Dresden, Techn. Univ., Diss., 1998.

Orientador: Alice Kimie Miwa Libardi / Coorientador: Tatiana Miguel Rodrigues de Souza / Banca: Elíris Cristina Rizziolli / Banca: Edivaldo Lopes dos Santos / Resumo: Intuitivamente, um ponto tem dimensão 0, uma reta tem dimensão 1, um plano tem dimensão 2 e um cubo tem dimensão 3. Porém, na geometria fractal encontramos objetos matemáticos que possuem dimensão fracionária. Esses objetos são denominados fractais cujo nome vem do verbo "frangere", em latim, que significa quebrar, fragmentar. Neste trabalho faremos um estudo sobre o conceito de dimensão, definindo dimensão topológica e dimensão de Hausdorff. O objetivo deste trabalho é, além de apresentar as definições de dimensão, também apresentar algumas aplicações da dimensão de Hausdorff na geometria fractal / Abstract: We know, intuitively, that the dimension of a dot is 0, the dimension of a line is 1, the dimension of a square is 2 and the dimension of a cube is 3. However, in the fractal geometry we have objects with a fractional dimension. This objects are called fractals whose name comes from the verb frangere, in Latin, that means breaking, fragmenting. In this work we will study about the concept of dimension, de ning topological dimension and Hausdor dimension. The purpose of this work, besides presenting the de nitions of dimension, is to show an application of the Hausdor dimension on the fractal geometry / Mestre

The theory of viscosity solutions was developed for certain types of nonlinear first-order and second-order partial differential equations. It has been particularly useful in describing the solutions of partial differential equations associated with deterministic and stochastic optimal control problems [16], [53]. In its classical formulation, see [16], the theory deals with solutions which are continuous functions. The concept of continuous viscosity solutions was further generalized in various ways to include discontinuous solutions with the definition of Ishii given in [71] playing a pivotal role. In this thesis we propose a new approach for the treatment of discontinuous solutions of first-order Hamilton-Jacobi equations, namely, by involving Hausdorff continuous interval valued functions. The advantages of the proposed approach are justified by demonstrating that the main ideas within the classical theory of continuous viscosity solutions can be extended almost unchanged to the wider space of Hausdorff continuous functions and the existing theory of discontinuous viscosity solutions is a particular case of that developed in this thesis in terms of Hausdorff continuous interval valued functions. Two approaches to numerical solutions for Hamilton-Jacobi equations are presented. The first one is a monotone scheme for Hamilton-Jacobi equations while the second is based on preserving total variation diminishing property for conservation laws. In the first approach, we couple the finite element method with the nonstandard finite difference method which is based on the Mickens’ rule of nonlocal approximation [9]. The scheme obtained in this way is unconditionally monotone. In the second approach, computationally simple implicit schemes are derived by using nonlocal approximation of nonlinear terms. Renormalization of the denominator of the discrete derivative is used for deriving explicit schemes of first or higher order. Unlike the standard explicit methods, the solutions of these schemes have diminishing total variation for any time step size. / Thesis (PhD (Mathematical Science))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted

Thesis advisor: Ian Biringer / We consider the space $\rootedH2$ of all complete hyperbolic surfaces without boundary with a basepoint equipped with the pointed Gromov-Hausdorff topology. Continuous paths within $\rootedH2$ arising from certain deformations on a hyperbolic surface and concrete geometric constructions are studied. These include changing some Fenchel-Nielsen parameters of a subsurface, pinching a simple closed geodesic to a cusp, and inserting an infinite strip along a proper bi-infinite geodesic. We then use these paths to show that $\rootedH2$ is path-connected and that it is locally weakly connected at points whose underlying surfaces are either the hyperbolic plane or hyperbolic surfaces of the first kind. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

L'attracteur de certains systèmes de fonctions itérées de similitudes dépendant de façon holomorphe d'un paramètre ainsi que l'ensemble de Julia de certaines familles holomorphes de polynômes hyperboliques sont mouvements holomorphes pour lesquels il a été montré que la réciproque de la dimension de Hausdorff peut s'écrire comme l'infimum d'une famille de fonctions harmoniques positives. Ces résultats motivent l'énonciation d'une conjecture concernant la variation de la dimension d'un mouvement holomorphe quelconque. Ce mémoire a pour objectif d'introduire les fonctions inf-harmoniques, d'étudier la littérature qui établit un lien entre ces dernières et les mouvements holomorphes et de dresser une piste de réponse à la conjecture.

<p> In this thesis we study extensive subcategories of various categories of Hausdorff spaces and continuous maps, and of Hausdorff uniform spaces and uniformly continuous maps, In particular, we obtain new methods to construct extensive subcategories which can be applied to many categories and give us an inclusive relationship between reflective subcategories of Haus and coreflective subcategories of Top. We consider perfect onto projectivity in those categories. The relationships between n-compact spaces and topologically complete spaces are discussed. </p> / Thesis / Doctor of Philosophy (PhD)

This dissertation contains work from the author's papers [35] and [36] with coauthor Eyvindur Palsson. The classic Mattila-Sjolin theorem shows that if a compact subset of $mathbb{R}^d$ has Hausdorff dimension at least $frac{(d+1)}{2}$ then its set of distances has nonempty interior. In this dissertation, we present a similar result, namely that if a compact subset $E$ of $mathbb{R}^d$, with $d geq 3$, has a large enough Hausdorff dimension then the set of congruence classes of triangles formed by triples of points of $E$ has nonempty interior. These types of results on point configurations with nonempty interior can be categorized as extensions and refinements of the statement in the well known Falconer distance problem which establishes a positive Lebesgue measure for the distance set instead of it having nonempty interior / Doctor of Philosophy / By establishing lower bounds on the Hausdorff dimension of the given compact set we can guarantee the existence of lots of triangles formed by triples of points of the given set. This type of result can be categorized as an extension and refinement of the statement in the well known Falconer distance problem which establishes that if a compact set is large enough then we can guarantee the existence of a significant amount of distances formed by pairs of points of the set

L'attracteur de certains systèmes de fonctions itérées de similitudes dépendant de façon holomorphe d'un paramètre ainsi que l'ensemble de Julia de certaines familles holomorphes de polynômes hyperboliques sont mouvements holomorphes pour lesquels il a été montré que la réciproque de la dimension de Hausdorff peut s'écrire comme l'infimum d'une famille de fonctions harmoniques positives. Ces résultats motivent l'énonciation d'une conjecture concernant la variation de la dimension d'un mouvement holomorphe quelconque. Ce mémoire a pour objectif d'introduire les fonctions inf-harmoniques, d'étudier la littérature qui établit un lien entre ces dernières et les mouvements holomorphes et de dresser une piste de réponse à la conjecture.

Cette thèse est consacrée à l'étude de certains processus aléatoires à valeurs dans les arbres continus. Nous définissons d'abord un cadre conceptuel pour cette étude, en construisant une topologie polonaise sur l'espace des R-arbres localement compacts, complets et munis d'une mesure borélienne localement finie. Cette topologie, dite de Gromov-Hausdorff-Prokhorov, permet alors la définition de processus de Markov à valeurs arbre. Nous donnons ensuite une nouvelle construction du processus d'élagage d'Abraham-Delmas-Voisin, qui est un exemple de processus qui prend ses valeurs dans les arbres de Lévy. Notre construction, qui dévoile une nouvelle structure généalogique des arbres de Lévy, est trajectorielle, et permet d'identifier explicitement les transitions du processus d'élagage. Nous appliquons cette description à l'étude de certains temps d'arrêt, comme le premier temps auquel le processus franchit une hauteur donnée. Nous décrivons le processus à cet instant grâce à une nouvelle décomposition de type spinal. Enfin, nous nous intéressons à la fragmentation d'Aldous-Pitman de l'arbre brownien d'Aldous. En particulier, nous étudions, à la suite d'Abraham et Delmas, l'effet de cette fragmentation sur les sous-arbres discrets de l'arbre brownien. Le nombre de coupures nécessaires avant d'isoler la racine, convenablement renormalisé, converge vers une variable aléatoire de Rayleigh ; nous donnons un théorème central limite qui précise les fluctuations autour de cette limite / In this thesis, we study continuum tree-valued processes. First, we define an abstract framework for these processes, by constructing a metric on the space of locally compact, complete R-trees, endowed with a locally finite Borel measure. This topology, called Gromov-Hausdorff-Prokhorov topology, allows for the definition of tree-valued Markov processes. We then give a new construction of the pruning process of Abraham-Delmas-Voisin, which is an example of a Lévy tree-valued process. Our construction reveals a new genealogical structure of Lévy trees. Furthermore, it is a path wise construction, which describes the transitions of the process explicitly. We apply this description to the study of certain stopping times, such as the first moment the process crosses a given height. We describe the process at that time through a new spinal decomposition. Finally, we focus on the Aldous-Pitman fragmentation of Aldous's Brownian tree. Following Abraham and Delmas, we study the effect of the fragmentation on discrete subtrees of the Brownian tree. The number of cuts needed to isolate the root, suitably renormalized, converges towards a Rayleigh-distributed random variable; we prove a Central Limit Theorem describing the fluctuations around this limit

Apresentamos duas propriedades genéricas para difeomorfismos conservativos da classe \'C POT.1\' sobre uma superfície compacta de dimensão dois. Obtemos uma limitação inferior para entropia topológica de difeomorfismos genéricos, e mostramos que tais difeomorfismos sempre possuem conjuntos invariantes fechados com órbitas densas e dimensão de Hausdorff dois / We present two generic properties of \'C POT.1\" area preserving diffeomorphisms of a two dimensional compact oriented surface. We obtain a lower bound for the topological entropy of a generic diffeomorphisms, and we show that such a diffeomorphism always has closed invariant sets with dense orbits and Hausdorff dimension two