The investigator's experience of teaching mathematics at a college of education since 1983 has reinforced his conviction that trainee students come to college with significant gaps, weaknesses and faults in their (mathematical) conceptual structures, probably as a result of shortcomings in the mathematics teaching to which they have been exposed. The theme of this investigation is thus a natural choice that appeared to be of immediate relevance to secondary school mathematics teaching. The analysis of the issue leads to a unified perspective: the problem is placed in a theoretical framework where Bruner [spiral curriculum], Ausubel [structured learning] and Skemp [relational understanding] are brought together. How the curriculum, textbooks and examination influence school mathematics teaching is examined in some depth and the consequences investigated. Two specific topics, viz. the generalised Pythagorean relation and absolute value are investigated in relation to published work, curriculum and textbooks, and each (topic) is presented as a unifying theme in secondary mathematics to standard 9 pupils. The classroom exercise is assessed to test the hypothesis that structured, integrated presentation around a spiral curriculum promotes "relational understanding". Analysis of results supports the hypothesis.

A thesis submitted to the Faculty of Engineering at the University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. / Particularly at the higher system voltages, certain faults have been attributed to VFTs. This work presents a full frequency domain mathematical model of the effects likely to be introduced by the application of a thin ferromagnetic coating to the surfaces of the busbars as a means of attenuating these transients. Experimental verification shows the model to perform accurately as an analysis tool for a continuous coated co-axial system. While comparison between the model developed here (based on a planar conductor) and the very accurate Bessel function solution is presented, it must be remembered that it is not feasible to include the effects of coatings in the latter solution. Hence it is used merely as a bench-mark for improvements to the general model. Consideration is also given to the transient skin effect, and this is shown to result in an increased attenuation of VFTs. detectable during high voltage experimentation. The frequency domain model is extended to that of a full travelling wave model for VFTs in GIS. The Fourier technique is used to move between the frequency and time domains. Laboratory results obtained at high voltage.show that this technique will introduce both risetime increase and peak magnitude attenuation of VFTs. As the most frequent faults directly attributable to disconnector operation are known to be faults to earth at the switch, it is noted that the technique proposed in this work is uniquely suited for use in a design strategy that requires totally robust disconnectors at the expense of more severe VFTs in the GIS. The model is used to determine a range of physical characteristics of materials suitable for use in this application. While the characteristics of the most suitable materials have therefore been established, it is concluded that extensive metallurgical experimentation remains before this technique may be economically applied. / Andrew Chakane 2018

The topological dynamics of continuous and noncontinuous dynamical systems are investigated. Various definitions of chaos are studied, as well as notions of stability. Results are obtained on asymptotically stable sets and the perturbation stability of such sets. The primary focus is on the traditional point sets of topological dynamics, including the chain recurrent set, omega-limit sets and attractors. The basic setting is that of a continuous function on a compact metric space, sometimes with additional properties on the space. The investigation includes results on the dynamical properties of typical continuous functions in the sense of Baire category. Results are also developed concerning dynamical systems involving quasi-continuous functions. An invariance property for the omega-limit sets of such functions is given. Omega-limit sets are characterized for Riemann integrable derivatives and derivatiyes which are continuous almost everywhere. Techniques used in the investigation and formulation of results include finding theorems which relate the rather disparate notions of dynamical properties and generalized continuity. In addition to dynamical systems, numerous other applications of generalized continuity are imoestigated. Techniques used include application of the Baire Category Theorem and the notion of semi-closure. For example, results are formulated concerning functions determined by dense sets, including separately continuous functions, thus generalizing the classical result for continuous functions on dense subsets of the domain. The uniform boundedness theorem is extended to functions which are not necessarily continuous, including various derivatives. The closed graph theorem is strictly generalized in two separate ways, and applications are presented using these generalizations. An invariance property of separately continuous functions is given. Cluster sets are studied in connection with separate continuity, and various results are presented concerning locally bounded functions.

The estimation of the stability of radar emissions is of considerable interest in the evaluation of radar clutter rejection performance and also for the general knowledge of the waveform required for the design of threat simulators. It should be stressed that for the estimation of clutter rejection capability, it is the stability of the entire waveform that is of general importance, although the stability of parameters such as phase, Pulse Repetition Interval (PRI) and amplitude are typically specified because of the ease in instrumenting the measurement. The parametric estimates are indeed the most useful in describing the characteristics of the waveform but not necessarily for evaluating clutter rejection performance. Two broad categories into which radar emissions can be subdivided are coherent and non-coherent RF. A great deal of confusion often surrounds the use of these terms, especially among those who measure radar emissions rather than those who build the radar sets. For the purposes of this paper, coherence will be defined in terms of the square root of the variance of the first pulse-to-pulse phase difference, Ï (Δθ ). For the case where Ï (Δθ) << 1 radian, the signal will be considered coherent. When the phase is uniformly distributed over 2Ï radians, the signal will be considered nonâ coherent. Since it is likely that, for most practical signals, the signal will be well within one of these two categories, ambiguity will be unlikely. If a radar emission is observed to be coherent, it implies that the radar uses this property for Moving Target Indication (MTI) processing. The performance of the MTI will probably, but not necessarily, depend on the pulse-to-pulse phase stability as the most critical parameter for this type of system. Alternatively, if the radar emission is observed to be non-coherent, it implies that if the radar has an MTI processor, it is likely that it is of the stored reference variety. The performance of the MTI will probably, but again not necessarily, depend on the pulse-to-pulse RF stability as the most critical parameter. The common thread between the two types of systems which indicates clutter rejection performance is the repeatability of adjacent pulse waveforms regardless of phase. This is not to imply that phase is not critical; it is important for determining the type of processor. The difference lies in the fact that for the internally coherent system, the phase information of the coherent reference oscillator is not observable as it is for the extremely coherent system. Hence, the only hint that such an emitter has an MTI processor is contained in the repeatability of adjacent pulse waveforms. This paper addresses the general problems of detecting coherence, estimating MTI performance, and estimating the phase stability, frequency stability and PRI stability using sample data derived from a system based on the IBM-PC. Both the analysis and radar waveform generation systems were implemented in software utilizing Microsoft Fortran and Microsoft C compilers. / Master of Science

Within undergraduate mathematics education, there are few validated instruments designed for large-scale usage. The Group Concept Inventory (GCI) was created as an instrument to evaluate student conceptions related to introductory group theory topics. The inventory was created in three phases: domain analysis, question creation, and field-testing. The domain analysis phase included using an expert consensus protocol to arrive at the topics to be assessed, analyzing curriculum, and reviewing literature. From this analysis, items were created, evaluated, and field-tested. First, 383 students answered open-ended versions of the question set. The questions were converted to multiple-choice format from these responses and disseminated to an additional 476 students over two rounds. Through follow-up interviews intended for validation, and test analysis processes, the questions were refined to best target conceptions and strengthen validity measures. The GCI consists of seventeen questions, each targeting a different concept in introductory group theory. The results from this study are broken into three papers. The first paper reports on the methodology for creating the GCI with the goal of providing a model for building valid concept inventories. The second paper provides replication results and critiques of previous studies by leveraging three GCI questions (on cyclic groups, subgroups, and isomorphism) that have been adapted from prior studies. The final paper introduces the GCI for use by instructors and mathematics departments with emphasis on how it can be leveraged to investigate their students' understanding of group theory concepts. Through careful creation and extensive field-testing, the GCI has been shown to be a meaningful instrument with powerful ability to explore student understanding around group theory concepts at the large-scale.

In South Africa access to mental healthcare resources is restricted for a number of reasons including language barriers that prevent suitable communication between mental healthcare professionals and African language speaking South Africans. The translation of psychometric tools into African languages has been identified as one method in improving access to psychological services for African language speakers. The Clinical Outcomes in Routine Evaluation - Outcome Measure (CORE-OM) has demonstrated its clinical utility within the United Kingdom (UK) National Healthcare Service (NHS) as a standardised psychotherapy outcome measure that evaluates the degree of psychological distress individuals present with at the start of psychotherapy treatment, and the degree of change that has been effected at the termination of therapy. A measure like the CORE-OM holds valuable clinical utility for the South African context. This thesis argues that the availability of a valid Xhosa version of the CORE-OM would allow for improved access to psychotherapy resources for Xhosa speaking individuals, and allow for the evaluation of the effectiveness of psychotherapy interventions conducted in Xhosa. The CORE-OM developers have provided a translation design and set of guidelines to standardise the translation of the CORE-OM into different languages. However this thesis argues that these guidelines are incomplete. Instead International Test Commission (ITC) guidelines are recommended as a culturally sensitive method to supplement current CORE-OM translation guidelines, in order to generate a valid Xhosa measure of distress. A mixed methods approach is applied which first investigates the construct equivalence and bias of the CORE-OM English version within a South African student population sample, both qualitatively and quantitatively, in order to establish the degree of adaptation required to generate a valid Xhosa version of distress. Next the CORE-OM English version is translated into Xhosa using the five-stage translation design prescribed by the CORE System Trust, supplemented by ITC guidelines. All changes made to the CORE-OM during translation into Xhosa are documented. The CORE-OM Xhosa version is then investigated for reliability and validity. This investigation reveals low internal reliability within the subjective wellbeing domain indicating that these items are less meaningful as depictions of distress within the Xhosa language. A reduced version of the CORE-OM demonstrates strong psychometric properties as a valid Xhosa measure of distress.

Indiana University-Purdue University Indianapolis (IUPUI) / As our population ages, treating bone and joint ailments is becoming increasingly important. Both osteoporosis, a bone disease characterized by a decreased density of mineral in bone, and osteoarthritis, a joint disease characterized by the degeneration of cartilage on the ends of bones, are major causes of decreased movement ability and increased pain. To combat these diseases, many treatments are offered, including drugs and exercise, and much biomedical research is being conducted. However, how can we get the most out of the research we perform and the treatment we do have? One approach is through computational analysis and mathematical modeling. In this thesis, quantitative methods of analysis are applied in different ways to two systems: osteoporosis and osteoarthritis. A mouse model simulating osteoporosis is treated with salubrinal and knee loading. The bone and cell data is used to formulate a system of differential equations to model the response of bone to each treatment. Using Particle Swarm Optimization, optimal treatment regimens are found, including a consideration of budgetary constraints. Additionally, an in vitro model of osteoarthritis in chondrocytes receives RNA silencing of Lrp5. Microarray analysis of gene expression is used to further elucidate the mode of regulation of ADAMTS5, an aggrecanase associated with cartilage degradation, by Lrp5, including the development of a mathematical model. The math model of osteoporosis reveals a quick response to salubrinal and a delayed but substantial response to knee loading. Consideration of cost effectiveness showed that as budgetary constraints increased, treatment did not start until later. The quantitative analysis of ADAMTS5 regulation suggested the involvement of IL1B and p38 MAPK. This research demonstrates the application of quantitative methods to further the usefulness of biomedical and biomolecular research into treatment and signaling pathways. Further work using these techniques can help uncover a bigger picture of osteoarthritis's mode of action and ideal treatment regimens for osteoporosis.

Indiana University-Purdue University Indianapolis (IUPUI) / Model-based drug development has become an essential element to potentially make drug development more productive by assessing the data using mathematical and statistical approaches to construct and utilize models to increase the understanding of the drug and disease. The modeling and simulation approach not only quantifies the exposure-response relationship, and the level of variability, but also identifies the potential contributors to the variability. I hypothesized that the modeling and simulation approach can: 1) leverage our understanding of pharmacokinetic-pharmacodynamic (PK-PD) relationship from pre-clinical system to human; 2) quantitatively capture the drug impact on patients; 3) evaluate clinical trial designs; and 4) identify potential contributors to drug toxicity and efficacy. The major findings for these studies included: 1) a translational PK modeling approach that predicted clozapine and norclozapine central nervous system exposures in humans relating these exposures to receptor binding kinetics at multiple receptors; 2) a population pharmacokinetic analysis of a study of sertraline in depressed elderly patients with Alzheimer’s disease that identified site specific differences in drug exposure contributing to the overall variability in sertraline exposure; 3) the utility of a longitudinal tumor dynamic model developed by the Food and Drug Administration for predicting survival in non-small cell lung cancer patients, including an exploration of the limitations of this approach; 4) a Monte Carlo clinical trial simulation approach that was used to evaluate a pre-defined oncology trial with a sparse drug concentration sampling schedule with the aim to quantify how well individual drug exposures, random variability, and the food effects of abiraterone and nilotinib were determined under these conditions; 5) a time to event analysis that facilitated the identification of candidate genes including polymorphisms associated with vincristine-induced neuropathy from several association analyses in childhood acute lymphoblastic leukemia (ALL) patients; and 6) a LASSO penalized regression model that predicted vincristine-induced neuropathy and relapse in ALL patients and provided the basis for a risk assessment of the population. Overall, results from this dissertation provide an improved understanding of treatment effect in patients with an assessment of PK/PD combined and with a risk evaluation of drug toxicity and efficacy.

Indiana University-Purdue University Indianapolis (IUPUI) / In recent decades, Petri nets (PNs) have been used to model traffic networks for different purposes, such as signal phase control, routing, and traffic flow estimation, etc. Because of the complex nature of traffic networks where both discrete and continuous dynamics come into play, the Hybrid Petri net (HPN) model becomes an important tool for the modeling and analysis of traffic networks. In Chapter 1 a brief historical summery about traffic systems control and then related work is mentioned followed by the major contributions in this research. Chapter 2 provides a theoretical background on Petri nets. In Chapter 3, we develop a HPN model for a single signalized intersection first, then we extend this model to study a simple traffic network that consists of two successive intersections. Time delays between different points of network are also considered in order to make the model suitable for analysis and simulation. In addition to HPN models, we also consider discrete Petri nets where their modeling simplicity enables the characterization of the occurrences of all events in the system. This discrete PN is particularly useful to give a higher-level representation of the traffic network and study its event occurrences and correlations. In Chapter 4, we build a discrete PN model to represent a traffic network with two successive intersections. However, we find that the model leads to unbounded places which cannot accurately reflect the dynamics of the traffic in terms of event occurrences. Hence, we introduce the Modified Binary Petri nets (MBPN) to overcome the limitation and resolve the confliction problem when we design our controllers. This MBPN model is a powerful tool and can be useful for the modeling and analysis of many other applications in traffic networks. Chapter 5 gives a summary for each chapter, provides conclusion and discusses future work for both discrete and hybrid Petri nets.

Indiana University-Purdue University Indianapolis (IUPUI) / There is a strong urge for advanced diagnosis method, especially in high power battery packs and high energy density cell design applications, such as electric vehicle (EV) and hybrid electric vehicle segment, due to safety concerns. Accurate and robust diagnosis methods are required in order to optimize battery charge utilization and improve EV range. Battery faults cause significant model parameter variation affecting battery internal states and output. This work is focused on developing diagnosis method to reliably detect various faults inside lithium-ion cell using electrochemical model based observer and fuzzy logic algorithm, which is implementable in real-time. The internal states and outputs from battery plant model were compared against those from the electrochemical model based observer to generate the residuals. These residuals and states were further used in a fuzzy logic based residual evaluation algorithm in order to detect the battery faults. Simulation results show that the proposed methodology is able to detect various fault types including overcharge, over-discharge and aged battery quickly and reliably, thus providing an effective and accurate way of diagnosing li-ion battery faults.