La Géométrie Discrète, comme Science de l'Informatique Théorique, étudie notamment les motifs linéaires tels que les primitives discrètes apparaissant dans les images : les droites discrètes, les segments discrets, les plans discrets, les morceaux de plans discrets par exemple. Dans ce travail, je me concentre tout particulièrement sur les diagrammes de Farey qui apparaissent lors de l'étude des primitives discrètes que sont les (m,n)-cubes, autrement dit les morceaux de plans discrets. J’étudie notamment la Combinatoire des droites formant les diagrammes de Farey, en établissant des formules exactes. Je montre alors que certaines méthodes utilisées auparavant ne permettront pas d'optimiser la Combinatoire des (m,n)-cubes. J'obtiens aussi une estimation asymptotique en utilisant la Théorie des Nombres Combinatoire. Puis, concernant les sommets apparaissant dans les diagrammes de Farey, j'obtiens une borne inférieure. J'analyse alors les stratégies déjà mises en place pour l'étude des $(m,n)$-cubes par les seuls diagrammes de Farey en deux dimensions. Afin d'obtenir de nouvelles bornes plus précises pour les $(m,n)$-cubes, une des seules méthodes actuellement existantes, est de proposer une généralisation de la notion de pré image d'un segment discret, à celle de pré image d'un $(m,n)$-cube, avec pour conséquence une nouvelle inégalité combinatoire sur le cardinal des (m,n)-cubes (inégalité qui pourrait même s'avérer être une égalité). Ainsi, nous introduisons la notion de diagramme de Farey en trois dimensions / Discrete Geometry, as Theoretical Computer Science, studies in particular linear patterns such as discrete primitives in images: the discrete lines, discrete segments, the discrete planes, pieces of discrete planes, for example. In this work, I particularly focused on Farey diagrams that appear in the study of the $ (m, n) $ - cubes, ie the pieces of discrete planes. Among others, I study the Combinatorics of the Farey lines forming diagram Farey, establishing exact formulas. I also get an asymptotic estimate using Combinatorial Number Theory. Then, I get a lower bound for the cardinality of the Farey vertices. After that, we analyze the strategies used in the literature for the study of (m, n)- cubes only by Farey diagrams in two dimensions. In order to get new and more accurate bounds for (m, n)- cubes, one of the few available methods, is to propose a generalization for the concept of preimage of a discrete segment for (m, n) - cube, resulting in a new combinatorial inequality. Thus, we introduce the notion Farey diagram in three dimensions

This dissertation consists of three parts. In the first part we consider the equidistribution of roots of quadratic congruences. The roots of quadratic congruences are known to be equidistributed. However,we establish a bound for the discrepancy of this sequence using a spectral method involvingautomorphic forms, especially Kuznetsov's formula, together with an Erdős-Turán inequality. Then we discuss the implications of our discrepancy estimate for the reducibility problem of arctangents of integers. In the second and third part of this dissertation we consider some aspects of Farey fractions. The set of Farey fractions of order at most [mathematical formula] is, of course, a classical object in Analytic Number Theory. Our interest here is in certain sumsets of Farey fractions. Also, in this dissertation we study Farey fractions by working in the quotient group Q/Z, which is the modern point of view. We first derive an identity which involves the structure of Farey fractions in the group ring of Q/Z. Then we use these identities to estimate the asymptotic magnitude of the size of the sumset [mathematical formula]. Our method uses results about divisors in short intervals due to K. Ford. We also prove a new form of the Erdős-Turán inequality in which the usual complex exponential functions are replaced by a special family of functions which are orthogonal in L²(R/Z). / text

Thesis (M.S.)--West Virginia University, 2003. / Title from document title page. Document formatted into pages; contains v, 30 p. Includes abstract. Includes bibliographical references (p. 29-30).

An introduction to periodic orbit techniques for deterministic dynamical systems is presented. The Farey map is considered as examples of intermittency in one-dimensional maps. The effect of intermittency on the Markov partition is considered. The Gauss map is shown to be related to the farey map by a simple transformation of trajectories. A method of calculating periodic orbits in the thermostated Lorentz gas is derived. This method relies on minimising the action from the Hamiltonian description of the Lorentz gas, as well as the construction of a generating partition of the phase space. This method is employed to examine a range of bifurcation processes in the Lorentz gas. A novel construction of the Sinai billiard is performed by using symmetry arguments to reduce two particles in a hard walled box to the square Sinai billiard. Infinite families of periodic orbits are found, even at the lowest order, due to the intermittency of the system. The contribution of these orbits is examined and found to be tractable at the lowest order. The number of orbits grows too quickly for consideration of any other terms in the periodic orbit expansion. A simple stochastic model for the diffusion in the Lorentz gas was constructed. The model produced a diffusion coefficient that was a remarkably good fit to more precise numerical calculations. This is a significant improvement to the Machta-Zwanzig approximation for the diffusion coefficient. We outline a general approach to constructing stochastic models of deterministic dynamical systems. This method should allow for calculations to be performed in more complicated systems.

Orientador: Reginaldo Palazzo Junior / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação / Made available in DSpace on 2018-08-18T16:51:18Z (GMT). No. of bitstreams: 1 Leskow_LucilaHelenaAllan_D.pdf: 2583405 bytes, checksum: 3161d9deabaa60a8965a9e3d20ff36aa (MD5) Previous issue date: 2011 / Resumo: Neste trabalho apresentamos como contribuição um novo conjunto de tesselações do plano hiperbólico construídas a partir de uma tesselação bem conhecida, a tesselação de Farey. Nestas tesselações a região de Dirichlet é formada por polígonos hiperbólicos de n lados, com n > 3. Explorando as características dessas tesselações, apresentamos alguns tipos possíveis de aplicações. Inicialmente, estudando a relação existente entre a teoria das frações contínuas e a tesselação de Farey, propomos um novo método de codificação de geodésicas. A inovação deste método está no fato de ser possível realizar a codificação de uma geodésica pertencente a PSL(2,Z) em qualquer uma das tesselações ou seja, para qualquer valor de n com n > 3. Neste método mostramos como é possível associar as sequências cortantes de uma geodésica em cada tesselação à decomposição em frações contínuas do ponto atrator desta. Ainda explorando as características dessas novas tesselações, propomos dois tipos de aplicação em teoria de codificação de fontes discretas. Desenvolvendo dois novos códigos para compactação de fontes (um código de árvore e um código de bloco), estes dois métodos podem ser vistos como a generalização dos métodos de Elias e Tunstall para o caso hiperbólico / Abstract: In this work we present as contribution a new set of tessellations of the hyperbolic plane, built from a well known tessellation, the Farey tessellation. In this set of tessellations the Dirichlet region is made of hyperbolic polygons with n sides where n > 3. While studying these tessellations and theirs properties, we found some possible applications. In the first one, while exploring the relationship between the continued fractions theory and the Farey tessellation we propose a new method for coding geodesics. Using this method, it is possible to obtain a relationship between the cutting sequence of a geodesic belonging to PSL(2,Z) in each tessellation and the continued fraction decomposition of its attractor point. Exploring the characteristics of these tessellations we also propose two types of applications regarding the discrete memoryless source coding theory, a fixed-to-variable code and a variable length-to-fixed code. These methods can be seen as a generalized version of the Elias and Tunstall methods for the hyperbolic case / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

Egyptian fractions are what we know as unit fractions that are of the form 1/n - with the exception, by the Egyptians, of 2/3. Egyptian fractions have actually played an important part in mathematics history with its primary roots in number theory. This paper will trace the history of Egyptian fractions by starting at the time of the Egyptians, working our way to Fibonacci, a geologist named Farey, continued fractions, Diophantine equations, and unsolved problems in number theory.

Radar systems have been very effective in gathering information in a battlefield, so that the tactical actions can be decided. On the contrary, self-protection systems have been developed to break this activity of radars, for which radar signals must be intercepted to be able to take counter measures on time. Ideally, interception should be done in a certain time with a 100% probability, but in reality this is not the case. To intercept radar signals in shortest time with the highest probability, a search strategy should be developed for the receiver. This thesis studies the conditions under which the intercept time increases and the probability of intercept decreases. Moreover, it investigates the performance of the search strategy of Clarkson with respect to these conditions, which assumes that a priori knowledge about the radars that will be intercepted is available. Then, the study identifies the cases where the search strategy of Clarkson may be not desirable according to tactical necessities, and proposes a probabilistic search strategy, in which it is possible to intercept radar signals with a specified probability in a certain time.

There are three main results in this dissertation. The first result is the construction of an abstract visual space for rational numbers mod1, based on the visual primitives, colour, and rational radial direction. Mathematics is performed in this visual notation by defining increasingly refined visual objects from these primitives. In particular, the existence of the Farey tree enumeration of rational numbers mod1 is identified in the texture of a two-dimensional animation. ¶ The second result is a new enumeration of the rational numbers mod1, obtained, and expressed, in abstract visual space, as the visual object coset waves of coset fans on the torus. Its geometry is shown to encode a countably infinite tree structure, whose branches are cosets, nZ+m, where n, m (and k) are integers. These cosets are in geometrical 1-1 correspondence with sequences kn+m, (of denominators) of rational numbers, and with visual subobjects of the torus called coset fans. ¶ The third result is an enumeration in time of the visual hierarchy of the discrete buds of the Mandelbrot boundary by coset waves of coset fans. It is constructed by embedding the circular Farey tree geometrically into the empty internal region of the Mandelbrot set. In particular, coset fans attached to points of the (internal) binary tree index countably infinite sequences of buds on the (external) Mandelbrot boundary.