O presente estudo teve como objetivo avaliar a resistência à fadiga cíclica flexural dos instrumentos de níquel- titânio, Hyflex CM (Coltène, EUA) e TF Adaptive (SybronEndo, EUA) em diferentes situações experimentais. Todas as limas que foram selecionadas possuíam conicidade 0,04 e diâmetro de ponta 35. Utilizou-se um dispositivo desenvolvido especificamente para realizar o ensaio flexural dinâmico. Os instrumentos TF Adaptive foram divididos em 3 grupos de acordo com o ângulo de curvatura do ensaio: 45º, 60º e 90º e cada grupo subdividido em 2 subgrupos de acordo com o tipo de movimento: rotação contínua e Adaptive. Cada subgrupo era composto por 15 instrumentos TF Adaptive, totalizando 90 instrumentos. Quinze instrumentos Hyflex CM formavam o grupo 4, no ensaio com ângulo de curvatura 90 graus e rotação contínua. A simulação foi realizada em canais artificiais de aço com ângulo de 45, 60, 90 graus e raio 5m m. O número de ciclos e o tempo em segundos até a fratura foram tabulados e analisados. Entretanto, a fadiga cíclica flexural foi significante maior nos três grupos em movimento Adaptive. E as limas TF Adaptive em seu próprio movimento tiveram maior número de ciclos e tempo até a fratura quando comparadas as Hyflex CM no ensaio de 90 graus. Portanto, conclui-se que o sistema Adaptive (limas TF Adaptive + movimento Adaptive) foi mais seguro à resistência á fadiga flexural, e no ensaio de 90 graus o sistema Adaptive foi mais resistente quando comparado com as limas Hyflex CM no movimento de rotação contínua. / The aim of this work to evaluate the cyclic fatigue resistance flexural the instruments of nickel-titanium, HyFlex CM (Coltène, USA) and TF Adaptive (SybronEndo, USA) in different experimental situations. All files that were selected had 0.04 taper and tip diameter 35. We used a device developed specifically to perform the dynamic flexural test. The TF Adaptive instruments were divided into 3 groups according to the angle of curvature of the test: 45, 60 and 90 and further divided in two subgroups according to the type of movement: continuous rotation and Adaptive. Each group consisted of 15 instruments TF Adaptive totaling 90 instruments. Fifteen HyFlex CM instruments formed the group 4 in the trial of bend angle 90 degrees and continuous rotation. The simulation was performed in artificial steel angled channels 45, 60, 90 degrees and radius 5m m. The number of cycles and the time in seconds until fracture were tabulated and analyzed. However, the cyclical flexural fatigue was significantly greater in the three groups Adaptive motion. And the TF Adaptive files on your own movement had a higher number of cycles and time to fracture when the HyFlex CM compared in 90-degree test. Therefore, it is concluded that the adaptive system (TF Adaptive Motion + Adaptive files) was safer resistance to flexural fatigue, and 90 degrees test Adaptive system is more resistant when compared with the HyFlex files into continuous.

Neste trabalho, estabelecemos uma equação que relaciona a curvatura de Ricci de uma variedade riemanniana M e as segundas formas fundamentais de duas folheações ortogonais de dimensões complementares, F e F, definidas em M. Usando essa equação, encontramos uma estimativa da curvatura média da folheação F e uma condição necessária e suficiente para que tal folheação seja totalmente geodésica. Mostramos também uma condição suficiente para que M seja localmente um produto riemanniano das folhas de F e F, se uma das folheações for totalmente umbílica. Por fim, provamos ainda uma fórmula integral válida para tais folheações. / In this work, we and an equation that relates the Ricci curvature of a riemannian manifold M and the second fundamental forms of two orthogonal foliations of complementary dimensions, F and F, defined on M. Using this equation, we and an estimate of the mean curvature of the foliation F and a necessary and suficient condition for the foliation F to be totally geodesic. We also show a suficient condition for the manifold M to be locally a riemannian product of the leaves of F and F, if one of the foliations is totally umbilical. Finally, we also prove an integral formula for such foliations.

Orientador: Cláudia Valéria Seullner Brandão / Resumo: Os gatos possuem papel de destaque como animais de estimação, assim como em estudos oftalmológicos comparativos. Pelo conhecimento dos autores, até o momento, a avaliação de bulbo de gatos utilizando tomógrafos e topógrafo associados, por meio dos sistemas de câmeras Scheimpflug duplas e um disco Plácido, não foi descrita na literatura. O objetivo deste estudo foi analisar a córnea em gatos considerando as curvaturas anterior e posterior, o poder total e espessura da córnea pelo sistema de duplo Scheimpflug e disco de plácido. Na ceratometria anterior (SimK) verificou-se valores de 38,37 ± 0,83D, ceratometria posterior -4,89±0,34D e poder corneal total médio 38,04 ± 0,94D. Quanto à espessura corneal central foi obtido valor médio de 617,34 ± 53,38 μm, superior aos encontrados em outros trabalhos utilizando a paquimetria ultrassônica. A utilização do sistema duplo Scheimpflug e disco de plácido foi efetiva em gatos domésticos e permitiu a obtenção de valores ainda não descritos na literatura para a espécie, tais como ceratometria anterior nas zonas da córnea, ceratometria corneal posterior, poder corneal total, volume corneal. / Abstract: Cats have an important role as pets and in ophthalmologic comparative studies. To the autors’ knowledge, this is the first study that evaluates cornea of cats using a device that combine tomography and topography, through dual Scheimpflug cameras and a Placid disc. The aim of this study was to evaluate anterior and posterior keratometry, total corneal power and pachymetry using dual Scheimpflug and Placid disc system. Concerning anterior keratometry (SimK), the value obtained was 38,37 ± 0,83D, posterior keratometry -4,89±0,34D and total corneal power 38,04 ± 0,94D. Regarding the pachymetry, the mean CCT value obtained was 617,34 ± 53,38 μm, superior to those find in other studies using ultrasonic pachymetry. The use of dual Scheimpflug and Placid disc system was effective in domestic cats and enabled the description of values that have not been described in the literature in felines yet, such as anterior keratomemetry in the corneal zones, posterior keratommetry, total corneal power, corneal volume. / Mestre

Freeform curve design has existed in various forms for at least two millennia, and is important throughout computer-aided design and manufacture. With the increasing importance of animation and robotics, coupled with the increasing power of computers, there is now interest in freeform motion design, which, in part, extends techniques from curve design, as well as introducing some entirely distinct challenges. There are several approaches to freeform motion construction, and the first step in designing freeform motions is to choose a representation. Unlike for curves, there is no "standard" way of representing freeform motions, and the different tools available each have different properties. A motion can be viewed as a continuously-varying pose, where a pose is a position and an orientation. This immediately presents a problem; the dimensions of rotations and translations are different, and it is not clear how the two can be compared, such as to define distance along a motion. One solution is to treat the rotational and translational components of a motion separately, but this is inelegant and clumsy. The philosophy of this thesis is that a motion is not defined purely by rotations and translations, but that the body following a motion is a part of that motion. Specifically, the part of the body that is accounted for is its inertia tensor. The significance of the inertia tensor is that it allows the rotational and translational parts of a motion to be, in some sense, compared in a dimensionally- consistent way. Using the inertia tensor, this thesis finds the form of kinetic energy in <;1'4, and also discusses extensions of the concepts of arc length and curvature to the space of motions, allowing techniques from curve fairing to be applied to motion fairing. Two measures of motion fairness are constructed, and motion fairing is the process of minimizing the measure of a motion by adjusting degrees of freedom present in the motion's construction. This thesis uses the geometric algebra <;1'4 in the generation offreeform motions, and the fairing of such motions. <;1'4 is chosen for its particular elegance in representing rigid-body transforms, coupled with an equivalence relation between elements representing transforms more general than for ordinary homogeneous coordinates. The properties of the algebra germane to freeform motion design and improvement are given, and two distinct frameworks for freeform motion construction and modification are studied in detail.

In this paper we prove structure results for the singular sets of stationary harmonic maps and mean curvature flows local to particular singularities. The original work is contained in Chapter 5 and Chapter 8. Chapters 1-5 are concerned with energy minimising maps and stationary harmonic maps. Chapters 6-8 are concerned with mean curvature flows and Brakke flows. In the case of stationary harmonic maps we consider a singularity at which the spine dimension is maximal, and such that the weak tangent map is homotopically non-trivial, and has minimal density amongst singularities of maximal spine dimen- sion. Local to such a singularity we show the singular set is a bi-Hölder continuous homeomorphism of the unit disk of dimension equal to the maximal spine dimension. A weak tangent map is translation invariant along a subspace, and invariant under dilations, so it completely defined by its values on a sphere. Such a map is said to be homotopically non-trivial if the mapping of a sphere into some target manifold cannot be deformed by a homotopy to a constant map. For an n-dimensional mean curvature flow we consider a singularity at which we can find a shrinking cylinder as a tangent flow, that collapses on an (n−1)-dimensional plane. Local to such a singularity we show that all singularities have such a cylindrical tangent, or else have lower Gaussian density than that of the shrinking cylinder. The subset of cylindrical singularities can be shown to be contained in a finite union of parabolic (n − 1)-dimensional Lipschitz submanifolds. In the case that the mean curvature flow arises from elliptic regularisation we can show that all singularities local to a cylindrical singularity with (n − 1)-dimensional spine are either cylindrical singularities with (n − 1)-dimensional spine, or contained in a parabolic Hausdorff (n − 2)-dimensional set.

In this thesis, we study the overfitting problem in supervised learning of classifiers from a geometric perspective. As with many inverse problems, learning a classification function from a given set of example-label pairs is an ill-posed problem, i.e., there exist infinitely many classification functions that can correctly predict the class labels for all training examples. Among them, according to Occam's razor, simpler functions are favored since they are less overfitted to training examples and are therefore expected to perform better on unseen examples. The standard technique to enforce Occam's razor is to introduce a regularization scheme, which penalizes some type of complexity of the learned classification function. Some widely used regularization techniques are functional norm-based (Tikhonov) techniques, ensemble-based techniques, early stopping techniques, etc. However, there is important geometric information in the learned classification function that is closely related to overfitting, and has been overlooked by previous methods. In this thesis, we study the complexity of a classification function from a new geometric perspective. In particular, we investigate the differential geometric structure in the submanifold corresponding to the estimator of the class probability P(y|x), based on the observation that overfitting produces rapid local oscillations and hence large mean curvature of this submanifold. We also show that our geometric perspective of supervised learning is naturally related to an elastic model in physics, where our complexity measure is a high dimensional extension of the surface energy in physics. This study leads to a new geometric regularization approach for supervised learning of classifiers. In our approach, the learning process can be viewed as a submanifold fitting problem that is solved by a mean curvature flow method. In particular, our approach finds the submanifold by iteratively fitting the training examples in a curvature or volume decreasing manner. Our technique is unified for both binary and multiclass classification, and can be applied to regularize any classification function that satisfies two requirements: firstly, an estimator of the class probability can be obtained; secondly, first and second derivatives of the class probability estimator can be calculated. For applications, where we apply our regularization technique to standard loss functions for classification, our RBF-based implementation compares favorably to widely used regularization methods for both binary and multiclass classification. We also design a specific algorithm to incorporate our regularization technique into the standard forward-backward training of deep neural networks. For theoretical analysis, we establish Bayes consistency for a specific loss function under some mild initialization assumptions. We also discuss the extension of our approach to situations where the input space is a submanifold, rather than a Euclidean space. / 2018-11-30T00:00:00Z

Neste trabalho apresentamos condições suficientes para a existência de gráficos de curvatura média constante (CMC) com bordo em dois planos paralelos. Também são feitas estimativas para a altura de superfícies CMC com vetor normal orientado para fora limitadas por um cilindro ou horocilindro. / In this work we present su cient existence conditions for constant mean curvature (CMC) graphs with boundary in two parallel planes. We also make height estimates for outwards-oriented CMC surfaces bounded by a cylinder or horocylinder.

La motivation initiale de cette thèse est l'étude d'une discrétisation volumique de surface (Chapitre 2) naturellement liée à la structure de varifold. Le point clé est qu'il est possible de munir d'une structure de varifold la plupart des objets utilisés pour représenter ou discrétiser des surfaces c'est-à-dire aussi bien des objets tels que les sous variétés ou les ensembles rectifiables que des objets tels que des nuages de points ou encore la discrétisation volumique proposée, ce qui permet d'étudier dans un cadre unifié une surface et sa discrétisation. Une difficulté essentielle est que, généralement, ces structures discrètes ne sont pas rectifiables, ce qui soulève la question suivante : comment assurer qu'un varifold, obtenu comme limite de discrétisations volumiques, soit une surface, au moins en un sens faible ? De façon plus précise : quelles conditions sur une suite de varifolds quelconques assurent que le varifold limite est rectifiable (Chapitre 3) ou encore qu'il est à variation première bornée (Chapitre 5) ? On obtient des conditions quantitatives assurant la rectifiabilité grâce à des énergies liées aux nombres beta de Jones. On s'intéresse ensuite à la régularité du varifold limite en termes de courbure (variation première). On a essayé de contrôler la variation première en utilisant des techniques de construction de mesures de type packing (Chapitre 4), une forme régularisée de la variation première d'un varifold. Cette régularisation permet de définir des énergies de Willmore approchées qui Gamma convergent dans l'espace des varifolds vers l'énergie de Willmore ainsi qu'une approximation de la courbure qui est testée numériquement dans le Chapitre 6 / The starting point of this work is the study of a volumetric surface discretization model naturally connected to the varifolds structure introduced in Chapter 2. The point is that not only the discretization we propose can be endowed with a structure of varifold but also a great part of objects used for surface representation and discretization (triangulation, cloud points, level sets etc.) so that we can use varifolds tools to study in some unified setting different ways of discretizing surfaces. An important point to overcome is that these structures are generally not rectifiable so that we address the following question: how to ensure that the limit of a sequence of such discrete surfaces is regular? More precisely, what conditions on a sequence of varifolds (not necessarily rectifiable nor with bounded variation) ensure that the limit varifold is rectifiable (Chapter 3) or has bounded first variation (Chapter 5)? We obtain quantitative conditions of rectifiability for variflods considering energies linked to Jones' beta numbers. We then address the question in terms of first variation (generalized curvature) of a limit varifold. We first try a packing measure construction of the first variation of a varifold V (Chapter 4), then we define a regularized form of the classical first variation, allowing us to exhibit an energetic condition ensuring that a limit of a sequence of varifolds has bounded first variation. We use this regularized form to build an approximate Willmore energy Gamma-converging in the class of varifolds to the Willmore energy. In Chapter 6, we test numerically a notion of approximate curvature derived from the regularized first variation

Seja K um grupo de Lie compacto agindo na esfera unitária S&#8319 por isometrias. Mostramos como uma cota superior para as curvaturas principais de uma órbita dessa ação pode ser usada (mas não é suficiente) para encontrar uma cota inferior para o diâmetro do espaço de órbitas S&#8319/K. Em seguida mostramos que existe uma órbita Kp com curvaturas principais majoradas por 4&#8730 14. / Let K be a compact Lie group acting on the unit sphere S&#8319 by isometries. We show how an upper bound on the principal curvatures of one orbit can be used (but is not sufficient) to obtain a lower bound for the diameter of the orbit space S&#8319/K. Then we show that there is an orbit Kp with principal curvatures bounded from above by 4&#8730 14.

A geometria de teias dedica-se ao estudo de invariantes locais para uma determinada configuração de folheações. Uma d-teia é uma coleção de folheações que estão em posição geral. Desta forma, uma d-teia plana, definida em \'R POT.2\' ou \'C POT.2\', nada mais é que uma família de d folheações por curvas. Apresentamos neste trabalho os principais conceitos da teoria clássica de teias, iniciada por W. Blaschke por volta de 1930, bem como uma abordagem atual utilizada no estudo de teias planas. São abordados dois tipos de problemas importantes na teoria: os problemas de linearização e de algebrização de teias. Provamos um resultado clássico no que concerne ao problema de linearização, e um resultado de algebrização de teias empregando métodos desenvolvidos mais recentemente / Web geometry is devoted to the study of local invariants of a certain configuration of foliations. A d-web is a collection of foliations in general position. Therefore, a d-web defined in \'R POT. 2\' or \'C POT. 2\' is just a family of d foliations by curves. We present in this work the main concepts of classical theory of webs, initiated by W. Blaschke around 1930, as well as newer methods used in the study of plane webs. We approach two important types of problems in the theory: problems of linearization and that of algebrization of webs. We prove a classical result concerning the linearization problem, and a result of algebrization of webs using recently developed methods