Neste trabalho apresentamos o modelo do semi-plano superior de Poincare da Geometria não-Euclidiana de Bolyai-Lobachevsky e mostramos que o bilhar 1f 1f hiperbólico no triângulo de ângulos O, ∏/3 e ∏/2 tem trajetórias densas, isto é, trajetórias que se aproximam com precisão arbitrária de qualquer ponto e direção dados. / In this work we present Poincare 's upper half-plane model of the non-Euclidean Geometry of Bolyai-Lobachevsky and show that the hyperbolic billiard on the tri- 7T 7T angle of angles O, ∏/3 and ∏/2 has dense orbits, i.e. trajectories coming arbitrarily close to any givcn point and direction.

The thesis reports the results of the research carried out towards the development of an electrooptic sensor for DC and extra low frequency electric field measurement. Available cubic electrooptic crystals were compared from the sensor sensitivity point of view. A new figure of merit was used taking into account the attenuation of the electric field in the dielectric crystal and its shape. The effect of optical activity in 23 cubic crystals was analyzed using the concept of Poincare sphere. The cubic crystals were further characterised for the charge relaxation time constant to estimate their performance in DC field measurements. Crystals of Bismuth Germanate and Lithium Niobate were identified as suitable materials for the DC field sensor. The selected crystals were found suitable at extra-low-frequencies. DC field measurements, without the rotation of the crystal, were possible only with Lithium Niobate. However, its performance was influenced to a great extent by the effect of stimulated conductivity. The quarter-wave plate and the crystal of Lithium Niobate were identified as the main sources of temperature instability. A new method of temperature compensation of the quarter-wave plate is proposed. Due to the temperature instability of Lithium Niobate, mainly attributed to the pyroelectric effect and natural birefringence, it is difficult to use the sensor in practical applications. The performance of the sensor is significantly affected by the presence of an external space charge. The proposed method of its elimination using an artificial extension of the sensing element did not reduce the space charge effect adequately. The response of the sensor in a space charge environment was found to be linear and independent of the space charge density. This enabled measurements of static fields in a unipolar environment. The direct field measurements in bipolar environment suffered from a drift which is intolerable in practical measurements. The minimum detectable electric field of this sensor in the frequency range from 1 to 200Hz was 1V/m, with a signal to noise ratio equal to 0dB and a resolution of 1V/m. The static field measurements were limited to measurements of pulses with a duration of 200s, due to a long term drift of photodetectors. The minimum detectable level of DC electric field was 2.4kV/m.

Several linear embeddings of the logistic equation, xn+1=axn(1-xn) are considered, the goal being to establish a connection between linear embedding and the Poincaré Functional Equation. In particular, we consider linear embedding schemes in a classical Hardy space.

[pt] Este trabalho apresenta em sua etapa inicial os conceitos básicos de polarização da luz descritos através da Esfera de Poincaré, Vetores de Jones, Parâmetros de Stokes, Matrizes de Mueller e Fórmula da Rotação de Rodrigues. Em seguida, as diversas técnicas utilizadas para as transformações dos SOPs (States of Polarization) são introduzidas. A partir destas etapas, é apresentada a seleção, o desenvolvimento e a realização prática de um sistema capaz de ativar o controle de polarização de sinais ópticos em uma fibra monomodo. Um segundo sistema capaz de controlar sinais ópticos multiplexados em frequencia é também realizado e apresentado. Ambos os sistemas são ativados através de uma ferramenta computacional dedicada baseada na linguagem de programação gráfica LabVIEW. / [en] This work presents initially a set of light polarization concepts using the Poincaré Sphere, Jone’s Vectors, Stoke’s Parameters, Mueller Matrices, Rodrigues’ Rotation Formula, and several SOPs (States of Polarization) transformations. Through these concepts and after a careful components selection, a SOP control system in monomode optical fiber is realized and presented. A second system able to implement the SOP control when multiplexed optical signals are employed is also realized and described. A dedicated computer tool using the software LabVIEW is developed to both systems.

Higher-dimensional Hamiltonian systems usually exhibit a mixed phase space in which regular and chaotic motion coexist. While regular trajectories are confined to regular tori, chaotic trajectories can be transported through a web of so called resonance channels which disrupt the regular structures. The focus of this thesis are time-discrete 4D symplectic maps which represent the lowest dimensional system for which the chaotic transport can circumvent regular tori. While the dynamics of 2D maps are well established, many fundamental questions are open for maps of dimension four and higher due to this property. In particular, the mechanism of the power-law trapping is unknown for these maps. In this thesis, the organization and hierarchy of the regular structures of 4D maps is uncovered and the slow chaotic transport close to these structures is examined. Specifically, this transport is shown to be organized by a set of overlapping resonance channels. The transport across these channels is found to be governed by partial transport barriers. For the transport along a channel a stochastic process including a drift is conjectured. Based on each of these two types of chaotic transport a possible mechanism for the power-law trapping in higher-dimensional systems is proposed.

Higher-dimensional Hamiltonian systems usually exhibit a mixed phase space in which regular and chaotic motion coexist. While regular trajectories are confined to regular tori, chaotic trajectories can be transported through a web of so called resonance channels which disrupt the regular structures. The focus of this thesis are time-discrete 4D symplectic maps which represent the lowest dimensional system for which the chaotic transport can circumvent regular tori. While the dynamics of 2D maps are well established, many fundamental questions are open for maps of dimension four and higher due to this property. In particular, the mechanism of the power-law trapping is unknown for these maps. In this thesis, the organization and hierarchy of the regular structures of 4D maps is uncovered and the slow chaotic transport close to these structures is examined. Specifically, this transport is shown to be organized by a set of overlapping resonance channels. The transport across these channels is found to be governed by partial transport barriers. For the transport along a channel a stochastic process including a drift is conjectured. Based on each of these two types of chaotic transport a possible mechanism for the power-law trapping in higher-dimensional systems is proposed.

Controlling the scrape-off layer (SOL) properties in order to limit divertor erosion and extend component lifetime will be crucial to successful operation of ITER and devices that follow, where intermittent thermal loads on the order of GW/m² are expected. Steady state transport in the edge region is generally turbulent with large, order unity, fluctuations and is convection dominated. Owing to the success of the past fifty years of progress in magnetically confining hot plasmas, in this work we examine convective transport phenomena in the SOL that occur in the relatively "slow", drift-ordered fluid limit, most applicable to plasmas near MHD equilibrium. Diamagnetic charge separation in an inhomogeneous magnetic field is the principal energy transfer mechanism powering turbulence and convective transport examined in this work. Two possibilities are explored for controlling SOL conditions. In chapter 2 we review basic physics underlying the equations used to model interchange turbulence in the SOL and use a subset of equations that includes electron temperature and externally applied potential bias to examine the possibility of suppressing interchange driven turbulence in the Texas Helimak. Simulated scans in E₀×B₀ flow shear, driven by changes in the potential bias on the endplates appears to alter turbulence levels as measured by the mean amplitude of fluctuations. In broad agreement with experiment negative biasing generally decreases the fluctuation amplitude. Interaction between flow shear and interchange instability appears to be important, with the interchange rate forming a natural pivot point for observed shear rates. In chapter 3 we examine the possibility of resonant magnetic perturbations (RMPs) or more generally magnetic field-line chaos to decrease the maximum particle flux incident on the divertor. Naturally occurring error fields as well as RMPs applied for stability control are known to cause magnetic field-line chaos in the SOL region of tokamaks. In chapter 3 2D simulations are used to investigate the effect of the field-line chaos on the SOL and in particular on its width and peak particle flux. The chaos enters the SOL dynamics through the connection length, which is evaluated using a Poincaré map. The variation of experimentally relevant quantities, such as the SOL gradient length scale and the intermittency of the particle flux in the SOL, is described as a function of the strength of the magnetic perturbation. It is found that the effect of the chaos is to broaden the profile of the sheath-loss coefficient, which is proportional to the inverse connection length. That is, the SOL transport in a chaotic field is equivalent to that in a model where the sheathloss coefficient is replaced by its average over the unperturbed flux surfaces. Both fully chaotic and the flux-surface averaged approximation of RMP application significantly lower maximum parallel particle flux incident on the divertor. / text

The Poincare-Miranda Theorem is a topological result about the existence of a zero of a function under particular boundary conditions. In this thesis, we explore proofs of the Poincare-Miranda Theorem that are discrete in nature - that is, they prove a continuous result using an intermediate lemma about discrete objects. We explain a proof by Tkacz and Turzanski that proves the Poincare-Miranda theorem via the Steinhaus Chessboard Theorem, involving colorings of partitions of n-dimensional cubes. Then, we develop a new proof of the Poincare-Miranda Theorem that relies on a polytopal generalization of Sperner's Lemma of Deloera - Peterson - Su. Finally, we extend these discrete ideas to attempt to prove the existence of a zero with the boundary condition of Morales.

Dans cette these, nous etudions d'abord le probleme des singularites eliminables des EDP elliptiques du second ordre; le cas modele etant $- \Delta u + cu \geq f$ sur $\Omega \backslash K$, avec $u \geq 0$ et $(\rm cap)_2((K))=0$. Nous démontrons aussi un principe du maximum fort pour l'operateur $-\Delta + a(x)$, avec un potentiel $a \in L^1$. Ces deux résultats utilisent plusieurs formulations de l'inegalite de Kato classique. Nous presentons ensuite quelques variantes de l'inegalite de Poincare, motives par une nouvelle caracterisation des espaces de Sobolev. Puis, nous nous interessons aux singularites topologiques des fonctions dans l'espace $W^(1,1)(S^2;S^1)$. A cet effet, nous etudions leur energie relaxee et la variation totale du jacobien. Finalement, nous considerons plusieurs proprietes des distributions de la forme $\sum_j((\delta_(p_j) - \delta_(n_j)))$, definies sur un espace metrique complet.

The author examines nonlinear aeroelastic responses of air vehicle systems. Herein, the governing equations for a cantilevered configuration are developed and the methods of analysis are explored. Based on the developed nonlinear bending-bending-torsion equations, internal resonance, which is possible in future air vehicles, and the possible cause of limit cycle oscillations of aircraft wings with stores are investigated. The nonlinear equations have three types of nonlinearities caused by wing flexibility, store geometry and aerodynamic stall, and retain up to third-order nonlinear terms. The internal resonance conditions are examined by the Method of Multiple Scales and demonstrated by time simulations. The effect of velocity change for various physical parameters and stiffness ratio is investigated through bifurcation diagrams derived from Poinar´e maps. The dominant factor causing limit cycle oscillations is the stiffness ratio between in-plane and out-of-plane motion.