Inaug.-diss.--Zürich.

Inaug.-diss.--Zürich.

Thesis--Copenhagen. / "Fortegnelse over større værker, som gentagne gange citeres": p. [x].

Thesis--Copenhagen. / "Fortegnelse over større værker, som gentagne gange citeres": p. [x].

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Thesis--Ph. D., University of Hong Kong. / Mimeographed.

Honors thesis (B.A.)-Ithaca College Dept. of Mathematics, 2008. / Title from abstract page. "April 15, 2008." includes abstract Includes bibliographical references (leaf 33). Also available in print form in the Ithaca College Archives.

M.Sc., Faculty of Science, University of the Witwatersrand, 2011 / Continued fractions have been extensively studied in number-theoretic ways. In this text, we will illuminate some of the geometric properties of contin- ued fractions by considering them as compositions of MÄobius transformations which act as isometries of the hyperbolic plane H2. In particular, we examine the geometry of simple continued fractions by considering the action of the extended modular group on H2. Using these geometric techniques, we prove very important and well-known results about the convergence of simple con- tinued fractions. Further, we use the Farey tessellation F and the method of cutting sequences to illustrate the geometry of simple continued fractions as the action of the extended modular group on H2. We also show that F can be interpreted as a graph, and that the simple continued fraction expansion of any real number can be can be found by tracing a unique path on this graph. We also illustrate the relationship between Ford circles and the action of the extended modular group on H2. Finally, our work will culminate in the use of these geometric techniques to prove well-known results about the relationship between periodic simple continued fractions and quadratic irrationals.

Hu, Jin. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 140-153). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

Made available in DSpace on 2014-06-11T19:27:09Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-05-10Bitstream added on 2014-06-13T19:55:25Z : No. of bitstreams: 1 pianoschi_ta_me_rcla.pdf: 4035413 bytes, checksum: 9b633c2e4923a1fff77a920decce8cca (MD5) / As funções de uma variável complexa podem ser estudadas como transformações no plano complexo. Esta abordagem, pouco explorada nas disciplinas de Variável Complexa dos cursos de graduação, mostra-se interessante pois permite a visualização e conecta este assunto às demais áreas da Matemática, por exemplo, vetores, cônicas, matrizes, entre outras. Nesta dissertação, as transformações no plano complexo são tratadas de duas formas diferentes. Na primeira, são estudadas as transformações de determinadas curvas no plano complexo enquanto que na segunda, considera-se as transformações de pontos do plano complexo os quais estão associados a uma cor definida segundo uma paleta de cores. Como aplicação deste último tratamento podese visualizar o Teorema Fundamental da Álgebra. A implementação computacional é feita utilizando os recursos gráficos do programa de geometria dinâmica GEOGEBRA © (www.geogebra.org) e do pacote gráfico ASYMPTOTE © (asymptote.sourceforge. net) ambos gratuitos (GNU Lesser General Public License) / The functions of a complex variable can be studied as transformations in the complex plane. This approach has been little explored in the disciplines of Variable Complex of undergraduate courses and it is interesting because it allows visualization and connects this subject to other areas of mathematics, e.g., vectors, conics, matrix, among others. In this dissertation, the transformations in the complex plane are treated in two different ways. In the first, they are studied as transformations of certain curves in the complex plane while in the second approach, it is considered the transformations of points of the complex plane which are associated with a color defined by a color palette. As an application of the latter approach one can visualize the Fundamental Theorem of Algebra. The computational implementation is made using the graphics capabilities of dynamic geometry program GEOGEBRA © (www.geogebra.org) and a vector graphic package ASYMPTOTE © (asymptote.sourceforge.net) both free (GNU Lesser General Public License)