Theses and Dissertations (Mathematics and Applied Mathematics)

Permanent URI for this collectionhttp://hdl.handle.net/2263/32239

Browse

Now showing 1 - 20 of 178

Completeness properties in the vector lattice C(X)
(University of Pretoria, 2024-07-12) Van der Walt, Jan Harm; Schwanke, Chris; odehyee.kwadwo@gmail.com; Afrane-Okese, Kwadwo Nyamedehyee
In this thesis we study certain vector lattice properties of the space $C(X)$ of continuous functions on a given topological space X. We show that C(X) is always relatively uniformly complete, and characterize those X for which C(X) is Dedekind complete. We characterise the bands and projection bands in C(X), for X a Tychonoff space, and characterize those Tychonoff spaces X for which C(X) has the projection property.
Finite element analysis of multi-dimensional and simplified models for beams and plates
(University of Pretoria, 2024-04-26) Van Rensburg, N.F.J.; rudidutoit071@gmail.com; Du Toit, Rudi
This dissertation is a literature study that investigates the validity of different linear models in application. Validity, in this context, refers to how well simplified lower-dimensional models, such as the Timoshenko beam and Reissner-Mindlin plate models, compare to more realistic higher-dimensional models, including two-dimensional beams and three-dimensional beams and plate models. All the models in this dissertation are special cases of a general vibration problem. First, the dissertation examines the existence and uniqueness of the general vibration problem. An example is used to explain the theory, which is then applied by proving that the assumptions are satisfied. Following this, the concept of modal analysis is introduced using an example, before delving into the general case. These results on modal analysis are crucial to the dissertation, as they explain that the solutions of the models will compare well if the eigenvalues and eigenfunctions of the models compare well. The dissertation then explores two theoretical results for the Finite Element Method (FEM). The initial result involves an analysis of an article on the convergence of the Galerkin Approximation. The findings of the article are reformulated as theorems with simplified notation for clearer presentation. Subsequently, the dissertation reviews results from a textbook regarding the convergence of eigenvalues and eigenfunctions in a general vibration problem when utilizing FEM. These results are adapted with updated notation and expanded upon for a more comprehensive explanation of the theory. Concerning the Timoshenko beam model, the dissertation investigates an article that presents a method to calculate the exact solutions of the eigenvalue problem. Two practical examples are provided to illustrate the application of this method. Additionally, the dissertation looks at an article that compares the theoretical results of the eigenvalue problem with an empirical study done by the authors. For the remaining models, the dissertation employs FEM to solve the eigenvalue problems. The boundary value problems of each model are rewritten as systems of ordinary differential equations in matrix form using FEM. The eigenvalue problems are then derived from this matrix representation. Piecewise Hermite cubic basis functions are used, and the solutions of the eigenvalue problems are approximated using MATLAB scripts. In investigating the validity of simplified models, the dissertation first considers an article comparing the Timoshenko beam model to a two-dimensional beam model. The authors' method of comparison is discussed, and their results are replicated with a higher degree of accuracy. The dissertation then extends this approach to assess the validity of a two-dimensional beam model and a Reissner-Mindlin plate model, using the three-dimensional model as a reference. The method to compare the models is the same as in the article: first, the eigenvalues are calculated, sorted, and matched by analyzing the corresponding mode shapes. The mode shapes are also used to identify eigenvalues specific to beam- and plate-type problems. The error can then be calculated. Different shapes of beams and plate models that are realistic in application are considered. The derivation and comparison of the two- and three-dimensional models are the main contributions of this dissertation.
Positive operators and their applications on ordered vector spaces
(University of Pretoria, 2023) Mabula, Mokhwetha D.; u17318450@tuks.co.za; Msibi, Mxolisi
A vector space X is called an ordered vector space if for any elements x, y, z ∈ X and α ∈ R+, x ⪯ y implies x + z ≤ y + z and 0 ≤ x implies 0 ≤ αx. If in addition, X is a lattice, that is if for a pair {x, y} the inf{x, y} and sup{x, y} exists, then X is a Riesz space (or a vector lattice). In this study, we discuss Banach lattices, ordered Banach spaces, operators on these spaces and their applications in economics, fixed-point theory, differential and integral equations.
Measures on Boolean Algebras
(University of Pretoria, 2023) Van der Walt, Jan Harm; Wortel, Marten; u17049637@tuks.co.za; Chamberlain, Tomas
This thesis deals with a number of related results on Boolean algebras. First, we prove the Stone Representation Theorem, which shows that every Boolean algebra is isomorphic to an algebra of sets, namely the clopen algebra of its Stone space. Then we prove the Loomis-Sikorski Theorem, which shows exactly how the Stone Representation Theorem may be extended to represent countable suprema and infima in terms of unions and intersections of sets. Finally, we discuss strictly positive measures. We provide a characterisation, in terms of intersection numbers and covering numbers, of those Boolean algebras which admit strictly positive measures, and we conclude by showing that a σ-complete Boolean algebra admits a strictly positive σ-additive measure if and only if it admits a strictly positive measure and it is weakly σ-distributive.
Modelling and analysis of plant-virus interaction in the co-infection of plants
(University of Pretoria, 2023) CHAPWANYA, MICHAEL; DUMONT, YVES; u13328043@tuks.co.za; Matusse, Americo J.
Co-infection is a simultaneous multiple parasitic infection within a host, and it is very common in humans and animals. Recently, thanks to molecular tools availability, co-infection has been detected in wild plants and crops. While in humans and animals, co-infection displays higher overall virulence and more severe symptoms, in plants, simultaneous infection can have different outcomes, from lower overall virulence with milder symptoms to higher overall virulence with more severe symptoms driving synergism. In particular, the co-infection driving synergism has threatened several crops. For instance, the co-infection of Beet Yellows Virus (BYV) and Beet Mosaic Virus (BtMV) leads to increased symptoms expression on Sugar Beet. The outbreak in Africa in 2011 of Maize Lethal Necrosis (MLND) as a synergistic interaction between Maize Chlorotic Mottle Virus (MCMV) and potyviruses has threatened the maize yield. Since not all mechanisms driving synergism are currently well known, that makes the study field and control strategies difficult. Mathematical modelling and analysis can help design central strategies or combine strategies to control disease. The aim of this thesis is to use a mathematical framework to develop our understanding of virus interaction driving synergistic co-infection in plants with particular focus on MLND. The mathematical framework follows from the construction of models, their theoretical analysis to the validation through numerical simulations and supplying insight into disease control. The first objective of this thesis is to provide a better understanding of disease dynamics driving synergistic co-infection with particular focus on potyviruses Sugarcane Mozaic Virus (SCMV) and MCMV dynamics driving to MLND and get more insight on disease control of MLND. The second objective is to access the impact of vectors dispersal on co-infection in crop and disease transmission dynamical with special focus on MLND and get more insight on crop protection. To address the first objective of this thesis, we develop a general crop-vector-borne disease temporal deterministic model for synergistic co-infection, with a particular focus on the knowledge we have on the viruses driving the MLND and the vector’s activity. The theoretical analysis of the model shows different thresholds driving the dynamics of the system: the well known basic reproduction number (BRN) and invasion reproduction number (IRN). The latter being essential for the emergence or not of the MLND. To address the second objective of this thesis, we allow vector dispersal by incorporating linear diffusion into the vector population. This model is formulated by partially degenerate reaction-diffusion systems in an unbounded domain. A particular type of solution of interest in this system is the traveling wave solutions. We assess different invasion scenarios depending on the threshold values. Overall, the models developed and analysed in this thesis show, through mathematical modelling, how we can get more understanding of virus interaction driving synergistic co-infection and we also highlight the importance of estimating the BRN and IRN as they summarize the whole dynamics of the system
Analysis of hyperbolic-type partial differential equations for non-fourier type heat conduction models
(University of Pretoria, 2023) Janse van Rensburg, N.F.J. (Nicolaas); rsieberhagen@nmisa.org; Sieberhagen, Rheinhardt Hendrik
Heat transfer modelling is routinely used to model the interaction between a heat source and a material specimen in applications such as additive manufacturing and medical surgery. The Fourier heat conduction model is well-known in the field of heat transfer, but in cases involving ultra-short heat pulses, or extremely small specimens, alternative models such as the Cattaneo-Vernotte \mbox{(C-V)} and dual-phase-lag (DPL) models are proposed. These two models are based on the concept of lagging responses (or lag times) in the heat flux and the temperature gradient. In 1982 an article appeared that reported on the existence of unwanted oscillations related to a so-called ``benchmark" problem that is based on the \mbox{C-V} model. This problem was studied and it was shown that the unwanted oscillations is the result of an ill-posed problem and not due to the choice of the numerical technique used to solve the problem. The problem was re-formulated to have a smooth initial condition and divided into auxiliary problems. It was solved using D'Alembert's and the finite element method, resulting in an oscillation-free solution. The theory and terminology of vibration analysis, \emph{e.g. overdamped and underdamped modes}, were incorporated into the Fourier, \mbox{C-V} and DPL heat conduction models. Weak variational formulations of these models, in terms of bilinear forms, were presented and the well-posedness of the model problems was established, based on a general existence result published in 2002. The modal analysis method was applied to the model problems and formal series solutions were derived. Convergence of the series solutions was proved in terms of the energy and inertia norms. This was used as a guideline to ensure accurate approximations for the series solutions of the model problems. Realistic lag time values were derived using modal analysis. This relied on the assumption that the solutions for the \mbox{C-V} and Fourier models will be the same after a sufficiently long time. The concept of a \emph{wane time} was introduced as the time instant at which the Fourier and \mbox{C-V} model predictions will correspond. This was proved with numerical experiments based on a continuous-heating model problem. Two model problems, based on single- and multi-pulse heating, were used to study aspects such as the contribution of overdamped and underdamped modes to the predicted temperature, the influence of the lag time values on the \mbox{C-V} and DPL model predictions, and the effect of heating parameters, \emph{e.g.} the duty ratio and the number of heating pulses on the model predictions. In conclusion, modal analysis proved to be successful in determining reliable lag times values and was effective for the numerical investigations into the properties of the solutions of the model problems. Future research should focus on investigating model problems that resemble reliable experimental techniques, thereby facilitating comparison of theory with practice.
Application of the finite element method to second order hyperbolic type partial differential equations
(University of Pretoria, 2023) Labuschagne, Madelein; Van Rensburg, Francois Nicolaas Janse; cdtikane@gmail.com; Tikane, Dipuo Constance
In this dissertation various models with variational forms similar to that of the wave equation are considered, i.e. second order hyperbolic type partial differential equations. These models include several linear vibration problems and heat conduction models taking phase-lag into account. Clearly numerical methods need to be used to solve these problems and the Finite Element Method (FEM) is used in this study. Before applying such a method, existence of a solution needs to be established. Therefore, a review of the work by Van Rensburg and Van der Merwe (2002) on general second order hyperbolic type problems was done. The results were not only presented, but additional remarks and a discussion which assists in applying the theory were also included. To obtain convergence results and error estimates when FEM is applied to the various models, general convergence results were presented. For this the article by Basson and Van Rensburg (2013) was used. The first model considered consists of two serially connected Timoshenko beams. One of the beams was modelled as embedded in an elastic material, while the other beam is either free or subjected to a prescribed external load. This model can be adapted for a single beam with di fferent loads on separate parts. To apply the convergence theory though it was necessary to use the double beam model, while a single beam model can be used when FEM is applied. This was demonstrated when these models were used to model a plant with a tap root system. In this biological application various things were investigated, including different forms of FEM, a comparison of the results for the static double beam and static single beam, and the dynamics of the beam. These experiments indicated that the two models compare well and gave insight into how the parameter modelling the resistance of the soil in influences key aspects of how the plant reacts due to external forces. Models for rigid bodies attached to beams were also investigated. The equations used to describe the dynamics of a beam with a tip body were derived, with special attention given to the interface conditions. Consequently, a model problem for an intermediate rigid body between two Timoshenko beams was investigated. Hyperbolic heat conduction models were also considered and the application to bio-heat transfer in skin was discussed. Specifically, a model from the work by Dekka and Dutta (2019) was investigated. Their approach to existence of solutions was scrutinized and it was found that their application of existence results from the 2002 article by Van Rensburg and Van der Merwe is incomplete. Due to this the exposition of the theory is improved in the dissertation. For all the mentioned models, the existence and uniqueness of a solution were obtained by defining the relevant function spaces and proving the required properties. Convergence was also established from the general convergence results and the systems of ordinary differential equations were obtained which can be used to obtain numerical approximations.
Exploring the decay parameter for the exponentially weighted moving average volatility methodology
(University of Pretoria, 2023) Van Vuuren, Gary; sibandasharmaine@gmail.com; Sibanda, Sharmaine Fanuel Promise
Volatility estimation is a crucial task for financial institutions, as it affects various aspects of their operations, such as risk management, capital allocation, investment strategy and derivative valuation. However, the traditional method of using equally weighted moving averages to estimate volatility can be inaccurate and incorrectly used, especially in volatile market conditions. It yields financial losses for financial institutions in that the volatility estimates do not correctly reflect financial markets in real time. In this dissertation, we implement the exponentially weighted moving average model instead, which assigns more weight to recent data than older data. We explore how the choice of the decay factor λ influences the performance of the exponentially weighted moving average model in different market scenarios. The optimal value of λ varies depending on the market volatility. We therefore demonstrate that the model can provide more reliable and timely volatility estimates than the equally weighted moving average model. These are useful for different applications in financial, such as Value at Risk or Expected Shortfall.
The theory of Gröbner bases and applications
(University of Pretoria, 2023) Messerschmidt, Miek; u13402260@tuks.co.za; Davies, Andrew Barnard
In this dissertation we explore the theory and applications of Gröbner bases, and the algorithms that allow us to compute them.
Fragmentation-coagulation equation with growth
(University of Pretoria, 2023) Banasiak, Jacek; Shindin, Sergey; u19402016@tuks.co.za; Poka, Wetsi D
The theory of fragmentation-coagulation equations began around 1916 with a series of papers by Smoluchowski on pure coagulation and since then continued to incorporate other processes into the model. The intention was to study the evolution of objects undergoing breakdown and/or merging. The scientific goals are to determine the conditions under which solutions exist, are unique and identify them accordingly. In this study, we considered the continuous fragmentation-coagulation equation with transport (decay or growth), subject to homogenous/McKendrick-von Foerster boundary condition in the latter case. The theory of semigroups of linear operators and, in particular, the Miyadera-Desch perturbation theorem are used to show the existence of semigroup solutions for the linear transport-fragmentation equation. We proved that the established semigroups have the moment improving property. The latter result plays a crucial role in the analysis of the complete transport-fragmentation-coagulation equation which is treated as a Lipschitz perturbation of the former linear problem. Under mild restrictions on the model coefficients, the existence of positive local classical solutions is established. Further, under additional conditions, their global in time existence is proved. Finally, a systematic technique is developed for obtaining closed-form solutions to continuous transport-fragmentation equations with homogenous boundary conditions and power-law coefficients. New solutions for the constant and linear decay/growth coefficients are presented. Furthermore, it is shown that the technique extends to some cases of the growth-fragmentation equation with the McKendrick-von Foerster boundary condition.
On groups with few 𝑝′-character degrees
(University of Pretoria, 2022) Madanha, Sesuai Yash; Rodrigues, Bernardo Gabriel; shaunmabena@gmail.com; Mabena, Lehlogonolo Shaun
Seitz’s theorem asserts that a finite group has exactly one non-linear irreducible character of degree greater than one if and only if the group is either an extraspecial 2-group or the group is isomorphic to a one-dimensional affine group over some field. An extension of Seitz’s theorem is Thompson’s celebrated theorem which states if the degrees of all non-linear irreducible characters of a group are divisible by a fixed prime 𝑝, then the group contains a normal 𝑝-complement. More recently, in 2020, as an extension to Thompson’s theorem, Giannelli, Rizo, and Schaeffer Fry showed that if the character degree set of a group 𝐺 contains only two 𝑝′-character degrees (where 𝑝 > 3 is a prime), then 𝐺 contains a normal subgroup 𝑁 such that 𝑁 has a normal 𝑝-complement and 𝐺/𝑁 has a normal 𝑝-complement. Moreover, 𝐺 is solvable. In this dissertation, we explore a variation of Thompson’s Theorem. We explore the structure of finite groups that have exactly one non-linear irreducible character whose degree is non-divisible by a fixed prime 𝑝. We call such groups (∗)-groups (𝑝 divides the order of the group). In 1998, Kazarin and Berkovich characterized the structure of (∗)-groups. We give a detailed proof of their work for solvable groups. Moreover, we produce a classification of (∗)-groups of order less than or equal to 100.
Application of network filtering techniques in finding hidden structures on the Johannesburg Stock Exchange
(University of Pretoria, 2023) Mare, Eben; yashin.gopi@gmail.com; Gopi, Yashin
Researchers from the field of econophysics have favoured the idea that financial markets are a complex adaptive system, consisting of entities that behave and interact in a diverse manner, leading to non-linear, emergent behaviour of the system. In the last twenty years, there has been an increasing focus on modelling complex adaptive systems using network theory. Correlation-based networks, where stocks are represented as entities in the network, and the relationships amongst the stocks are based on the strength of the co-movements of the stocks, have been widely studied. Network filtering tools, such as the Minimal Spanning Tree (MST), and the Planar Maximally Filtered Graph (PMFG), have been useful to attenuate the impact of noise in these networks, thereby allowing important macroscopic and mesoscopic structures to emerge. One of the main benefits of the PMFG is that it is accompanied by a hierarchical clustering algorithm called the Directed Bubble Hierarchical Tree (DBHT). This method has the benefit of being fully unsupervised in that it does not require the user to decide a priori on the number of clusters that the data should be split into. These techniques have been applied here to analyse the complex interactions amongst stocks on the Johannesburg Stock Exchange. A structure emerged in which shares from similar ICB sectors tended to cluster together. However, the so-called Rand Hedge shares, and shares which exhibited low liquidity, tended to override the sector effect and clustered together. From a dynamic perspective, the MST and PMFG seemed to shrink during market crashes, while the Basic Materials sector was typically the most important or central sector over time. Over the long-term, the DBHT divided the stocks in the South African stock market into six clusters. This technique was compared to other popular hierarchical clustering algorithms, and the amount of economic information that each method extracted was quantified. The most recent PMFG and DBHT showed a changed structure as compared to the long-term data, highlighting that the way that market participants view South African shares can change over time.
Reaction diffusion modelling of bacteria colonies : microbial quiescence and chemotaxis
(University of Pretoria, 2022) Chapwanya, Michael; phindile.dumani@up.ac.za; Dumani, Phindile
The physiological structure of microbial communities in natural environments is typically a response to changes in internal and external conditions. External conditions may include the availability or depletion of growth-limiting nutrients, and presence of inhibiting or toxic substances while internal conditions may include cell to cell interactions. We present and investigate spatio-temporal bacterioplankton-nutrient-chemoattractant-chemorepellent interaction models that take into account the quiescent stage and chemotaxis. We establish conditions under which microbial population oscillations (boom-and-bust) may occur. In the study, we observe that population oscillations occur when the switching of states is dependent on the active microbial density in the environment or cell-to-cell interaction. Furthermore, in the case where dormant cells are neglected or disregarded, oscillations are not observed in the bacterioplankton population dynamics. It is well known from experiments that colonies of \emph{Escherichia coli} and \emph{Salmonella typhimurium} exhibit various patterns, therefore, we establish the existence of traveling wavefronts for the proposed reaction-diffusion model. Using the theory of monotone wave fronts for cooperative and partially degenerate reaction-diffusion systems, we show that the minimal wave speed coincides with the spreading speed. However, that is for the constant switch rates case. Whereas, for the switch functions dependent on the chemo-attractant concentration we numerically observe that the model admits non-monotonic traveling wave profiles. Moreover, we demonstrate that neglecting dormant cells overestimates the spreading speed of the colony. Numerical results also indicate the importance of the quiescent stage in the speed of spread. Microbial populations depend on their environment but can also modify it. Instead of breaking down complex nutrients for their growth, microbes can exhibit negative local or global behaviour by engineering the environment in ways that are detrimental to their proliferation. A reaction diffusion model system consisting of active and inactive microbial population in a harsh environment accounting for the directed movement and switch of cells to dormancy at high concentration is studied. Results show an essential mechanism generating oscillating patterns in microbial populations under environmental stress. Bifurcation analysis of the model and the interplay between Turing and Hopf instability is discussed. Theoretical and numerical investigation of the proposed model is presented to provide insight into the conditions that may lead to the extinction of the microbial population -- ecological suicide. A qualitative study of the proposed numerical schemes is presented. Numerical simulations are provided to support our theoretical observations.
A guide to the Rado graph : exploring structural and logical properties of the Rado graph
(University of Pretoria, 2023) Kellerman, Ruaan; u16004231@tuks.co.za; Michau, Michelle
The Rado graph, denoted R, is the unique (up to isomorphism) countably infinite random graph. It satisfies the extension property, that is, for two finite sets U and V of vertices of R there is a vertex outside of both U and V connected to every vertex in U and none in V . This property of the Rado graph allows us to prove quite a number of interesting results, such as a 0-1-law for graphs. Amongst other things, the Rado graph is partition regular, non-fractal, ultrahomogeneous, saturated, resplendent, the Fra´ıss´e-limit of the class of finite graphs, a non-standard model of the first-order theory of finite graphs, and has a complete decidable theory. We classify the definable subgraphs of the Rado graph and prove results for finite graphs that satisfy a restricted version of the extension property. We also mention some parallels between the rationals viewed as a linear order and the Rado graph.
Analysis of the vibration of flexible structures
(University of Pretoria, 2022) Stapelberg, Belinda; u17026742@tuks.co.za; Blecher, Tatiana Monique
Research on vibrations of flexible structures is ongoing in engineering and applied mathematics fields. Flexible structures in practice can be considered as systems of interconnected rod-like components. This dissertation consists largely of a literature study on some mathematical models for flexible structures and structural components, and includes existence theory and finite element analysis of these models and their solutions. These models include beam models such as the Timoshenko theory, Euler-Bernoulli theory, as well as recently published work on a so-called locally linear Timoshenko rod. The multi-dimensional wave equation is also used to illustrate the application of some of the theory in this dissertation.
Applications of direct and inverse limits in analysis
(University of Pretoria, 2022-11-28) Van der Walt, Jan Harm; De Jeu, Marcel; sjvdwvanamstel@gmail.com; Van Amstel, Walt
In this dissertation, we use the categorical notions of direct and inverse limits to solve certain problems in analysis; in particular, in the field of vector lattices. Chapter 1 provides a general overview and motivation of the problems we will focus on. Specifically, these are a decomposition theorem for C(X) spaces that are order dual spaces, and the problem of existence of free objects in certain categories of locally convex structures. The connecting thread between these two disparate problems will be our extensive and fundamental use of direct and inverse limits in their solutions. Chapter 2 deals with the first of these two problems. After settling some preliminaries, the first few sections of Chapter 2 develops the basic theory of direct and inverse limits in categories of vector lattices. This includes results on existence, permanence properties, as well as some examples. After this, we give some results on the duality between direct and inverse limits. In particular, we will show that the order (continuous) dual of a direct limit of vector lattices is an inverse limit of order (continuous) duals, and (under more strict conditions) the order (continuous) dual of an inverse limit of vector lattices is a direct limit of order (continuous) duals. The rest of Chapter 2 deals with applications of this duality theory in various contexts, among these will be our desired decomposition result for certain C(X) spaces, which is formulated in terms of an inverse limit. Chapter 3 starts with some further preliminaries we need in order to define certain categories of algebraic structures, normed structures, and locally convex structures forming the setting of this chapter. After this, we cover some material from universal algebra to prove the existence of free objects in these algebraic categories. We use the existence of these algebraic free objects to expand upon the existing literature regarding certain `free objects' in categories of normed structures. As we shall detail below, these are not bona fide free objects in our sense of the term. Inverse limits re-enter the picture at this point: We will prove a general categorical result involving inverse limits that allows us to use our results for categories of normed structures to obtain genuine free objects in categories of locally convex structures. The abstract material in Chapter 3 will be interspersed with some concrete examples chosen from two particular cases. We conclude Chapter 3 by giving concrete descriptions of two free objects in certain categories of locally convex structures whose existence was proven using our general abstract methods.
Beyond the tanh method-looking for explicit travelling wave solutions to partial differential equations
(University of Pretoria, 2022) Banasiak, Jacek; kkhhaannyy@gmail.com; Maqele, Khanyisani
In this work, we focus on a general procedure for finding exact travelling wave solutions for evolution equations with polynomial nonlinearites. Mathematically, looking for travelling wave solutions is asking the question whether a given PDE has solutions invariant under a Galilean transformation; in such a case, it can be reduced to an ODE. We discuss the existence of travelling wave solutions by using phase plane analysis. We show that popular methods such as the tanh-method, G0/G-method and many more are special cases of the presented approach. Analytical solutions to several examples of nonlinear equations are illustrated. In the application, we use the Maple program to compute solutions to nonlinear systems of equations.
Invariant solutions of the Black-Scholes equation
(University of Pretoria, 2022) Mokhwetha, Mabula; Maluleke, Gaza; makobakmothiba@gmail.com; Mothiba, Makoba Melidah Kholofelo
In this study, we discuss derivatives, Lie symmetries and invariant solutions of the Black Scholes equation. We combine the Lie group methods with the Adomian decomposition method to solve the Black and Scholes equation via the heat equation. We further discuss several examples to illustrate the theory in this study
Semi-order units in vector lattices
(University of Pretoria, 2021) Van der Walt, Jan Harm; Wortel, Marten; u16124970@tuks.co.za; Chitanga, Painos
The space C(X) of real-valued continuous functions on a topological space X is a vector lattice and a locally convex topological vector space, but what is the interaction between these structures? In the case of a compact space X, the norm and the order are closely related to one another. Indeed, one may define the norm through the order structure. We aim to generalize these results to a non-compact space. Let X be a Tychonoff space and consider C(X) equipped with the compact-open topology. We will establish a relationship between this topology and the order structure on C(X)
The use of personal response systems to renegotiate the didactical contract in tertiary mathematics education
(University of Pretoria, 2020) Harding, Ansie; Louw, Ina; karin.bothma@up.ac.za; Bothma, Karin
Challenges experienced by first year students transitioning from secondary to tertiary mathematics education are examined through the lens of the didactical contract or agreement between the lecturer and students that is founded on beliefs about mutual obligations. First year students’ fundamental beliefs about the nature of mathematics and mathematics teaching/learning must be challenged to renegotiate the didactical contract at tertiary level. Through pedagogy aimed at self-directed learning personal response systems (PRS) are periodically used to make students aware of their own learning and their responsibility for learning. A Likert scale questionnaire is administered at the beginning of the students’ first year to gauge their beliefs about mathematics and mathematics teaching/learning and again at the end of the first semester (or term) to observe possible changes in beliefs and hence the didactical contract. The intervention consists of PRS sessions or so called Time-out sessions, regularly incorporated into the traditional transmission mode lecture to create a student-centred learning environment, aimed at influencing students’ beliefs. Questionnaire data is quantified and compared for the two surveys. There is evidence of a shift towards students taking ownership of their learning and a renegotiation of the didactical contract. Qualitative data generated by focus group interviews confirms the role of the PRS sessions in student beliefs.

Browse

Recent Submissions