This thesis aims to acquire, deepen and promote understanding of computing techniques for high dimensional scattered data approximation with radial basis functions. The main contributions of this thesis include sufficient conditions for the sovability of compactly supported radial basis functions with different shapes, near points preconditioning techniques for high dimensional interpolation systems with compactly supported radial basis functions, a heterogeneous hierarchical radial basis function interpolation scheme, which allows compactly supported radial basis functions of different shapes at the same level, an O(N) algorithm for constructing hierarchical scattered data set andan O(N) algorithm for sparse kernel summation on Cartesian grids. Besides the main contributions, we also investigate the eigenvalue distribution of interpolation matrices related to radial basis functions, and propose a concept of smoothness matching. We look at the problem from different perspectives, giving a systematic and concise description of other relevant theoretical results and numerical techniques. These results are interesting in themselves and become more interesting when placed in the context of the bigger picture. Finally, we solve several real-world problems. Presented applications include 3D implicit surface reconstruction, terrain modelling, high dimensional meteorological data approximation on the earth and scattered spatial environmental data approximation.

Some metals, such as Ti-6Al-4V, have a high elongation to failure when strained at certain rates and temperatures. Superplastic forming is the utilisation of this property, and it can be used to form thin, geometrically complex components. Superplastic forming is a slow process, and this is one of the reasons why it is an expensive manufacturing process. Localised thinning occurs if the specimen is strained too quickly, and components with locally thin wall thickness fail prematurely. The goal of this study is to find a technique that can be used to minimise the forming time while limiting the minimum final thickness. The superplastic forming process is investigated with the finite element method. The finite element method requires a material model which describes the superplastic behaviour of the metal. Several material models are investigated in order to select a material model that can show localised thinning at higher strain rates. The material models are calibrated with stress-strain data, grain size-time data and strain rate sensitivity-strain data. The digitised data from literature is for Ti-6Al-4V with three different initial grain sizes strained at different strain rates at 927 C. The optimisation of the forming time is done with an approximate optimisation algorithm. This algorithm involves fitting a metamodel to simulated data, and using the metamodels to find the optimum instead of using the finite element model directly. One metamodel is fitted to the final forming time results, and another metamodel is fitted to the final minimum thickness results. A regressive radial basis function method is used to construct the metamodels. The interpolating radial basis function method proved to be unreliable at the design space boundaries due to non-smooth finite element results. The non-smooth results are due to the problem being path dependent. The final forming time of the superplastic forming of a rectangular box was successfully minimised while limiting the final minimum thickness. The metamodels predicted that allowing a 4% decrease in the minimum allowable thickness (1.0 mm to 0.96 mm) and a 1 mm gap between the sheet and the die corner the forming time is decreased by 28.84%. The finite element verification indicates that the final minimum thickness reduced by 3.8% and that the gap between the sheet and the die corner is less than 1 mm, resulting in the forming time being reduced by 28.81%. / Dissertation (MEng)--University of Pretoria, 2017. / Mechanical and Aeronautical Engineering / MEng / Unrestricted

Light-matter interactions lead to a variety of interesting phenomena, for example photosynthesis which is a process fundamental to life on earth. There exists many different spectroscopic methods to measure light-matter interactions, for example UV/Vis spectroscopy, that can provide information about electronically excited states. However, numerical methods and theory are important to model and gain understanding of these experiments. Quantum chemistry provides that understanding, giving the possibility to numerically calculate molecular properties like excitation energies. The aim of this thesis was to implement a reduced-space algorithm in Dalton, to solve an eigenvalue equation obtained by response theory, for the calculation of excitation energies of molecular systems. There already was a similar algorithm in Dalton, that was able to perform these calculations. However, in a different module of Dalton used mainly for complex response theory, an algorithm to obtain eigenvalues was missing. The new implementation was similar to the existing one, except for the division of the reduced space into even and odd parts used in the complex response module. The thesis starts with a quick introduction of light-matter interactions and proceeds with a description of many-body theory, including numerical methods used in that field. In the end of the theoretical part, the eigenvalue equation, used to calculate excitation energies, is derived. In the following section, the reduced-space algorithm is described. In the end of the thesis, numerical results obtained with the algorithm are presented, including a small basis set and method study. The comparison with the existing implementation of the similar algorithm verified the successful implementation of the algorithm presented in this thesis.

Let S31(?) be the bivariate C1-cubic spline space over a triangulated quadrangulation ?. In this paper, an explicit representation of a locally supported basis of S31(?) is given using the interpolation conditions at vertices.

A research report submitted to the SCHOOL OF ACCOUNTING Faculty of commerce, law and management University of the Witwatersrand In partial fulfilment of the requirements for the degree of Master of Commerce / This study assesses the ability of Jansen, Ramnath & Yohn (2012) diagnostic, which is based on the relationship between the change in the asset turnover ratio and profit margin ratio, to distinguish between those firms suspected of manipulating reported financial figures by means of earnings management (EM) and firms that have not attempted earnings management. The study aims to determine whether, as suggested by Jansen et al (2012), the change in the asset turnover ratio and profit margin ratio as well as the direction of the change, can potentially indicate EM. In addition, the study aims to determine whether this new, simplistic diagnostic is incrementally useful to discretionary accruals in identifying EM. The sample of suspected EM firms was obtained from a study conducted by Rabin & Negash (2012), using kernel density estimation (Lahr, 2014). The results of this research suggest that Jansen et al.’s (2012) diagnostic is a useful indicator for identifying firms that might have manipulated reported financial figures through the use of earnings management. The study however shows that, due to weaknesses in either the diagnostic, in that it is limited in its ability to identify EM through sales, or in the method used to obtain the sample, this diagnostic is not incrementally useful to discretionary accruals models in identifying EM. Instead it should be used in conjunction with other models. / MT2017

Mesh free modeling techniques are a promising alternative to traditional meshed methods for solving computational fluid dynamics problems. These techniques aim to solve for the field variable using solely the values of nodes and therefore do not require the generation of a mesh. This results in a process that can be much more reliably automated and is therefore attractive. Radial basis functions (RBFs) are one type of "meshless" method that has shown considerable growth in the past 50 years. Using these RBFs to directly solve a partial differential equation is known as Kansa's method and has been used to successfully solve many flow problems. The problem with Kansa's method is that there is no formal guarantee that its solution matrix will be non-singular. More recently, an expansion on Kansa's method was proposed that incorporates the boundary and PDE operators into the solution of the field variable. This method, known as Hermitian method, has been shown to be non-singular provided certain nodal criteria are met. This work aims to perform a comparison between Kansa and Hermitian methods to aid in future selection of a method. These two methods were used to solve steady and transient one-dimensional convection-diffusion problems. The methods are compared in terms of accuracy (error) and computational complexity (conditioning number) in order to evaluate overall performance. Results suggest that the Hermitian method does slightly outperform Kansa method at the cost of a more ill-conditioned collocation matrix.

Gender differences in emotion regulation (ER) are well documented, but studies have often relied on behavioral and self-report data. Less is known about gender differences in the neural basis of ER. This systematic review aims to fill this gap and investigate gender differences in the neural basis of ER. The systematic search process ended in eight articles, using either structural or functional neuroimaging methods while investigating the neural correlates of ER using either an ER task to manipulate ER or assessed trait ER with questionnaires. The studies either used reappraisal or expressive suppression as ER strategies. The results were partly inconsistent, but most studies demonstrate the involvement of areas within the prefrontal cortex in ER. Males activated areas involved in cognitive control while females activated emotion-focused areas involved in emotional processing. There is disagreements among researchers whether more activity in the prefrontal cortex represent more effort during ER (and how it does so), as well as whetherless activity represent less effort or more efficient regulation. These insights may help us understand each other better. Future research is needed to address if activation within the prefrontal cortex reflects more or less efficiency when regulating emotions.

Learning algorithms based on the Fourier transform attempt to learn functions by approximating the largest coefficients of their Fourier representations. Nearly all previous work in Fourier-based learning has been in the theoretical realm, where properties of the transform have made it possible to prove many interesting learnability results. The real-world usefulness of Fourier-based methods, however, has not been thoroughly explored. This thesis explores methods for the practical application of Fourier-based learning. The primary contribution of this thesis is a new search algorithm for finding the largest coefficients of a function's Fourier representation. Although the search space is exponentially large, empirical results demonstrate that only a small fraction of the space needs to be explored to find the largest coefficients. Furthermore, the algorithm is applicable to a much wider range of learning scenarios than previous approaches. Results of learning real-world problems with algorithms based on this search technique are also presented. The accuracies of the Fourier-based learning methods are not particularly impressive, however, and analysis and empirical results suggest why the Fourier representation may be a poor choice for typical real-world learning problems. Finally, this thesis shows that the search algorithm can be generalized to explore any basis of functions. Furthermore, it can search multiple bases simultaneously. This greatly enhances the learning techniques, and empirical results demonstrate significantly improved accuracy over the Fourier-based approach.

We present a method for determining the reduced Gröbner basis with respect to a given admissible term order of order type ! of the intersection ideal of an infinite sequence of polynomial ideals. As an application we discuss the Lagrange type interpolation on algebraic sets and the 'approximation' of the ideal I of an algebraic set by zero dimensional ideals, whose affine Hilbert functions converge towards the affine Hilbert function of I.

Cardiovascular diseases (CVD) are the number one causes of death in the world. According to the world Health Organization (WHO) 17.5 million people died from cardiovascular disease in 2005, representing 30 % of all global deaths . Of these deaths, 7.6 million were due to heart attacks and 5.7 million due to stroke. If current trends are allowed to continue, by 2015 an estimated 20 million people will die annually from cardiovascular disease. The trends are similar in the United States where on an average 1 person dies every 37 seconds due to CVD. In 2008 an estimated 770,000 Americans will experience a new heart attack (coronary stenosis) and 600,000 will experience a first stroke. Although the exact causes of cardiovascular disease are not well understood, hemodynamics has been long thought to play a primary role in the progression of cardiovascular disease and stroke. There is strong evidence linking the fluid mechanical forces to the transduction mechanisms that trigger biochemical response leading to atherosclerosis or plaque formation. It is hypothesized that the emergence of abnormal fluid mechanical stresses which dictate the cell mechanotransduction mechanisms and lead to disease progression is dependent on the geometry and compliance of arteries, and pulsatility of blood flow. Understanding of such hemodynamic regulation in relation to atherosclerosis is of significant clinical importance in the prediction and progression of heart disease as well as design of prosthetic devices such as stents. The current work will systematically study the effects of compliance and complex geometry and the resulting fluid mechanical forces. The objective of this work is to understand the relationship of fluid mechanics and disease conditions using both experimental and computational methods where (a) Compliance effects are studied in idealized stenosed coronary and peripheral arteries using Digital Particle Image Velocimetry (DPIV), (b) Complex geometric effects of stented arteries with emphasis on its design parameters is investigated using CFD, Also (c) a novel method to improve the accuracy of velocity gradient estimation in the presence of noisy flow fields such as in DPIV where noise is inherently present is introduced with the objective to improve accuracy in the estimation of WSS, which are of paramount hemodynamic importance. The broad impact of the current work extends to the understanding of fundamental physics associated with arterial disease progression which can lead to better design of prosthetic devices, and also to better disease diagnostics. / Ph. D.