Orientadores: Carlos Eduardo Durán Fernandez, Márcio Antônio de Faria Rosa / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Cientifica / Made available in DSpace on 2018-08-16T14:48:33Z (GMT). No. of bitstreams: 1 ManoOtero_Diego_M.pdf: 1221591 bytes, checksum: 33ae6e3b671523a9602f3398e14d4fb7 (MD5) Previous issue date: 2010 / Resumo: A motivação inicial deste trabalho foi tentar relacionar os conceitos de geometria de Finsler com situações físicas que temos uma certa dependência de direções no nosso espaço. Apresentamos o conceito do cálculo variacional em variedades e sua relação com as geodésicas. Estudamos também o operador laplaciano ?? para espaços de Minkowski, que generaliza o caso Euclideano, e mais especificamente o problema...Observação: O resumo, na íntegra poderá ser visualizado no texto completo da tese digital / Abstract: The initial motivation of this study was to try to relate the concepts of Finsler geometry with physical situations where we have a certain dependence on the directions of our space. We introduce the concept of variational calculus on manifolds and their relationship with the geodesics. We also studied the Laplacian operator ?? in Minkowski space, which generalizes the Euclidean case, and more specifically the problem ...Note: The complete abstract is available with the full electronic digital thesis or dissertations. / Mestrado / Geometria / Mestre em Matemática

In this work we study the existence and uniqueness of solutions of a boundary value problem for equations of non-homogeneous wave. We will use the Faedo - Galerkin method to ensure the existence of solutions and also prove the exponential decay of the solution. / Neste trabalho estudaremos a existência e unicidade de soluções de um problema de valor de fronteira para equações da onda não homogênea. Usaremos o método de Faedo-Galerkin para garantir a existência de soluções e também provaremos o decaimento exponencial da solução do problema.

Neste trabalho estudamos a equação de ondas do tipo p-Laplaciano \'u IND. tt\' - \'DELTA\' IND.p u + \'(- \'DELTA\' POT. alpha\' u IND. t\' = \' [u] POT.q - 2 u, definida num domínio limitado limitado do \'R POT. n\', com 2 \' > ou = \' p < q e 0 < \' alpha\' < 1. Utilizando o método de Faedo-Galerkin provamos a existência de soluções fracas globais para dados iniciais pequenos. Para essas soluções estudamos também o decaimento polinomial da energia associada. A questão da não existência de soluções globais é considerada para o caso em que a energia inicial do sistema é negativa / In this work we study the p-Laplacian wave equation \'u IND. tt\' - \' DELTA\' IND p u + \'(- \'DELTA\' POT. \'alpha\' \' u IND. t\' = \'[u] POT. q - 2 u, defined in a bounded domain of \'R POT n\', with 2 \'> or =\' p < q and 0 < \' alpha\' < 1. By using the Faedo-Galerkin method we prove the existence of weak global solutions for small initial data. We also study the polynomial decay of the associate energy. The blow-up of solutions in finite time is considered for negative initial energy

While the formalism of isolated horizons is known for some time, only quite recently the near horizon solution of Einstein's equations has been found in the Bondi-like coordinates by Krishnan in 2012. In this framework, the space-time is regarded as the characteristic initial value problem with the initial data given on the horizon and another null hypersurface. It is not clear, however, what ini- tial data reproduce the simplest physically relevant black hole solution, namely that of Kerr-Newman which describes stationary, axisymmetric black hole with charge. Moreover, Krishnan's construction employs the non-twisting null geodesic congruence and the tetrad which is parallelly propagated along this congruence. While the existence of such tetrad can be easily established in general, its explicit form can be very difficult to find and, in fact it has not been provided for the Kerr-Newman metric. The goal of this thesis was to fill this gap and provide a full description of the Kerr-Newman metric in the framework of isolated horizons. In the theoretical part of the thesis we review the spinor and Newman-Penrose formalism, basic geometry of isolated horizons and then present our results. Thesis is complemented by several appendices.

In this thesis, the stability of the family of subextremal Kerr-Newman space- times is studied in the case of linear scalar perturbations. That is, nondegenerate energy bounds (NEB) and integrated local energy decay (ILED) results are proved for solutions of the wave equation on the domain of outer communications. The main obstacles to the proof of these results are superradiance, trapping and their interaction. These difficulties are surmounted by localising solutions of the wave equation in phase space and applying the vector field method. Miraculously, as in the Kerr case, superradiance and trapping occur in disjoint regions of phase space and can be dealt with individually. Trapping is a high frequency obstruction to the proof whereas superradiance occurs at both high and low frequencies. The construction of energy currents for superradiant frequencies gives rise to an unfavourable boundary term. In the high frequency regime, this boundary term is controlled by exploiting the presence of a large parameter. For low superradiant frequencies, no such parameter is available. This difficulty is overcome by proving quantitative versions of mode stability type results. The mode stability result on the real axis is then applied to prove integrated local energy decay for solutions of the wave equation restricted to a bounded frequency regime. The (ILED) statement is necessarily degenerate due to the trapping effect. This implies that a nondegenerate (ILED) statement must lose differentiability. If one uses an (ILED) result that loses differentiability to prove (NEB), this loss is passed onto the (NEB) statement as well. Here, the geometry of the subextremal Kerr-Newman background is exploited to obtain the (NEB) statement directly from the degenerate (ILED) with no loss of differentiability.

The diploma thesis deals with pressure pulsations in pipeline system with dynamic muffler. There is presented original geometry of side-branch resonator. Pressure pulsations are solved by a created mathematical model, numerical simulations and verified by an experimental approach. The influence of dynamic and bulk viscosity is involved in derived governing equations. A system of nonlinear equations is solved by genetic algorithm and frequency dependent relationship of bulk viscosity of air is determined afterwards. The correct function of used pressure sensors is tested. The processing of experimental data is performed by the fast Fourier transform with coherent sampling. Finally, a comparison of analytical, numerical and experimental approaches is introduced for different geometric variants of presented muffler.

Dans cette thèse, nous étudions les équations d’ondes non-linéaires discrètes dans des réseaux finis arbitraires. C’est un modèle général, où le Laplacien continu est remplacé par le Laplacien de graphe. Nous considérons une telle équation d’onde avec une non-linéarité cubique sur les nœuds du graphe, qui est le modèle φ4 discret, décrivant un réseau mécanique d’oscillateurs non-linéaires couplés ou un réseau électrique où les composantes sont des diodes ou des jonctions Josephson. L’équation d’onde linéaire est bien comprise en termes de modes normaux, ce sont des solutions périodiques associées aux vecteurs propres du Laplacien de graphe. Notre premier objectif est d’étudier la continuation des modes normaux dans le régime non-linéaire et le couplage des modes en présence de la non-linéarité. En inspectant les modes normaux du Laplacien de graphe, nous identifions ceux qui peuvent être étendus à des orbites périodiques non-linéaires. Il s’agit des modes normaux dont les vecteurs propres du Laplacien sont composés uniquement de {1}, {-1,+1} ou {-1,0,+1}. Nous effectuons systématiquement une analyse de stabilité linéaire (Floquet) de ces orbites et montrons le couplage des modes lorsque l’orbite est instable. Ensuite, nous caractérisons tous les graphes pour lesquels il existe des vecteurs propres du Laplacien ayant tous leurs composantes dans {-1,+1} ou {-1,0,+1}, en utilisant la théorie spectrale des graphes. Dans la deuxième partie, nous étudions des solutions périodiques localisées spatialement. En supposant une condition initiale de grande amplitude localisée sur un nœud du graphe, nous approchons l’évolution du système par l’équation de Duffing pour le nœud excité et un système linéaire forcé pour le reste du réseau. Cette approximation est validée en réduisant l’équation φ4 discrète à l’équation de Schrödinger non-linéaire de graphes et par l’analyse de Fourier de la solution numérique. Les résultats de cette thèse relient la dynamique non-linéaire à la théorie spectrale des graphes. / In this thesis, we study the discrete nonlinear wave equations in arbitrary finite networks. This is a general model, where the usual continuum Laplacian is replaced by the graph Laplacian. We consider such a wave equation with a cubic on-site nonlinearity which is the discrete φ4 model, describing a mechanical network of coupled nonlinear oscillators or an electrical network where the components are diodes or Josephson junctions. The linear graph wave equation is well understood in terms of normal modes, these are periodic solutions associated to the eigenvectors of the graph Laplacian. Our first goal is to investigate the continuation of normal modes in the nonlinear regime and the modes coupling in the presence of nonlinearity. By inspecting the normal modes of the graph Laplacian, we identify which ones can be extended into nonlinear periodic orbits. They are normal modes whose Laplacian eigenvectors are composed uniquely of {1}, {-1,+1} or {-1,0,+1}. We perform a systematic linear stability (Floquet) analysis of these orbits and show the modes coupling when the orbit is unstable. Then, we characterize all graphs for which there are eigenvectors of the graph Laplacian having all their components in {-1,+1} or {-1,0,+1}, using graph spectral theory. In the second part, we investigate periodic solutions that are spatially localized. Assuming a large amplitude localized initial condition on one node of the graph, we approximate its evolution by the Duffing equation. The rest of the network satisfies a linear system forced by the excited node. This approximation is validated by reducing the discrete φ4 equation to the graph nonlinear Schrödinger equation and by Fourier analysis. The results of this thesis relate nonlinear dynamics to graph spectral theory.

Micro-assembly is of crucial importance in industry nowadays. Nevertheless, currently applied processes require improvements. Indeed, when dealing with the assembly of submillimetric components, usually neglected surface forces disturb the manipulation task. They are responsible for the component sticking to the gripper, because of downscaling laws. A promising strategy to tackle adhesion consists in working without contact. The present dissertation is focused on contactless handling with acoustic levitation.<p>The advantages of contactless handling, the physical principles suitable for levitation and their applications are detailed. The opportunity for new handling strategies are shown. Acoustic levitation appears as the most fitted principle for micro-assembly. The elements to model acoustic forces are analysed and performances of existing modellings are assessed. A general numerical model of acoustic forces is implemented and theoretically validated with literature benchmarks. A fully automated modular levitator prototype is designed and used to experimentally validate the implemented numerical model. Specific instrumentations and protocols are developed for the acoustic force measurements.<p>The numerical model is finally applied to the real levitator. Modelling results are used to support experimental observations: the optimisation of the levitator resonance, the influence of the reflector shape, the dynamical study of the component oscillations, the stability with lateral centring forces and rotation torques, the component insertion and extraction from the levitator, the effect of pressure harmonics on the acoustic forces, and the manipulation of non spherical components. Acoustic forces are experimentally measured and a very good agreement with the modellings is obtained. Consequently, the implemented simulation tool can successfully be applied to a complex manipulation task with a component of any shape in a real levitator. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

The thesis consists of two parts, a theoretical part and an applied part, and in addition an Appendix. Except for a very short chapter in the applied part and the appendix we only present previously unknown results leading to a very concise style. In the theoretical part we study rates of decay for vector-valued functions and semigroups of operators depending on a real and positive variable. Under boundedness assumptions on the function/semigroup itself and under analytic extendability assumptions of its Laplace transform/resolvent across the imaginary axis we provide (almost) sharp rates of decay. Our results improve known results in this very active area of research. In the second part of the thesis we apply our results to specific examples (from the field of PDEs): local energy decay for wave equations on exterior domains, energy decay for damped wave equations on bounded domains and decay for a viscoelastic boundary damping model for sound waves. Many more examples can be found in the vast literature.:Part 1 Quantified Tauberian theorems and decay of C0-semigroups 1 Decay of vector-valued functions 2 Optimal decay for C0-semigroups on Hilbert spaces Part 2 Applications: decay of waves 3 Local decay for waves in exterior domains 4 Waves on a square with constant damping on a strip 5 A viscoelastic boundary damping model

Lineare hyperbolische partielle Differentialgleichungen in homogenen Medien, beispielsweise die Wellengleichung, die die Ausbreitung und die Streuung akustischer Wellen beschreibt, können im Zeitbereich mit Hilfe von Randintegralgleichungen formuliert werden. Im ersten Hauptteil dieser Arbeit stellen wir eine effiziente Möglichkeit vor, numerische Approximationen solcher Gleichungen zu implementieren, wenn das Huygens-Prinzip nicht gilt. Wir nutzen die Faltungsquadraturmethode für die Zeitdiskretisierung und eine Galerkin-Randelement-Methode für die Raumdiskretisierung. Mit der Faltungsquadraturmethode geht eine diskrete Faltung der Faltungsgewichte mit der Randdichte einher. Bei Gültigkeit des Huygens-Prinzips konvergieren die Gewichte exponentiell gegen null, sofern der Index hinreichend groß ist. Im gegenteiligen Fall, das heißt bei geraden Raumdimensionen oder wenn Dämpfungseffekte auftreten, kann kein Verschwinden der Gewichte beobachtet werden. Das führt zu Schwierigkeiten bei der effizienten numerischen Behandlung. Im ersten Hauptteil dieser Arbeit zeigen wir, dass die Kerne der Faltungsgewichte in gewisser Weise die Fundamentallösung im Zeitbereich approximieren und dass dies auch zutrifft, wenn beide bezüglich der räumlichen Variablen abgeleitet werden. Da die Fundamentallösung zudem für genügend große Zeiten, etwa nachdem die Wellenfront vorbeigezogen ist, glatt ist, schließen wir Gleiches auch in Bezug auf die Faltungsgewichte, die wir folglich mit hoher Genauigkeit und wenigen Interpolationspunkten interpolieren können. Darüber hinaus weisen wir darauf hin, dass zur weiteren Einsparung von Speicherkapazitäten, insbesondere bei Langzeitexperimenten, der von Schädle et al. entwickelte schnelle Faltungsalgorithmus eingesetzt werden kann. Wir diskutieren eine effiziente Implementierung des Problems und zeigen Ergebnisse eines numerischen Langzeitexperimentes. Im zweiten Hauptteil dieser Arbeit beschäftigen wir uns mit Transmissionsproblemen der Wellengleichung im Freiraum. Solche Probleme werden gewöhnlich derart behandelt, dass der Freiraum, wenn nötig durch Einführen eines künstlichen Randes, in ein unbeschränktes Außengebiet und ein beschränktes Innengebiet geteilt wird mit dem Ziel, eventuelle Inhomogenitäten oder Nichtlinearitäten des Materials vollständig im Innengebiet zu konzentrieren. Wir werden eine Lösungsstrategie vorstellen, die es erlaubt, die aus der Teilung resultierenden Teilprobleme so weit wie möglich unabhängig voneinander zu behandeln. Die Kopplung der Teilprobleme erfolgt über Transmissionsbedingungen, die auf dem ihnen gemeinsamen Rand vorgegeben sind. Wir diskutieren ein Kopplungsverfahren, das auf verschiedene Diskretisierungsschemata für das Innen- und das Außengebiet zurückgreift. Wir werden insbesondere ein explizites Verfahren im Innengebiet einsetzen, im Gegensatz zum Außengebiet, bei dem wir ein auf ein Mehrschrittverfahren beruhendes Faltungsquadraturverfahren nutzen. Die Kopplung erfolgt nach der Strategie von Johnson und Nédélec, bei der die direkte Randintegralmethode zum Einsatz kommt. Diese Strategie führt auf ein unsymmetrische System. Wir analysieren das diskrete Problem hinsichtlich Stabilität und Konvergenz und unterstreichen die Einsatzfähigkeit des Kopplungsalgorithmus mit der Durchführung numerischer Experimente.