Thesis (doctoral)--Universität Bern, 1909.

Thesis--Universität Bern, 1914.

Thesis(M.S.)--Auburn University, 2006. / Abstract (p.34-35). Vita. Includes bibliographic references.

Thesis(M.S.)--Auburn University, 2006. / Abstract. Vita. Includes bibliographic references (p.49).

An affine transformation such as T(P)=Q is a locus of an affine conic. Any affine conic can be produced from this incidence construction. The affine type of conic (ellipse, parabola, hyperbola) is determined by the invariants of T, the determinant and trace of its linear part. The purpose of this thesis is to obtain a corresponding classification in the hyperbolic plane of conics defined by this construction.

This study is to improve the coupling efficiency between 980nm high-power pump laser diode and single-mode fiber. In this study, we use the third generation of fiber grinding machine which is designed by Cheng Shiu University, professor Ying-Chien Tsai. This machine is fully automatic. we use it to fabricate the hyperbola microlenses. The advantages about hyperbola microlenses structure are a single-step fabrication, grinding steps to simplify, reduce the grinding time and will greatly reduce the offset of fiber. In the fusing procedure, the slight arc fusion was mainly applied for fine polishing merely instead of reshaping for the reason that the fabricated hyperbola fiber endface was very close to the ideal shape. The fabrication reproducibility and yield increase, and can reduce the cost of grinding. The fiber end shape is similar to the math on the hyperboloid, and the length of the axis of the hyperboloid profile shows a hyperbola. By mathematical properties of hyperbola, we derivation the parameter of radius of curvature for hyperbola microlenses. The definition of the radius of curvature of the hyperbolic vertex and the mode field diameter (the MFD) = 4.2£gm point of intersection with the hyperbola, the characteristics of the formation of this three o'clock round the curvature is the radius of curvature we have said. The radius of curvature (R) is a semi-consistent axial length (a) and two progressive line angle (£c) function, it means we can through the control of ¡§a¡¨ and £c to control the R, but £c is fixed after grinding process. So we choose control parameter ¡§a¡¨ by fusing process, via control ¡§a¡¨ to achieve the purpose of the control R. By various fusing parameters to adjust the gain of ¡§a¡¨, we can control the R in an ideal 2.6-2.8£gm. This process indeed improves the coupling efficiency. This method gives a low offset of the fiber it easier for more than 80%. And larger offset of the fiber by this method can achieve to 70% even 80%.

In this investigation an attempt was made to determine how learners and teachers use computers in the teaching and learning of hyperbolic graphs in Mathematics. A comprehensive literature study showed that there are many benefits in using computers to study Mathematics. The investigation was done in two phases. In the first phase, a questionnaire was given to learners. The second phase involved interviewing learners and teachers. Findings indicate that learners and teachers enjoy using computers in the teaching and learning of Mathematics. Analysis of the results shows that the use of computers in teaching and learning of Mathematics, in particular the teaching and learning of hyperbolic graphs is beneficial. / Mathematical Sciences / M.Sc. (Mathematics Education)

Title: Utilization of the internet by teaching conics at high school Author: Bc. Věra Effenberger Department: Department of Mathematics Education Supervisor: RNDr. Jana Hromadová, Ph.D. Supervisor's e-mail address: Jana.Hromadova@mff.cuni.cz Abstract This diploma thesis is dealing with conics' problems. It is mainly destined for high school (or university) teachers of descriptive geometry and for students too. It can by used in aid of education of conics or by self-study, because it includes many of illustrative pictures and dynamic applets made in the program GeoGebra, which support the written theoretical text. In the work are enumerated definitions, properties and various constructions of individual conics. Further there is their origin as an intersection of a right circular cone (as the case may be of a right circular cylinder) with a plane, their osculating circle and conjugate diameters. Compilation of examples constitutes an addition of this work. The examples have various difficulty and also can serve as a control over got knowledge. Keywords: ellipse, hyperbola, parabola, foci, tangents, normals, construction of conics

[pt] A ideia principal desta dissertação é apresentar as cônicas e demonstrar suas equações cartesianas bem como suas propriedades reflexivas. O trabalho está focado em abordar tais propriedades reflexivas com o auxílio do software GeoGebra. / [en] The main idea of this dissertation is to present the conics and demonstrate their cartesian equations and their reflective properties. The work is focused on addressing such reflective properties with the aid of software GeoGebra.

A research project submitted in partial fulfilment of the requirements of the degree of Masters in Science Education (Mathematics Education) University of the Witwatersrand Johannesburg South Africa May 2016 / This study investigated learners’ mathematical discourse on the hyperbola using commognitive theory, with particular focus on the use of words, narratives, routines and visual mediators. Data was collected by means of task-based interviews with nine Grade 10 and 11 learners from a township school in Johannesburg, South Africa. An analytical tool, named the Discourse Profile of the Hyperbola, was adapted from the Arithmetic Discourse Profile of Ben-Yahuda et al (2005) and was used to analyse learners’ mathematical discourse. The study focused on three representations of the hyperbola, namely, the formulae (equation); the graph and the table. Learners’ views and definition of the asymptote, in relation to the graph, emerged as a central theme in the analysis. The analysis also focused on the mismatch between what is said and what is done by learners, for example most learners sketched the graph of a hyperbola showing a vertical asymptote yet talked as if there is no vertical asymptote. Most routines were ritualized, for example learners failed to link iconic and symbolic mediators they had used in responding to tasks. However, there were traces of exploratory routines from a few learners, evidenced by links between equations, and identifying the hyperbola from unfamiliar tasks. While a few learners used literate words, colloquial word use was dominant. The discourse of learners was found to be visual. For example, some reasoned that an equation with a fraction represents a hyperbola while an equation not expressed in standard form does not represent a hyperbola. Some learner narratives are not endorsed by the community of mathematicians.