There is no abstract available for this thesis.

Thesis (M.Sc.)--Victoria University of Wellington, 2009. / Includes bibliographical references.

A dissertation submitted to the Faculty of Education in partial fulfilment of the requirements for the degree of Masters of Education in the Department of Mathematics, Science and Technology Education at the University Of Zululand, South Africa, 2015 / This research aims to establish the level of students’ cognitive skills using GeoGebra, and investigates whether GeoGebra as a technological tool helps in improving poor performance in respect of Euclidean geometry or geometry of the circle. Students’ interests, in learning about circle geometry in mathematics, are also being tested. GeoGebra is an innovative, dynamic mathematics software which integrates algebra, geometry and calculus to aid students during the learning process. The specific sample in this research consists of 112 Grade 11 secondary school learners within the UMkhanyakude district, Hlabisa circuit, under the Empembeni and Ezibayeni wards. During this research, GeoGebra and the concept of circle geometry were introduced to students. Afterwards, students had to answer several geometry of the circle questions, entailing key theorems as prescribed by the National Mathematics pacesetter for Grade 11 and Grade 12. As students answered the above questions, they solved problems and conducted discussions among themselves. At the end, students were individually required to answer questionnaires which consisted of 15 closed items relating to views on GeoGebra and its impact on Euclidean geometry and mathematics, as well as three open-ended questions which asked learners about their reflections on the application of GeoGebra. The above methods provided a strong base to explore whether GeoGebra as a tool helps students in the learning process. The results showed that students endorsed the use of GeoGebra as a technological tool in the teaching of Euclidean geometry. Some students even suggested that GeoGebra be used in other mathematical topics. Students overall enjoyed the use of GeoGebra, finding it user-friendly and a highly significant learning motivator.

<p> In this thesis, we shall solve the classical Dirichlet problem for a ball in n-dimensional Euclidean space, and then point out that the classical Dirichlet problem is not always solvable. Following Wiener and Brelot, we then introduce a generalized Dirichlet problem for any bounded region in n-dimensional Euclidean space and establish necessary and sufficient conditions for its solution. We show that the solution of the generalized Dirichlet problem coincides with the solution of the classical Dirichlet problem whenever the latter exists. Finally, we characterize those regions for which the classical Dirichlet problem is solvable by considering the boundary behaviour of those functions for which the generalized problem is solvable.</p> / Thesis / Master of Science (MSc)

The teaching and learning of Euclidean geometry in black South African schools is characterised by teachers' poor knowledge of Euclidean geometry and the use of traditional taaching methods that encourage rote learning. In the light of this, the study investigated the extent to which learners perform in Process-Based Instruction. Process-Based Instruction is charactarised by the gradual transfer of instruction from the teacher to the learner. In Process-Based Instruction learners are expected to communicate thought,form concepts and master theorems by drawing, 'showed with and used hands', talking and writing. Learners' performance is assessed by rubrics and is analysed using graphs. The findings of the empirical investigation revealed that learners of both grades 8 and 9 cannot communicate thought by any media (talking, writing, 'showed with and used hands' and drawing). A few learners formed a few concepts and mastered theorems. On the basis of the findings and the shortcomings of the study recommendations are made. / Educational studies / M. Ed. (Didactics)

Text in English, abstract in English and Afrikaans / The teaching and learning of Euclidean geometry in black South African schools is characterised by teachers' poor knowledge of Euclidean geometry and the use of traditional teaching methods that encourage rote learning. In the light of this, the study investigated the extent to which learners perform in Process-Based Instruction. Process-Based Instruction is characterised by the gradual transfer of instruction from the teacher to the learner. In Process-Based Instruction learners are expected to communicate thought, form concepts and master theorems by drawing, 'showed with and used hands', talking and writing. Learners' performance is assessed by rubrics and is analysed using graphs. The findings of the empirical investigation revealed that learners of both grades 8 and 9 cannot communicate thought by any media (talking, writing 'showed with and used hands' and drawing). A few learners formed a few concepts and mastered theorems. On the basis of the findings and the shortcomings of the study recommendations are made. / Die onderrig van Euclediaanse Meetkunde in swart Suid-Afrikaanse skole word gekenmerk aan onderwysers se swak kennis van die vak, asook die gebruik van onderrigmetodes wat op die leerder se geheue staatmaak. Hierdie studie ondersoek die sukses van leerders se prestasie in proses-gebasseerde ondenig. Proses-gebaseerde onderig word gekenmerk aan die geleidelike oordra van kennis, van die onderwyser na die leerder. Daar word van leerders verwag om gedagtes oor te dra, konsepte (begrippe) te verstaan en stellings onder die knie te kry deur middel van sketse, die gebruik van gebare, verbale verduideliking en skriftelike werk. Leerders se prestasie word geevalueer deur die gebruik van klassifisering in kolomme (rubrics) en word ge-analiseer deur gralieke. Die bevinding van die empieriese ondersoek het getoon dat leerders van sowel graad 8 as graad 9 nie gedagtes kan oordra deur enige van die gemelde metodes nie. 'n Paar leerders het wel enkele begrippe verstaan en kon stellings bemeester. Na aanleiding van die bevindings van die studie en die tekortkominge wat daardeur opgespoor is, word sekere voorstelle gemaak. / Didactics / M. Ed. (Didactics)

Part I discusses the problem of determining the set of intersection points, with corresponding multiplicities, of two algebraic plane curves. We derive an algorithm based on the Euclidean Algorithm for polynomials and show how to use it to find the intersection points of two given curves. We also show that an easy proof of Bézout’s Theorem follows. We then discuss how, for curves with rational coefficients, this algorithm can bemodified to find the intersection points with coordinates expressed in terms of algebraic extensions of the rational numbers. Part II deals with the problem of determining the (monic) integer transfinite diameter of a given real interval. We show how this problem relates to the problem of determining the structure of the spectrum of normalised leading coefficients of polynomials with integer coefficients and all roots in the given interval. We then find dense regions of this spectrum for a number of intervals and discuss algorithms for finding discrete subsets of the spectrum for the interval [0,1]. This leads to an improvement in the known upper bound for the integer transfinite diameter. Finally, we discuss the connection between the infimum of the spectrum and the monic integer transfinite diameter.

The purpose of this paper is to explore some of the properties of the Cantor set and to extend the idea of this set to metric spaces, in general, and to other sets of real numbers and sets in N-dimensional Euclidean space, in particular.

Diffusion tensor imaging (DTI) is a relatively new technology of magnetic resonance imaging, which enables us to observe the insight structure of the human body in vivo and non-invasively. It displays water molecule movement by a 3×3 diffusion tensor at each voxel. Tensor field processing, visualisation and tractography are all based on the diffusion tensors. The accuracy of estimating diffusion tensor is essential in DTI. This research focuses on exploring the potential improvements at the tensor estimation of DTI. We analyse the noise arising in the measurement of diffusion signals. We present robust methods, least median squares (LMS) and least trimmed squares (LTS) regressions, with forward search algorithm that reduce or eliminate outliers to the desired level. An investigation of the criterion to detect outliers is provided in theory and practice. We compare the results with the generalised non-robust models in simulation studies and applicants and also validated various regressions in terms of FA, MD and orientations. We show that the robust methods can handle the data with up to 50% corruption. The robust regressions have better estimations than generalised models in the presence of outliers. We also consider the multiple tensors problems. We review the recent techniques of multiple tensor problems. Then we provide a new model considering neighbours' information, the Bayesian single and double tensor models using neighbouring tensors as priors, which can identify the double tensors effectively. We design a framework to estimate the diffusion tensor field with detecting whether it is a single tensor model or multiple tensor model. An output of this framework is the Bayesian neighbour (BN) algorithm that improves the accuracy at the intersection of multiple fibres. We examine the dependence of the estimators on the FA and MD and angle between two principal diffusion orientations and the goodness of fit. The Bayesian models are applied to the real data with validation. We show that the double tensors model is more accurate on distinct fibre orientations, more anisotropic or similar mean diffusivity tensors. The final contribution of this research is in covariance tensor estimation. We define the median covariance matrix in terms of Euclidean and various non-Euclidean metrics taking its symmetric semi-positive definiteness into account. We compare with estimation methods, Euclidean, power Euclidean, square root Euclidean, log-Euclidean, Riemannian Euclidean and Procrustes median tensors. We provide an analysis of the different metric between different median covariance tensors. We also provide the weighting functions and define the weighted non-Euclidean covariance tensors. We finish with manifold-valued data applications that improve the illustration of DTI images in tensor field processing with defined non-weighted and weighted median tensors. The validation of non-Euclidean methods is studied in the tensor field processing. We show that the root square median estimator is preferable in general, which can effectively exclude outliers and clearly shows the important structures of the brain. The power Euclidean median estimator is recommended when producing FA map.

acase@tulane.edu / 1 / Fang Sun