Student Number : 8800092K - MSc research report - School of Education - Faculty of Science / This research report explored the strategies which grade 12 learners employ to solve geometric problems. The purpose of this research was to gain an understanding of how grade 12 learners begin to solve geometric problems involving cyclic quadrilateral and tangent theorems. A case study method was used as the main research method. The study employed the van Hiele level’s of geometric thought as a method for categorising learners levels of understanding. Data about the strategies which learners recruit to solve geometric problems were gathered using learner-based tasks, semi-structured interviews and document analysis. From the data gathered, the following patterns emerged: learners incorrect use of theorems to solve geometrical problems; learners base their responses on the visual appearance of the diagram; learners “force “ a solution when one is not available; learners’ views of proof. Each of these aspects is discussed. The report concludes that learners strategies to solving geometric problems are based largely on the manner in which educators approach the solving of geometrical problems.

Many digital watermarking schemes have been proposed for copyright protection recently due to the rapid growth of multimedia data distribution. Robustness is one of the crucial important issues in watermarking. But, most of traditional digital watermarking schemes is normally not to resist both geometric distortion and signal processing attacks well. There are two different types of solutions to resisting geometrical attacks: nonblind and blind methods. With the noblind approach, due to availability of the original image, the problem can be resolved with a good solution by elective search between the geometrically attacked and unattacked image. The blind solution, which does not use the original image in watermark extraction, is obviously more challenging. In this research, we propose a blind watermarking scheme which based on histogram property. So that, we propose a novel scheme to define the lattice structure of color space of host image for embedding watermark data. We utilize the histograms of various properties that calculated from the host image, and partition each histogram space into several divisions with dynamic interval. The number of pixels of each division is equal. And then we embed watermark data by modifying distribution of each division. The experimented results present the algorithm is robust to resist common geometric attacks and high quality JPEG compression at the same time.

In the thesis, geometric programming is considered as a numerical optimization technique. The problem of minimizing the integral square error of a system characterized by a second order plant with proportional- integral-derivative (PID) controller is investigated. Constraints are imposed upon the state of the system In order to obtain feasible solutions and conditions that are amenable to the geometric programming technique. The application of geometric programming requires the use of approximation procedures to eliminate untenable conditions in the objective and constraint functions. The techniques utilized render solutions that are easily obtainable, usually amounting to solving a set of linear equations and requiring no differentiation of terms. In addition, there is rapid convergence to an optimum. The accuracy of the results is dependent upon the validity of the approximations. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

In the setting of Riemannian manifolds with nonnegative Ricci curvature, we provide geometric inequalities as consequences of the Monotonicity Formulas holding along the flow of the level sets of the p-capacitary potential. The work is divided into three parts. (1) In the first part, we describe the asymptotic behaviour of the p-capactitary potential in a natural class of Riemannian manifolds. (2) The second part is devoted to the proof of our Monotonicity-Rigidity Theorems. (3) In the last part, we apply the Monotonicity Theorems to obtain geometric inequalities, focusing on the Extended Minkowski Inequality.

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No / This title provides detailed description of how Partial Differential Equations are used in the field of geometric design, and supplies clear and concise explanations of how to implement the techniques described. It also offers extensive discussions (with examples) or practical applications of Partial Differential Equations in geometric design.

No / A methodology for shape morphing using partial differential equation (PDE) surfaces is presented in this work. The use of the PDE formulation shows how shape morphing can be based on a boundary-value approach by which intermediate shapes can be created. Furthermore, the mathematical properties of the method give rise to several alternatives in which morphing one shape into another can be achieved. Three of these alternatives are presented here. The first one is based on the gradual variation of the weighted sum of the boundary conditions for each surface, the second one consists of varying the Fourier mode for which the PDE is solved whilst the third results from a combination of the first two. Examples showing the efficiency of these methodologies are presented. Thus, it is shown that the PDE based approach for morphing, when combined with a parametric variation of the boundary conditions, is capable of obtaining smooth intermediate surfaces automatically.

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