No description available.

No description available.

If the discovery of fire makes human re-define the darkness, then the invention of computer can be said to change the the view of people admiring the technology. Today, the computer which evolves from ancient abacus, not only can perform arbitrary precision computation, but also support symbolic computation, programming language, and data and function visualization. The Mathematica is such a mathematical software which can handle problems in every field. This thesis will explore how to solve the problems of equations, inequalities and optimization, and how to get solutions which can¡¦t easy get from pen and paper by Mathematica.

No description available.

No description available.

No description available.

This thesis describes the development of a model of cerebral atrophic change associated with neurodegeneration. Neurodegenerative diseases such as Alzheimer's dementia present a significant health problem within the elderly population. Effective treatment relies upon the early detection of anatomic change, and the subsequent differential diagnosis of the disorder from other closely related neurological conditions. Importantly, this also includes the investigation of the relationship between atrophic change and cognitive function. In unison with the growth in neuroimaging technology, myriad methodologies have been developed since the first quantitative measures of atrophic change were deduced via manual tracing. Subsequently, automated region of interest analysis, segmentation, voxel-based morphometry and non-linear registration have all been used to investigate atrophy. These methods commonly report findings of ventricular enlargement and temporal lobe change in AD and other dementias. Whilst these results are accurate indicators of atrophy, they are largely non-specific in their diagnostic utility. In addition, the aforementioned methods have been employed to discern change observed at discrete intervals during a disease process. In order to gain a greater understanding of the temporal characteristics of changes that occur as a result of atrophy, a deformation modelling method that allows the continuous tracking of these changes in a cohort of AD patients and elderly control subjects is presented in this thesis. Deformation modelling involves non-linear registration of images to investigate the change that is apparent between two or more images. The non- linear registration results are analysed and presented via three metrics: local volume loss (atrophy); volume (CSF) increase; and translation (interpreted as representing collapse of cortical structures). Changes observed in the analyses in this thesis are consistent with results from neuro-anatomical studies of AD. Results using the more traditional methods of analysis are presented for comparative purposes.

This thesis describes the development of a model of cerebral atrophic change associated with neurodegeneration. Neurodegenerative diseases such as Alzheimer's dementia present a significant health problem within the elderly population. Effective treatment relies upon the early detection of anatomic change, and the subsequent differential diagnosis of the disorder from other closely related neurological conditions. Importantly, this also includes the investigation of the relationship between atrophic change and cognitive function. In unison with the growth in neuroimaging technology, myriad methodologies have been developed since the first quantitative measures of atrophic change were deduced via manual tracing. Subsequently, automated region of interest analysis, segmentation, voxel-based morphometry and non-linear registration have all been used to investigate atrophy. These methods commonly report findings of ventricular enlargement and temporal lobe change in AD and other dementias. Whilst these results are accurate indicators of atrophy, they are largely non-specific in their diagnostic utility. In addition, the aforementioned methods have been employed to discern change observed at discrete intervals during a disease process. In order to gain a greater understanding of the temporal characteristics of changes that occur as a result of atrophy, a deformation modelling method that allows the continuous tracking of these changes in a cohort of AD patients and elderly control subjects is presented in this thesis. Deformation modelling involves non-linear registration of images to investigate the change that is apparent between two or more images. The non- linear registration results are analysed and presented via three metrics: local volume loss (atrophy); volume (CSF) increase; and translation (interpreted as representing collapse of cortical structures). Changes observed in the analyses in this thesis are consistent with results from neuro-anatomical studies of AD. Results using the more traditional methods of analysis are presented for comparative purposes.

In this thesis, two distinct problems in data-driven computational science are considered. The main problem of interest is the multiobjective optimization problem, where the tradeoff surface (called the Pareto front) between multiple conflicting objectives must be approximated in order to identify designs that balance real-world tradeoffs. In order to solve multiobjective optimization problems that are derived from computationally expensive blackbox functions, such as engineering design optimization problems, several methodologies are combined, including surrogate modeling, trust region methods, and adaptive weighting. The result is a numerical software package that finds approximately Pareto optimal solutions that are evenly distributed across the Pareto front, using minimal cost function evaluations. The second problem of interest is the closely related problem of multivariate interpolation, where an unknown response surface representing an underlying phenomenon is approximated by finding a function that exactly matches available data. To solve the interpolation problem, a novel algorithm is proposed for computing only a sparse subset of the elements in the Delaunay triangulation, as needed to compute the Delaunay interpolant. For high-dimensional data, this reduces the time and space complexity of Delaunay interpolation from exponential time to polynomial time in practice. For each of the above problems, both serial and parallel implementations are described. Additionally, both solutions are demonstrated on real-world problems in computer system performance modeling. / Doctor of Philosophy / Science and engineering are full of multiobjective tradeoff problems. For example, a portfolio manager may seek to build a financial portfolio with low risk, high return rates, and minimal transaction fees; an aircraft engineer may seek a design that maximizes lift, minimizes drag force, and minimizes aircraft weight; a chemist may seek a catalyst with low viscosity, low production costs, and high effective yield; or a computational scientist may seek to fit a numerical model that minimizes the fit error while also minimizing a regularization term that leverages domain knowledge. Often, these criteria are conflicting, meaning that improved performance by one criterion must be at the expense of decreased performance in another criterion. The solution to a multiobjective optimization problem allows decision makers to balance the inherent tradeoff between conflicting objectives. A related problem is the multivariate interpolation problem, where the goal is to predict the outcome of an event based on a database of past observations, while exactly matching all observations in that database. Multivariate interpolation problems are equally as prevalent and impactful as multiobjective optimization problems. For example, a pharmaceutical company may seek a prediction for the costs and effects of a proposed drug; an aerospace engineer may seek a prediction for the lift and drag of a new aircraft design; or a search engine may seek a prediction for the classification of an unlabeled image. Delaunay interpolation offers a unique solution to this problem, backed by decades of rigorous theory and analytical error bounds, but does not scale to high-dimensional "big data" problems. In this thesis, novel algorithms and software are proposed for solving both of these extremely difficult problems.

The study took the form of action research situated in a case study. The participants consisted of sixteen Grade 11 learners who attend a non-profit tutoring organisation called Ikamva Youth on Saturday mornings and who volunteered to learn mathematics with mathematical software. Data were collected from the learners using a structured questionnaire, journals in which learners were encouraged to record their daily experiences and by studying the learners’ own written work during the research period. Moreover, the research closed with a focus group session. The study primarily described three aspects, viz. the degree to which learners are able to work with the strands of strategic competence and adaptive reasoning with particular emphasis on interpretation and application, knowledge production and justification and social collaboration; barriers in terms of working with these strands in a digital environment that may interfere with the learning process; and, the personal (affective) response of the students to the availability of technology. Findings suggest that the participants experienced difficulties in accessing these particular strands of mathematical knowledge, and subsequently expressed the desire to revert back to a place where the teacher assumes a more direct teaching style and where the focus of mathematical learning is on technique. In other words, learners preferred routine expertise, whilst appearing to lack in adaptive expertise. With respect to the second aspect of the research, it is suggested that one of the most prominent barriers to integrating technology into mathematics does not necessarily relate to adjustments in terms of the instrumental use of software and the computer environment, but seems to stem from the learners’ own epistemologies and beliefs about the nature of what constitutes effective mathematical teaching and learning. In paying attention to the voice of the learners it surfaced that the learners tend to associate computers more strongly with a cultural tool for entertainment than with mathematical learning. Although the study points out that implementing an interactive curriculum integrated with technology into a classroom with learners from low socio-economic backgrounds is not necessarily straightforward, it does suggest that with more frequent exposure certain learners can develop a propensity for working within a cognitively rich problem-solving context and effectively employ the mathematical software as an extension of their own thinking. This study adds to broader research on the role of technology in learning by reporting that the learners felt that the contextualisation of mathematics, followed by the ability to manipulate the graph themselves were the main contributors to their mathematical learning. The learners disregarded the visualisation effect of the computer as they felt that it had very little impact on their mathematical learning.