The Maple V package Vessiot is an extensive set of procedures for performing computations in variational and tensor calculus. Vessiot is an extension of a previous package, Helmholtz, which was written by Cinnamon Hillyard for performing operations in the calculus of variations. The original set of commands included standard operators on differential forms, Euler-Lagrange operators, the Lie bracket operator, Lie derivatives, and homotopy operators. These capabilities are preserved in Vessiot, and enhanced so as to function in a multiple coordinate frame context. In addition, a substantial number of general tensor operations have been added to the package. These include standard algebraic operations such as the tensor product, contraction, raising and lowering of indices, as well covariant and Lie differentiation. Objects such as connections, the Riemannian curvature tensor, and Ricci tensor and scalar may also be easily computed. A synopsis of the command syntax appears in Appendix A on pages 194 through 225, and a complete listing of the Maple procedural code is given in Appendix B, beginning on page 222.

Thesis (M.S.)--University of Michigan, 1960.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-01-06T13:40:54Z No. of bitstreams: 1 davidsonmendesferreiradepaula.pdf: 1155298 bytes, checksum: 9e0a8999b7a791620c60d74d75c4915e (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-02-07T13:21:42Z (GMT) No. of bitstreams: 1 davidsonmendesferreiradepaula.pdf: 1155298 bytes, checksum: 9e0a8999b7a791620c60d74d75c4915e (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-02-07T14:08:17Z (GMT) No. of bitstreams: 1 davidsonmendesferreiradepaula.pdf: 1155298 bytes, checksum: 9e0a8999b7a791620c60d74d75c4915e (MD5) / Made available in DSpace on 2017-02-07T14:08:17Z (GMT). No. of bitstreams: 1 davidsonmendesferreiradepaula.pdf: 1155298 bytes, checksum: 9e0a8999b7a791620c60d74d75c4915e (MD5) Previous issue date: 2016-08-18 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O conceito de Limite é uma das ferramentas fundamentais no ensino de cálculo diferencial e integral no ensino superior, mas que normalmente não é lecionado no ensino básico, embora esse tema tenha feito parte dos livros do ensino médio por um longo tempo. Este trabalho visa mostrar a importância de se abordar esse assunto nesse nível, como um elo que une o ensino médio e a graduação, pois, o conteúdo ensinado até o ensino médio, não é completamente eficiente para estudar matemática mais avançada. Ele é composto por vários planos de aula que tratam desde a noção inicial de limite, passa por derivada, e finda nas somas de Riemann e noções de integral, além de uma síntese sobre esses assuntos. Temas como máximos e mínimos de funções, áreas e volumes, tornam a introdução do tema menos impactante e abrem caminho para resolvermos problemas mais avançados. / The Limit concept is one of the fundamental tools in teaching differential and integral calculus in higher education although it is not usually taught at basic education level, although this theme has been part of high school books for a long time. This work aims to show the importance to deal with Limit at this level, so it can work as a link between high school and higher education, once the content taught at elementary and high school are not really efficient for the study of more advanced mathematics. This work consists of several lesson plans that deal from the initial concept of limit, go through derivative, and end in Riemann’s sums and integral notions, as well as offer an overview of such issues. Topics such as maximum and minimum of the functions and also areas and volume make the introduction of the topic less impactful and pave the way to the solution of more advanced problems.

D.Phil. / The research problematised the learning of mathematics in South African high schools in a Pedagogical Content Knowledge context. The researcher established that while at best, teachers may command mathematics content knowledge, or pedagogic knowledge, that command proves insufficient in leveraging the learning of mathematics and differentiation. Teachers' awareness of their learners' errors and misconceptions on a mathematics topic is critical in developing appropriate pedagogical content knowledge. The researcher argues that the study of learner errors in mathematics affords educators critical knowledge of the learners' Zones of Proximal Development. The space where learners experience misconceptions as they attempt to assign meaning to new mathematical ideas to which they may or may not have obtained semiotic mediation. In their Zones of Proximal Development learners may harbour concept images that are incompetition with established mathematical knowledge.Educators need to study and understand those concept images (amateur or alternative conceptions), and how learners come to have them, if they are to help learners learn mathematics better. Besides the socio-cultural v1ew, the study presumed that the misconceptions formed by learners in mathematicsmay also beexplained within a constructivist perspective of learning. The constructivist perspective of learning assumes that learners interpret new knowledge on the basis of the knowledge they already have. However, some of the knowledge that learners construct though meaningful to them may be full of misconceptions. This may occur through overgeneralisation of prior knowledge to new situations. The researcher presumed that the ideas that learners have of particular mathematical concepts were concept images they construct. Though some of the concept images may be deficient or defective from a mathematics expert's point of view, they are still used by the learners to learn new mathematics concepts and to solve mathematics problems. The lack of success in mathematics that results in the application of erratic concept images ultimately leads to unsuccessful learning of mathematics with the danger of snowballing if there are no practicable interventions. Differentiation is a new topic in the South African mathematics curriculum and most teachers and learners have registered problems in teaching and learning it. Hence it was imperative to do research on this topic from an angle of learner errors on that topic. The significance of the study is that this research isolated the differentiation learner errors and misconceptions that teachers can focus on for the improvement of learning and achievement in the topic of introductory differentiation. The research focused on the nature of errors and misconceptions learners have on introductory differentiation as exhibited in their 2008 examination scripts. It sought to identify, categorise (form a database) and discuss the errors and their conceptual links. A typology of errors and misconceptions in introductory calculus was constructed. The study mainly used qualitative methods to collect and analyse data. Content analysis techniques were used to analyse the data on the basis of a conceptual framework of mathematics and calculus errors obtained from literature. One thousand Grade 12, Mathematics Paper 1 examination scripts from learners of both sexes emanating from diverse social backgrounds provided data for the study. The unit of analysis was students' errors in written responses to differentiation examination items.

In this dissertation we prove the Completeness, Soundness and Compactness of the Monadic Hybrid Calculus MHC and we prove its expressive equivalence to the Monadic Predicate Calculus MPC. The Monadic Hybrid Calculus MHC is a new system that is based on the (propositional) modal logic S5. It is “Hybrid” in the sense that it includes quantifier free MPC and therefore, unlike S5, allows free individual constants. The main innovation in this system is the elimination of bound variables. In MHC, upper case letters denote properties and lower case letters denote individuals. Universal quantification is represented by square brackets, [], and existential quantification is represented by angled brackets, 〈〉. Thus, All Athenians are Greek and mortal is formalized as [A](G∧M), Some mortal Greeks are Athenians as 〈M∧G〉A, and Socrates is mortal and Athenian as s(M∧A). We give the formal syntax and the formal semantics of [MHC] and give Beth-style Tableau Rules (Inference Rules). In these rules, if [P]Q is on the right then we select a new constant [v] and we add [vP] on left, vQ on the right, and we cancel the formula. If [P]Q is on the left then we select a pre-used constant p and split the tree. We add pP on the right of one branch and pQ on the left of the other branch. We treat 〈P〉Q similarly. Our Completeness proof uses induction on formulas down a path in the proof tree. Our Soundness proof uses induction up a path. To prove that MPC is logically equivalent to the Monadic Predicate Calculus, we present algorithms that transform formulas back and forth between these two systems. Compactness follows immediately. Finally, we examine the pragmatic usage of the Monadic Hybrid Calculus and we compare it with the Monadic Predicate Calculus using natural language examples. We also examine the novel notions of the Hybrid Predicate Calculus along with their pragmatic implications. / Graduate / 0800 / 0984

Magister Scientiae Dentium - MSc(Dent) / The epidemiology of periodontal disease in the Republic of South Africa has received only scant attention in the past and consequently the available information is limited. The present study was therefore planned with the primary goal being to establish base-line information regarding periodontal disease in a portion of the population. The adolescent age group was selected as the target for the survey in that destructive periodontal disease (periodontitis) probably commences in many instances during the teenage years and therefore the group would be the one most likely to derive maximum benefit from preventive care and simple treatment measures which could be realistically provided by existing community dental health services. To translate the result into practicality a simple method of treatment needs estimation was also incorporated. In view of the diverse nature of the inhabitants of the Republic of South Africa and in keep with previously conducted studies, the presedt survey was confined to a single ethnic group. The population comprised all 3 .684 white pupils in Standard VIII attending the 34 schools in the Cape Peninsula during 1977. A random sample of 500 was selected for investigation. The average age of the sample was 15 years 9 months and the two sexes were equally represented. Only 7,2% were classified in the lower grade socio-economic class and thus were considered not to have a significant effect on the results. METHOD A team of three, consisting of the author and two assistants, visited each school. Portable equipment included a reclining chair, lighting, compressed air, and hand instruments. The investigation began with a questionnaire to establish the attitude to and experience of symptoms, prevention, and treatment of periodontal disease within the sample. Each subject was then examined and at each of 12 sites, on the 8 incisors and 4 first molars, recordings were made of plaque, gingivitis, supragingival calculus, subgingival calculus, and loss of attachment (periodontitis) according to defined criteria. A standard statistical package was used to analyse the recordings. RESULTS The questionnaire: This showed that almost all the subjects (98%) were interested in the prevention and treatment of periodontal disease in order to achieve and maintain oral health. Not with standing this.The overall prevalence of plaque was 97% and the mean Plaque Index (Pl.I) was 0,94 with 75% of the subjects having a mean Pl.I=0,5. The site prevalence data revealed that out of 12 sites, on average, 4 had Pl.I~O, 4 had Pl.I~l, and 4 had Pl.I~2. In the maxilla the molar sites had the higher plaque levels, whilst in the mandible the incisor sites had higher plaque levels. The sex-specific data showed the males to have higher mean plaque levels than the females, but in 50% of sample with a mean PI.I 0,5 to 1,45 there was ) had had any appurtenant treatment. The overall prevalence of plaque was 97% and the mean Plaque Index (Pl.I) was 0,94 with 75% of the subjects having a mean Pl.I=0,5. The site prevalence data revealed that out of 12 sites, on average, 4 had Pl.I~O, 4 had Pl.I~l, nd 4 had Pl.I~2. In the maxilla the molar sites had the higher plaque levels, whilst in the mandible the incisor sites had higher plaque levels. The sex-specific data showed the males to have higher mean plaque levels than the females, but in 50% of sample with a mean PI.I 0,5 to 1,45 there was no difference.

Thesis (Ph.D. (Applied Mathematics)) -- University of Limpopo, 2021 / Vovk and Shafer, [41], introduced game-theoretic framework for probability in mathematical finance. This is a new trend in financial mathematics in which no probabilistic assumptions on the space of price paths are made. The only assumption considered is the no-arbitrage opportunity widely accepted by the financial mathematics community. This approach rests on game theory rather than measure theory. We deal with various properties and constructions of quadratic variation for model-free càdlàg price paths and integrals driven by such paths. Quadratic variation plays an important role in the analysis of price paths of financial securities which are modelled by Brownian motion and it is sometimes used as the measure of volatility (i.e. risk). This work considers mainly càdlàg price paths rather than just continuous paths. It turns out that this is a natural settings for processes with jumps. We prove the existence of partition independent quadratic variation. In addition, following assumptions as in Revuz and Yor’s book, the existence and uniqueness of the solutions of SDEs with Lipschitz coefficients, driven by model-free price paths is proven. / National Research Foundation (NRF)

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This thesis is concerned with the calculus of alternating differential forms on a manifold. It establishes that this calculus is of a purely algebraic nature by developing its precise analogue for an arbitrary commutative algebra with unit over a field. / Thesis / Master of Science (MSc)

This study examined the effects of computational labs in Business Calculus classes used at a single, private institution on student outcomes of conceptual understanding of calculus and attitudes towards mathematics. The first manuscript addresses the changes in conceptual understanding through multiple-method research design, a quantitative survey given pre and post study and qualitative student comments, found no significant gains in conceptual knowledge as measured by a concept inventory, however, student comments revealed valuable knowledge demonstrated through reflection on and articulation of how specific calculus concepts could be used in real world applications. The second manuscript presents results to the effects on attitudes toward mathematics, studied through multiple-method research design, using a quantitative survey given at two intervals, pre and post, and analysis of student comments, which showed that students that participated in the labs had a smaller decline in attitude, although not statistically significant, than students that did not complete the labs and the labs were most impactful on students that had previously taken calculus; student comments overwhelmingly demonstrate that students felt and appreciated that the labs allowed them to see how calculus could be applied outside the classroom. Overall students felt the labs were beneficial in the development of advantageous habits, taught some a skill they hope to further develop and study, and provided several recommendations for improvement in future implementation. Collectively, this research serves as a foundation for the effectiveness of computational tools employed in general education mathematics courses, which is not currently a widespread practice. / Doctor of Philosophy / Students from a variety of majors often leave their introductory calculus courses without seeing the connections and utility it may have to their discipline and may find it uninspiring and boring. To address these issues, there is a need for educators to continue to develop and research potentially positive approaches to impacting students' experience with calculus. This study discusses a method of doing so, by studying students' understanding of and attitude toward calculus in a one-semester Business Calculus course using computational labs to introduce students to calculus concepts often in context of a business scenario. No significant gains in conceptual knowledge were found as measured by a concept inventory; however, student comments revealed valuable knowledge demonstrated through articulation of how specific calculus concepts could be used in real world applications. Students that participated in the labs also had a smaller decline in attitude than students that did not complete the labs. Student comments overwhelmingly demonstrate that students felt and appreciated that the labs allowed them to see how calculus could be applied outside the classroom. The labs were most impactful on students that had previously taken calculus. Overall students felt the labs were beneficial in the development of advantageous habits such as persistence, utilizing resources, and precision, introduced them to coding, a skill they hope to further develop and study, and students provided several recommendations for improvement in future implementation. This research provides a foundation for the effectiveness of computational tools used in general education mathematics courses.