Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

Learning to drive: from hazard detection to hazard handling

Hazard perception has been found to correlate with crash involvement, and has thus been suggested as the most likely source of any skill gap between novice and experienced drivers. The most commonly used method for measuring hazard perception is to evaluate the perception-reaction time to filmed traffic events. It can be argued that this method lacks ecological validity and may be of limited value in predicting the actions drivers’ will take to hazards encountered. The first two studies of this thesis compare novice and experienced drivers’ performance on a hazard detection test, requiring discrete button press responses, with their behaviour in a more dynamic driving environment, requiring hazard handling ability. Results indicate that the hazard handling test is more successful at identifying experience-related differences in response time to hazards. Hazard detection test scores were strongly related to performance on a driver theory test, implying that traditional hazard perception tests may be focusing more on declarative knowledge of driving than on the procedural knowledge required to successfully avoid hazards while driving. One in five Irish drivers crash within a year of passing their driving test. This suggests that the current driver training system does not fully prepare drivers for the dangers they will encounter. Thus, the third and fourth studies in this thesis focus on the development of two simulator-based training regimes. In the third study participants receive intensive training on the molar elements of driving i.e. speed and distance evaluation. The fourth study focuses on training higher order situation awareness skills, including perception, comprehension and projection. Results indicate significant improvement in aspects of speed, distance and situation awareness across training days. However, neither training programme leads to significant improvements in hazard handling performance, highlighting the difficulties of applying learning to situations not previously encountered.

Reasoning about trust, entrepreneurial services, and the virtual team

This Portfolio of Exploration (PoE) tracks a transformative learning developmental journey that is directed at changing meaning making structures and mental models within an innovation practice. The explicit purpose of the Portfolio is to develop new and different perspectives that enable the handling of new and more complex phenomena through self transformation and increased emotional intelligence development. The Portfolio provides a response to the question: ‘What are the key determinants that enable a Virtual Team (VT) to flourish where flourishing means developing and delivering on the firm’s innovative imperatives?’ Furthermore, the PoE is structured as an investigation into how higher order meaning making promotes ‘entrepreneurial services’ within an intra-firm virtual team, with a secondary aim to identify how reasoning about trust influence KGPs to exchange knowledge. I have developed a framework which specifically focuses on the effectiveness of any firms’ Virtual Team (VT) through transforming the meaning making of the VT participants. I hypothesized it is the way KGPs make meaning (reasoning about trust) which differentiates the firm as a growing firm in the sense of Penrosean resources: ‘inducement to expand and a limit of expansion’ (1959). Reasoning about trust is used as a higher order meaning-making concept in line with Kegan’s (1994) conception of complex meaning making, which is the combining of ideas and data in ways that transform meaning and implicates participants to find new ways of knowledge generation. Simply, it is the VT participants who develop higher order meaning making that hold the capabilities to transform the firm from within, providing a unique competitive advantage that enables the firm to grow.

Development and characterization of novel transgenic techniques for the study of neuronal circuitry

Modern neuroscience relies heavily on sophisticated tools that allow us to visualize and manipulate cells with precise spatial and temporal control. Transgenic mouse models, for example, can be used to manipulate cellular activity in order to draw conclusions about the molecular events responsible for the development, maintenance and refinement of healthy and/or diseased neuronal circuits. Although it is fairly well established that circuits respond to activity-dependent competition between neurons, we have yet to understand either the mechanisms underlying these events or the higher-order plasticity that synchronizes entire circuits. In this thesis we aimed to develop and characterize transgenic mouse models that can be used to directly address these outstanding biological questions in different ways. We present SLICK-H, a Cre-expressing mouse line that can achieve drug-inducible, widespread, neuron-specific manipulations in vivo. This model is a clear improvement over existing models because of its particularly strong, widespread, and even distribution pattern that can be tightly controlled in the absence of drug induction. We also present SLICK-V::Ptox, a mouse line that, through expression of the tetanus toxin light chain, allows long-term inhibition of neurotransmission in a small subset (<1%) of fluorescently labeled pyramidal cells. This model, which can be used to study how a silenced cell performs in a wildtype environment, greatly facilitates the in vivo study of activity-dependent competition in the mammalian brain. As an initial application we used this model to show that tetanus toxin-expressing CA1 neurons experience a 15% - 19% decrease in apical dendritic spine density. Finally, we also describe the attempt to create additional Cre-driven mouse lines that would allow conditional alteration of neuronal activity either by hyperpolarization or inhibition of neurotransmission. Overall, the models characterized in this thesis expand upon the wealth of tools available that aim to dissect neuronal circuitry by genetically manipulating neurons in vivo.