Dissertations / Theses on the topic 'Transformation de Möbius'

The aim of this work is the study of arbitrary mobius transformations by means of simple complex transformations, namely: the Translation, the Rotation, the Homotetia (Contraction and Dilatation) and Inversion. The results obtained were applied in circles and straight line. At the end, we give the the alternative of studying mobius transformations via matrices.
O objetivo deste trabalho é estudar transformações de Möbius arbitrárias por meio de transformações complexas mais simples, a saber: a Translação, a Rotação, a Homotetia (Contração e Dilatação) e a Inversão. Os resultados obtidos foram aplicados em círculos e retas. No final, damos a alternativa de estudar transformações de Möbius via matrizes.

La fonction de Möbius est définie par$$\mu(n)= \begin{cases} 1 & \textit{si $n=1$},\\ (-1)^k& \textit{si n est le produit de k nombres premiers distincts,}\\ 0 & \textit{si n contient un facteur carré. } \end{cases}$$Nous avons démontré que pour $x \ge \exp( 10^9) $ et $h=x^{1-\frac{1}{16000}}$, il existe dans chaque intervalle $[x-h,x]$ des entiers $n_1$ avec $\mu(n_1)=1$ et des entiers $n_2$ avec $\mu(n_2)=-1$.\\Ce résultat est une conséquence d'un résultat plus général.\\Pour $x \ge \exp(4\times 10^6)$, $\frac{1}{\sqrt{\log x}} \le \theta \le \frac{1}{2000}$, $h=x^{1-\theta}$ et $Q=(x/h)^{\frac{1}{20}}$, nous avons \\$$\sum_{q \leq Q} \log(Q/q)\sum_{\chi mod q}^*\left| \sum_{x.-h\le n \le x} \mu(n) \chi(n) \right| \leq 10^{20} h \theta \log(x) \exp( \frac{-1}{300 \theta}); $$la somme $\sum^*$ portant sur les caractères primitifs sauf l'éventuel caractère exceptionnel.\\Et en particulier pour $x \ge \exp( 10^9)$,$$ \left | \sum_{x.-x^{1-\frac{1}{16000}}\le n \le x} \mu(n) \right | \le \frac{1}{100} x^{1-\frac{1}{16000}}.\\$$
The Möbius function is defined by$$\mu(n)= \begin{cases} 1 & \textit{if $n=1$},\\ (-1)^k& \textit{if n is a product of k distinct prime numbers,}\\ 0 & \textit{if n contains a square factor. } \end{cases}$$We demonstrate that for $x \ge \exp( 10^9) $ and $h=x^{1-\frac{1}{16000}}$, it exists in each interval $[x-h,x]$ integers $n_1$ with $\mu(n_1)=1$ and integers $n_2$ with $\mu(n_2)=-1$.\\This result is a consequence of a more general result. \\For $x \ge \exp(4\times 10^6)$, $\frac{1}{\sqrt{\log x}} \le \theta \le \frac{1}{2000}$, $h=x^{1-\theta}$ et $Q=(x/h)^{\frac{1}{20}}$, we have \\ $$\sum_{q \leq Q} \log(Q/q)\sum_{\chi mod q}^*\left| \sum_{x-h \le n \le x} \mu(n) \chi(n) \right| \leq 10^{20} h \theta \log(x) \exp( \frac{-1}{300 \theta}); $$the sum $\sum^*$ relating to primitive characters except for possible exceptional character.\\And in particular for $x \ge \exp( 10^9)$,$$\left | \sum_{x-.x^{1-\frac{1}{16000}}\le n \le x} \mu(n) \right | \le \frac{1}{100} x^{1-\frac{1}{16000}}.$$

Thesis advisor: Jan R. Engelbrecht
Thesis advisor: Renato E. Mirollo
Many phenomena in nature that involve ordering in time can be understood as collective behavior of coupled oscillators. One paradigm for studying a population of self-sustained oscillators is the Kuramoto model, where each oscillator is described by a phase variable, and interacts with other oscillators through trigonometric functions of phase differences. This dissertation studies $N$ identical Kuramoto oscillators in a general form \[ \dot{\theta}_{j}=A+B\cos\theta_{j}+C\sin\theta_{j}\qquad j=1,\dots,N, \] where coefficients $A$, $B$, and $C$ are symmetric functions of all oscillators $(\theta_{1},\dots,\theta_{N})$. Dynamics of this model live in group orbits of M\"obius transformations, which are low-dimensional manifolds in the full state space. When the system is a phase model (invariant under a global phase shift), trajectories in a group orbit can be identified as flows in the unit disk with an intrinsic hyperbolic metric. A simple criterion for such system to be a gradient flow is found, which leads to new classes of models that can be described by potential or Hamiltonian functions while exhibiting a large number of constants of motions. A generalization to extended phase models with non-identical couplings gives rise to richer structures of fixed points and bifurcations. When the coupling weights sum to zero, the system is simultaneously gradient and Hamiltonian. The flows mimic field lines of a two-dimensional electrostatic system consisting of equal amounts of positive and negative charges. Bifurcations on a partially synchronized subspace are discussed as well
Thesis (PhD) — Boston College, 2017
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Physics

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Set a generalization of Möbius transformation and build a theory of inductive that may be an n-dimensional hyperbolic space. This theory allows for the inductive starting with n = 1, together with the extension notion of the Poincaré build a chain groups GM(n) transformation Möbius and spaces hyperbolic H2 members. We will see explicit formulas for the Poincaré bisectors in size 2. And may on models of hiperbolic space ball these bisectors coincide with the isometric spheres of isometries. We will be using explicit formulas of bissectors, to ge youself an algorithm, the DAFC, to obtain generators for Fuchsianos groups, which will be our study group.
Definir uma generalização do conceito de transformação de Möbius e construir uma teoria indutiva do que venha a ser um espaço hiperbólico de dimensão n. Essa teoria indutiva nos permite que se iniciando com n = 1, juntamente com a noção de extensão de Poincaré, construir uma cadeia de grupos GM(n) de transformação de Möbius e os espaços hiperbólicos H2 associados. Veremos fórmulas explícitas para os bissetores de Poincaré em dimensão 2. E que nos modelos de bola do espaço hiperbólico, esses bissetores coincidem com as esferas isométricas das isometrias. Iremos usar fórmulas explícitas dos bissetores, para obter-se um algoritmo, o DAFC, para obtenção de geradores para grupos Fuchsianos, que será nosso grupo em estudo.

Motivated by the theory of Kleinian groups and by the theory of continued fractions, we study semigroups of Möbius transformations. Like Kleinian groups, semigroups have limit sets, and indeed each semigroup is equipped with two limit sets. We find that limit sets have an internal structure with features similar to the limit sets of Kleinian groups and the Julia sets of iterates of analytic functions. We introduce the notion of a semidiscrete semigroup, and find that this property is akin to the discreteness property for groups. We study semigroups of Möbius transformations that fix the unit disc, and lay the foundations of a theory for such semigroups. We consider the composition sequences generated by such semigroups, and show that every such composition sequence converges pointwise in the open unit disc to a constant function whenever the identity element does not lie in the closure of the semigroup. We establish various results that have counterparts in the theory of Fuchsian groups. For example we show that aside from a certain exceptional family, any finitely-generated semigroup S is semidiscrete precisely when every two-generator semigroup contained in S is semidiscrete. We show that the limit sets of a nonelementary finitely-generated semidiscrete semigroup are equal (and non-trivial) precisely when the semigroup is a group. We classify two-generator semidiscrete semigroups, and give the basis for an algorithm that decides whether any two-generator semigroup is semidiscrete. We go on to study finitely-generated semigroups of Möbius transformations that map the unit disc strictly within itself. Every composition sequence generated by such a semigroup converges pointwise in the open unit disc to a constant function. We give conditions that determine whether this convergence is uniform on the closed unit disc, and show that the cases where convergence is not uniform are very special indeed.

Dans cette thèse j’étudie l’impact de la géométrie de micro et nano-domaines biologiques sur les propriétés d'électrodiffusion, ceci à l'aide des équations aux dérivées partielles de Poisson-Nernst-Planck. Je considère des domaines non-triviaux ayant une forme cuspide ou elliptique. Mon objectif est de développer des modèles ainsi que des méthodes mathématiques afin d'étudier les caractéristiques électriques de ces nano/micro-domaines, et ainsi mieux comprendre comment les signaux électriques sont modulés à ces échelles. Dans la première partie j’étudie le voltage à l'équilibre pour un électrolyte dans un domaine borné, et ayant un fort excès de charges positives. Je montre que le premier temps de sortie dans une boule chargée dépend de la surface et non du volume. J’étudie ensuite la géométrie composées d'une boule à laquelle est attachée un domaine cuspide. Je construis ensuite une solution asymptotique pour le voltage dans les cas 2D et 3D et je montre qu’ils sont donnés au premier ordre par la même expression. Enfin, j’obtiens la même conclusion en considérant une géométrie formée d'une ellipse, dont je construis une solution asymptotique du voltage en 2D et 3D. La seconde partie porte sur la modélisation de la compartimentalisation électrique des épines dendritiques. A partir de simulations numériques, je mets en évidence le lien entre la polarisation de concentration dans l'épine et sa géométrie. Je compare ensuite mon modèle à des données de microscopie. Je développe une méthode de déconvolution pour extraire la dynamique rapide du voltage à partir des données de microscopie. Enfin j’estime la résistance du cou et montre que celle-ci ne suit pas la loi d'Ohm
In this PhD I study how electro-diffusion within biological micro and nano-domains is affected by their shapes using the Poisson-Nernst-Planck (PNP) partial differential equations. I consider non-trivial shapes such as domains with cusp and ellipses. Our goal is to develop models, as well as mathematical tools, to study the electrical properties of micro and nano-domains, to understand better how electrical neuronal signaling is regulated at those scales. In the first part I estimate the steady-state voltage inside an electrolyte confined in a bounded domain, within which we assume an excess of positive charge. I show the mean first passage time in a charged ball depends on the surface and not on the volume. I further study a geometry composed of a ball with an attached cusp-shaped domain. I construct an asymptotic solution for the voltage in 2D and 3D and I show that to leading order expressions for the voltage in 2D and 3D are identical. Finally, I obtain similar conclusion considering an elliptical-shaped domain for which I construct an asymptotic solution for the voltage in 2D and 3D. In the second part, I model the electrical compartmentalization in dendritic spines. Based on numerical simulations, I show how spines non-cylindrical geometry leads to concentration polarization effects. I then compare my model to experimental data of microscopy imaging. I develop a deconvolution method to recover the fast voltage dynamic from the data. I estimate the neck resistance, and we found that, contrary to Ohm's law, the spine neck resistance can be inversely proportional to its radius

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O objetivo deste trabalho é ampliar os conhecimentos sobre números complexos já adquiridos no ensino médio. Diversas formas de representação e propriedades operatórias são abordadas. Para este fim, primeiramente, os números complexos são definidos a partir do conceito de matrizes quadradas de ordem 2, e portanto, serão definidos como pares ordenados de números reais. Na sequência, a partir da apresentação geométrica dos conceitos e operações, é estudado o plano complexo estendido, as Transformações de Möbius e a Projeção Estereográfica.
The objective of this paper is to extend the concepts of complex numbers already acquired in high school. Many forms of representation and operative properties are used. For that, first, the complex numbers are defined from the concept of square matrices of order 2, and will therefore be defined as ordered pairs of real numbers. Following, from the geometric presentation of concepts and operations, it is studied the extended complex plane, the Möbius Transformations and the Stereographic Projection.

We consider scalar integrable lattice equations which arise as the natural discrete counterparts to KdV-type PDEs. Several results are reported. We identify a new and natural connection between the ‘Schwarzian’ (Möbius invariant) integrable lattice systems and the Möbius group itself. The lattice equation in some sense describes dynamics of fixed-points as they change under composition between transformations. A classification result is given for lattice equations which are linear but also consistent on the cube. Such systems lie outside previous classification schemes. New Bäcklund transformations (BTs) for some known integrable lattice equations are given. As opposed to the natural auto-BT inherent in every such equation, these BTs are of two other kinds. Specifically, it is found that some equations admit additional auto-BTs (with Bäcklund parameter), whilst some pairs of apparently distinct equations admit a BT which connects them. Adler’s equation has come to hold the status of ‘master equation’ among the integrable lattice equations. Solutions of this equation are derived which are associated with 1-cycles and 2-cycles of the BT. They were the first explicit solutions written for Adler’s equation. We also apply the BT to the 1-cycle solution in order to construct a soliton-type solution.

I denna uppsats presenteras grundläggande delar av hyperbolisk geometri. Uppsatsen är indelad i två kapitel. I första kapitlet studeras Möbiusavbildningar på Riemannsfären. Andra kapitlet presenterar modellen av hyperbolisk geometri i övre halvplanet H, skapad av Poincaré på 1880-talet.

Huvudresultatet i uppsatsen är Gauss – Bonnét´s sats för hyperboliska trianglar.

In this thesis we present fundamental concepts in hyperbolic geometry. The thesis is divided into two chapters. In the first chapter we study Möbiustransformations on the Riemann sphere. The second part of the thesis deal with hyperbolic geometry in the upper half-plane. This model of hyperbolic geometry was created by Poincaré in 1880.

The main result of the thesis is Gauss – Bonnét´s theorem for hyperbolic triangles.

The purpose of this thesis is to study the effects of hyperbolic transformations on the cubic that is determined by locus of centroids of the equilateral triangles in H² whose base coincides with the line y=0, and whose common vertex is at the origin. The derivation of the formulas within this work are based on the Poincaré disk model of H², where H² is understood to mean the hyperbolic plane. The thesis explores the properties of both the untransformed cubic (the original locus of centroids) and the transformed cubic (the original cubic taken under a linear fractional transformation).

On étudie deux problèmes en géométrie algébrique. Le premier est la comparaison des géométries tropicale et algébrique. En particulier, on compare les configurations d'incidence, la règle de Cramer et les résultants. On présente la notion de construction géométrique et on traduit, sous plusieurs restrictions, des théorèmes classiques d'incidence dans le contexte tropical, comme le théorème de Pappus, de Fano ou de Cayley-Bacharach. La deuxième partie traite des hypercercles. Ces courbes ont été introduites par Andradas, Recio et Sendra et sont employées dans le problème de trouver des reparamétrisations de courbes rationnelles avec des coefficients algébriquement optimaux à partir d'une paramétrisation donnée. On étude la variété de Weil dans le cas paramétrique (hyperquadriques), la géométrie des hypercercles et on donne une méthode pour obtenir des reparamétrisations optimales en utilisant uniquement des reparamétrisations affines de la courbe
This thesis deals with two problems in algebraic geometry. The first one is the comparison of tropical and algebraic geometry. In particular, we study the relationship between incidence configurations, Cramer's rule and the notion of resultant. We introduce the notion of geometric construction and we transfer, under some assumptions, classical incidence theorems to the tropical framework, such as Pappus, Fano of Cayley-Bacharach theorems. The second part relates to hypercircles. These curves where introducedby Andradas, Recio and Sendra, that are used in the problem of computing reparametrizations of rational curves with optimal algebraic coefficients from a given non optimal parametrization. We study the Weil variety in the parametric case (hyperquadric), the geometry of hypercircles and we provide an algorithm to compute an optimal reparametrization using only affine reparametrization of the curve

Les fonctions de treillis, apparaissent être des outils essentiels en recherche opérationnelle. Elles ouvrent en effet de nouveaux champs d'application en théorie des jeux coopératifs, et en aide à la décision (les jeux sont dans ce cas des capacités, ou mesures floues). Cette thèse a pour objet l'investigation de concepts de solutions pour les jeux définis sur des structures générales de coalitions. À cette fin, nous proposons plusieurs généralisations et axiomatisations de la valeur de Shapley pour les jeux multi-choix, les jeux à actions combinées, et les jeux réguliers. L'indice d'interaction quantifie la véritable contribution d'une coalition par rapport à toutes ses sous-coalitions. Mathématiquement, il s'agit d'un prolongement de la valeur de Shapley. Nous proposons des axiomatisations de l'indice d'interaction de Shapley pour les jeux bi-coopératifs, ainsi que des procédés calculatoires permettant de déterminer l'opérateur d'interaction et son inverse.

Cette thèse est rédigée par articles. Les articles sont rédigés en anglais et le reste de la thèse est rédigée en français.
Le travail établit une équivalence en termes de puissance entre les tests basés sur la alpha-distance de covariance et sur le critère d'indépendance de Hilbert-Schmidt (HSIC) avec fonction caractéristique de distribution de probabilité stable d'indice alpha avec paramètre d'échelle suffisamment petit. Des simulations en grandes dimensions montrent la supériorité des tests de distance de covariance et des tests HSIC par rapport à certains tests utilisant les copules. Des simulations montrent également que la distribution de Pearson de type III, très utile et moins connue, approche la distribution exacte de permutation des tests et donne des erreurs de type I précises. Une nouvelle méthode de sélection adaptative des paramètres d'échelle pour les tests HSIC est proposée. Trois simulations, dont deux sont empruntées de l'apprentissage automatique, montrent que la nouvelle méthode de sélection améliore la puissance des tests HSIC. Le problème de tests d'indépendance entre deux vecteurs est généralisé au problème de tests d'indépendance mutuelle entre plusieurs vecteurs. Le travail traite aussi d'un problème très proche à savoir, le test d'indépendance sérielle d'une suite multidimensionnelle stationnaire. La décomposition de Möbius des fonctions caractéristiques est utilisée pour caractériser l'indépendance. Des tests généralisés basés sur le critère d'indépendance de Hilbert-Schmidt et sur la distance de covariance en sont obtenus. Une équivalence est également établie entre le test basé sur la distance de covariance et le test HSIC de noyau caractéristique d'une distribution stable avec des paramètres d'échelle suffisamment petits. La convergence faible du test HSIC est obtenue. Un calcul rapide et précis des valeurs-p des tests développés utilise une distribution de Pearson de type III comme approximation de la distribution exacte des tests. Un résultat fascinant est l'obtention des trois premiers moments exacts de la distribution de permutation des statistiques de dépendance. Une méthodologie similaire a été développée pour le test d'indépendance sérielle d'une suite. Des applications à des données réelles environnementales et financières sont effectuées.
The main result establishes the equivalence in terms of power between the alpha-distance covariance test and the Hilbert-Schmidt independence criterion (HSIC) test with the characteristic kernel of a stable probability distribution of index alpha with sufficiently small scale parameters. Large-scale simulations reveal the superiority of these two tests over other tests based on the empirical independence copula process. They also establish the usefulness of the lesser known Pearson type III approximation to the exact permutation distribution. This approximation yields tests with more accurate type I error rates than the gamma approximation usually used for HSIC, especially when dimensions of the two vectors are large. A new method for scale parameter selection in HSIC tests is proposed which improves power performance in three simulations, two of which are from machine learning. The problem of testing mutual independence between many random vectors is addressed. The closely related problem of testing serial independence of a multivariate stationary sequence is also considered. The Möbius transformation of characteristic functions is used to characterize independence. A generalization to p vectors of the alpha -distance covariance test and the Hilbert-Schmidt independence criterion (HSIC) test with the characteristic kernel of a stable probability distributionof index alpha is obtained. It is shown that an HSIC test with sufficiently small scale parameters is equivalent to an alpha -distance covariance test. Weak convergence of the HSIC test is established. A very fast and accurate computation of p-values uses the Pearson type III approximation which successfully approaches the exact permutation distribution of the tests. This approximation relies on the exact first three moments of the permutation distribution of any test which can be expressed as the sum of all elements of a componentwise product of p doubly-centered matrices. The alpha -distance covariance test and the HSIC test are both of this form. A new selection method is proposed for the scale parameter of the characteristic kernel of the HSIC test. It is shown in a simulation that this adaptive HSIC test has higher power than the alpha-distance covariance test when data are generated from a Student copula. Applications are given to environmental and financial data.

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.