Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (p. 101-104).

In this thesis we describe how to estimate the distance spanned in the pants graph by a train track splitting sequence on a surface, up to multiplicative and additive constants. If some moderate assumptions on a splitting sequence are satisfied, each vertex set of a train track in it will represent a vertex of a graph which is naturally quasi-isometric to the pants graph; moreover the splitting sequence gives an edge-path in this graph so, more precisely, our distance estimate holds between the extreme points of this path. The present distance estimate is inspired by a result of Masur, Mosher and Schleimer for distances in the marking graph. However, we can apply their line of proof only after some manipulation of the splitting sequence: a rearrangement, changing the order the elementary moves are performed in, so that the ones producing Dehn twists are brought together; and then an untwisting, which suppresses the majority of these latter moves to give a new sequence, which does not end with the same track as before, but does not include any portion that is almost stationary in the pants graph. The required distance is then, up to constants, the number of splits occurring in the untwisted sequence. A consequence of our main theorem together with a result of Brock is that, given a pseudo-Anosov self-diffeomorphism ψ of a surface S, the maximal splitting sequence introduced by Agol gives us an estimate for the hyperbolic volume of the mapping torus built from S and ψ. There are also some interesting consequences for the hyperbolic volume of a solid torus minus a closed braid, via a machinery employed by Dynnikov and Wiest.

Neste trabalho tem por objetivo a construção de conjugações suaves de pontos fixos hiperbólicos com condições de não ressonância. Por tanto, inicialmente são apresentados alguns conceitos básicos sobre espaços de Banach e alguns resultados de equações diferenciais ordinárias em espaços de Banach e sistemas dinâmicos, apresentamos o teorema de Hartman Grobman como motivação inicial de Linearização. Apresentamos também vários exemplos como motivação para estudar o Teorema de Sternberg para contrações hiperbólicas, o principal resultado estudado nesta dissertação para contrações hiperbólicas / This work has the objetive of building smooth conjugations of hyperbolic fixed points with non-resonance conditions. So, first we present some basics of Banach spaces and some results of ordinary differential equations in Banach spaces and dynamical systems, we present the theorem of Hartman Grobman as original motivation for linearization . We also present several examples as motivation to study the Sternberg theorem for hyperbolic contractions, as main result studied in this dissertation

Neste trabalho, mostramos que os bilhares hiperbólicos construídos originalmente por Bussolari- Lenci têm a propriedade de Bernoulli. Tais bilhares não satisfazem as técnicas standard de Wojtkowski-Markarian-Donnay-Bunimovich para bilhares focalizadores hiperbólicos, a qual requer que o diâmetro da mesa do bilhar seja de mesma ordem que o maior raio de curvatura ao longo da componente focalizadora. Nossa prova, utiliza um teorema ergódico local que nos diz que sob certas condições, existe um conjunto de medida total do espaço de fase do bilhar tal que cada ponto desse conjunto possui uma vizinhança contida (mod 0) em uma componente Bernoulli da aplicação do bilhar. / In this work, we show that hyperbolic billiards constructed originally by Bussolari-Lenci has the Bernoulli property. These billiards do not satisfy the standard Wojtkowski-Markarian-Donnay- Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane, which requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. Our proof employs a locally ergodic theorem which says that under a few conditions, there exists a full measure set of the billiard phase space such that each of its points has a neighborhood contained, up to a zero measure set, in one Bernoulli component of the billiard map.

This study is concerned with the development and implementation of a preconditioner for a set of hyperbolic partial differential equations resulting from a new 5-moment closure for the prediction of gas flows both in and out of local equilibrium. This new 5-moment closure offers a robust and efficient system of first-order hyperbolic partial differential equations that has proven to provide an accurate treatment of one-dimensional gases, both in and for significant departures from local thermodynamic equilibrium. However, numerical computations using this model have proven to be difficult as a result of a singularity in the closing flux of the system. This also causes infinitely large wavespeeds in the system. The main goal of this work is to mitigate these numerical issues. Since the solution of a hyperbolic system is characterized by the waves of the system, one could suggest to scale these wavespeeds to remove the arbitrarily large speeds without altering the solution of the system. To accomplish this, this work starts with a detailed study of the behaviour of the system’s wavespeeds, given by the eigenvalues of the flux Jacobian of the system. Since, it is not possible to solve for these eigenvalues explicitly, it is suggested to approximate them by interpolation between the few states at which these waves can be solved for explicitly. With an estimate for the wavespeeds, the nature of the singularity in the system can be analyzed mathematically. The results of this mathematical analysis are used to develop a preconditioner matrix to remove the singularity from the model. To implement the proposed preconditioned model numerically, a centred-difference scheme with artificial dissipation is proposed. A dual-time-stepping strategy is developed and implemented with implicit Euler time marching for both physical and pseudo time iteration. This dual-time treatment allows the preconditioned system to remain applicable to time-accurate problems and is found to greatly increase the robustness of the solution of the steady-state problems. Solutions to several canonical problems for both continuum and non-equilibrium flow are computed and comparisons are made to classical models.

Cette thèse s'inscrit dans la théorie des cartes planaires aléatoires, active depuis une quizaine d'années, et plus précisément dans l'étude de modèles de nature hyperbolique.Dans un premier temps, nous nous intéressons à un modèle de triangulations aléatoires dynamiques basé sur les flips d'arêtes, et nous montrons une borne inférieure sur le temps de mélange de ce modèle.Dans la suite, l'objet d'étude principal est une famille de triangulations aléatoires hyperboliques, appelées PSHT. Il s'agit de variantes de la triangulation uniforme du plan (UIPT), qui ont été introduites en 2014 par Nicolas Curien. Nous commençons par établir un résultat de limite d'échelle quasi-critique : si on renormalise les distances tout en faisant tendre le paramètre d'hyperbolicité vers sa valeur critique, les triangulations étudiées convergent vers un espace métrique aléatoire appelé plan brownien hyperbolique. Nous étudions également des propriétés métriques fines des PSHT et du plan brownien hyperbolique, et notamment la structure de leurs géodésiques infinies. Nous présentons aussi de nouvelles propriétés de la frontière de Poisson des PSHT.Enfin, nous nous intéressons à un autre modèle naturel de cartes aléatoires hyperboliques : les cartes causales surcritiques, qui sont construites à partir d'arbres de Galton--Watson surcritiques, en ajoutant des arêtes entre sommets de même hauteur. Nous établissons des résultats d'hyperbolicité métrique, ainsi que des propriétés de la marche aléatoire sur ces cartes, dont un résultat de vitesse positive. Certaines des propriétés obtenues sont robustes, et peuvent se généraliser à n'importe quelle carte planaire contenant un arbre de Galton--Watson surcritique. / This thesis falls into the theory of random planar maps, which has been active in the last fifteen years, and more precisely into the study of hyperbolic models.We are first interested in a model of dynamical random triangulations based on edge-flips, where we prove a lower bound on the mixing time.In the rest of this thesis, the main objects that we study are the random hyperbolic triangulations called PSHT. These are hyperbolic variants of the Uniform Infinite Planar Triangulation (UIPT), and were introduced by Nicolas Curien in 2014. We first establish a near-critical scaling limit result: if we let the hyperbolicity parameter go to its critical value at the same time as the distances are renormalized, the PSHT converge to a random metric space that we call the hyperbolic Brownian plane. We also study precise metric properties of the PSHT and of the hyperbolic Brownian plane, such as the structure of their infinite geodesics. We obtain as well new properties of the Poisson boundary of the PSHT.Finally, we are interested in another natural model of hyperbolic random maps: supercritical causal maps, which are obtained from supercritical Galton--Watson trees by adding edges between vertices at the same height. We establish metric hyperbolicity results about these maps, as well as properties of the simple random walk (including a positive speed result). Some of the properties we obtain are robust, and may be generalized to any planar map containing a supercritical Galton--Watson tree.

Over the past several decades our understanding and meticulous characterization of the transient and spatial properties of materials evolved rapidly. The results present an exciting field for discovery, and craft materials to control and reshape light that we are just beginning to fathom. State-of-the-art nano-deposition processes, for example, can be utilized to build stratified waveguides made of thin dielectric layers, which put together result in a material with effective abnormal dispersion. Moreover, materials once deemed well known are revealing astonishing properties, v.gr. chalcogenide glasses undergo an atomic reconfiguration when illuminated with electrons or photons, this ensues in a temporal modification of its permittivity and permeability which could be used to build new Photonic Integrated Circuits.. This work revolves around the characterization and model of heterogeneous and time-varying materials and their applications, revisits Maxwell's equations in the context of nonlinear space- and time-varying media, and based on it introduces a numerical scheme that can be used to model waves in this kind of media. Finally some interesting applications for light confinement and beam transformations are shown.

The aim of this dissertation is to provide a critique of the idea that skepticism was the driving force in the development of early modern thought. Historian of philosophy Richard Popkin introduced this thesis in the 1950s and elaborated on it over the next five decades, and recent scholarship shows that it has become an increasingly accepted interpretation. I begin with a study of the relevant historical antecedents—the ancient skeptical traditions of which early modern thinkers were aware—Pyrrhonism and Academicism. Then I discuss the influence of skepticism on three pre-Cartesians: Francisco Sanches, Michel de Montaigne, and Pierre Charron. Basing my arguments on an informed understanding of both ancient Greek skepticism and some of the writings of these philosophers, I contend that it is inaccurate to predominantly characterize Sanches, Montaigne, and Charron as skeptics. To support his thesis about the singular influence of skepticism on early modern thought, Popkin says that René Descartes’ metaphysical philosophy was formed as a response to a skeptical threat and that Descartes ultimately conceded to the force of skepticism. He also argues that David Hume was a Pyrrhonist par excellence. I disagree with Popkin’s claims. I argue that Descartes was not as deeply affected by skepticism as Popkin suggests and that it is inaccurate to characterize Hume as a Pyrrhonist. By offering this critique, I hope to make clear to the readers two things: first, that Popkin’s thesis, though it is both enticing and generally accepted by many scholars, is questionable with regard to its plausibility; second, that the arguments I present in this dissertation reveal that further research into the role of skepticism in early modern philosophy is in order.

The purpose of this dissertation is to address various geometric aspects of second-order scalar hyperbolic partial differential equations in two independent variables and one dependent variable F(x, y, u, u_x, u_y, u_xx, u_xy, u_yy )= 0 (1) We find a characterization of hyperbolic Darboux integrable equations at level k (1) in terms of the vanishing of the generalized Laplace invariants and provide an invariant characterization of various cases in the Goursat general classification of hyperbolic Darboux integrable equations (1). In particular we give a contact invariant characterization of equations integrable by the methods of general and intermediate integrals. New relative invariants that control the existence of the first integrals of the characteristic Pfaffian systems are found and used to obtain an invariant characterization for the class of -Gordon equations. A notion of a hyperbolic Darboux system is introduced and we show by examples that the classical Laplace transformation is just a special case of a diffeomorphism of hyperbolic Darboux systems. We also construct new examples of homomorphisms between certain hyperbolic systems. We characterize Monge-Ampere equations and explicitly exhibit two invariants whose vanishing is a necessary and sufficient condition for the equation to be of the Monge-Ampere type. The solution to the inverse problem of the calculus of variations for hyperbolic equations (1) in terms of the generalized Laplace invariants is presented. We also obtain some partial results on symplectic conservation laws. We characterize, up to contact equivalence, some classical equations using the generalized Laplace invariants. These results contain characterizations of the wave, Liouville, Klein-Gordon, and certain types of Euler-Poisson equations.

In the models analyzed in this paper, there exists an incumbent politician with one objective, two choices, and voters who remember the past differently. The politician's primary goal is to get reelected, which is done by maximizing the number of votes on the day of election. The politician can increase their chances of reelection if they influence the state of the economy over time and ensure the economy is in its 'best' state on the days leading up to the election. In conducting this research, I wanted to study how different rates of memory decay influences the choices the politician makes during the course of their term. Also, I wanted to explore how long a politician would wait to have an election if that were a choice they could make. I found that voters who remember more of the past place a greater constraint on the incumbent leading to moderate fluctuations in the economy and frequent elections.