In der vorliegenden Arbeit wird ein neues Tensorformat vorgestellt und eingehend analysiert. Das hierarchische Format verwendet einen binären Baum, um den Tensorraum der Ordnung d mit einer geschachtelten Unterraumstruktur zu versehen. Der Speicheraufwand für diese Darstellung ist von der Größenordnung O(dnr + dr^3), wobei n den Speicheraufwand in den Ansatzräumen kennzeichnet und r ein Rangparameter ist, der durch die Dimensionen der geschachtelten Unterräume bestimmt wird. Das hierarchische Format umfasst verschiedene Standardformate zur Tensordarstellung wie das kanonische oder r-Term-Format und die Unterraum-/Tucker-Darstellung. Die in dieser Arbeit entwickelte zugehörige Arithmetik inklusive mehrerer Approximationsmethoden basiert auf stabilen Methoden der Linearen Algebra, insbesondere die Singulärwertzerlegung und die QR-Zerlegung sind von zentraler Bedeutung. Die rechnerische Komplexität ist hierbei O(dnr^2+dr^4). Die lineare Abhängigkeit von der Ordnung d des Tensorraumes ist hervorzuheben. Für die verschiedenen Approximationsmethoden, deren Effizienz und Effektivität für die Anwendbarkeit des neuen Formates entscheidend sind, werden qualitative und quantitative Fehlerabschätzungen gezeigt. Umfassende numerische Experimente mit einem Fokus auf den Approximationsmethoden bestätigen zum einen die theoretischen Resultate und belegen die Stärken der neuen Tensordarstellung, zeigen aber zum anderen auch weitere, eher überraschende positive Eigenschaften der mit FastHOSVD bezeichneten schnellsten Kürzungsmethode. / In this dissertation we present and a new format for the representation of tensors and analyse its properties. The hierarchical format uses a binary tree in order to define a hierarchical structure of nested subspaces in the tensor space of order d. The strorage requirements are O(dnr+dr^3) where n is determined by the storage requirements in the ansatz spaces and r is a rank parameter determined by the dimensions of the nested subspaces. The hierarchichal representation contains the standard representation like canonical or r-term representation and subspace or Tucker representation. The arithmetical operations that have been developed in this work, including several approximation methods, are based on stable Linear Alebra methods, especially the singular value decomposition (SVD) and the QR decomposition are of importance. The computational complexity is O(dnr^2+dr^4). The linear dependence from the order d of the tensor space is important. The approximation methods are one of the key ingredients for the applicability of the new format and we present qualitative and quantitative error estimates. Numerical experiments approve the theoretical results and show some additional, but unexpected positive aspects of the fastest method called FastHOSVD.

<p>This thesis presents quantum chemical calculations, applications of the response function formalism recently implemented within the framework of density functional theory (DFT) by our research group. The purpose of the calculations is to assess the performance of this perturbative approach to determining heavy atom effects on magnetic resonance parameters. Relativistic corrections can be generated by spin-orbit interactions or by scalar relativistic effects due to high velocity electrons in the atomic core region of heavy atoms. In this work, the evaluation of nuclear magnetic resonance (NMR) parameters is considered, the nuclear shielding tensor and the indirect nuclear spin-spin coupling tensor. For series of homologous compounds, it is found that both types of corrections to these parameters are increasing in size upon substitution of a constituent atom by a heavier element, but that their relative importance is system dependent. The obtained results are compatible with the ones provided by electron correlated <em>ab initio</em> methods, and a qualitative agreement with experimentally determined parameters is overall achieved. The methodology presented in this thesis aims to be a practical approach which can be applied in the study of molecular properties of large systems.</p><p>This thesis also addresses the calculation of hyperfine coupling constants, and evaluates a novel approach to the treatment of spin-polarization in spin restricted calculations without the spin contamination associated with spin unrestricted calculations.</p>

What is in common between the Kepler problem, a Hydrogen atom and a rotating black- hole? These systems are described by different physical theories, but much information about them can be obtained by separating an appropriate Hamilton-Jacobi equation. The separation of variables of the Hamilton-Jacobi equation is an old but still powerful tool for obtaining exact solutions. The goal of this thesis is to present the theory and application of a certain type of conformal Killing tensor (hereafter called concircular tensor) to the separation of variables problem. The application is to spaces of constant curvature, with special attention to spaces with Euclidean and Lorentzian signatures. The theory includes the general applicability of concircular tensors to the separation of variables problem and the application of warped products to studying Killing tensors in general and separable coordinates in particular. Our first main result shows how to use these tensors to construct a special class of separable coordinates (hereafter called Kalnins-Eisenhart-Miller (KEM) coordinates) on a given space. Conversely, the second result generalizes the Kalnins-Miller classification to show that any orthogonal separable coordinates in a space of constant curvature are KEM coordinates. A closely related recursive algorithm is defined which allows one to intrinsically (coordinate independently) search for KEM coordinates which separate a given (natural) Hamilton-Jacobi equation. This algorithm is exhaustive in spaces of constant curvature. Finally, sufficient details are worked out, so that one can apply these procedures in spaces of constant curvature using only (linear) algebraic operations. As an example, we apply the theory to study the separability of the Calogero-Moser system.

In der vorliegenden Arbeit wird ein neues Tensorformat vorgestellt und eingehend analysiert. Das hierarchische Format verwendet einen binären Baum, um den Tensorraum der Ordnung d mit einer geschachtelten Unterraumstruktur zu versehen. Der Speicheraufwand für diese Darstellung ist von der Größenordnung O(dnr + dr^3), wobei n den Speicheraufwand in den Ansatzräumen kennzeichnet und r ein Rangparameter ist, der durch die Dimensionen der geschachtelten Unterräume bestimmt wird. Das hierarchische Format umfasst verschiedene Standardformate zur Tensordarstellung wie das kanonische oder r-Term-Format und die Unterraum-/Tucker-Darstellung. Die in dieser Arbeit entwickelte zugehörige Arithmetik inklusive mehrerer Approximationsmethoden basiert auf stabilen Methoden der Linearen Algebra, insbesondere die Singulärwertzerlegung und die QR-Zerlegung sind von zentraler Bedeutung. Die rechnerische Komplexität ist hierbei O(dnr^2+dr^4). Die lineare Abhängigkeit von der Ordnung d des Tensorraumes ist hervorzuheben. Für die verschiedenen Approximationsmethoden, deren Effizienz und Effektivität für die Anwendbarkeit des neuen Formates entscheidend sind, werden qualitative und quantitative Fehlerabschätzungen gezeigt. Umfassende numerische Experimente mit einem Fokus auf den Approximationsmethoden bestätigen zum einen die theoretischen Resultate und belegen die Stärken der neuen Tensordarstellung, zeigen aber zum anderen auch weitere, eher überraschende positive Eigenschaften der mit FastHOSVD bezeichneten schnellsten Kürzungsmethode. / In this dissertation we present and a new format for the representation of tensors and analyse its properties. The hierarchical format uses a binary tree in order to define a hierarchical structure of nested subspaces in the tensor space of order d. The strorage requirements are O(dnr+dr^3) where n is determined by the storage requirements in the ansatz spaces and r is a rank parameter determined by the dimensions of the nested subspaces. The hierarchichal representation contains the standard representation like canonical or r-term representation and subspace or Tucker representation. The arithmetical operations that have been developed in this work, including several approximation methods, are based on stable Linear Alebra methods, especially the singular value decomposition (SVD) and the QR decomposition are of importance. The computational complexity is O(dnr^2+dr^4). The linear dependence from the order d of the tensor space is important. The approximation methods are one of the key ingredients for the applicability of the new format and we present qualitative and quantitative error estimates. Numerical experiments approve the theoretical results and show some additional, but unexpected positive aspects of the fastest method called FastHOSVD.

This thesis presents quantum chemical calculations, applications of the response function formalism recently implemented within the framework of density functional theory (DFT) by our research group. The purpose of the calculations is to assess the performance of this perturbative approach to determining heavy atom effects on magnetic resonance parameters. Relativistic corrections can be generated by spin-orbit interactions or by scalar relativistic effects due to high velocity electrons in the atomic core region of heavy atoms. In this work, the evaluation of nuclear magnetic resonance (NMR) parameters is considered, the nuclear shielding tensor and the indirect nuclear spin-spin coupling tensor. For series of homologous compounds, it is found that both types of corrections to these parameters are increasing in size upon substitution of a constituent atom by a heavier element, but that their relative importance is system dependent. The obtained results are compatible with the ones provided by electron correlated ab initio methods, and a qualitative agreement with experimentally determined parameters is overall achieved. The methodology presented in this thesis aims to be a practical approach which can be applied in the study of molecular properties of large systems. This thesis also addresses the calculation of hyperfine coupling constants, and evaluates a novel approach to the treatment of spin-polarization in spin restricted calculations without the spin contamination associated with spin unrestricted calculations. / QC 20101122

published_or_final_version / Psychiatry / Doctoral / Doctor of Philosophy

published_or_final_version / Diagnostic Radiology / Doctoral / Doctor of Philosophy

Diffusion Tensor Magnetic Resonance Imaging (DTI) is a recently-developed technique that can image in vivo the white matter pathways of the central nervous system. This study used 12-direction diffusion-weighted MRI data from nine stroke patients acquired as part of a three-year stroke rehabilitation study coordinated by the Movement Neuroscience Laboratory at the University of Auckland. DTI was used to investigate corticospinal connectivity. From the FA maps, it is found that in those patients whose motor connectivity has been compromised by the stroke to the extent that no motor evoked potential (MEP) can be elicited from a selected affected muscle group, the asymmetry in mean FA values in the posterior limbs of the internal capsules (PLICs) is correlated with functional recovery as measured by the Fugl-Meyer clinical score. Using probabilistic tractography in the contralesional hemisphere produced CST location and somatotopy results that were consistent with those of previous studies. However, in the ipsilesional hemisphere, connectivity results were highly variable. A measure of change in symmetry of mean connectivity is found to correlate with functional recovery as measured by change in FM score. This supports previous work which has correlated CST integrity and functional improvement and it supports the theory that functional recovery after stroke depends on the extent to which motor CNS symmetry can be regained in the new post-stroke architecture. It also suggests that the movement of the fMRI activations occurs in such a way as to make the most of the preserved white matter connectivity.

Acoustic emissions (AE) can provide information relating to the internal state of a deforming rock sample during laboratory testing and have been utilised to quantify damage progression for time-dependent failure modeling. However, the underlying physical mechanisms that produce AE in different materials and their evolution during the process of damage localisation are not fully understood, particularly in porous media. In order to investigate the sources of laboratory acoustic emissions, a moment tensor inversion was applied to data from triaxial compression experiments on Aue granite and Clashach sandstone. The moment tensor inversion was verified for granite, by comparison with results obtained using a more simplistic source analysis technique. In the non-porous Aue granite, AE sources exhibited a predominantly tensile behaviour in the early stages of AE activity. However, shear sources become dominant in the vicinity of the peak stress. In contrast, during deformation of the Clashach sandstone, which has a significant pre-existing porosity, AE sources are dominated by a collapse signature and generally involve a notable shear component. AE that have a predominantly shear mechanism are also a major contributor to the microscale deformation imaged by the technique, and dominate during shear localisation. A combination of correlation analysis and source analysis was used to elucidate the temporal and spatial evolution of the AE source mechanisms involved in the localisation process, as well as during a temporary hiatus in the progression to failure. The results support the concept that the cascade to failure requires the simultaneous involvement of a range of micromechanical behaviours to maintain the progression of localised damage, and eventual formation of a fault. Localisation of collapse mechanisms was not observed until the final approach to failure. Finally, AE sources produced during brittle deformation of the Clashach sandstone were characterised in detail and compared to microstructural observations representing the integrated effect of all times up to the end of the test, and including smaller structures that may have been formed insufficiently dynamically to produce AE. Equivalent focal mechanisms for these events are presented and the relative proportions of their volumetric and shear components considered. The results indicate that AE sources display a wide spectrum of micromechanical behaviour that is consistent with microstructural observations, indicating that AE mechanisms are representative of ongoing deformation processes within the sandstone. It is argued that moment tensor inversion of acoustic emissions is a powerful tool for elucidating the micromechanical evolution of damage, during the brittle deformation of rock.

Background: For Diffusion Tensor Imaging (DTI) to become a clinically useful tool in the detection of traumatic brain injury (TBI) and prediction of functional outcome, a reliable method enabling the identification of likely injury in individual patients needs to be developed. Objective: To explore different methods of analysing DTI measures to determine if individual TBI patients can be differentiated from a group of non-brain injured controls and if so, how these differences are associated with cognitive function. Method: 4 participants with TBI and 11 control participants were scanned using DTI and completed a battery of neuropsychological tests. The DTI measures of Fractional Anisotropy (FA) and Mean Diffusivity (MD) in the uncinate fasciculus were compared across individual TBI patients and a control group using 3 different methods of analysis. Results: The comparison of mean FA/MD from individual TBI patients with the overall mean FA/MD of the control group revealed that some TBI patients had lower values of FA whilst others had increased MD. This difference in FA may be associated with deficits in measures of attention. The histogram curves and cumulative frequency plots for individual TBI patients and the controls revealed subtle yet potentially significant differences in the distribution of FA/MD. However at this stage these differences could not be associated with cognitive function. Conclusion: Initial findings indicate that individual TBI patients can be differentiated from a control group using different methods with differing degrees of sensitivity. These differences may be related to cognitive function but further research is warranted before firm conclusions can be drawn.