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The computation of curvature quantities over discrete geometry is often required when processing geometry composed of meshes. Curvature information is often important for the purpose of shape analysis, feature recognition and geometry segmentation. In this paper we present a method for accurate estimation of curvature on discrete geometry especially those composed of meshes. We utilise a method based on fitting a continuous surface arising from the solution of the Biharmonic equation subject to suitable boundary conditions over a 1-ring neighbourhood of the mesh geometry model. This enables us to accurately determine the curvature distribution of the local area. We show how the curvature can be computed efficiently by means of utilising an analytic solution representation of the chosen Biharmonic equation. In order to demonstrate the method we present a series of examples whereby we show how the curvature can be efficiently computed over complex geometry which are represented discretely by means of mesh models.

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PEREIRA, José Ilhano da Silva. Hipersuperfícies mínimas de R4 com curvatura de Gauss-Kronecker nula. 2017. 44 f. Dissertação (Mestrado em Matemática) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-10-02T15:01:31Z No. of bitstreams: 1 2017_dis_jispereira.pdf: 596580 bytes, checksum: 3c2c1a16d4ce273bfb7c246f7926c01a (MD5) / Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, Estou devolvendo a Dissertação de JOSÉ ILHANO DA SILVA PEREIRA, pois há alguns erros a serem corrigidos. Os mesmos seguem listados a seguir. 1- FOLHA DE APROVAÇÃO (substitua a folha de aprovação, por outra que não contenha as assinaturas dos membros da banca examinadora) 2- NUMERAÇÃO INDEVIDA (a numeração indevida de página que aparece na folha de aprovação deve ser retirada) 3- RESUMO (retire o recuo de parágrafo presente no resumo e no abstract) 4- PALAVRAS-CHAVE (apenas o primeiro elemento de cada palavra-chave deve começar com letra maiúscula, assim reescreva as palavras-chave como no exemplo a seguir: Hipersuperfícies mínimas) 5- SUMÁRIO (Os títulos dos capítulos principais, que aparecem no sumário e no interior do trabalho, devem estar em caixa alta (letra maiúscula). Ex.: 2 PRELIMINARES 2.1 Tensores 6 – REFERÊNCIAS (retire o conjunto de “citações” à autores que aparece no final das referências bibliográficas, pois elas fogem ao padrão ABNT para a página das referências) Atenciosamente, on 2017-10-04T17:50:58Z (GMT) / Submitted by Andrea Dantas (pgmat@mat.ufc.br) on 2017-10-23T19:57:28Z No. of bitstreams: 1 2017_dis_jispereira.pdf: 333124 bytes, checksum: 37989a2f3787d5914a0c0553afd4e89f (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2017-11-01T12:35:13Z (GMT) No. of bitstreams: 1 2017_dis_jispereira.pdf: 333124 bytes, checksum: 37989a2f3787d5914a0c0553afd4e89f (MD5) / Made available in DSpace on 2017-11-01T12:35:13Z (GMT). No. of bitstreams: 1 2017_dis_jispereira.pdf: 333124 bytes, checksum: 37989a2f3787d5914a0c0553afd4e89f (MD5) Previous issue date: 2017-08-25 / This work does study the complete minimal hypersurfaces in the Euclidean space R4 , with Gauss-Kronecker curvature identically zero. Our main result is to prove that if f: M3 → R4 is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature boun-ded from below, then f(M3) splits as a Euclidean product L2 × R , where L2 is a complete minimal surface in R3 with Gaussian curvature bounded from below. Moreover, we show a result about the Gauss-Kronecker curvature of f, without any assumption on the scalar curvature. / Este trabalho tem como objetivo estudar as hipersuperfícies mínimas em R4, com curvatura de Gauss-Kronecker identicamente zero. Como resultado principal provamos que se f : M3 → R4 é uma hipersuperfície mínima com curvatura de Gauss-Kronecker identicamente zero, segunda forma fundamental não se anulando em nenhum ponto e curvatura escalar limitada inferiormente, então f(M3) se decompõe como um produto euclidiano do tipo L2 × R , onde L2 é uma superfície mínima de R3 com curvatura Gaussiana limitada inferiormente. Finalmente, apresentamos um resultado sobre a curvatura de Gauss-Kronecker de f sem nenhuma hipótese sobre a curvatura escalar.

[pt] A geometria diferencial provê um conjunto de medidas invariantes sob a ação de um grupo de transformações, em particular rígidas, afins e projetivas. Os invariantes por transformações rígidas são usados em quase todas as aplicações de computação gráfica e modelagem geométrica. O caso afim, por ser mais geral, permite estender essas ferramentas. Neste trabalho, propriedades geométricas são apresentadas no caso de superfícies paramétricas ou implícitas, em particular, a métrica afim, os vetores co-normal e normal afins e as curvaturas Gaussiana e média afins. Alguns resultados usuais de geometria Euclidiana, como a fórmula de Minkowski, são estendidos para o caso afim. Esse estudo permite definir estimadores das estruturas afins no caso de isossuperfícies. Porém, um cálculo direto dessas estruturas resulta em um grande número de operações e instabilidade numérica. Uma redução geométrica é proposta, obtendo fórmulas mais simples e mais estáveis numericamente. As propriedades geométricas incorporadas no Marching Cubes são analisadas e discutidas. / [en] Differential Geometry provides a set of measures invariant under a set of transformations, in particular rigid, affine, and projective. The invariants by rigid motions are using almost all applications of computer graphics and geometric modeling. The affine case, since it is more general, allows to extend these tools. In this work, geometric properties are presented in the case of parametric or implicit surfaces, in particular the affine metric, the conormal and normal vectors, and the affine Gaussian and mean curvatures. Some usual results of Euclidean geometry, as the Minkowski formula, are extended for the affine case. This study allows to define estimators of affines structure in the case of isosurfaces. Although, the direct calculation of these structures greatly increases the number of operations and numerical instabilities. A geometrical reduction is proposed obtaining a much simpler and numerical stabler formulae. The geometrical properties are incorporated in the Marching Cubes algorithms, then they are analyzed and discussed.

<p>The curvatures in Finsler geometry can be defined in similar ways as in Riemannian geometry. However, since there are fewer restrictions on the metrics, many geometric quantities arise in Finsler geometry which vanish in the Riemannian case. These quantities are generally known as non-Riemannian quantities and interact with the curvatures in controlling the global geometrical and topological properties of Finsler manifolds. In the present work, we study general weighted Ricci curvatures which combine the Ricci curvature and the S-curvature, and define a weighted flag curvature which combines the flag curvature and the T -curvature. We characterize Randers metrics of almost isotropic weighted Ricci curvatures and show the general weighted Ricci curvatures can be divided into three types. On the other hand, we show that a proper open forward complete Finsler manifold with positive weighted flag curvature is necessarily diffeomorphic to the Euclidean space, generalizing the Gromoll-Meyer theorem in Riemannian geometry.</p>

Any Riemannian manifold possesses a minimal solution of the heat equation for the Dirichlet Laplacian, called the heat kernel. During the last decades many authors investigated geometric properties of the manifold such that its heat kernel fulfills a so-called Gaussian upper bound. Especially compact and non-compact manifolds with lower bounded Ricci curvature have been examined and provide such Gaussian estimates. In the compact case it ended even with integral Ricci curvature assumptions. The important techniques to obtain Gaussian bounds are the symmetrization procedure for compact manifolds and relative Faber-Krahn estimates or gradient estimates for the heat equation, where the first two base on isoperimetric properties of certain sets. In this thesis, we generalize the existing results to the following. Locally uniform integral bounds on the negative part of Ricci curvature lead to Gaussian upper bounds for the heat kernel, no matter whether the manifold is compact or not. Therefore, we show local isoperimetric inequalities under this condition and use relative Faber-Krahn estimates to derive explicit Gaussian upper bounds. If the manifold is compact, we can even generalize the integral curvature condition to the case that the negative part of Ricci curvature is in the so-called Kato class. We even obtain uniform Gaussian upper bounds using gradient estimate techniques. Apart from the geometric generalizations for obtaining Gaussian upper bounds we use those estimates to generalize Bochner’s theorem. More precisely, the estimates for the heat kernel obtained above lead to ultracontractive estimates for the heat semigroup and the semigroup generated by the Hodge Laplacian. In turn, we can formulate rigidity results for the triviality of the first cohomology group if the amount of curvature going below a certain positive threshold is small in a suitable sense. If we can only assume such smallness of the negative part of the Ricci curvature, we can bound the Betti number by explicit terms depending on the generalized curvature assumptions in a uniform manner, generalizing certain existing results from the cited literature. / Jede Riemannsche Mannigfaltigkeit besitzt eine minimale Lösung für die Wärmeleitungsgleichung des zur Mannigfaltigkeit gehörigen Dirichlet-Laplaceoperators, den Wärmeleitungskern. Während der letzten Jahrzehnte fanden viele Autoren geometrische Eigenschaften der Mannigfaltigkeiten unter welchen der Wärmeleitungskern eine sogenannte Gaußsche obere Abschätzung besitzt. Insbesondere bestizen sowohl kompakte als auch nichtkompakte Mannigfaltigkeiten mit nach unten beschränkter Ricci-Krümmung solche Gaußschen Abschätzungen. Im kompakten Fall reichten bisher sogar Integralbedingungen an die Ricci-Krümmung aus. Die wichtigen Techniken, um Gaußsche Abschätzungen zu erhalten, sind die Symmetrisierung für kompakte Mannigfaltigkeiten und relative Faber-Krahn- und Gradientenabschätzungen für die Wärmeleitungsgleichung, wobei die ersten beiden auf isoperimetrischen Eigenschaften gewisser Mengen beruhen. In dieser Arbeit verallgemeinern wir die bestehenden Resultate im folgenden Sinne. Lokal gleichmäßig beschränkte Integralschranken an den Negativteil der Ricci-Krümmung ergeben Gaußsche obere Abschätzungen sowohl im kompakten als auch nichtkompakten Fall. Dafür zeigen wir lokale isoperimetrische Ungleichungen unter dieser Voraussetzung und nutzen die relativen Faber-Krahn-Abschätzungen für eine explizite Gaußsche Schranke. Für kompakte Mannigfaltigkeiten können wir sogar die Integralschranken an den Negativteil der Ricci-Krümmung durch die sogenannte Kato-Bedingung ersetzen. In diesem Fall erhalten wir gleichmäßige Gaußsche Abschätzungen mit einer Gradientenabschätzung. Neben den geometrischen Verallgemeinerungen für Gaußsche Schranken nutzen wir unsere Ergebnisse, um Bochners Theorem zu verallgemeinern. Wärmeleitungskernabschätzungen ergeben ultrakontraktive Schranken für die Wärmeleitungshalbgruppe und die Halbgruppe, die durch den Hodge-Operator erzeugt wird. Damit können wir Starrheitseigenschaften für die erste Kohomologiegruppe zeigen, wenn der Teil der Ricci-Krümmung, welcher unter einem positiven Level liegt, in einem bestimmten Sinne klein genug ist. Wenn der Negativteil der Ricci-Krümmung nicht zu groß ist, können wir die erste Betti-Zahl noch immer explizit uniform abschätzen.

Spintronics is the study of mutual dependence of magnetization and electron transport, which forms a complementary picture in ferromagnetic (FM) materials. Recently, spintronics based on antiferromagnetic (AF) materials has been suggested. However, a systematic study is not yet available, and a complementary picture of the AF dynamics with electron transport is highly desired. By developing a microscopic theory, we predict the occurrence of spintronic phenomena both in bulk AF texture and on the interface of AF with normal metals. For the bulk, we find that the electron dynamics becomes adiabatic when the local staggered field is varying slowly over space and time, by which the spin-motive force and the reactive spin-transfer torque (STT) are derived as reciprocal effects. While the former generates a pure spin voltage across the texture, the latter can be used to drive AF domain wall and trigger spin wave excitation with lower current densities compared to FM materials. For the interface, by calculating how electrons scatter off a normal metal -antiferromagnet heterostructure, we derive the pumped spin and staggered spin currents in terms of the staggered order parameter, the magnetization, and their rates of change; the reactions of an incident spin current on the antiferromagnet is derived as STTs. These effects are applicable to both compensated and uncompensated interfaces with a similar order of magnitude. In contrast to FM materials, the direction of spin pumping is controlled by the circular polarization of driving microwave; and conversely, the chirality of AF spin wave is tunable by the direction of spin accumulation. / text

Our tactile perceptual experiences occur when we interact, actively and passively, with environmental objects and surfaces. Previous research has demonstrated that active manual exploration enhances the tactile perception of object shape. Nevertheless, the factors that contribute to this enhancement are not well understood. The present study evaluated the ability of 14 older adults to discriminate curved surfaces by actively feeling objects with a single index finger and by passively feeling objects that moved relative to a restrained finger. The curvature discrimination thresholds obtained for passive-dynamic touch were significantly lower than those that occurred during active-dynamic touch. This result demonstrates that active exploratory movements of the hand and fingers do not necessarily lead to the best curvature discrimination performance; rather, performance was best in the current study when dynamic tactile stimulation occurred in the absence of active movement. The results of the present study also clarify those obtained by Norman et al. (2013), who found that active-dynamic touch was superior to static touch -- the current findings extend this previous research and indicate that passive-dynamic touch can yield performance that is even higher than what is obtained for active-dynamic touch.

<p>An explicit algebraic Reynolds stress model (EARSM) based ona pressure strain rate model including terms tensoriallynonlinear in the mean velocity gradients is developed in orderto improve predictions for .ows with strong curvature and/orrotation. This work has been carried out in the context of acollaborative international project on high-lift aerodynamics.For 2D mean .ows the nonlinear terms can easily be accountedfor in the model formulation. This is not the case for 3D mean.ows and approximations making the 2D and 3D mean .owformulations consistent are suggested. The proposed EARSM, theparent-EARSM and the corresponding di.erential Reynolds stressmodels (DRSM) are tested for spanwise rotating channel .ow andaxially rotating pipe .ow. The model predictions are comparedto experimental and DNS data. The nonlinear extensions areshown to have a signi.cant e.ect on the .ow predictions,somewhat less pronounced for the DRSM though. The turbulentdi.usion modelling in the EARSM computations is important forthe rotating pipe. It is shown that by using a Daly and Harlowdi.usion model, turbulence levels in good agreement withexperiments and DRSM can be achieved. However, by using asimpler e.ective eddy viscosity based di.usion model theturbulence kinetic energy levels are drastically overpredicted.Finally the proposed EARSM is tested on a standard high-liftcon.guration. The EARSM predictions are compared withexperiments and the predictions made by the standard K - ωtwo-equation model.</p><p><b>Descriptors:</b>Turbulence model, nonlinear modelling,streamline curvature, high-lift aerodynamics.</p>