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In this thesis, a new smoothing penalty algorithm is introduced to solve a mathematical program with equilibrium constraints (MPEC). By smoothing the exact penalty function, an MPEC is reformulated as a series of subprograms which belong to a class of MPECs with simple linear complementarity constraints. To deal with the subproblems, a hybrid algorithm is proposed, which combines the active set algorithm, the 6-active search algorithm and the PSQP algorithm. It is shown that the smoothing penalty algorithm converges globally to a M-stationary point of MPEC under weak conditions. Supervisor: Dr. Jane Ye (Department of Mathematics and Statistics) Co-Supervisor: Dr. Wu-Sheng Lu (Department of Electrical and Computer Engineering)

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We call two increasing sequences of positive integers {aᵢ}, {b(j)} a "Ramanujan Pair" if the following identity holds: (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI). The goal of this investigation is to determine all Ramanujan Pairs. Although this goal was not completely reached, we have determined all pairs for which the first term a₁ ≥ 5 and have proved that any Ramanujan Pair which begins with a₁ = m, where 1 ≤ m ≤ 4, aside from the known pairs, would have to branch off the first Euler identity with {aᵢ} = {i + m - 1}, {b(j)} = {j m}. A great deal of computing was done to discover the proofs given here. The search methods used and their programs are discussed in detail. Beyond these results, we have found all finite Ramanujan Pairs. Finally, modular Ramanujan Pairs (where the coefficients in the identity are reduced modulo n) are also examined.

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]. / Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 2003.

The purpose of this research study is to propose guidelines for providing a positive user experience for pupils using Dr Math®. User experience was found to have a positive impact on the acceptance and adoption of a product. Thus the proposed guidelines contribute in maximizing the adoption and acceptance of Dr Math® among pupils. This study begins with an introductory chapter that describes the problem that forms the basis for this research. The chapter defines the objectives that this study is intended to achieve in order to accomplish its ultimate goal. The methodology followed to conduct this research study as well as its scope are also defined here. The results from a preliminary survey revealed that despite its potential accessibility, Dr Math® has a low adoption rate. However, when compared to other mobile learning (m-learning) applications for mathematics learning, Dr Math® is more popular. Thus Dr Math® was selected as a case for study. Chapter 2 of this study provides a detailed description of Dr Math® as a local mobile application for mathematics learning. It was found that the affordability and accessibility of Dr Math® did not necessarily imply a high adoption rate. There are various possible barriers to its low adoption. User experience (UX), which is the focus of this study, is one of them. Thus, a subsequent chapter deals with UX. Chapter 3 discusses UX, its scope, components and definition and places particular emphasis on its significance in the success of any product. The chapter also highlights the characteristics of a positive UX and the importance of designing for this outcome. In Chapter 4, a discussion and justification of the methodology used to conduct this research is discussed. This study primarily employs a qualitative inductive approach within an interpretivism paradigm. An exploratory single case study was used to obtain an in-depth analysis of the case. Data was collected using Dr Math® log files as a documentary source. Gathered data was then analysed and organized into themes and categories using qualitative content analysis as outlined in Chapter 5. Also the findings obtained from the results, which are mainly the factors that were found to have an impact on the user interaction with Dr Math®, are presented here. The identified factors served as a basis from which the guidelines presented in Chapter 6 were developed. Chapter 7 presents the conclusions and recommendations of the research. From both theoretical and empirical work, it was concluded that Dr Math® has the potential to improve mathematics learning in South Africa. Its adoption rate, however, is not satisfying: hence, the investigation of the factors impacting on the user interaction with Dr Math®, from which the proposed guidelines are based.

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The reconstruction of a function and its derivative from a set of measured samples is a fundamental operation in visualisation. Multiresolution techniques, such as wavelet signal processing, are instrumental in improving the performance and algorithm design for data analysis, filtering and processing. This dissertation explores the possibilities of combining traditional multiresolution analysis and processing features of wavelets with the design of appropriate filters for reconstruction of sampled data. On the one hand, a multiresolution system allows data feature detection, analysis and filtering. Wavelets have already been proven successful in these tasks. On the other hand, a choice of discrete filter which converges to a continuous basis function under iteration permits efficient and accurate function representation by providing a “bridge” from the discrete to the continuous. A function representation method capable of both multiresolution analysis and accurate reconstruction of the underlying measured function would make a valuable tool for scientific visualisation. The aim of this dissertation is not to try to outperform existing filters designed specifically for reconstruction of sampled functions. The goal is to design a wavelet filter family which, while retaining properties necessary to preform multiresolution analysis, possesses features to enable the wavelets to be used as efficient and accurate “building blocks” for function representation. The application to visualisation is used as a means of practical demonstration of the results. Wavelet and visualisation filter design is analysed in the first part of this dissertation and a list of wavelet filter design criteria for visualisation is collated. Candidate wavelet filters are constructed based on a parameter space search of the BC-spline family and direct solution of equations describing filter properties. Further, a biorthogonal wavelet filter family is constructed based on point and average interpolating subdivision and using the lifting scheme. The main feature of these filters is their ability to reconstruct arbitrary degree piecewise polynomial functions and their derivatives using measured samples as direct input into a wavelet transform. The lifting scheme provides an intuitive, interval-adapted, time-domain filter and transform construction method. A generalised factorisation for arbitrary primal and dual order point and average interpolating filters is a result of the lifting construction. The proposed visualisation filter family is analysed quantitatively and qualitatively in the final part of the dissertation. Results from wavelet theory are used in the analysis which allow comparisons among wavelet filter families and between wavelets and filters designed specifically for reconstruction for visualisation. Lastly, the performance of the constructed wavelet filters is demonstrated in the visualisation context. One-dimensional signals are used to illustrate reconstruction performance of the wavelet filter family from noiseless and noisy samples in comparison to other wavelet filters and dedicated visualisation filters. The proposed wavelet filters converge to basis functions capable of reproducing functions that can be represented locally by arbitrary order piecewise polynomials. They are interpolating, smooth and provide asymptotically optimal reconstruction in the case when samples are used directly as wavelet coefficients. The reconstruction performance of the proposed wavelet filter family approaches that of continuous spatial domain filters designed specifically for reconstruction for visualisation. This is achieved in addition to retaining multiresolution analysis and processing properties of wavelets.