Esta dissertação tem como objetivo discutir sob quais condições uma G- estrutura é integrável. Primeiro apresentam-se fibrados principais, vetoriais e outras estruturas a elas associados como torção, espaços verticais, espaços horizontais e conexões. Depois apresentam-se a definição de G-estrutura, de integrabilidade de G-estruturas, com exemplos e as respectivas versões de integrabilidade e equivalência de G-estruturas. Finalmente, são descritas condições mais gerais que garantem a integrabilidade de G-estruturas. / This dissertation aims to discuss what are the conditions for the inte- grability of a G-structure. We begin presenting principal bundles, vectoer bundles, associated bundles and other structures related to them like torsion, vertical spaces, horizontal spaces and connections. After this, we present the definition of G-structure, integrability os G-structures with examples ans respectives versions of integrabilities and the equivalence of G-estructures. Finally, we describe more general conditions that ensure the integrability of G-structures.

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CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Esta tese està dividida em quatro partes. Na primeira delas estudaremos pontos crÃticos do funcional curvatura escalar total restrito ao espaÃo das mÃtricas de curvatura escalar constante e volume unitÃrio. Provaremos que sob certas condiÃÃes integrais convenientes os pontos crÃticos de tal funcional sÃo variedades de Einstein provando assim a conjectura dos pontos crÃticos neste caso. Na segunda parte, veremos duas estimativas para o primeiro autovalor do Laplaciano de uma variedade compacta com curvatura de Ricci limitada por baixo por uma constante. As estimativas que obtemos melhoram a estimativa correspondente provada por Li e Yau (1980). Na terceira parte, estamos interessados em estimar o diÃmetro de hipersuperfÃcies mÃnimas da esfera. A estimativa que encontramos depende apenas do primeiro autovalor do Laplaciano da hipersuperfÃcie considerada. Para superfÃcies imersas na esfera de dimensÃo trÃs, obtemos uma estimativa ligeiramente melhor do que a obtida no caso de dimensÃo alta. Na Ãltima parte, introduzimos o conceito de variedade de energia constante e provamos que a esfera e o toro sÃo as Ãnicas superfÃcies que tÃm energia constante. Em dimensÃo mais alta a situaÃÃo à bem diferente uma vez que o produto de uma esfera por qualquer variedade compacta tem energia constante. Entretanto, se impusermos uma condiÃÃo sobre a curvatura de Ricci, à possÃvel caracterizar a esfera tambÃm neste caso. Em seguida, aplicamos as informa-ÃÃes obtidas ao estudo de hipersuperfÃcies da esfera provando alguns resultados de rigidez desde que a hipersuperfÃcie tenha energia constante. / This thesis is divided into four parts. In the first one we study the critical points of the total scalar curvature functional restricted to the space of metrics with constant scalar curvature and volume one. We shall prove that under certain suitable integral conditions the critical points of such functional are Einstein manifolds proving this way the critical point equation conjecture in this case. In the second part, we will provide an estimate for the first eigenvalue of the Laplacian of a compact manifolds with Ricci curvature bounded from below by a constant. The estimate we obtain improves the corresponding estimate proved by Li and Yau (1980). In the third part, we are interested in to estimate the diameter of minimal hypersurfaces of the sphere. The estimate we get depends only on the first eigenvalue of the Laplacian of the considered hypersurface. For immersed surfaces on the three dimensional sphere, we obtain an estimate slightly better than the one obtained in the case of higher dimension. In the last part, we introduce the concept of manifolds with constant energy and prove that the sphere and the torus are the only compact surfaces that have constant energy. For higher dimension, the situation is very different sine the product of the sphere with any compact manifold has constant energy. Nevertheless, if we impose a condition over the Ricci curvature it is possible to characterize the sphere also in this case. After that, we apply the informations obtained to the study of hypersurfaces of the sphere proving some rigidity results provided that the hypersurfaces has constant energy.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / In this work we approach four research lines, where we began with the study of isometrically immersed hypersurfaces in a horoball. Next we studied Liouville type theorems in a complete Riemannian manifold for general operators. After we studied hypersurfaces f-minimal closed on a manifold with density, and nally we studied properly embedded minimal hypersurfaces with free boundary in a n-dimensional compact Riemannian manifold. Continuing, we obtain under a more general class operator than '-Laplacian, a Liouville type theorem for a complete Riemannian manifold, so that, prove a classication theorem for Killing graph of a foliation. Firstly, we are going to assume a weak maximum principle and that immersion is contained in a horoball, i.e., the set of bounded above Bussemann functions . We obtain an estimate for the highest quotient of r-curvatures. Moreover, under certain conditions on sectional curvature and assuming that the immersion is contained in a horoball, we forced the validity of the weak maximum principle and obtain the same estimates. Next, we establish a Choi-Wang type estimate for the rst eigenvalue of the weighter Laplacian on spaces with density in responding partially to Yau's conjecture for the rst eigenvalue weighter Laplacian for spaces with density, and moreover, we obtain an inequality Poincare type. With the estimates obtained, we establish an estimate of volume for a closed surface immersed in a space with density. Still following the study of spaces with density, we obtain a type Hientze-Karcher inequality for a compact manifold with nonempty boundary , so that, we obtain that if holds the equality than the manifold is isometric to a Euclidian ball. As consequence, we obtain under same conditions that if the f-mean curvature satisfy a bounded below than the manifold is isometric to a Euclidian ball. Finally, we obtain an estimate for the nonzero rst Steklov eigenvalue, where we are giving a answer partial to a conjecture by Fraser and Li. Moreover, as a consequence we establish an estimate for the total length of the boundary of the properly embedded minimal surfaces with free boundary in terms of its topology, thus, we proved the same when the surface is embedded in the Euclidean ball 3-dimensional. / Neste trabalho, abordamos quatro linhas de estudo, onde iniciamos com o estudo de hipersuperfcies isometricamente imersas sobre uma horobola. Em seguida estudamos Teoremas tipo Liouville para uma variedade Riemanniana completa em operadores mais gerais que o Laplaciano. Alem disso, estudamos hipersuperfcies f-mÃnimas fechadas em uma variedade com densidade e, por fim, estudamos hipersuperfÃcies mÃnimas com bordo livre, propriamente imersas em uma variedade Riemanniana compacta n-dimensional. Primeiramente, assumindo um princpio do maximo fraco e que a imersÃo està contida em uma horobola, i.e., um conjunto em que a funcÃo de Busemann à limitada superiormente, obtemos uma estimativa para o supremo do quociente das r-Ãsimas curvaturas. AlÃm disso, sob certas condiÃÃes sobre as curvaturas seccionais e assumindo que a imersÃo està contida em uma horobola, forÃamos a validade do princÃpio do mÃximo fraco e obtemos as mesmas estimativas. Prosseguindo, obtemos, para um operador mais geral que o '-Laplaciano, um teorema tipo-Liouville para uma variedade Riemanniana completa. Como aplicaÃÃo provamos um teorema de classificaÃÃo para grÃficos de Killing de uma folheaÃÃo. Em seguida, estabelecemos uma estimativa tipo Choi e Wang para o primeiro autovalor do f-Laplaciano em espaÃos com densidade, no sentido de responder parcialmente à conjectura de Yau para o primeiro autovalor do Laplaciano; alÃm disso, obtemos uma desigualdade tipo Poincarà para esse operador. Com a estimativa obtida, pudemos estabelecer uma estimativa de volume para uma superfÃcie fechada mergulhada em um espaÃo com densidade. Ainda seguindo o estudo de espaÃos com densidade, obtemos uma desigualdade tipo Heintze-Karcher para uma variedade compacta com bordo e verificamos que, se vale a igualdade, entÃo a variedade à isomÃtrica a uma bola Euclidiana. Como consequÃncia, obtemos que, nas mesmas condiÃÃes, e se a f-curvatura mÃdia satisfizer uma certa limitaÃÃo inferior, entÃo a variedade ainda à isometrica a uma bola Euclidiana. Finalmente, obtemos uma estimativa para o primeiro autovalor de Steklov, dando uma resposta parcial a uma conjectura devida a Fraser e Li. AlÃm disso, como consequÃncia, estabelecemos uma estimativa para o comprimento do bordo de uma superfÃcie mÃnima, compacta e propriamente megulhada com bordo livre em termos de sua topologia; assim, provamos o mesmo resultado quando a superfÃcie està mergulhada em uma bola Euclidiana 3-dimensional.

Cette thèse s'inscrit dans le domaine de l'analyse harmonique et plus exactement, des estimations à poids. Un intérêt particulier est porté aux estimations Lp à poids des transformées de Riesz sur des variétés Riemanniennes complètes ainsi qu'à l'optimalité des résultats en terme de la puissance de la caractéristique des poids. On obtient un premier résultat (en terme de la linéarité et de la non dépendance de la dimension) sur des espaces pas nécessairement de type homogène, lorsque p = 2 et la courbure de Bakry-Emery est positive. On utilise pour cela une approche analytique en exhibant une fonction de Bellman concrète. Puis, en utilisant des techniques stochastiques et une domination éparse, on démontre que les transformées de Riesz sont bornées sur Lp, pour p ∈ (1, +∞) et on déduit également le résultat précèdent. Enfin, on utilise un changement élégant dans la preuve précèdente pour affaiblir l'hypothèse sur la courbure et la supposer minorée. / The topics addressed in this thesis lie in the field of harmonic analysis and more pre- cisely, weighted inequalities. Our main interests are the weighted Lp-bounds of the Riesz transforms on complete Riemannian manifolds and the sharpness of the bounds in terms of the power of the characteristic of the weights. We first obtain a linear and dimensionless result on non necessarily homogeneous spaces, when p = 2 and the Bakry-Emery curvature is non-negative. We use here an analytical approach by exhibiting a concrete Bellman function. Next, using stochastic techniques and sparse domination, we prove that the Riesz transforms are Lp-bounded for p ∈ (1, +∞) and obtain the previous result for free. Finally, we use an elegant change in the precedent proof to weaken the condition on the curvature and assume it is bounded from below.

Thesis (Ph.D, Civil Engineering) -- Queen's University, 2011-01-31 20:52:11.162 / One of the most severe hazards for buried pipelines, which are sometimes referred to as lifelines due to their essential role in delivering vital resources, is the hazard due to Permanent Ground Deformation (PGD). Earthquake induced PGD can be caused by surface faulting, landslides and seismic settlement. In this thesis, the behaviour of buried pipelines subject to normal faulting has been experimentally investigated through a series of centrifuge tests performed on both continuous and jointed pipelines. Both pipe and soil displacements were measured using image analysis. Signal processing techniques were then developed to filter this data so as to enable the calculation of curvature and other aspects of the response from the observed pipe deformations. First, a series of centrifuge tests was conducted on continuous pipelines of varying materials, representing a wide range of pipe stiffness relative to the soil and investigating the effect of pipe stiffness relative to the soil on soil-pipe interaction. The experimentally derived p-y curves at different locations along the pipe were compared to the recommended soil-pipe interaction models in the relevant guidelines. These p-y curves showed that the central shearing region was not captured well with independent soil springs. The response of the pipelines predicted by the ALA (2001) guideline, however, was shown to match the experimental data within 50%. Two new simplified design approaches were then developed. The first features calculations based on simplified pressure distributions. The second featured peak curvature normalized using a characteristic length, ipipe, the distance from peak to zero moment. A series of centrifuge tests using brittle pipes was also performed. The pipes were buried at three different depths, and the post-failure fracture angle of the pipe was measured to be used as an input for design of liners. Based on the experimental data, a computationally efficient approach was developed to estimate the initial fracture angle which occurs immediately after the pipe breaks. The last series of centrifuge tests was conducted on jointed pipelines with five different joint stiffnesses to investigate the flexural behaviour of jointed pipelines under normal faulting. Based on the observed pipe response, a simplified kinematic model was proposed to estimate the maximum joint rotation for a given geometry, pipe segment length, and the magnitude of the imposed ground displacement. / Ph.D

The vibrational contribution to nonlinear optics (NLO) properties is the mean goal of study of this Thesis. The vibrational contribution can be split in the nuclear relaxation contribution and the curvature contribution. Nuclear relaxation contribution is caused by the changes in the equilibrium geometry induced by the electric field, while the perturbation on the potential energy surface curvature generates the curvature contribution. The general results of the present Thesis are: the development of new methodology to the calculation of the vibrational contribution, the implementation of the new developed methodology in order to facilitate the routine calculation of the NLO properties, the systematic investigation of the importance of vibrational contribution for several chemical systems / El principal objecte d’estudi d’aquesta tesi doctoral és la contribució vibracional a les propietats òptiques no lineals. La contribució vibracional es pot dividir en la contribució de relaxació nuclear i la contribució de curvatura. La contribució de relaxació nuclear és deguda als canvis en la geometria d’equilibri induïts pel camp elèctric incident, i la contribució de curvatura de la superfície de l’energia potencial induïda pel camp òptic genera la contribució de curvatura. En la primera secció de la tesi s’ha desenvolupat una metodologia per calcular les contribucions de relaxació nuclear i de curvatura a les propietats elèctriques. En la segona secció es presenten un conjunt de programes codificats en FORTRAN90 que permeten el càlcul analític i numèric de les hiperpolaritats vibracionals estàtiques i dinàmiques. En la tercera secció s’estudia l’efecte de les funcions de base i la correlació electrònica en el càlcul de les propietats òptiques no lineals

Complex patterns arise in many extended nonlinear nonequilibrium systems in physics, chemistry and biology. Information extraction from these complex patterns is a challenge and has been a main subject of research for many years. We study patterns in Rayleigh-Benard convection (RBC) acquired from our laboratory experiments to develop new characterization techniques for complex spatio-temporal patterns. Computational homology, a new topological characterization technique, is applied to the experimental data to investigate dynamics by quantifying convective patterns in a unique way. The homology analysis is used to detect symmetry breakings between hot and cold flows as a function of thermal driving in experiments, where other conventional techniques, e.g., curvature and wave-number distribution, failed to reveal this asymmetry. Furthermore, quantitative information is acquired from the outputs of homology to identify different spatio-temporal states. We use this information to obtain a reduced dynamical description of spatio-temporal chaos to investigate extensivity and physical boundary effects in RBC. The results from homological analysis are also compared to other dimensionality reduction techniques such as Karhunen-Loeve decomposition and Fourier analysis.

Deposition of nanoscale thickness films is ubiquitous in micro- and nano-scale device manufacturing. Current techniques such as spin-coating and chemical vapor deposition are designed to create only uniform thin films, and can be wasteful in material consumption. They lack the ability to adaptively prescribe desired film thickness profiles. This dissertation presents a novel inkjet-based zero-waste polymer deposition process referred to as Jet and Coat of Adaptive Sustainable Thin Films or J-CAST. The core of this process is built on an experimentally validated multi-scale fluid evolution model, based on extensions of lubrication theory. This model involves a nano-scale fluid film sandwiched between two flat plates: a compliant superstrate and a rigid substrate, with spatial topography on both surfaces. Accounting for the flexural elasticity of the compliant superstrate, and describing the temporal evolution of the fluid film in the presence of different boundary conditions reveals that instead of seeking process equilibrium, non-equilibrium transients should be exploited to guide film deposition. This forms the first core concept behind the process. This concept also enables robust full-wafer processes for creation of uniform films as well as nanoscale films with prescribed variation of thickness at mm-scale spatial wavelengths. The use of inkjets enables zero-waste adaptive material deposition with the preferred drop volumes and locations obtained from an inverse optimization formulation. This forms the second core concept behind the process. The optimization is based on the prescribed film thickness profile and typically involves >100,000 integer parameters. Using simplifying approximations for the same, three specific applications have been discussed - gradient surfaces in combinatorial materials science and research, elliptical profiles with ~10km radius of curvature for X-ray nanoscopy applications and polishing of starting wafer surfaces for mitigation of existing nanotopography. In addition, the potential of extending the demonstrated process to high throughput roll-roll systems has also been mentioned by modifying the model to incorporate the compliance of the substrate along with that of the superstrate. / text

Im Fokus dieser Arbeit steht die Entwicklung und Realisierung einer Nutzerschnittstelle für die computergestützte Bearbeitung von Oberflächen dreidimensionaler Designobjekte, die durch herkömmliche CAD-Systeme nur unzureichend unterstützt werden können. Es werden Methoden und Algorithmen erarbeitet, um die Interaktion des Benutzers direkt auf Basis der für das Design relevanten 3D Objektoberflächen zu ermöglichen. Den Ausgangspunkt bildet das Proxy-Konzept des Forschungsfeldes zur Mensch-Maschine-Interaktion, in welchem physische Stellvertreterobjekte zur Manipulation virtueller Objekte genutzt werden. Im ersten Teil der Arbeit wird dieses Konzept um eine auf die Oberfläche des bearbeiteten Objektes bezogene Interaktion und Designfunktionalität erweitert. Dazu wird das Konzept eines CAD-Proxys entwickelt und verschiedene Umsetzungen und Anwendungen erläutert. Dabei werden die CAD-Proxys in drei Typen unterschieden. Der exakte CAD-Proxy ermöglicht es, ein virtuelles Objekt mit einem formgleichen, physischen Gegenstück zu bearbeiten. Beim abstrakten CAD-Proxy wird gezeigt, wie auch ein anders geformtes Gegenstück zum Einsatz kommen kann. Der augmentierte CAD-Proxy erlaubt die Anzeige des Designs direkt auf dem physischen Gegenstück. Nutzerstudien belegen, dass die CAD-Proxys zur Erstellung eines Designs auf einer Oberfläche deutlich besser geeignet sind als eine klassische Nutzerschnittstelle. Dies gilt insbesondere für anspruchsvolle Designaufgaben, etwa im künstlerischen Design. Im zweiten Teil der Arbeit wird die Verarbeitung von Linienzügen im Gestaltungsprozess eingeführt, welche das Konzept der aktiven Konturen auf Oberflächen erweitern. Diese Linienzüge werden auf die Form der Oberfläche bezogen verarbeitet und beachten gleichzeitig wichtige Merkmale, wie bspw. die Kanten des Objektes. Es wird ein neues Verfahren vorgestellt, mit dem aktive Konturen auch auf schlecht vernetzten Oberflächen eingesetzt werden können und gleichzeitig Formen flexibler repräsentieren können als bisherige Ansätze. Für den Bezug der Designlinien zur Oberfläche wird eine neue Datenstruktur eingeführt, die ein geometrisches Nachbarschaftsproblem löst. Es wird gezeigt, dass diese Struktur auf den getesteten Objekten mindestens um Faktor 14 schneller ist als alternative Ansätze und auch direkt auf andere Problemstellungen anwendbar ist. Der dritte Teil enthält die Betrachtungen zum Formverständnis der Designobjekte, welche für die Designlinien und die Modellierung der Designobjekte benötigt werden. Dabei kommt die Krümmungsberechnung als etabliertes Werkzeug derartiger Analysen zum Einsatz. Hier wird eine neue Betrachtung erarbeitet, die den Einfluss unterschiedlicher Faktoren auf die Genauigkeit der Krümmungsschätzung und auf die Eignung verschiedener Schätzungsverfahren erstmals umfassend untersucht. Durch die Kombination existierender Verfahren kann das Verfahren der Shape Operator Approximation entwickelt werden, welches Eigenschaften wie Schätzungsgenauigkeit, Performanz und eine einfache Formulierung bekannter Verfahren vereint.