A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2013. / Continued fractions have been extensively studied in number theoretic ways. These continued fractions are expressed as compositions of M¨obius maps in the Picard group PS L(2;C) that act, by Poincar´e’s extension, as isometries on H3. We investigate the Picard group with its generators and derive the fundamental domain using a direct method. From the fundamental domain, we produce an ideal octahedron, O0, that generates the Farey tessellation of H3. We explore the properties of Farey neighbours, Farey geodesics and Farey triangles that arise from the Farey tessellation and relate these to Ford spheres. We consider the Farey addition of two rationals in R as a subdivision of an interval and hence are able to generalise this notion to a subdivision of a Farey triangle with Gaussian Farey neighbour vertices. This Farey set allows us to revisit the Farey triangle subdivision given by Schmidt [44] and interpret it as a theorem about adjacent octahedra in the Farey tessellation of H3. We consider continued fraction algorithms with Gaussian integer coe cients. We introduce an analogue of Series [45] cutting sequence across H2 in H3. We derive a continued fraction expansion based on this cutting sequence generated by a geodesic in H3 that ends at the point in C that passes through O0.

No description available.

Thesis (doctoral)--Universitate Litteraria Friderica Guilelma, 1835. / Vita.

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This report examines the theory of continued fractions and how their use enhances the secondary mathematics curriculum. The use of continued fractions to calculate best approximants of real numbers is justified geometrically, and famous irrational numbers are described as continued fractions. Periodic continued fractions and other applications of continued fractions are also discussed. / text

No description available.

Globally there is a shortage in supply of rural health professionals. Tele-health was developed to help reduce distance barriers for health professionals in geographically isolated locations seeking continued medical education (CME). E-health includes all health information and health care delivered electronically. Further, telehealth defines all telecommunication technology used to transfer health information and health care. Tele-health comes in a variety of forms including videoconferencing. The use of videoconferencing to deliver medical education for rural health professionals is expanding area for health education. Videoconferencing provides a flexible method for rural GP registrars to access CME without travel. However, there is limited knowledge surrounding the use and success of telehealth for medical education. Each new telehealth project requires evaluation in order to ensure effectiveness of future programs. This study was based on one General Practice Education Year One (GPEP-1) registrar training program trialling videoconferencing as a method of teaching rurally distant registrars. A qualitative health research thematic analysis was undertaken. All individuals involved in the study were included in this research including; three offsite registrars (who teleconferenced as they lived remotely from the seminars), three onsite registrars (who attended the face to face seminars), and three facilitators. Three focus group interviews were conducted to collect data, one for each respective group. The focus groups accounted for the participant’s individual and collective experience of teleconferencing. Seven main themes emerged from the focus groups including; group culture, facilitation, teleconferenced education, technology, personal priorities, travel, and rural/remote. These themes were further sorted into two groups; reasons for switching to teleconferencing and factors that determine the success or failure of a telehealth CME. The factors that influenced the offsite registrars reasons to switch to teleconferencing included; personal priorities, travel, and rural/ remote. Further, the other four themes determined the success or failure of the teleconferenced training including; group culture, facilitation, teleconferenced education, and technology. All seven themes were relevant for the offsite registrars and onsite registrars, however, for the facilitators only the factors that determined the success of teleconferencing were applicable. Overall, the pilot study there was no difference in the educational experience provided. This study outlined the key factors that are required for effective teleconferenced training including; face to face bonding prior to teleconferencing, a skilled facilitator that provides active facilitation, a well organised structure, good quality technology, and regular checkups with the learners teleconferencing to ensure satisfaction. Additional research is required to replicate this study to compare this teleconferenced training with further teleconferenced training.

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Continued fractions have been extensively studied in number theoretic ways. In this text we will consider continued fraction expansions with partial quotients that are in Z = f x : x 2 Zg and where = 2 cos( q ); q 3 and with 1 < < 2. These continued fractions are expressed as the composition of M obius maps in PSL(2;R), that act as isometries on H2, taken at 1. In particular the subgroups of PSL(2;R) that are studied are the Hecke groups G . The Modular group is the case for q = 3 and = 1. In the text we show that the Hecke groups are triangle groups and in this way derive their fundamental domains. From these fundamental domains we produce the v-cell (P0) that is an ideal q-gon and also tessellate H2 under G . This tessellation is called the -Farey tessellation. We investigate various known -continued fractions of a real number. In particular, we consider a geodesic in H2 cutting across the -Farey tessellation that produces a \cutting sequence" or path on a -Farey graph. These paths in turn give a rise to a derived -continued fraction expansion for the real endpoint of the geodesic. We explore the relationship between the derived -continued fraction expansion and the nearest - integer continued fraction expansion (reduced -continued fraction expansion given by Rosen, [25]). The geometric aspect of the derived -continued fraction expansion brings clarity and illuminates the algebraic process of the reduced -continued fraction expansion.

<p> This thesis examines continued fraction expansions of the square root of nonsquare positive integers of periods one to six, and shows their relationships with Newton's method of approximation. It also contains known results concerning continued fractions.</p> / Thesis / Master of Science (MSc)