Predicting active material utilisation in LiFePO4 electrodes using a multi-scale mathematical model

A mathematical model is developed to simulate the discharge of a LiFePO4 cathode. This model contains 3 size scales, which match with experimental observations present in the literature on the multi-scale nature of LiFePO4 material. A shrinking-core is used on the smallest scale to represent the phase-transition of LiFePO4 during discharge. The model is then validated against existing experimental data and this validated model is then used to investigate parameters that influence active material utilisation. Specifically the size and composition of agglomerates of LiFePO4 crystals is discussed, and we investigate and quantify the relative effects that the ionic and electronic conductivities within the oxide have on oxide utilisation. We find that agglomerates of crystals can be tolerated under low discharge rates. The role of the electrolyte in limiting (cathodic) discharge is also discussed, and we show that electrolyte transport does limit performance at high discharge rates, confirming the conclusions of recent literature.

Advanced mathematical modelling and experimental investigation of new torque arms for shaft-mounted drives

Shaft-mounted gearboxes are widely used in industry. The torque arm that holds the reactive torque on the housing of the gearbox, if properly positioned creates the reactive force that lifts the gearbox and unloads the bearings of the output shaft. The shortcoming of these torque arms is that if the gearbox is reversed the direction of the reactive force on the torque arm changes to opposite and added to the weight of the gearbox overloads the bearings shortening their operating life. In this paper, a new patented design of torque arms that develop a controlled lifting force and counteract the weight of the gearbox regardless of the direction of the output shaft rotation is described. Several mathematical models of the conventional and new torque arms were developed and verified experimentally on a specially built test rig that enables modelling of the radial compliance of the gearbox bearings and elastic elements of the torque arms. Comparison showed a good agreement between theoretical and experimental results.

Mathematical modelling of chronic wound healing

Chronicwounds fail to proceed through an orderly process to produce anatomic and functional integrity and are a significant socioeconomic problem. There is much debate about the best way to treat these wounds. In this thesis we review earlier mathematical models of angiogenesis and wound healing. Many of these models assume a chemotactic response of endothelial cells, the primary cell type involved in angiogenesis. Modelling this chemotactic response leads to a system of advection-dominated partial differential equations and we review numerical methods to solve these equations and argue that the finite volume method with flux limiting is best-suited to these problems. One treatment of chronic wounds that is shrouded with controversy is hyperbaric oxygen therapy (HBOT). There is currently no conclusive data showing that HBOT can assist chronic wound healing, but there has been some clinical success. In this thesis we use several mathematical models of wound healing to investigate the use of hyperbaric oxygen therapy to assist the healing process - a novel threespecies model and a more complex six-species model. The second model accounts formore of the biological phenomena but does not lend itself tomathematical analysis. Bothmodels are then used tomake predictions about the efficacy of hyperbaric oxygen therapy and the optimal treatment protocol. Based on our modelling, we are able to make several predictions including that intermittent HBOT will assist chronic wound healing while normobaric oxygen is ineffective in treating such wounds, treatment should continue until healing is complete and finding the right protocol for an individual patient is crucial if HBOT is to be effective. Analysis of the models allows us to derive constraints for the range of HBOT protocols that will stimulate healing, which enables us to predict which patients are more likely to have a positive response to HBOT and thus has the potential to assist in improving both the success rate and thus the cost-effectiveness of this therapy.

Early childhood teachers' mathematical content knowledge

The process of becoming numerate begins in the early years. According to Vygotskian theory (1978), teachers are More Knowledgeable Others who provide and support learning experiences that influence children’s mathematical learning. This paper reports on research that investigates three early childhood teachers mathematics content knowledge. An exploratory, single case study utilised data collected from interviews, and email correspondence to investigate the teachers’ mathematics content knowledge. The data was reviewed according to three analytical strategies: content analysis, pattern matching, and comparative analysis. Findings indicated there was variation in teachers’ content knowledge across the five mathematical strands and that teachers might not demonstrate the depth of content knowledge that is expected of four year specially trained early years’ teachers. A significant factor that appeared to influence these teachers’ content knowledge was their teaching experience. Therefore, an avenue for future research is the investigation of factors that influence teachers’ content numeracy knowledge.

A mathematical model of chlamydial infection incorporating movement of chlamydial particles

We present a spatiotemporal mathematical model of chlamydial infection, host immune response and spatial movement of infectious particles. The re- sulting partial differential equations model both the dynamics of the infection and changes in infection profile observed spatially along the length of the host genital tract. This model advances previous chlamydia modelling by incorporating spatial change, which we also demonstrate to be essential when the timescale for movement of infectious particles is equal to, or shorter than, the developmental cycle timescale. Numerical solutions and model analysis are carried out, and we present a hypothesis regarding the potential for treatment and prevention of infection by increasing chlamydial particle motility.

Towards a model for the analysis and description of mathematical knowledge and understanding

Mathematics education literature has called for an abandonment of ontological and epistemological ideologies that have often divided theory-based practice. Instead, a consilience of theories has been sought which would leverage the strengths of each learning theory and so positively impact upon contemporary educational practice. This research activity is based upon Popper’s notion of three knowledge worlds which differentiates the knowledge shared in a community from the personal knowledge of the individual, and Bereiter’s characterisation of understanding as the individual’s relationship to tool-like knowledge. Using these notions, a re-conceptualisation of knowledge and understanding and a subsequent re-consideration of learning theories are proposed as a way to address the challenge set by literature. Referred to as the alternative theoretical framework, the proposed theory accounts for the scaffolded transformation of each individual’s unique understanding, whilst acknowledging the existence of a body of domain knowledge shared amongst participants in a scientific community of practice. The alternative theoretical framework is embodied within an operational model that is accompanied by a visual nomenclature with which to describe consensually developed shared knowledge and personal understanding. This research activity has sought to iteratively evaluate this proposed theory through the practical application of the operational model and visual nomenclature to the domain of early-number counting, addition and subtraction. This domain of mathematical knowledge has been comprehensively analysed and described. Through this process, the viability of the proposed theory as a tool with which to discuss and thus improve the knowledge and understanding with the domain of mathematics has been validated. Putting of the proposed theory into practice has lead to the theory’s refinement and the subsequent achievement of a solid theoretical base for the future development of educational tools to support teaching and learning practice, including computer-mediated learning environments. Such future activity, using the proposed theory, will advance contemporary mathematics educational practice by bringing together the strengths of cognitivist, constructivist and post-constructivist learning theories.