Thesis (Ph.D.)--Ohio University, November, 2004. / Title from PDF t.p. Includes bibliographical references (p. 168-177)

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Civil Engineering, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Bibliography: leaves 207-213. / by Warren Buckler Powell. / Ph.D.

Recently,1 an algorithm has been derived for the explicit determination of an induced SL (n+2,R) Lie invariance transformation group for a completely integrable 2n - dimensional dynamical system defined on IR2n from that known for a free particle system with n degree of freedom. 2 In particular, the universal transitive Lie invariance transformation group for both the isotropic harmonic oscillator3 and the anharmonic oscillator4 (quartic potential) has been obtained by this algorithm. Further,5 it has been shown in theory and by example that a complete set of functionally independent constants of motion corresponds to an abelian subalgebra of the induced SL (n+2,R) group. In this work, preparations necessary to apply this algorithm to the 3-dimensional classical Kepler problem have been made. A brief explanation of the algorithm and its relation to the Kepler problem in given in Chapter I. The preparations including the identification of a suitable parametric form unifying the solution completely and simplify are given in subsequent chapters and Appendix I. They are followed in Appendix II by a paper6 containing the actually application involving the extension of the algorithm to arbitrarily reparameterized system. We should mention that only conservative Hamiltonian systems are treated in this thesis.

The Lesotho Junior Secondary Examination Analysis (2009 and 2010) revealed that students performance in Mathematics in general and Transformation geometry of rotation in particular was generally poor. Only a few number of students that sat for the final Form C Examination passed. This study employed the van Hiele’s levels of learning to investigate and describe the difficulties students have in the learning of rotational transformation geometry. Both a written test and interview were used to solicit information regarding students’ difficulties. This information was collected from 90 students from Qaoling Secondary School in Maseru district in Lesotho. Findings from the study revealed that students had difficulties in identifying and naming transformation of rotation, finding the centre, angle of rotation and locating the exact image of a rotated figure after rotation. Also, they had greater difficulties when using transformation to do proof. The analysis showed that students mostly had difficulties at the level of Abstraction and Deduction. This gave an indication that the vast majority of the students in Form C are reasoning at the lowest two levels of the van Hiele’s model which are Visualization and Description. For these students’ difficulties to be curbed, the analysis demonstrated amongst others that teachers needed to use Manipulative materials and Information Communication Technology (ICT) during the process of teaching and learning. Manipulative materials provide experience in which students can transfer their understanding smoothly from one concept to another. / Mathematics Education / M. Ed. (Mathematics Education)

by Johnson, Tin-Chak Pang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references. / Acknowledgments / Abstract / List of figures / List of tables / Chapter 1. --- Introduction --- p.1-1 / Chapter 1.1 --- Introduction --- p.1-1 / Chapter 1.2 --- Introduction to asynchronous system --- p.1-5 / Chapter 1.2.1 --- Motivation --- p.1-5 / Chapter 1.2.2 --- Hazards --- p.1-7 / Chapter 1.2.3 --- Classes of Asynchronous circuits --- p.1-8 / Chapter 1.3 --- Introduction to Transform Coding --- p.1-9 / Chapter 1.4 --- Organization of the Thesis --- p.1-16 / Chapter 2. --- Asynchronous Design Methodologies --- p.2-1 / Chapter 2.1 --- Introduction --- p.2-1 / Chapter 2.2 --- Self-timed system --- p.2-2 / Chapter 2.3 --- DCVSL Methodology --- p.2-4 / Chapter 2.3.1 --- DCVSL gate --- p.2-5 / Chapter 2.3.2 --- Handshake Control --- p.2-7 / Chapter 2.4 --- Micropipeline Methodology --- p.2-11 / Chapter 2.4.1 --- Summary of previous design --- p.2-12 / Chapter 2.4.2 --- New Micropipeline structure and improvements --- p.2-17 / Chapter 2.4.2.1 --- Asymmetrical delay --- p.2-20 / Chapter 2.4.2.2 --- Variable Delay and Delay Value Selection --- p.2-22 / Chapter 2.5 --- Comparison between DCVSL and Micropipeline --- p.2-25 / Chapter 3. --- Self-timed Multipliers --- p.3-1 / Chapter 3.1 --- Introduction --- p.3-1 / Chapter 3.2 --- Design Example 1 : Bit-serial matrix multiplier --- p.3-3 / Chapter 3.2.1 --- DCVSL design --- p.3-4 / Chapter 3.2.2 --- Micropipeline design --- p.3-4 / Chapter 3.2.3 --- The first test chip --- p.3-5 / Chapter 3.2.4 --- Second test chip --- p.3-7 / Chapter 3.3 --- Design Example 2 - Modified Booth's Multiplier --- p.3-9 / Chapter 3.3.1 --- Circuit Design --- p.3-10 / Chapter 3.3.2 --- Simulation result --- p.3-12 / Chapter 3.3.3 --- The third test chip --- p.3-14 / Chapter 4. --- Current-Sensing Completion Detection --- p.4-1 / Chapter 4.1 --- Introduction --- p.4-1 / Chapter 4.2 --- Current-sensor --- p.4-2 / Chapter 4.2.1 --- Constant current source --- p.4-2 / Chapter 4.2.2 --- Current mirror --- p.4-4 / Chapter 4.2.3 --- Current comparator --- p.4-5 / Chapter 4.3 --- Self-timed logic using CSCD --- p.4-9 / Chapter 4.4 --- CSCD test chips and testing results --- p.4-10 / Chapter 4.4.1 --- Test result --- p.4-11 / Chapter 5. --- Self-timed ICT processor architecture --- p.5-1 / Chapter 5.1 --- Introduction --- p.5-1 / Chapter 5.2 --- Comparison of different architecture --- p.5-3 / Chapter 5.2.1 --- General purpose Digital Signal Processor --- p.5-5 / Chapter 5.2.1.1 --- Hardware and speed estimation : --- p.5-6 / Chapter 5.2.2 --- Micropipeline without fast algorithm --- p.5-7 / Chapter 5.2.2.1 --- Hardware and speed estimation : --- p.5-8 / Chapter 5.2.3 --- Micropipeline with fast algorithm (I) --- p.5-8 / Chapter 5.2.3.1 --- Hardware and speed estimation : --- p.5-9 / Chapter 5.2.4 --- Micropipeline with fast algorithm (II) --- p.5-10 / Chapter 5.2.4.1 --- Hardware and speed estimation : --- p.5-11 / Chapter 6. --- Implementation of self-timed ICT processor --- p.6-1 / Chapter 6.1 --- Introduction --- p.6-1 / Chapter 6.2 --- Implementation of Self-timed 2-D ICT processor (First version) --- p.6-3 / Chapter 6.2.1 --- 1-D ICT module --- p.6-4 / Chapter 6.2.2 --- Self-timed Transpose memory --- p.6-5 / Chapter 6.2.3 --- Layout Design --- p.6-8 / Chapter 6.3 --- Implementation of Self-timed 1-D ICT processor with fast algorithm (final version) --- p.6-9 / Chapter 6.3.1 --- I/O buffers and control units --- p.6-10 / Chapter 6.3.1.1 --- Input control --- p.6-11 / Chapter 6.3.1.2 --- Output control --- p.6-12 / Chapter 6.3.1.2.1 --- Self-timed Computational Block --- p.6-13 / Chapter 6.3.1.3 --- Handshake Control Unit --- p.6-14 / Chapter 6.3.1.4 --- Integer Execution Unit (IEU) --- p.6-18 / Chapter 6.3.1.5 --- Program memory and Instruction decoder --- p.6-20 / Chapter 6.3.2 --- Layout Design --- p.6-21 / Chapter 6.4 --- Specifications of the final version self-timed ICT chip --- p.6-22 / Chapter 7. --- Testing of Self-timed ICT processor --- p.7-1 / Chapter 7.1 --- Introduction --- p.7-1 / Chapter 7.2 --- Pin assignment of Self-timed 1 -D ICT chip --- p.7-2 / Chapter 7.3 --- Simulation --- p.7-3 / Chapter 7.4 --- Testing of Self-timed 1-D ICT processor --- p.7-5 / Chapter 7.4.1 --- Functional test --- p.7-5 / Chapter 7.4.1.1 --- Testing environment and results --- p.7-5 / Chapter 7.4.2 --- Transient Characteristics --- p.7-7 / Chapter 7.4.3 --- Comments on speed and power --- p.7-10 / Chapter 7.4.4 --- Determination of optimum delay control voltage --- p.7-12 / Chapter 7.5 --- Testing of delay element and other logic cells --- p.7-13 / Chapter 8. --- Conclusions --- p.8-1 / Bibliography / Appendices

An overview of the historical developments of the paradigms of classical mechanics, the free particle, oscillator and the Kepler problem, is given ito (in terms of) their conserved quantities. Next, the orbits of the three paradigms are found from quadratic forms. The quadratic forms are constructed using first integrals found by the application of Poisson's theorem. The orbits are presented ito expanding surfaces defined by the quadratic forms. The Lie and Noether symmetries of the paradigms are investigated. The free particle is discussed in detail and an overview of the work done on the oscillator and Kepler problem is given. The Lie and Noether theories are compared from various aspects. A technical description of Lie groups and algebras is given. This provides a basis for a discussion of the historical development of the paradigms of mechanics ito their group properties. Lastly the paradigms are discussed ito of Quantum Mechanics. / Thesis (M.Sc.)-University of Natal, 1996.

Thesis (MSc (Applied Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: This thesis provides an introduction to the statistical modeling technique known as graphical models. Since graph theory and probability theory are the two legs of graphical models, these two topics are presented, and then combined to produce two examples of graphical models: Bayesian Networks and Markov Random Fields. Furthermore, the max-sum, sum-product and junction tree algorithms are discussed. The graphical modeling technique is then applied to the specific problem of finding coplanar points in stereo images, taken with an uncalibrated camera. Although it is discovered that graphical models might not be the best method, in terms of speed, to use for this appliation, it does illustrate how to apply this technique in a real-life problem. / AFRIKAANSE OPSOMMING: Hierdie tesis stel die leser voor aan die statistiese modelerings-tegniek genoemd grafiese modelle. Aangesien grafiek teorie en waarskynlikheidsleer die twee bene van grafiese modelle is, word hierdie areas aangespreek en dan gekombineer om twee voorbeelde van grafiese modelle te vind: Bayesian Netwerke en Markov Lukrake Liggaam. Die maks-som, som-produk en aansluitboom algoritmes word ook bestudeer. Nadat die teorie van grafiese modelle en hierdie drie algoritmes afgehandel is, word grafiese modelle dan toegepas op ’n spesifieke probleem— om punte op ’n gemeenskaplike vlak in stereo beelde te vind, wat met ’n ongekalibreerde kamera geneem is. Alhoewel gevind is dat grafiese modelle nie die optimale metode is om punte op ’n gemeenskaplike vlak te vind, in terme van spoed, word die gebruik van grafiese modelle wel ten toongestel met hierdie praktiese voorbeeld. / National Research Foundation (South Africa)

The Lesotho Junior Secondary Examination Analysis (2009 and 2010) revealed that students performance in Mathematics in general and Transformation geometry of rotation in particular was generally poor. Only a few number of students that sat for the final Form C Examination passed. This study employed the van Hiele’s levels of learning to investigate and describe the difficulties students have in the learning of rotational transformation geometry. Both a written test and interview were used to solicit information regarding students’ difficulties. This information was collected from 90 students from Qaoling Secondary School in Maseru district in Lesotho. Findings from the study revealed that students had difficulties in identifying and naming transformation of rotation, finding the centre, angle of rotation and locating the exact image of a rotated figure after rotation. Also, they had greater difficulties when using transformation to do proof. The analysis showed that students mostly had difficulties at the level of Abstraction and Deduction. This gave an indication that the vast majority of the students in Form C are reasoning at the lowest two levels of the van Hiele’s model which are Visualization and Description. For these students’ difficulties to be curbed, the analysis demonstrated amongst others that teachers needed to use Manipulative materials and Information Communication Technology (ICT) during the process of teaching and learning. Manipulative materials provide experience in which students can transfer their understanding smoothly from one concept to another. / Mathematics Education / M. Ed. (Mathematics Education)

Transformation geometry has been neglected in our schools because teachers are often not proficient enough to teach it, as it was not part of the syllabus during their training. The study investigates effective ways of teaching transformation geometry in grade 9, applying van Hiele’s theory (1986) of geometry teaching and learning and Kilpatrick’s model of mathematical proficiency. The teaching programme activities require consistent use of physical manipulatives by the teacher for effective teaching, learning and understanding of geometric concepts. The type of study is a case study. Data collection tools are: - baseline evaluation, teacher & learner interviews (pre & post programme intervention) and observation (pre & post) during the implementation of the teaching programme. Results were analysed both qualitatively and quantitatively. My research findings show some improvement of learner performance after the application of the programme. Baseline evaluation shows that some learners attained below and above 30%. Interviews showed that some learners had problems before the implementation of the programme and some problems were eliminated by the use of the programme activities and learning progression was evident. Learner performance showed that learners had acquired some knowledge and critical thinking and reasoning skills, reflection skills, communication through LOLT improved, commitment to activities of the programme and teaching practice had improved. Learner performance showed that a learner can be in two different levels at the same time. Consistent use of manipulatives resulted in effective teaching and learning of geometry in grade 9. The results of this research support other researchers’ views of teaching geometry using van Hiele’s theory (1986) and Kilpatrick et al. (2001). Shaw (2002) argues that teaching geometry with manipulatives enhances conceptual understanding by the learner. In my opinion, it also promotes immediate intervention by the teacher as soon as the learner picks an incorrect object. The project enhanced and improved levels of communication between the learner, teacher and others in the classroom.

Orientador: Claudina Izepe Rodrigues / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T16:14:16Z (GMT). No. of bitstreams: 1 Franco_MarciaCristinaLemosGuimaraes_M.pdf: 20497304 bytes, checksum: c28b5c2e4a775ec0c3f67197069a584f (MD5) Previous issue date: 2015 / Resumo: Neste trabalho apresentamos um estudo sobre grupos, transformações geométricas e isometrias no plano. Apresentamos o teorema da classificação das isometrias no plano, o teorema de Leonardo que classifica os grupos de simetria de ornamentos limitados e o teorema da classificação dos grupos de frisos. Propomos sequências de atividades para a Educação Básica envolvendo as isometrias e a identificação do grupo de simetria de um ornamento limitado e de um friso. Além disso, as atividades sugeridas apresentam intuitivamente a ideia da estrutura algébrica de grupos. Finalizamos este trabalho relatando como ocorreu a aplicação de três das sequências sugeridas, os procedimentos adotados e os resultados obtidos / Abstract: We present a study of groups, geometric transformations and isometries in the plane. Introducing the classification theorem of isometries in the plane, the Leonardo theorem that classifies symmetry groups of limited ornaments and the classification theorem of friezes groups. We propose a sequence of activities for the basic education involving isometry and identification of symmetry group of limited ornaments and friezes. Moreover some of the suggested activities provide an intuitive idea of the algebraic structure of groups. We end this paper by reporting on the manner in which the application of three of the suggested sequences occurred, the procedures that were adopted and the results that were obtained / Mestrado / Matemática em Rede Nacional / Mestra