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>

The key of presented diploma thesis is to find a way to take shape of roads into account when performing network analysis. Methods of mathematical representation of shape of lines are discussed; the curvature is chose for later use. Then the relation between road's curvature and speed is discussed - the model for setting the speed is suggested. Important part of the thesis is focused on implementation in GIS. Designed model is implemented by Python in the ArcGIS for Desktop. Due to testing on specific examples, it was confirmed, that the taking the shape characteristics into account provides high précising of network analysis.

We adress an obstacle problem for (graphical) mean curvature flow with Dirichlet boundary conditions. Using (an adapted form of) the standard implicit time-discretization scheme we derive the existence of distributional solutions satisfying an appropriate variational inequality. Uniqueness of this flow and asymptotic convergence towards the stationary solution is proven.

A teoria das transforma»ções de superfícies de curvatura constante começou, no fim do século XIX, com o trabalho [3] de A.V. Bäcklund e, em seguida, recebeu importantes contribuições por parte de diversos geômetras, entre eles, L. Bianchi e C. Guichard (veja, por exemplo, [5, 6, 7, 17]). Nessa dissertação apresentamos alguns dos mais importantes resultados desse tópico da geometria diferencial que estão relacionados às superfícies de curvatura média (ou gaussiana não nula) constante. Tais superfícies estão associadas a soluções de equações diferenciais parciais de segunda ordem e não lineares. A interpretação analítica da teoria das transformações de superfícies de curvatura constante nos capacita obter soluções dessas equações diferenciais parciais a partir de uma outra dada, mediante integração de um sistema de equações diferenciais, chamado transformação de Bäcklund. Então, os teoremas de permutabilidade fornecem uma \"fórmula de superposição\" para a construção algébrica de novas soluções / The theory on transformations of surfaces with constant curvature begins, in the late nineteen century, with the article [3] of A.V. Bäcklund and, after, received important contributions from various geometricians, among others, L. Bianchi and C. Guichard (see, for example, [5, 6, 7, 17]). In this dissertation we outline some of the most important results on the theory of surfaces of constant mean (or gaussian) curvature. Such surfaces are associated to the solutions of nonlinear partial differential equations of second order. The analytic interpretation of the theory on transformations of constant curvature surfaces provides a method of obtaining, from a given solution of these partial differential equations, a new solution of the same equation, by integrating a system of differential equations, called Bäcklund transformation. Then, the permutability theorems give a \"superposition formula\" to construct, algebraically, new solutions

O objetivo deste trabalho e desenvolver um instrumento computadorizado para análise da superfície anterior da córnea humana, gerando para isto, um mapa topográfico bidimensional em código de cores. Utilizando a córnea como um espelho esférico convexo, projeta-se anéis luminosos sobre sua superfície e aquisiciona-se com uma câmera a imagem por ela refletida. Esta imagem e digitalizada e armazenada em um microcomputador para posterior processamento. Através de técnicas de computação gráfica e processamento de imagens digitais, extrai-se da imagem as informações necessárias para construção da topografia desejada. Por fim, a topografia e apresentada em forma de mapas coloridos, sendo cada cor associada a uma determinada dioptria, transmitindo ao médico oftalmologista uma noção exata da superfície da córnea do paciente em análise. / The main goal of this work is to develop a computerized instrument for the analysis of the anterior portion of the human cornea, which displays its result in topographic color maps. Approximating the cornea to a spherical convex mirror, and by projecting a known pattern over it, the reflected image is captured and stored. By means of computer graphics technology and image processing, the necessary information for mathematical calculations is extracted. The resulting maps are color-coded in accordance to the degree of power of each corneal region, that is, to the diopter value. The ophthalmologist can then make important diagnostics and surgery tactics from the analysis of these topographic maps.

Wo, Weifeng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 99-102). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

by Fan Xuqian. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 99-100). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- The Theorem of Efimov --- p.7 / Chapter 2.1 --- The Idea of the Proof of the Efimov's Theorem --- p.8 / Chapter 2.2 --- Proof of the Efimov's Main Lemma --- p.12 / Chapter 2.3 --- Proof of Lemma 2.3 --- p.48 / Chapter 2.4 --- Proof of Lemma 2.4 --- p.52 / Chapter 3 --- Isometric Immersion into R3 of Complete Surfaces with Negative Curvature --- p.62 / Chapter 3.1 --- The Sketch of the Proof of Theorem 3.1 --- p.66 / Chapter 3.2 --- Proof of Lemma 3.4 --- p.75 / Chapter 3.3 --- Proof of Lemma 3.5 --- p.76 / Chapter 3.4 --- Proof of Lemma 3.6 --- p.86 / Chapter 3.5 --- Proof of Lemma 3.7 --- p.89 / Chapter 3.6 --- The Geometric Properties of the Immersion --- p.95