This thesis is a study of circle packings for arbitrary combinatorial tori in the geometric setting of affine tori. Certain new tools needed for this study, such as face labels instead of the usual vertex labels, are described. It is shown that to each combinatorial torus there corresponds a two real parameter family of affine packing labels. A construction of circle packings for combinatorial fundamental domains from affine packing labels is given. It is demonstrated that such circle packings have two affine side-pairing maps, and also that these side-pairing maps depend continuously on the two real parameters.

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<p>A closed Riemann surface which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. Accola showed that the trigonal morphism is unique for Riemann surfaces of genus g ≥ 5. This thesis will characterize the Riemann surfaces of genus 4 wiht non-unique trigonal morphism. We will describe the structure of the space of cyclic trigonal Riemann surfaces of genus 4.</p> / Report code: LiU-Tek-Lic-2004:54. The electronic version of the printed licentiate thesis is a corrected version where errors in the calculations have been corrected. See Errata below for a list of corrections.

A closed Riemann surface which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. Accola showed that the trigonal morphism is unique for Riemann surfaces of genus g ≥ 5. This thesis characterizes the cyclic trigonal Riemann surfaces of genus 4 with non-unique trigonal morphism using the automorphism groups of the surfaces. The thesis shows that Accola’s bound is sharp with the existence of a uniparametric family of cyclic trigonal Riemann surfaces of genus 4 having several trigonal morphisms. The structure of the moduli space of trigonal Riemann surfaces of genus 4 is also characterized. Finally, by using the same technique as in the case of cyclic trigonal Riemann surfaces of genus 4, we are able to deal with p-gonal Riemann surfaces and show that Accola’s bound is sharp for p-gonal Riemann surfaces. Furthermore, we study families of p-gonal Riemann surfaces of genus (p − 1)2 with two p-gonal morphisms, and describe the structure of their moduli space.

A closed Riemann surface which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. Accola showed that the trigonal morphism is unique for Riemann surfaces of genus g ≥ 5. This thesis will characterize the Riemann surfaces of genus 4 wiht non-unique trigonal morphism. We will describe the structure of the space of cyclic trigonal Riemann surfaces of genus 4. / <p>Report code: LiU-Tek-Lic-2004:54. The electronic version of the printed licentiate thesis is a corrected version where errors in the calculations have been corrected. See Errata below for a list of corrections.</p>

On compact Riemann surfaces, the Laplacian $\Delta$ has a discrete, non-negative spectrum of eigenvalues $\{\lambda_{i}\}$ of finite multiplicity. The spectrum is intrinsically linked to the geometry of the surface. In this work, we consider surfaces of constant negative curvature with a large symmetry group. It is not possible to explicitly calculate the eigenvalues for surfaces in this class, so we combine group theoretic and analytical methods to derive results about the spectrum. In particular, we focus on the Bolza surface and the Klein quartic. These have the highest order symmetry groups among compact Riemann surfaces of genera 2 and 3 respectively. The full automorphism group of the Bolza surface is isomorphic to $\mathrm{GL}_{2}(\mathbb{Z}_{3})\rtimes\mathbb{Z}_{2}. We analyze the irreducible representations of this group and prove that the multiplicity of $\lambda_{1}$ is 3, building on the work of Jenni, and identify the irreducible representation that corresponds to this eigenspace. This proof relies on a certain conjecture, for which we give substantial numerical evidence and a hopeful method for proving. We go on to show that $\lambda_{2}$ has multiplicity 4.

<p>A compact Riemann surface can be realized as a quotient space $\mathcal{U}/\Gamma$, where $\mathcal{U}$ is the sphere $\Sigma$, the euclidian plane $\mathbb{C}$ or the hyperbolic plane $\mathcal{H}$ and $\Gamma$ is a discrete group of automorphisms. This induces a covering $p:\mathcal{U}\rightarrow\mathcal{U}/\Gamma$.</p><p>For each $\Gamma$ acting on $\mathcal{H}$ we have a polygon $P$ such that $\mathcal{H}$ is tesselated by $P$ under the actions of the elements of $\Gamma$. On the other hand if $P$ is a hyperbolic polygon with a side pairing satisfying certain conditions, then the group $\Gamma$ generated by the side pairing is discrete and $P$ tesselates $\mathcal{H}$ under $\Gamma$.</p>

A compact Riemann surface can be realized as a quotient space $\mathcal/\Gamma$, where $\mathcal$ is the sphere $\Sigma$, the euclidian plane $\mathbb$ or the hyperbolic plane $\mathcal$ and $\Gamma$ is a discrete group of automorphisms. This induces a covering $p:\mathcal\rightarrow\mathcal/\Gamma$. For each $\Gamma$ acting on $\mathcal$ we have a polygon $P$ such that $\mathcal$ is tesselated by $P$ under the actions of the elements of $\Gamma$. On the other hand if $P$ is a hyperbolic polygon with a side pairing satisfying certain conditions, then the group $\Gamma$ generated by the side pairing is discrete and $P$ tesselates $\mathcal$ under $\Gamma$.

Orientador: Reginaldo Palazzo Junior / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-09T06:02:14Z (GMT). No. of bitstreams: 1 Bastos_JeffersonLuizRocha_D.pdf: 792249 bytes, checksum: 7a08577655f1a651a653df4d98e29e62 (MD5) Previous issue date: 2007 / Resumo: Tendo como base trabalhos recentes que associam o desempenho de sistemas de comunicação digital ao gênero de uma superfície compacta de Riemann, este trabalho tem como objetivo propor uma integração entre modulação e codificação de canal, tendo como base o gênero da superfície. Para atingir tais objetivos, nossa proposta é a seguinte: fixado um gênero g (g = 0,1,2,3), encontrar curvas com este gênero e fazer uma análise dos parâmetros dos códigos associados a esta curva, a fim de se obter uma modulação e um sub-código desta modulação para ser utilizado na codificação de canal / Abstract: Based on recent research showing that the performance of bandwidth efficent communication systems also depends on the genus of a. compact Riemann surface in which the communication channel is embedded, this study aims at proposing a combined form of modulation and coding technique when only the genus of a surface is given to the communication system designeI. To achieve this goal, the following procedure is proposed. Knowing that the channel is embedded in a surface of genus g, find algebraic curves with the given genus which will give rise to the modulation system, an (n, n, 1) type of code, and from this find the best (n, k, d) subcode, to be employed in the aforementioned combined formo Keywords: Riemann surface, algebraic curves, Goppa codes, modulation / Doutorado / Engenharia de Computação / Doutor em Engenharia Elétrica

Nesta dissertação, definimos a geometria hiperbólica usando o disco de Poincaré (D2) e o semiplano superior (H2) com as respectivas propriedades. Além disso, apresentamos algumas funções e relações importantes da geometria hiperbólica; conceituamos as superfícies de Riemann, analisando suas propriedades e representações; estudamos o espaço de Teichmüller com a devida decomposição em calças. Esses temas são ferramentas necessárias para atingir o objetivo da dissertação: definir as coordenadas de Fenchel Nielsen como um sistema de coordenadas locais do espaço de Teichmüller Tg. / In this dissertation, we defined the hyperbolic geometry using the Poincares disk (D2) and upper half-plane (H2) with its properties. Besides, we presented some functions and important relations of the hyperbolic geometry; we conceptualize the Riemann surfaces, analyzing its properties and representations; we studied the Teichmüller Space with proper decomposition pants. These themes are essential tools to reach the goal of the work: The definition of the Fenchel Nielsen coordenates as local coordinate system of the Teichmüller space Tg.