In the first two Chapters I outline the theory and background of separation of variables as an ansatz for solving fundamental partial differential equations (pdes) in Mathematical Physics. Two fundamental approaches will be highlighted, and more modern approaches discussed. In Chapter 3 I calculate the general trace-free conformal Killing tensor defined in Euclidean space - from the sum of symmetric tensor products of conformal Killing vectors. In Chapter 4 I determine the subcases with rotational symmetry and recover known examples pertaining to classical rotational coordinates. In Chapter 5 I obtain the induced action of the conformal group on the space of trace-free conformal Killing tensors. In Chapter 6 I use the invariants of trace-free conformal Killing tensors under the action of the conformal group to characterize, up to equivalence, the symmetric R-separable webs in E^3 that permit conformal separation of variables of the fundamental pdes in Mathematical Physics. In Chapter 7 the asymmetric R-separable metrics are obtained via a study of the separability conditions for the conformally invariant Laplace equation.

In this dissertation, we introduce and analyse a scheme for the fast evaluation of integrals stemming from boundary element methods including discretisations of the classical single and double layer potential operators. Our method is based on the parametrisation of boundary elements in terms of a d-dimensional parameter tuple. We interpret the integral as a real-valued function f depending on d parameters and show that f is smooth in a d-dimensional box. A standard interpolation of f by polynomials leads to a d-dimensional tensor which is given by the values of f at the interpolation points. This tensor may be approximated in a low rank tensor format like the canonical format or the hierarchical format. The tensor approximation has to be done only once and allows us to evaluate interpolants in O(dr(m+1)) operations in the canonical format, or O(dk³ + dk(m + 1)) operations in the hierarchical format, where m denotes the interpolation order and the ranks r, k are small integers. In particular, we apply an efficient black box scheme in the hierarchical tensor format in order to adaptively approximate tensors even in high dimensions d with a prescribed (but heuristic) target accuracy. By means of detailed numerical experiments, we demonstrate that highly accurate integral values can be obtained at very moderate costs.

<p>Identifying a lower-dimensional latent space for representation of high-dimensional observations is of significant importance in numerous biomedical and machine learning applications. In many such applications, it is now routine to collect data where the dimensionality of the outcomes is comparable or even larger than the number of available observations. Motivated in particular by the problem of predicting the risk of impending diseases from massive gene expression and single nucleotide polymorphism profiles, this dissertation focuses on building parsimonious models and computational schemes for high-dimensional continuous and unordered categorical data, while also studying theoretical properties of the proposed methods. Sparse factor modeling is fast becoming a standard tool for parsimonious modeling of such massive dimensional data and the content of this thesis is specifically directed towards methodological and theoretical developments in Bayesian sparse factor models.</p><p>The first three chapters of the thesis studies sparse factor models for high-dimensional continuous data. A class of shrinkage priors on factor loadings are introduced with attractive computational properties, with operating characteristics explored through a number of simulated and real data examples. In spite of the methodological advances over the past decade, theoretical justifications in high-dimensional factor models are scarce in the Bayesian literature. Part of the dissertation focuses on exploring estimation of high-dimensional covariance matrices using a factor model and studying the rate of posterior contraction as both the sample size & dimensionality increases. </p><p>To relax the usual assumption of a linear relationship among the latent and observed variables in a standard factor model, extensions to a non-linear latent factor model are also considered.</p><p>Although Gaussian latent factor models are routinely used for modeling of dependence in continuous, binary and ordered categorical data, it leads to challenging computation and complex modeling structures for unordered categorical variables. As an alternative, a novel class of simplex factor models for massive-dimensional and enormously sparse contingency table data is proposed in the second part of the thesis. An efficient MCMC scheme is developed for posterior computation and the methods are applied to modeling dependence in nucleotide sequences and prediction from high-dimensional categorical features. Building on a connection between the proposed model & sparse tensor decompositions, we propose new classes of nonparametric Bayesian models for testing associations between a massive dimensional vector of genetic markers and a phenotypical outcome.</p> / Dissertation

Eshelby-type inclusion problems of an infinite or a finite homogeneous isotropic elastic body containing an arbitrary-shape inclusion prescribed with an eigenstrain and an eigenstrain gradient are analytically solved. The solutions are based on a simplified strain gradient elasticity theory (SSGET) that includes one material length scale parameter in addition to two classical elastic constants. For the infinite-domain inclusion problem, the Eshelby tensor is derived in a general form by using the Green’s function in the SSGET. This Eshelby tensor captures the inclusion size effect and recovers the classical Eshelby tensor when the strain gradient effect is ignored. By applying the general form, the explicit expressions of the Eshelby tensor for the special cases of a spherical inclusion, a cylindrical inclusion of infinite length and an ellipsoidal inclusion are obtained. Also, the volume average of the new Eshelby tensor over the inclusion in each case is analytically derived. It is quantitatively shown that the new Eshelby tensor and its average can explain the inclusion size effect, unlike its counterpart based on classical elasticity. To solve the finite-domain inclusion problem, an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET are proposed and utilized. The solution for the disturbed displacement field incorporates the boundary effect and recovers that for the infinite-domain inclusion problem. The problem of a spherical inclusion embedded concentrically in a finite spherical body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. It is demonstrated through numerical results that the newly obtained Eshelby tensor can capture the inclusion size and boundary effects, unlike existing ones. Finally, a homogenization method is developed to predict the effective elastic properties of a heterogeneous material using the SSGET. An effective elastic stiffness tensor is analytically derived for the heterogeneous material by applying the Mori-Tanaka and Eshelby’s equivalent inclusion methods. This tensor depends on the inhomogeneity size, unlike what is predicted by existing homogenization methods based on classical elasticity. Numerical results for a two-phase composite reveal that the composite becomes stiffer when the inhomogeneities get smaller.

The North Anatolian Fault (NAF) is one of the world&rsquo / s largest active continental strikeslip faults, and forms the northern margin of the Anatolian plate. Although its geologic and geomorphologic features are well defined, crustal deformation and associated seismicity around central segment of the NAF is relatively less-known. In this study, we analyzed locations and focal mechanisms of 172 events with magnitude &ge / 3, which are recorded by 39 broadband seismic stations deployed by the North Anatolian Passive Seismic Experiment (2005-2008). Distribution of the events shows that the local seismicity in the area is widely distributed, suggesting a widespread continental deformation, particularly in the southern block. For the entire data set, P- and S- arrival times are picked and events are relocated using the HYPOCENTER program. Then, relocated events which have a good azimuthal coverage with a maximum gap of 120&deg / and at least 13 P- wave readings are selected and 1-D inversion algorithm, VELEST, is used to derive the 1-D seismic velocity model of the region. The final model with updated locations is later put together to the FOCMEC program, to obtain focal mechanisms solutions. In this step, an iterative scheme is applied by increasing the number of data errors. To obtain more unique solutions, first motions of P and SH v phases are used along with SH/P amplitude ratios. Resultant 109 well-constrained focal mechanisms later used to perform stress tensor inversion across the region. Our focal mechanisms suggest a dominant strike-slip deformation along two major fault sets in the region. In the east, E-W trending splays (Ezinepazari, Almus, and La&ccedil / in Kizilirmak) show right-lateral strike-slip motion similar to the NAF whereas in the west, N-S trending faults (Dodurga, Eldivan) show left lateral strike-slip motion. Overall, stress orientations are found as: maximum principal stress, &sigma / 1, is found to be subhorizontal striking NW-SE, the intermediate principle stress, &sigma / 2, is vertically orientated and the minimum principal stress, &sigma / 3, is found to be NE &ndash / SW striking, consistent with the strike-slip regime of the region.

The aim of this thesis is to develop a methodology for obtaining mass properties of a vehicle using specific test rig. Investigated mass properties are the mass, location of center of gravity and the inertia tensor. Accurate measurement of mass properties of vehicles is crucial for vehicle dynamics research. The test rig consists of a frame on which the vehicle is fixed and which is suspended from the ceiling of the laboratory using steel cables. Mass and location of center of gravity are measured using the data from the test rig in equilibrium position and basic static equations. Inertia tensor is measured using the data from dynamical response of the system. For this purpose an identification routine which employs prediction error method is developed using the built&ndash / in functions from the System Identification Toolbox of MATLAB&reg / . The experiment was also simulated using Simmechanics Toolbox of MATLAB&reg / . Identification code is verified using the results of the experiment simulations for various cases.

A complete scientific understanding of human nature requires delineation of the neurobiological characteristics underlying the unique features of the human mind. This effort can be facilitated by comparing the human brain with the brains of other living primate species. Humans are more susceptible to neurodegenerative diseases than other primate species, including our closest living primate relatives, the chimpanzees. Comparing age-related changes in brain structure between humans and non-human primates could, therefore, potentially shed light on the neurological basis of this human vulnerability. Further, human brains are lateralized with specialized cognitive and behavioral functions. Comparing the magnitude of hemispheric asymmetries in brain structure between humans and non-human primates can probe insights into this human specific capability and learn more about human evolution. Diffusion weighted MRI protocols were developed for different species, taking into account their neuroanatomical differences. For Chimpanzees, a multi-shot DWI sequence was developed and compared with a single-shot DWI sequence to determine which provided a better quality diffusion data free of acquisition related artifacts. Different simulation techniques were used to evaluate the effect of segmentation-related motion artifact (ghosting) on the multi-shot DTI data. Although both protocols generated high-resolution diffusion MRI data with correctable susceptibility-induced distortions, the single-shot protocol enables the acquisition of the high-resolution diffusion MRI data freed of ghosting and with twice the signal-to-noise ratio (SNR), for the same scan duration. The acquired chimpanzee and macaque diffusion data were used to compare the magnitude of microstructural asymmetries and age-related decline of brain white matter with those in humans. Hemispheric asymmetry results show a pattern of strong leftward asymmetry in human DTI indices that differs markedly from the chimpanzee (multi-shot data) and the rhesus macaque patterns involving both rightward and leftward asymmetries. The magnitude of leftward asymmetry increased for chimpanzees scanned with single-shot DTI sequence. Region of interest analyses within the corpus callosum revealed a significant age-related increase in fractional anisotropy (FA) in the genu for chimpanzees (multi-shot data) and no significant change in any region for macaques. Additionally, voxel-wise analysis using Tract Based Spatial Statistics (TBSS) revealed widespread age-related FA increases for chimpanzees (multi-shot data) and weak age-related decreases in FA for macaques across most white matter tracts. Overall, results from these multi-shot data analyses suggest that rhesus monkeys show age-related decreases in white matter integrity that parallel changes found in humans, whereas chimpanzees show age-related increases in white matter integrity. On the contrary, the single-shot data results for chimpanzees revealed no significant relationship between age and the different DTI indices. These noteworthy species differences may help to explain the unique features of the human mind and why humans are more susceptible to neurodegenerative diseases. Furthermore, these studies demonstrate the need for complementary histological studies of white matter microstructure in humans, chimpanzees and macaques to clarify the cellular and molecular basis of these findings.

<p>This thesis deals with Computational Fluid Dynamics (CFD)simulations of the flow around delta wings at high angles ofattack. These triangular wings, mainly used in militaryaircraft designs, experience the formation of two vortices ontheir lee-side at large angles of attack. The simulation ofthis vortical flow by solving the Navier-Stokes equations isthe subject of this thesis. The purpose of the work is toimprove the understanding of this flow and contribute to thedesign of such a wing by developing methods that enable moreaccurate and efficient CFD simulations.</p><p>Simulations of the formation, burst and disappearance of thevortices while the angle of attack is changing are presented.The structured flow solver NSMB has been used to get thetime-dependent solutions of the flow. Both viscous and inviscidresults of a 70°-swept delta wing pitching in anoscillatory motion are reported. The creation of the dynamiclift and the hysteresis observed in the history of theaerodynamic forces are well reproduced.</p><p>The second part of the thesis is focusing on automatic meshrefinement and its influence on simulations of the delta wingleading-edge vortices. All the simulations to assess the gridquality are inviscid computations performed with theunstructured flow solver EDGE. A first study reports on theeffects of refining thewake of the delta wing. A70°-swept delta wing at a Mach number of 0.2 and an angleof attack of 27° where vortex breakdown is present abovethe wing, is used as testcase. The results show a strongdependence on the refinement, particularly the vortex breakdownposition, which leads to the conclusion that the wake should berefined at least partly. Using this information, a grid for thewing in the wind tunnel is created in order to assess theinfluence of the tunnel walls. Three sensors for automatic meshrefinement of vortical flows are presented. Two are based onflow variables (production of entropy and ratio of totalpressures) while the third one requires an eigenvalue analysisof the tensor of the velocity gradients in order to capture theposition of the vortices in the flow. These three vortexsensors are successfully used for the simulation of the same70° delta wing at an angle of attack of 20°. Acomparison of the sensors reveals the more local property ofthe third one based on the eigenvalue analysis. This lattertechnique is applied to the simulation of the wake of a deltawing at an angle of attack of 20°. The simulations on ahighly refined mesh show that the vortex sheet shed from thetrailing-edge rolls up into a vortex that interacts with theleading-edge vortex. Finally the vortex-detection technique isused to refine the grid around a Saab Aerosystems UnmannedCombat Air Vehicle (UCAV) configuration and its flight dynamicscharacteristics are investigated.</p><p><b>Key words:</b>delta wing, high angle of attack, vortex,pitching, mesh refinement, UCAV, vortex sensor, tensor ofvelocity gradients.</p>

<p>This is a thesis written for a master's degree at the Computer Vision Laboratory, University of Linköping. An abstract outer product is defined and used as a bridge to reach 2:nd and 4:th order tensors. Some applications of these in geometric analysis of range data are discussed and illustrated. In idealized setups, simple geometric objects, like spheres or polygons, are successfully detected. Finally, the generalization to n:th order tensors for storing and analysing geometric information is discussed.</p>

Thesis (Ph.D)--Biomedical Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Hu, Xiaoping; Committee Member: Brummer, Marijn; Committee Member: Duong, Tim; Committee Member: Keilholz, Shella; Committee Member: Schumacher, Eric. Part of the SMARTech Electronic Thesis and Dissertation Collection.