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Results from combinatorial topology have shown that certain combinatorial lemmas are equivalent to certain topologocal fixed point theorems. For example, Sperner’s lemma about labelings of triangulated simplices is equivalent to the fixed point theorem of Brouwer. Moreover, since Sperner’s lemma has a constructive proof, its equivalence to the Brouwer fixed point theorem provides a constructive method for actually finding the fixed points rather than just stating their existence. The goal of this research project is to develop a combinatorial analogue for the Poincare ́-Birkhoff fixed point theorem.

The birefringence of one-dimension PMMA surface gratings on a gold film substrate is investigated. The grating served as a coupler to facilitate the incoming light coupled to surface plasmon wave (SPW) which possesses high propagation wave vector. Since surface plasmon waves(SPWs) have a special dispersion relation, the birefringence £Gneff (£Gneff =nx-ny, where grating k-vector is along x axis) of this structure is relatively large and can be changed from positive to negative by changing the operation wavelength. The obtained the four Stokes parameters at 515nm and 633nm are marked on the Poincare sphere. £Gneff is 2£k/7 and -£k/8 at 515nm and 633nm respectively. Further, by changing the form factor of PMMA gratings, we found that the maximum £Gneff occurred when PMMA stripe width : air gap=1:1.

This report discusses two examples of the use of Poincare disc models and their different relationships to Euclidean geometry. The topics include light reflection in hyperbolic geometry and the Hyperbolic Pythagorean Theorem; all in relation to the Poincare unit disc and Poincare upper half plane model. / text

In the first part of this thesis, the design of a compass robot is explored by considering its components and their interaction with each other. Three components including robot's structure, gear and motor are interacting during design process to achieve better performance, higher stability and lower cost. In addition, the modeling of the system is upgraded by considering the torque-velocity constraint in the motor. Adding this constraint of DC motor make the interaction of different components more complicated since it affects the gear and walking dynamics. After achieving the design method, different actuators (motor+ gear+ batteries) are selected for a given structure and the their performance is compared in the terms of cost, efficiency and their effect on the walking stability. In the second part of the thesis, structural optimization of the compass robot with stability constraint is investigated. The stability of a compass robot as a hybrid system is analyzed by Poincare map. Including stability analysis in the optimization process, makes it very complicated. In addition, the objective function of the system has to be evaluated in the convergent limit cycle. Different methods are examined to solve this problem. Limit cycle convergence is the best solution among the existing methods. By adding convergence constraint to the optimization, in addition of making the stability analysis valid, it helps the optimization estimates the correct objective function in each iteration. Finally, the optimization process is improved in two steps. The first step is using a predictive model in the optimization which covers the stable domain so that one does not need to check the stability of walking in each iteration. The Support Vector Domain Description (SVDD) approach which is applied to establish the stable domain, improve the decreases the optimization time. Another important step to upgrade the optimization is developing a computational algorithm which obtains the convergent limit cycle and its fixed-point in a short time. This algorithm speeds up the optimization time tremendously and allows the optimization search in a broader area. Combining SVDD approach in combination with Fixed-Point Finder Algorithm improve the optimization in the terms of time and broader area for search.

This thesis studies Ricci solitons of cohomogeneity one and uniform Poincaré inequalities for differentials on Riemann surfaces. In the two summands case, which assumes that the isotropy representation of the principal orbit consists of two inequivalent Ad-invariant irreducible summands, complete steady and expanding Ricci solitons have been detected numerically by Buzano-Dancer-Gallaugher-Wang. This work provides a rigorous construction thereof. A Lyapunov function is introduced to prove that the Ricci soliton metrics lie in a bounded region of an associated phase space. This also gives an alternative construction of non-compact Einstein metrics of non-positive scalar curvature due to Böhm. It is explained how the asymptotics of the Ricci flat trajectories induce Böhm's Einstein metrics on spheres and other low dimensional spaces. A numerical study suggests that all other Einstein metrics of positive scalar curvature which are induced by the generalised Hopf fibrations occur in an entirely non-linear regime of the Einstein equations. Extending the theory of cohomogeneity one steady and expanding Ricci solitons, an estimate which allows to prescribe the growth rate of the soliton potential at any given time is shown. As an application, continuous families of Ricci solitons on complex line bundles over products of Fano Kähler Einstein manifolds are constructed. This generalises work of Appleton and Stolarski. The method also applies to the Lü-Page-Pope set-up and allows to cover an optimal parameter range in the two summands case. The Ricci soliton equation on manifolds foliated by torus bundles over products of Fano Kähler Einstein manifolds is discussed. A rigidity theorem is obtained and a preserved curvature condition is discovered. The cohomogeneity one initial value problem is solved for m-quasi-Einstein metrics and complete metrics are described. L^p-Poincaré inequalities for k-differentials on closed Riemann surfaces are shown. The estimates are uniform in the sense that the Poincaré constant only depends on p ≥1, k ≥ 2 and the genus γ ≥ 2 of the surface but not on its complex structure. Examples show that the analogous estimate for 1-differentials cannot be uniform. This part is based on joint work with Melanie Rupflin.

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.

Made available in DSpace on 2014-08-13T14:50:49Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-01-28Bitstream added on 2014-08-13T18:00:25Z : No. of bitstreams: 1 000759961.pdf: 1328359 bytes, checksum: 563dde7eb5f2c1f263d919181e3cfbc7 (MD5) / Neste trabalho, realizamos o estudo global de sistemas diferenciais polinomiais planares quadráticos e cúbicos com duas circunferências não-conc êntricas como curvas algébricas invariantes. Apresentamos todos os possíveis retratos de fase dos campos vetoriais polinomiais associados a tais sistemas no disco de Poincaré. Mostramos que existem 3 classes de equivalência topológica para o caso quadrático e 19 classes de equivalência topológica para o caso cúbico. Como uma consequência deste estudo, provamos que estes sistemas diferenciais polinomiais não apresentam ciclos limites. / In this work, we perform a global study of quadratic and cubic planar polynomial differential systems having two nonconcentric circles as invariant algebraic curves. We give all possible global phase portraits on the Poincar´e disk of the polynomial vector fields associated to these systems. We show that there exist 3 topological equivalent classes for quadratic cases and 19 topological equivalent classes for cubic ones. As a consequence, we prove that these polynomial differential systems have no limit cycles.

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.