The present dissertation develops an invariant framework for 3D gesture comparison studies. 3D gesture comparison without Lagrangian models is challenging not only because of the lack of prediction provided by physics, but also because of a dual geometry representation, spatial dimensionality and non-linearity associated to 3D-kinematics. In 3D spaces, it is difficult to compare curves without an alignment operator since it is likely that discrete curves are not synchronized and do not share a common point in space. One has to assume that each and every single trajectory in the space is unique. The common answer is to assert the similitude between two or more trajectories as estimating an average distance error from the aligned curves, provided that the alignment operator is found. In order to avoid the alignment problem, the method uses differential geometry for position and orientation curves. Differential geometry not only reduces the spatial dimensionality but also achieves view invariance. However, the nonlinear signatures may be unbounded or singular. Yet, it is shown that pattern recognition between intrinsic signatures using correlations is robust for position and orientation alike. A new mapping for orientation sequences is introduced in order to treat quaternion and Euclidean intrinsic signatures alike. The new mapping projects a 4D-hyper-sphere for orientations onto a 3D-Euclidean volume. The projection uses the quaternion invariant distance to map rotation sequences into 3D-Euclidean curves. However, quaternion spaces are sectional discrete spaces. The significance is that continuous rotation functions can be only approximated for small angles. Rotation sequences with large angle variations can only be interpolated in discrete sections. The current dissertation introduces two multi-scale approaches that improve numerical stability and bound the signal energy content of the intrinsic signatures. The first is a multilevel least squares curve fitting method similar to Haar wavelet. The second is a geodesic distance anisotropic kernel filter. The methodology testing is carried out on 3D-gestures for obstetrics training. The study quantitatively assess the process of skill acquisition and transfer of manipulating obstetric forceps gestures. The results show that the multi-scale correlations with intrinsic signatures track and evaluate gesture differences between experts and trainees.

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Almost contact metric submersions constitute a class of Riemannian submersions whose total space is an almost contact metric manifold. Regarding the base space, two types are studied. Submersions of type I are those whose base space is an almost contact metric manifold while, when the base space is an almost Hermitian manifold, then the submersion is said to be of type II. After recalling the known notions and fundamental properties to be used in the sequel, relationships between the structure of the fibres with that of the total space are established. When the fibres are almost Hermitian manifolds, which occur in the case of a type I submersions, we determine the classes of submersions whose fibres are Kählerian, almost Kählerian, nearly Kählerian, quasi Kählerian, locally conformal (almost) Kählerian, Gi-manifolds and so on. This can be viewed as a classification of submersions of type I based upon the structure of the fibres. Concerning the fibres of a type II submersions, which are almost contact metric manifolds, we discuss how they inherit the structure of the total space. Considering the curvature property on the total space, we determine its corresponding on the fibres in the case of a type I submersions. For instance, the cosymplectic curvature property on the total space corresponds to the Kähler identity on the fibres. Similar results are obtained for Sasakian and Kenmotsu curvature properties. After producing the classes of submersions with minimal, superminimal or umbilical fibres, their impacts on the total or the base space are established. The minimality of the fibres facilitates the transference of the structure from the total to the base space. Similarly, the superminimality of the fibres facilitates the transference of the structure from the base to the total space. Also, it is shown to be a way to study the integrability of the horizontal distribution. Totally contact umbilicity of the fibres leads to the asymptotic directions on the total space. Submersions of contact CR-submanifolds of quasi-K-cosymplectic and quasi-Kenmotsu manifolds are studied. Certain distributions of the under consideration submersions induce the CR-product on the total space. / Mathematical Sciences / D. Phil. (Mathematics)

Orientador: Glaucia Amorim Faria / Resumo: Considerando que a intensidade de infestação (I.I.%) é um importante dado sobre o dano causado pela Diatraea saccharalis em cana-de-açúcar e que existem poucos trabalhos na literatura relacionados ao tamanho de ótimo de parcela para este tipo de amostragem, o objetivo deste trabalho foi estimar o tamanho ótimo de parcela em hectares e número de colmos que deverá ser utilizado no processo de amostragem de modo que represente a intensidade de infestação causada pelo ataque da D. saccharalis em cana-de-açúcar. Para os cálculos relativos ao tamanho da área a ser amostrada foram utilizados quatro métodos para o cálculo do tamanho de parcela: método de inspeção visual da curvatura máxima, método da máxima curvatura modificado, modelo linear segmentado com platô e modelo quadrático segmentado com platô. Para os cálculos referentes ao número de entrenós foi utilizado o método da estimativa da suficiência amostral. O método da máxima curvatura modificado foi o que proporcionou melhores resultados. De acordo com os resultados encontrados neste trabalho, podemos concluir que o número mínimo a ser amostrado é o de 36 entrenós por hectare e a área máxima a ser amostrada é a de 27,5 hectares. / Abstract: Infestation intensity (II%) is an important data on the damage caused by Diatraea saccharalis in sugarcane. There are few studies in the literature related to the optimal plot size for this type of sampling. The objective of this work is to estimate the optimal plot size in hectares and number of stems to be used in the sampling process to represent the intensity of infestation caused by D. saccharalis attack on sugarcane. For the calculation of the size of the minimum sampled area, four methods were used: 1. visual inspection method of maximum curvature; 2. modified maximum curvature method; 3. segmented linear model with plateau and; 4. quadratic segmented model with plateau. For the calculations referring to the number of internodes, the method of estimating the sample adequacy was used. The modified maximum curvature method presented the best results. According this study, the minimum number to be sampled is 36 trains per hectare and the maximum area to be sampled is 27.5 hectares. / Mestre

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The main goal of this dissertation is to present some results on minimal hypersurfaces in the Euclidean space related to the stability operator. Initially, we will present the demonstrations of the formulas of first and second variations of area and also the demonstration of the Simons inequality. These results (which are basic results of the theory) will be used later. Next we will present the proof of the do Carmo-Peng s theorem showing that a complete stable minimal hypersurface immersed in the Euclidean space with finite L2 norm of the second fundamental form is a hyperplane. We will include in this dissertation a similar result with the L3 norm of the second fundamental form. This last result was proved by Li-Wei in the case where the hypersurface has dimension 3, but we note that proof applies to 3≤n≤7. We will conclude by presenting some results on non-stable minimal hypersurfaces in R^3 due to Fischer-Colbrie and Lopez-Ros. In particular, we will show that the catenoid and Enneper s surface are the only minimal complete orientable surfaces with index equal to one. / O objetivo principal desta dissertação é apresentar alguns resultados importantes sobre hipersuperfícies mínimas no espaço Euclidiano relacionados com o operador de estabilidade. Inicialmente, apresentaremos as demonstrações das fórmulas da primeira e da segunda variações da área bem como a demonstração da desigualdade de Simons. Estes resultados, que são básicos da teoria, serão usados posteriormente. Em seguida, apresentaremos a demonstração do teorema de do Carmo-Peng, o qual assegura que uma hipersuperfície mínima completa estável imersa no espaço Euclidiano com a norma L2 da segunda forma fundamental finita é um hiperplano. Incluiremos na dissertação um resultado análogo com a norma L3 da segunda forma fundamental. Este último resultado foi provado por Li-Wei no caso em que a hipersuperfície tem dimensão 3, mas notamos que a demonstração se aplica para 3≤n≤7. Concluiremos apresentando alguns resultados sobre hipersuperfícies mínimas não estáveis no R^3 obtido por Fischer-Colbrie e López-Ros. Em particular, mostraremos que o catenóide e a superfície de Enneper são as únicas superfícies mínimas completas e orientadas com índice igual a um.

In this work we prove two theorems that characterize the hypersurfaces in the unitary sphere of dimension n+1. The first result, obtained by H. Alencar and M. do Carmo, classifies hypersurfaces with constant mean curvature in the sphere. This result was published in April 1994 in Proceedings of The American Mathematical Society, volume 120, number 4 with the title Hypersurfaces with Constant Mean Curvature. The second result was obtained by Li Haizhong in the article Hypersurfaces with Constant Scalar Curvature in Space Forms, published in 1996 in the journal Mathematisch Annalen, volume 305. The theorem of Li Haizhong characterizes hypersurfaces with constant scalar curvature in the sphere. We prove the theorem of Li Haizhong using the results obtained by H. Alencar and M. do Carmo. / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Nesta dissertação apresentamos dois teoremas que caracterizam as hipersuperfícies na esfera unitária de dimensão n+1. O primeiro resultado, obtido por H. Alencar e M. do Carmo, classifica as hipersuperfícies com curvatura média constante na esfera. Este resultado foi publicado em abril de 1994 no Proceedings of The American Mathematical Society, volume 120, número 4 com o título Hypersurfaces With Constant Mean Curvature.O segundo resultado provado nesta dissertação foi obtido por Li Haizhong no artigo Hypersurfaces With Constant Scalar Curvature in Spaces Forms, publicado em 1996 no Mathematische Annalen, volume 305. O Teorema de Li Haizhong caracteriza as hipersuperfícies com curvatura escalar constante na esfera. Demonstraremos o Teorema de Li Haizhong utilizando os resultados obtidos por H. Alencar e M. do Carmo.

The goal of this dissertation is to present a R. Reilly's demonstration of the theorem of Alexandrov . The theorem states that The only compact hypersurfaces, conected, of constant mean curvature, immersed in Euclidean space are spheres. The theorem of Alexandrov was proved by A. D. Alexandrov in the article Uniqueness Theorems for Surfaces in the Large V, published in 1958 by Vestnik Leningrad University, volume 13, number 19, pages 5 to 8. In his demonstration, Alexandrov used the famous Principle of tangency, introduced by him in that article. In the year 1962, M. Obata shown in Certain Conditions for a Riemannian Manifold to be isometric With the Sphere, published by the Journal of Mathematical Society of Japan, volume 14, pages 333 to 340, that a Riemannian Manifold M, compact, connected and without boundary, is isometric to a sphere, since the Ricci curvature of M satisfies certain lower bound. This theorem solves the problem of finding manifolds that reach equality in the estimate of Lichnerowicz for the first eigenvalue. In 1977, R. Reilly, in the article Applications of the Hessian operator in a Riemannian Manifold, published in Indianna University Mathematical Journal, volume 23, pages 459 to 452, showed a generalization of the Obata theorem for compact manifolds with boundary. As an example of the technique developed in this demonstration, he presents a new demonstration of the theorem of Alexandrov. This demonstration, as well as the techniques involved are the object of study of this work. / Conselho Nacional de Desenvolvimento Científico e Tecnológico / O objetivo desta dissertação é apresentar uma demonstração de R. Reilly para o Teorema de Alexandrov. O teorema estabelece que As únicas hipersuperfícies compactas, conexas, de curvatura média constante, mergulhadas no espaço Euclidiano são as esferas. O teorema de Alexandrov foi provado por A. D. Alexandrov no artigo Uniqueness Theorems for Surfaces in the Large V, publicado em 1958 pela Vestnik Leningrad University, volume 13, número 19, páginas 5 a 8. Em sua demonstração, Alexandrov usou o famoso Princípio de Tangência, introduzido por ele no citado artigo. No ano de 1962, M. Obata demonstrou em Certain Conditions for a Riemannian Manifold to be Isometric With a Sphere, publicado pelo Journal of Mathematical Society of Japan, volume 14, páginas 333 a 340, que uma variedade Riemanniana M, compacta, conexa e sem bordo, é isométrica a uma esfera, desde que a curvatura de Ricci de M satisfaça determinada limitação inferior. Este teorema resolve o problema de encontrar as variedades que atingem a igualdade na estimativa de Lichnerowicz para o primeiro autovalor. Em 1977, R. Reilly, no artigo Applications of the Hessian Operator in a Riemannian Manifold, publicado no Indianna University Mathematical Journal, volume 23, páginas 459 a 452, demonstrou uma generalização do Teorema de Obata para variedades compactas com bordo. Como exemplo da técnica desenvolvida nesta demonstração, ele apresenta uma nova demonstração do Teorema de Alexandrov. Esta demonstração, bem como as técnicas envolvidas, são o objeto de estudo deste trabalho.

EVANGELISTA, I. S. Compact almost Ricci soliton, critical metrics of the total scalar curvature functional and p-fundamental tone estimates. 2017. 75 f. Tese (Doutorado em Matemática) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-07-10T12:41:32Z No. of bitstreams: 1 2017_tese_isevangelista.pdf: 618771 bytes, checksum: 7e4bb8d9fd8825ef347e309171075037 (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-07-10T14:06:18Z (GMT) No. of bitstreams: 1 2017_tese_isevangelista.pdf: 618771 bytes, checksum: 7e4bb8d9fd8825ef347e309171075037 (MD5) / Made available in DSpace on 2017-07-10T14:06:18Z (GMT). No. of bitstreams: 1 2017_tese_isevangelista.pdf: 618771 bytes, checksum: 7e4bb8d9fd8825ef347e309171075037 (MD5) Previous issue date: 2017-07-04 / The present thesis is divided in three different parts. The aim of the first part is to prove that a compact almost Ricci soliton with null Cotton tensor is isometric to a standard sphere provided one of the following conditions associated to the Schouten tensor holds: the second symmetric function is constant and positive; two consecutive symmetric functions are non null multiple or some symmetric function is constant and the quoted tensor is positive. The aim of the second part is to study the critical metrics of the total scalar curvature funcional on compact manifolds with constant scalar curvature and unit volume, for simplicity, CPE metrics. It has been conjectured that every CPE metric must be Einstein. We prove that the Conjecture is true for CPE metrics under a suitable integral condition and we also prove that it suffices the metric to be conformal to an Einstein metric. In the third part we estimate the p-fundamental tone of submanifolds in a Cartan-Hadamard manifold. First we obtain lower bounds for the p-fundamental tone of geodesic balls and submanifolds with bounded mean curvature. Moreover, we provide the p-fundamental tone estimates of minimal submanifolds with certain conditions on the norm of the second fundamental form. Finally, we study transversely oriented codimension one C 2-foliations of open subsets Ω of Riemannian manifolds M and obtain lower bounds estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental tone of Ω. / A presente tese está dividida em três partes diferentes. O objetivo da primeira parte é provar que um quase soliton de Ricci compacto com tensor de Cotton nulo é isométrico a uma esfera canônica desde que uma das seguintes condições associadas ao tensor de Schouten seja válida: a segunda função simétrica é constante e positiva; duas funções simétricas consecutivas são múltiplas, não nulas, ou alguma função simétrica é constante e o tensor de Schouten é positivo. O objetivo da segunda parte é estudar as métricas críticas do funcional curvatura escalar total em variedades compactas com curvatura escalar constante e volume unitário, por simplicidade, métricas CPE. Foi conjecturado que toda métrica CPE deve ser Einstein. Prova-se que a conjectura é verdadeira para as métricas CPE sob uma condição integral adequada e também se prova que é suficiente que a métrica seja conforme a uma métrica Einstein. Na terceira parte, estima-se o p-tom fundamental de subvariedades em uma variedade tipo Cartan-Hadamard. Primeiramente, obtém-se estimativas por baixo para o p-tom fundamental de bolas geodésicas e em subvariedades com curvatura média limitada. Além disso, obtém-se estimativas do p-tom fundamental de subvariedades mínimas com certas condições sobre a norma da segunda forma fundamental. Por fim, estudam-se folheações de classe C 2 transversalmente orientadas de codimensão 1 de subconjuntos abertos Ω de variedades riemannianas M e obtêm-se estimativas por baixo para o ínfimo da curvatura média das folhas em termos do p-tom fundamental de Ω.

The primary aim of this research was to investigate the bending behaviour of helically wound steel cables of various types (i.e. normal spiral strands, sheathed spiral strands and locked coil cables) in the presence of friction and to propose more efficient computational models for their analysis under combined tension and bending. The proposed model fully takes into account interwire contact forces both in the radial direction (point contact between wires in different layers) and hoop direction (line contact within the wires in the same layer). Extensive theoretical parametric studies have been undertaken on a variety of cable constructions covering a wide range of geometrical and material parameters. Explicit formulations have been developed for the smooth transition of the bending stiffness from no-slip to full slip regimes, as a function of cable curvature. Based on these formulations, it is now possible to calculate the relative displacements of the wires, as well as the tensile, bending and hoop stresses in the individual wires of the cable. Furthermore, bending stiffness of the cable is shown to decrease by a factor of 2 to 16, depending upon the friction coefficient between wires and the type of cable construction. Wherever possible, the theoretical results have been compared with experimental results from the available literature and are found in very good agreement with them. A simple method for the determination of the bending stiffness of large diameter multi-layered cable has been developed. The simplified method is further shown to provide estimates of the bending stiffness which are very close to those calculated by the original theory, allowing hand calculations for an easier use in industry. The proposed formulations have been extended to cater for the effects of external hydrostatic pressure on sheathed spiral strands in deep water applications. These forces are shown to have a great influence on the pattern of interwire contact forces and hence the interlayer slippage between the wires in the strand. Numerical results have been obtained and analysed for three different 127 mm diameter strands with lay angles of 12°, 18° and 24° respectively, experiencing a wide range of external hydrostatic pressures of 0 to 2,000 metres. The significant increase in normal contact force between wires is shown to suppress the slippage of wires in the cable. However, the no-slip and full slip values of the effective bending stiffness of the cable is shown to be independent of the level of hydrostatic pressure. A theoretical model is also proposed for estimating wire kinematics, pattern of interwire slippage, contact forces as well as the flexural rigidity of locked coil cables with outer layers made of shaped wires. In order to validate this model, numerical results are reported for two different locked coil cables. It is shown that the shaped wires in the outer layers of locked coil cables play an important role in the distribution of contact forces, slip initiation and cable unwinding.