Thesis (Ph. D.)--University of Oregon, 2006. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 146-147). Also available for download via the World Wide Web; free to University of Oregon users.

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Motivated by various higher dimensional theories in high-energy-physics and cosmology, we consider the local and global isometric embeddings of pseudo-Riemannian manifolds into manifolds of higher dimensions. We provide the necessary background in general relativity, topology and differential geometry, and present the technique for local isometric embeddings. Since an understanding of the local results is key to the development of global embeddings, we review some local existence theorems for general pseudo-Riemannian embedding spaces. In order to gain insight we recapitulate the formalism required to embed static spherically symmetric space-times into fivedimensional Einstein spaces, and explicitly treat some special cases, obtaining local and isometric embeddings for the Reissner-Nordstr¨om space-time, as well as the null geometry of the global monopole metric. We also comment on existence theorems for Euclidean embedding spaces. In a recent result, it is claimed (Katzourakis 2005a) that any analytic n-dimensional space M may be globally embedded into an Einstein space M × F (F an analytic real-valued one-dimensional field). As a corollary, it is claimed that all product spaces are Einsteinian. We demonstrate that this construction for the embedding space is in fact limited to particular types of embedded spaces. We analyze this particular construction for global embeddings into Einstein spaces, uncovering a crucial misunderstanding with regard to the form of the local embedding. We elucidate the impact of this misapprehension on the subsequent proof, and amend the given construction so that it applies to all embedded spaces as well as to embedding spaces of arbitrary curvature. This study is presented as new theorems. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2007.

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Information visualization is a powerful tool for understanding large datasets. However, many commonly-used techniques in information visualization are not C^1 smooth, i.e. when represented as a function, they are either discontinuous or have a discontinuous first derivative. For example, histograms are a non-smooth visualization of density. Not only are histograms non-smooth visually, but they are also non-smooth over their parameter space, as they change abruptly in response to smooth change of bin width or bin offset. For large data visualization, histograms are commonly used in place of smooth alternatives, such as kernel density plots, because histograms can be constructed from data cubes, allowing histograms to be constructed quickly for large datasets. Another example of a non-smooth technique in information visualization is the commonly-used transition approach to animation. Although transitions are designed to create smooth animations, the transition technique produces animations that have velocity discontinuities if the target is changed before the transition has finished. The smooth and efficient zooming and panning technique also shares this problem---the animations produced are smooth while in-flight, but they have velocity discontinuities at the beginning and end and of the animation as well as velocity discontinuities when interrupted. This dissertation applies ideas from signal processing to construct smooth alternatives to these non-smooth techniques. To visualize density for large datasets, we propose BLOCs, a smooth alternative to data cubes that allows kernel density plots to be constructed quickly for large datasets after an initial preprocessing step. To create animations that are smooth even when interrupted, we present LTI animation, a technique that uses LTI filters to create animations that are smooth, even when interrupted. To create zooming and panning animations that are smooth, even when interrupted, we generalize signal processing systems to Riemannian manifolds, resulting in smooth, efficient, and interruptible animations. / Ph. D. / Information visualization is a powerful tool for understanding large datasets. However, many commonly-used techniques in information visualization are not smooth, and this lack of smoothness can be problematic, especially for interactive visualizations of large datasets. This dissertation considers several visualization techniques that are currently non-smooth and introduces smooth alternatives. It is argued that these smooth alternatives are superior to the existing non-smooth techniques.

This thesis is divided into three part, the first part concerns metric spaces and specically length spaces where the existence of shortest path between points is the main focus. In the second part, an example of a length space, the Riemannian geometry will be given. Here both a classical approach to Riemannian geometry will be given together with specic results when considered as a metric space. In the third part, the Finsler geometry will be examined both with a classical approach and trying to deal with it as a metric space.

We introduce Riemannian First-Passage Percolation (Riemannian FPP) as a new model of random differential geometry, by considering a random, smooth Riemannian metric on R^d . We are motivated in our study by the random geometry of first-passage percolation (FPP), a lattice model which was developed to model fluid flow through porous media. By adapting techniques from standard FPP, we prove a shape theorem for our model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one.In differential geometry, geodesics are curves which locally minimize length. They need not do so globally: consider great circles on a sphere. For lattice models of FPP, there are many open questions related to minimizing geodesics; similarly, it is interesting from a geometric perspective when geodesics are globally minimizing. In the present study, we show that for any fixed starting direction v, the geodesic starting from the origin in the direction v is not minimizing with probability one. This is a new result which uses the infinitesimal structure of the continuum, and for which there is no equivalent in discrete lattice models of FPP.

Until the twentieth century, theories of elastic rods and shells arose from collections of geometric and mechanical assumptions and approximations. These theories often lacked internal consistency and were appropriate for highly proscribed and sometimes unknown geometries and deformation sizes. The pioneering work of Truesdell, Antman, and others converted mechanical intuition into rigorous mathematical statements about the physics and mechanics of rods and shells. The result is the modern, geometrically exact theory of finite deformations of rods and shells.In the latter half of the twentieth century, biomechanics became a major focus of both experimental and theoretical mechanics. The genesis of residual stress by non-elastic growth has significant impact on the shape and mechanical properties of soft tissues. Inspired by the geometry of blood vessels and adopting a formalism found in elasto-plasticity, mechanicians have produced rigorous and applied results on the effect of growth on finite elastic deformations of columns and hollow tubes. Less attention has been paid to shells.A theory of growing elastic plates has been constructed in the context of linear elasticity. It harnessed many results in the theory of Riemann surfaces and has produced solutions that are surprisingly similar to experimental observations. Our intention is to provide a finite-deformation alternative by combining growth with the geometrically exact theory of shells. Such a theory has a clearer and more rigorous foundation, and it is applicable to thicker structures than is the case in the current theory of growing plates.This work presents the basic mathematical tools required to construct this alternative theory of finite elasticity of a shell in the presence of growth. We make clear that classical elasticity can be viewed in terms of three-dimensional Riemannian geometry, and that finite elasticity in the presence of growth must be considered in this way. We present several examples that demonstrate the viability and tractability of this approach.

This thesis suggests an approach to compute the short-time behaviour of the hypoelliptic heat kernel corresponding to sub-Riemannian structures on unimodular Lie groups of type I, without previously holding a closed form expression for this heat kernel. Our work relies on the use of classical non-commutative harmonic analysis tools, namely the Generalized Fourier Transform and its inverse, combined with the Trotter product formula from the theory of perturbation of semigroups. We illustrate our main results by computing, to our knowledge, a first expression in short-time for the hypoelliptic heat kernel on the Engel and the Cartan groups, for which there exist no closed form expression. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2011-10-08 01:32:32.896

Brain-computer interfaces are an emerging technology that could provide channels for communication and control to severely disabled people suffering from locked-in syndrome. It has been found that motor imagery can be detected and classified from EEG signals. The motivation of the present work was to compare several algorithms for motor imagery classification in EEG signals as well as to test several novel algorithms. The algorithms tested included the popular method of common spatial patterns (CSP) spatial filtering followed by linear discriminant analysis (LDA) classification of log-variance features (CSP+LDA). A second set of algorithms used classification based on concepts from Riemannian geometry. The basic idea of these methods is that sample spatial covariance matrices (SCMs) of EEG epochs belong to the Riemannian manifold of symmetric positive-definite (SPD) matrices and that the tangent space at any SPD matrix on the manifold is a finite-dimensional Euclidean space. Riemannian classification methods tested included minimum distance to Riemannian mean (MDRM), tangent space LDA (TSLDA), and Fisher geodesic filtering followed by MDRM classification (FGDA). The novel algorithms aimed to combine the CSP method with the Riemannian geometry methods. CSP spatial filtering was performed prior to sample SCM calculation and subsequent classification using Riemannian methods. The novel algorithms were found to improve classification accuracy as well as reduce the computational costs of Riemannian classification methods for binary, synchronous classification on BCI competition IV dataset 2a. / Master of Science