptual growth pattern was also exhibited in the ways the PMTs justified each of their further generalizations, as they were progressively able to see the general proof through particular proofs, and hence justify their deductive generalization of Viviani’s theorem. This study has also shown that the phenomenon of looking back (folding back) at their prior explanations assisted the PMTs to extend their logical explanations to the general equi-sided polygon. This development of a logical explanation (proof) for the general case after looking back and carefully analysing the statements and reasons that make up the proof argument for the prior particular cases (i.e. specific equilateral convex polygons), namely pentagon, rhombus and equilateral triangle, emulates the ‘discovery’ function of proof. This suggests that the ‘explanatory’ function of proof compliments and feeds into the ‘discovery’ function of proof. Experimental exploration in a dynamic geometry context provided students with a heuristic counterexample to their initial conjectures that caused internal cognitive conflict and surprise to the extent that their cognitive equilibrium became disturbed. This paved the way for conceptual change to occur through the modification of their postulated conjecture generalizations. Furthermore, this study has shown that there exists a close link between generalization and justification. In particular, justifications in the form of logical explanations seemed to have helped the students to understand and make sense as to why their generalizations were always true, but through considering their justifications for their earlier generalizations (equilateral triangle, rhombus and pentagon) students were able to make their generalization to any convex equi-sided polygon on deductive grounds. Thus, with ‘deductive’ generalization as shown by the students, especially in the final stage, justification was woven into the generalization itself. In conclusion, from a practitioner perspective, this study has provided a descriptive analysis of a ‘guided approach’ to both the further constructions and justifications of generalizations via an evolutionary process, which mathematics teachers could use as models for their own attempts in their mathematics classrooms.
Convective heat and mass transfer in boundary layer flow through porous media saturated with nanofluids.
(2016) Haroun, Nageeb Abdallah Hamid.; Sibanda, Precious.
The thesis is devoted to the study of flow, heat and mass transfer processes, and crossdiffusion effects in convective boundary layer flows through porous media saturated with nanofluids. Of particular interest is how nanofluids perform as heat transfer fluids compared to traditional fluids such as oil and water. Flow in different geometries and subject to various source terms is investigated. An important aspect of the study and understanding of transport processes is the solution of the highly non-linear coupled differential equations that model both the flow and the heat transportation. In the literature, various analytical and numerical methods are available for finding solutions to fluid flow equations. However, not all these methods give accurate solutions, are stable, or are computationally efficient. For these reasons, it is important to constantly devise numerical schemes that work more efficiently, including improving the performance of existing schemes, to achieve accuracy with less computational effort. In this thesis the systems of differential equations that describe the fluid flow and other transport processes were solved numerically using both established and recent numerical schemes such as the spectral relaxation method and the spectral quasilinearization method. These spectral methods have been used only in a limited number of studies. There is therefore the need to test and prove the accuracy and general application of the methods in a wider class of boundary value problems. The accuracy, convergence, and validity of the solutions obtained using spectral methods, have been established by careful comparison with solutions for limiting cases in the published literature, or by use of a different solution method. In terms of understanding the physically important variables that impact the flow, we have inter alia, investigated the significance of different fluid and physical parameters, and how changes in these parameters affect the skin friction coefficient, the heat and mass transfer rates and the fluid properties. Some system parameters of interest in this study include the nanoparticle volume fraction, the Hartmann number, thermal radiation, Brownian motion, the heat generation, the Soret and Dufour effects, and the Prandtl and Schmidt number. The dependency of the heat, mass transfer and skin friction coefficients on these parameters has been quantified and discussed. In this thesis, we show that nanofluids have a significant impact on heat and mass transfer processes compared with traditional heat transfer fluids.
A critical analysis of technology adoption in teaching by in-service teachers in Botswana during the Covid-19 pandemic.
(2022) Mafa, Rodnie Kgalemelo.; Govender, Desmond Wesley.
Information and Communication Technologies (ICTs) have been growing immensely for over the years in almost every sector of the economy. Even though this is the case, a lot still needs to be achieved in the education sector. Facilitators in Botswana schools are still depending and relying on the traditional teaching pedagogies and are far from benefiting from the rewards ICTs are bringing to the classroom to enhance teaching and learning. This study critically analysed technology adoption and use in teaching by in-service teachers in Botswana during the Covid - 19 era. It also analysed the ICT infrastructure found and available in Botswana schools and assessed the facilitator's skills, knowledge confidence and their perceptions on ICT integration in teaching and learning. A mixed methods (quantitative and qualitative) approach was adopted and used to collect data for the study. The major findings of the study revealed that the integration of ICT teaching in Botswana schools during the Covid- 19 era showed some challenges in teaching and learning and yet was also beneficial. These challenges were influenced by several factors such as lack of ICT skills, lack of self-confidence in the usage of technology by teachers and lack of appropriate JCT tools for use in schools. The study recommends that all schools (government and public) should be equipped and installed with the right ICTs supporting infrastructure, resources, and that all teachers be trained on the right pedagogies on the integration of ICTs in teaching and learning. In addition, the schools-based curriculums should be designed and developed in a way that accommodates ICT integration in teaching all subjects areas taught in schools.
Design model for integrating learning management systems and massive open online courses on a digital e-Learning platform: implications for Zimbabwean universities.
(2019) Rugube, Talent Tapiwa.; Chibaya, Colin.; Govender, Desmond Wesley.
Available affordances for learning provide opportunities for advanced technology-enhanced teaching and learning. Digital learning environments can make relevant learning content available to students using existing infrastructure. This creates an environment which requires different learning management systems (LMS) to interact with, and exchange information. Increasing use of mobile devices, digital learning platforms, LMS, and massive open online courses (MOOCs), has necessitated integration design approaches. However, ignorance of resources offered and discouragement and frustrations arising from the economic situation in Zimbabwe regarding regulated access to electronic services make automation of teaching processes a great challenge. In this thesis, a design model for integrating LMS and MOOCs on a digital learning platform is proposed. From an e-learning point of view, the study contributes to the working of e-learning management systems through automation process of uploading content to LMS. From a computer science point of view, the study contributes to software engineering principles where it puts together three different platforms; LMS, MOOCs and digital learning platforms under one design. Methodologically, the study uses design science research (DSR) framework with software modelling language to address challenges in teaching and learning. This study describes how the Technology Adoption Model (TAM) and Task-Technology Fit (TTF) model can be used together with DSR in relation to design model evaluation. A software modeling language was used to create the logical designs, which were evaluated using experimental design approach. Software engineering experts and lecturers were invited to validate proposed logical designs. The key deliverables of the study include requirements specifications for the design model for integrated learning management systems, as well as the logical designs for the design model. The design model, as per requirements specification and the evaluation thereof, are based on TAM and TTF. The hybrid model proposed was further validated using structural equation modeling via the partial least squares and path modeling. In our views, the interventions of integration work would support decision making, which influences choices made by policy makers when taking decisions about higher education technological infrastructure.
Embedding theorems and finiteness properties for residuated structures and substructural logics
(2008) Hsieh, Ai-Ni.; Raftery, James Gordon.
Paper 1. This paper establishes several algebraic embedding theorems, each of which asserts that a certain kind of residuated structure can be embedded into a richer one. In almost all cases, the original structure has a compatible involution, which must be preserved by the embedding. The results, in conjunction with previous findings, yield separative axiomatizations of the deducibility relations of various substructural formal systems having double negation and contraposition axioms. The separation theorems go somewhat further than earlier ones in the literature, which either treated fewer subsignatures or focussed on the conservation of theorems only. Paper 2. It is proved that the variety of relevant disjunction lattices has the finite embeddability property (FEP). It follows that Avron’s relevance logic RMImin has a strong form of the finite model property, so it has a solvable deducibility problem. This strengthens Avron’s result that RMImin is decidable. Paper 3. An idempotent residuated po-monoid is semiconic if it is a subdirect product of algebras in which the monoid identity t is comparable with all other elements. It is proved that the quasivariety SCIP of all semiconic idempotent commutative residuated po-monoids is locally finite. The lattice-ordered members of this class form a variety SCIL, which is not locally finite, but it is proved that SCIL has the FEP. More generally, for every relative subvariety K of SCIP, the lattice-ordered members of K have the FEP. This gives a unified explanation of the strong finite model property for a range of logical systems. It is also proved that SCIL has continuously many semisimple subvarieties, and that the involutive algebras in SCIL are subdirect products of chains. Paper 4. Anderson and Belnap’s implicational system RMO can be extended conservatively by the usual axioms for fusion and for the Ackermann truth constant t. The resulting system RMO is algebraized by the quasivariety IP of all idempotent commutative residuated po-monoids. Thus, the axiomatic extensions of RMO are in one-to-one correspondence with the relative subvarieties of IP. It is proved here that a relative subvariety of IP consists of semiconic algebras if and only if it satisfies x (x t) x. Since the semiconic algebras in IP are locally finite, it follows that when an axiomatic extension of RMO has ((p t) p) p among its theorems, then it is locally tabular. In particular, such an extension is strongly decidable, provided that it is finitely axiomatized.
Evolving dynamic fitness measures for genetic programming.
(2018) Ragalo, Anisa Waganda.; Pillay, Nelishia.
This research proposes dynamic fitness measure genetic programming (DFMGP). DFMGP modifies the conventional genetic programming (GP) approach: rather than applying a single fitness measure individually throughout GP, a different fitness measure (or combination of fitness measures) is applied on each GP generation. A detailed review of the fitness measures used in GP is presented. The review demonstrates that different fitness measures were introduced to overcome different shortcomings, e.g. escaping local optima, reducing bloat, thereby improving on the performance of the GP algorithm. A subsequent analysis of the fitness measures shows that there is no universal “best” fitness measure; rather, different fitness measures are appropriate for different problems. The literature also anticipates that applying different fitness measures at different points of the GP problem solving process should be more effective then applying a single fitness measure throughout the algorithm. Hence the case for DFMGP. Selecting the fitness measures to apply on each GP generation is in itself a combinatorial optimization problem: the study investigates two approaches to serve this purpose, namely, a genetic algorithm and genetic programming. The genetic algorithm (GA) derives a sequence of fitness measures to be applied, while GP produces an arithmetic function combining the fitness measures. The performance of DFMGP applying the evolved fitness measure sequences and DFMGP applying the evolved fitness measure combinations is compared to the conventional GP approach on a number of benchmark and complex, real-world problems. DFMGP is found to be more effective than standard GP. The study also reveals that both the sequences and arithmetic combinations of the fitness measures are effective when applied to problem instances different from those used to derive them. Hence, the sequences and arithmetic combinations are reusable, whereby simpler problems are used for derivation, and DFMGP applying the derived fitness measures is then used to solve more complex problems. Therefore the time necessary for the derivations is reduced. An analysis of the evolved sequences and arithmetic combinations of the fitness measures shows that fitness measures applied in the preliminary DFMGP generations support exploration while those applied in later DFMGP generations support exploitation. GP search is a constant balance between exploration and exploitation, with the former being more suited to the preliminary generations, and the latter, later generations. DFMGP’s performance advantage over standard GP is therefore justified by the premise that the fitness measure used on each generation supports the more suitable search in the on-going phase of GP. DFMGP applying the fitness measure combinations derived by GP is also found to perform better than DFMGP applying the fitness measure sequences derived by the GA. The former approach facilitates combining explorative and exploitative fitness measures on some of the DFMGP generations, whereby rather than simply switching between exploration and exploitation, the fitness measure can drive the two processes to occur simultaneously when required. Hence it follows that GP searching the space of fitness measure combinations is the preferred approach to generating dynamic fitness measures for DFMGP. Overall, the study reveals the effectiveness of DFMGP when applied to benchmark and real-world problems. Future work will look at a priori detecting the properties of complex problems, such that simpler problems with similar properties can be used to derive better dynamic fitness measures for DFMGP.
Exact models for radiating relativistic stars.
(2007) Rajah, Suryakumari Surversperi.; Maharaj, Sunil Dutt.
In this thesis, we seek exact solutions for the interior of a radiating relativistic star undergoing gravitational collapse. The spherically symmetric interior spacetime, when matched with the exterior radiating Vaidya spacetime, at the boundary of the star, yields the governing equation describing the gravitational behaviour of the collapsing star. The investigation of the model hinges on the solution of the governing equation at the boundary. We first examine shear-free models which are conformally flat. The boundary condition is transformed to an Abel equation and several new solutions are generated. We then study collapse with shear in geodesic motion. Two classes of solutions are generated which are regular at the stellar centre. Our treatment extends the results of Naidu et al (2006) which had the undesirable feature of a singularity at the centre of the star. In an attempt to find more general models, we transform the fundamental equation to a Riccati equation. Two general classes of solution are found and are used to study the thermal evolution in the causal theory of thermodynamics. These solutions are shown to reduce to the Friedmann dust solution in the absence of heat flow. Furthermore, we obtain new categories of solutions for the case of gravitational collapse with expansion, shear and acceleration of the stellar fluid. This is achieved by transforming the boundary condition into a Riccati equation. In special cases the Bernoulli equation is regained. The solutions are given in terms of elementary functions and they permit the investigation of the physical features of radiative stellar collapse.

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