A modern and straight forward summary of the necessary tools andconcepts needed to understand and work with gauge theory in a fibre bundle formalism. Due to the aim of being a quick but thorough introductionfull derivations are rarely included, but references to such are given wherethey have been omitted. General Relativity, although being a geometrical theory, in the sense that the gravitational force is described by the curvature of space-time, may not be derived from geometry like the other fundamental forces as in Yang-Mills theory. Thus, a possibility of unification lies in a geometrical derivation of gravity from gauge principles. By applying the presented formalism to the case of Gravity such a derivationis pursued along the lines of nonlinear realizations of the gauge group.

<p> Both the theories of differential geometry and of Lie groups and their algebras have been invaluable to the physicist. In the theory of fibre bundles and in the symplectic formulation of mechanics, these fields coalesce to provide a rich structure that enables her/him to obtain a more unified overview of the modern theories in physics. In this short work, we introduce this structure and examine its consequences in general relativity and quantum mechanics.</p> / Thesis / Master of Science (MSc)

Engineering and Physical Science Research Council (EPSRC) / This thesis describes the development of the two-frequency Planar Doppler Velocimetry (2n-PDV) flow measurement technique. This is modification of the Planar Doppler Velocimetry (PDV) technique that allows the measurement of up to three components of the flow velocity across a plane defined by a laser light sheet. The 2n-PDV technique reduces the number of components required to a single CCD camera and iodine cell from the two CCDs in conventional PDV. This removes the error sources associated with the misalignment of the two camera images and polarisation effects due to the beam splitters used in conventional PDV. The construction of a single velocity component 2n-PDV system is described and measurements made on the velocity field of a rotating disc and an axisymmetric air jet. The system was then modified to make 3D velocity measurements using coherent imaging fibre bundles to port multiple views to a single detector head. A method of approximately doubling the sensitivity of the technique was demonstrated using the measurements made on the velocity field of the rotating disc and was shown to reduce the error level in the final orthogonal velocity components by ~40 to 50%. Error levels of between 1.5ms-1 and 3.1ms-1 depending upon observation direction are demonstrated for a velocity field of ±34ms-1. The factors that will influence the selection of a viewing configuration when making 3D PDV measurements are then investigated with the aid of a computer model. The influence of the observation direction, the magnitude of the flow velocity, and the transformation to orthogonal velocity components are discussed. A new method using additional data in this transformation is presented and experimental results calculated using four-measured velocity components are compared to those found conventionally, using only three components. The inclusion of additional data is shown to reduce the final error levels by up to 25%.

This thesis describes the development of the two-frequency Planar Doppler Velocimetry (2n-PDV) flow measurement technique. This is modification of the Planar Doppler Velocimetry (PDV) technique that allows the measurement of up to three components of the flow velocity across a plane defined by a laser light sheet. The 2n-PDV technique reduces the number of components required to a single CCD camera and iodine cell from the two CCDs in conventional PDV. This removes the error sources associated with the misalignment of the two camera images and polarisation effects due to the beam splitters used in conventional PDV. The construction of a single velocity component 2n-PDV system is described and measurements made on the velocity field of a rotating disc and an axisymmetric air jet. The system was then modified to make 3D velocity measurements using coherent imaging fibre bundles to port multiple views to a single detector head. A method of approximately doubling the sensitivity of the technique was demonstrated using the measurements made on the velocity field of the rotating disc and was shown to reduce the error level in the final orthogonal velocity components by ~40 to 50%. Error levels of between 1.5ms-1 and 3.1ms-1 depending upon observation direction are demonstrated for a velocity field of ±34ms-1. The factors that will influence the selection of a viewing configuration when making 3D PDV measurements are then investigated with the aid of a computer model. The influence of the observation direction, the magnitude of the flow velocity, and the transformation to orthogonal velocity components are discussed. A new method using additional data in this transformation is presented and experimental results calculated using four-measured velocity components are compared to those found conventionally, using only three components. The inclusion of additional data is shown to reduce the final error levels by up to 25%.

This thesis describes the development of a Mach-Zehnder interferometric filter based planar Doppler velocimetry (MZI-PDV) flow measurement technique. The technique uses an entirely new optical system, an unbalanced MZI incorporating glass blocks for wavefront-matching, to replace the iodine cell currently used in conventional PDV. The free spectral range of the interferometric filter can be selected by adjusting the optical path difference of the MZI. This allows the velocity measurement range, sensitivity and resolution to be varied. This system offers no restricts to the choice of laser wavelength of operation which is not the case with most techniques. Two techniques to process the interference fringe images are presented. The first uses the shift of the fringe pattern to determine the Doppler frequency shift along profiles. The second provides a full-field measurement by normalising the received light intensity at each pixel in the image. With the single camera MZI-PDV scheme, exact alignment of the two output images on the active area of the camera is automatic. This eliminates the pixel-matching problem in conventional two camera PDV systems. The technique allows the measurement of up to three components of the flow velocity across a plane defined by a laser light sheet. The construction of a single velocity component MZI-PDV system that incorporates a phase-locking system designed to stabilise the filter is described. Measurements are made on the velocity field of a rotating disc with maximum velocities of ~±70ms-1 and an axis-symmetric air jet (with a nozzle diameter of 20mm) with an exit velocity of ~85ms-1. Standard deviations in the measured velocities were found to be about 2.9 and 2ms-1 for the two processing methods respectively. The system was then modified to make 3-component velocity measurements using imaging fibre bundles to port multiple views to a single detector head, and the standard deviation of the velocity error is around ±3ms-1 for a maximum velocity of ~±30ms-1 in the field of view. The factors that will affect the quality of the interference fringe image are investigated including polarisation sensitivity of the two beam splitters and flatness of the optical components. The inclination angle and the optical path deviation have little effect on the contrast of the interference fringes since collimated light beams, rather than divergent ones, are used in the interferometer.

Desenvolvemos aqui um estudo das formulações alternativas-equivalentes da Relatividade Geral, baseada no formalismo de conexões de Ashtekar. Iniciamos discutindo a estrutura matemática necessária de fibrados e conexões, e a teoria de sistemas Hamiltonianos vinculados. Em seguida, damos uma breve introdução ao formalismo métrico de Einstein e então passamos ao formalismo geometrodinâmico canônico (formalismo ADM). Introduzimos as transformações no espaço de fase que geram as formulações alternativas, de forma generalizada tal que possamos obter ambas as variáveis complexas de Ashtekar ou as variáveis reais de Barbero, ou mesmo qualquer forma intermediária por meio do parâmetro de Immirzzi. / We develop here a study of the alternative-equivalent formulations of General Relativity, based on Ashtekars connexion formalism. We begin discussing the mathematical structure needed of fibre bundles and connexions, and the theory of constrained Hamiltonian systems. Next, we give a brief introduction for Einsteins metric formalism and then we pass to the canonical geometrodynamic formalism (ADM formalism). We introduce the transformations of the phase space which generate the alternative formulations, in a generalized form such that we can obtain both Ashtekars complex variables or Barberos real variables, or even any intermediary form by using the Immirzzi parameter.

Desenvolvemos aqui um estudo das formulações alternativas-equivalentes da Relatividade Geral, baseada no formalismo de conexões de Ashtekar. Iniciamos discutindo a estrutura matemática necessária de fibrados e conexões, e a teoria de sistemas Hamiltonianos vinculados. Em seguida, damos uma breve introdução ao formalismo métrico de Einstein e então passamos ao formalismo geometrodinâmico canônico (formalismo ADM). Introduzimos as transformações no espaço de fase que geram as formulações alternativas, de forma generalizada tal que possamos obter ambas as variáveis complexas de Ashtekar ou as variáveis reais de Barbero, ou mesmo qualquer forma intermediária por meio do parâmetro de Immirzzi. / We develop here a study of the alternative-equivalent formulations of General Relativity, based on Ashtekars connexion formalism. We begin discussing the mathematical structure needed of fibre bundles and connexions, and the theory of constrained Hamiltonian systems. Next, we give a brief introduction for Einsteins metric formalism and then we pass to the canonical geometrodynamic formalism (ADM formalism). We introduce the transformations of the phase space which generate the alternative formulations, in a generalized form such that we can obtain both Ashtekars complex variables or Barberos real variables, or even any intermediary form by using the Immirzzi parameter.

This thesis describes the development of a Mach-Zehnder interferometric filter based planar Doppler velocimetry (MZI-PDV) flow measurement technique. The technique uses an entirely new optical system, an unbalanced MZI incorporating glass blocks for wavefront-matching, to replace the iodine cell currently used in conventional PDV. The free spectral range of the interferometric filter can be selected by adjusting the optical path difference of the MZI. This allows the velocity measurement range, sensitivity and resolution to be varied. This system offers no restricts to the choice of laser wavelength of operation which is not the case with most techniques. Two techniques to process the interference fringe images are presented. The first uses the shift of the fringe pattern to determine the Doppler frequency shift along profiles. The second provides a full-field measurement by normalising the received light intensity at each pixel in the image. With the single camera MZI-PDV scheme, exact alignment of the two output images on the active area of the camera is automatic. This eliminates the pixel-matching problem in conventional two camera PDV systems. The technique allows the measurement of up to three components of the flow velocity across a plane defined by a laser light sheet. The construction of a single velocity component MZI-PDV system that incorporates a phase-locking system designed to stabilise the filter is described. Measurements are made on the velocity field of a rotating disc with maximum velocities of ~±70ms-1 and an axis-symmetric air jet (with a nozzle diameter of 20mm) with an exit velocity of ~85ms-1. Standard deviations in the measured velocities were found to be about 2.9 and 2ms-1 for the two processing methods respectively. The system was then modified to make 3-component velocity measurements using imaging fibre bundles to port multiple views to a single detector head, and the standard deviation of the velocity error is around ±3ms-1 for a maximum velocity of ~±30ms-1 in the field of view. The factors that will affect the quality of the interference fringe image are investigated including polarisation sensitivity of the two beam splitters and flatness of the optical components. The inclination angle and the optical path deviation have little effect on the contrast of the interference fringes since collimated light beams, rather than divergent ones, are used in the interferometer.

<p>We introduce Hypoelliptic Diffusion Maps (HDM), a novel semi-supervised machine learning framework for the analysis of collections of anatomical surfaces. Triangular meshes obtained from discretizing these surfaces are high-dimensional, noisy, and unorganized, which makes it difficult to consistently extract robust geometric features for the whole collection. Traditionally, biologists put equal numbers of ``landmarks'' on each mesh, and study the ``shape space'' with this fixed number of landmarks to understand patterns of shape variation in the collection of surfaces; we propose here a correspondence-based, landmark-free approach that automates this process while maintaining morphological interpretability. Our methodology avoids explicit feature extraction and is thus related to the kernel methods, but the equivalent notion of ``kernel function'' takes value in pairwise correspondences between triangular meshes in the collection. Under the assumption that the data set is sampled from a fibre bundle, we show that the new graph Laplacian defined in the HDM framework is the discrete counterpart of a class of hypoelliptic partial differential operators.</p><p>This thesis is organized as follows: Chapter 1 is the introduction; Chapter 2 describes the correspondences between anatomical surfaces used in this research; Chapter 3 and 4 discuss the HDM framework in detail; Chapter 5 illustrates some interesting applications of this framework in geometric morphometrics.</p> / Dissertation

Yes / Recent developments in optical coherence tomography (OCT) technology enable direct enface visualisation of retinal nerve fibre bundle (RNFB) loss in glaucoma. However, the optimum depth at which to visualise RNFBs across the retina is unknown. We aimed to evaluate the range of depths and optimum depth at which RNFBs can be visualised across the retina in healthy eyes. The central ± 25° retina of 10 healthy eyes from 10 people aged 57–75 years (median 68.5 years) were imaged with spectral domain OCT. Slab images of maximum axial resolution (4 μm) containing depth‐resolved attenuation coefficients were extracted from 0 to 193.5 μm below the inner limiting membrane (ILM). Bundle visibility within 10 regions of a superimposed grid was assessed subjectively by trained optometrists (n = 8), according to written instructions. Anterior and posterior limits of RNFB visibility and depth of best visibility were identified for each grid sector. Effects of retinal location and individual eye on RNFB visibility were explored using linear mixed modelling with likelihood ratio tests. Intraclass correlation coefficient (ICC) was used to measure overall agreement and repeatability of grading. Spearman’s correlation was used to measure correlation between depth range of visible RNFBs and retinal nerve fibre layer thickness (RNFLT). Retinal location and individual eye affected anterior limit of visibility (χ2(9) = 58.6 and 60.5, both p < 0.0001), but none of the differences exceeded instrument resolution, making anterior limit consistent across the retina and different eyes. Greater differences were observed in the posterior limit of visibility across retinal areas (χ2(9) = 1671.1, p < 0.0001) and different eyes (χ2(9) = 88.7, p < 0.0001). Optimal depth for visualisation of RNFBs was around 20 µm below the ILM in most regions. It varied slightly with retinal location (χ2(9) = 58.8, p < 0.0001), but it was not affected by individual eye (χ2(9) = 10.7, p = 0.29). RNFB visibility showed good agreement between graders (ICC 0.89, 95%CI 0.87–0.91), and excellent repeatability (ICC 0.96–0.99). Depth range of visible RNFBs was highly correlated with RNFLT (ρ = 0.9, 95%CI: 0.86–0.95). The range of depths with visible RNFBs varies markedly across the healthy retina, consistently with RNFLT. To extract all RNFB information consistently across the retina, slab properties should account for differences across retinal locations and between individual eyes. / This work was supported by a College of Optometrists Research Fellowship (JD).