Major transportation and land use problems are experienced in the Menlyn precinct in Tshwane. Current infrastructure allows motor vehicles to dominate public transport and pedestrian thoroughfare. Pedestrians and motor vehicles are placed in danger and traffic problems occur when taxis and buses make use of road lanes and sidewalks to stop. The interaction and superposition of these problems are manifested as a fracture in the urban framework, disconnecting people, environments and different transport modes from one another and the rest of Tshwane. The aim of this design proposal is to address the fractured experience in the Menlyn area. The project should respond to the poor connectivity and transport problems currently experienced in Menlyn and reinforce Menlyn as an important node within Tshwane. It is argued that a Platform Building at Menlyn could integrate the various transport systems present in the Menlyn precinct with the envisioned underground Gautrain station. / Dissertation (MArch(Prof))--University of Pretoria, 2012. / Architecture / unrestricted

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Intra-cellular sodium (23Na) concentration is directly related to cellular health. Thus, sodium magnetic resonance imaging (MRI) can provide metabolic information on tissue health that a routine clinical (proton) MRI cannot. 23Na-MRI could be a valuable tool to assist physicians in the diagnosis, prognosis, and monitoring of a variety of pathologies. However, due to factors that include quantum mechanical limitations and biological restrictions, the signal-to-noise ratio (SNR) of a sodium scan is much lower than that of a standard proton scan, which limits the practicality of 23Na-MRI in a clinical setting. This project looks to improve the viability of 23Na-MRI and focuses on an often overlooked facet of MRI development, the radio frequency (RF) coil. Fractal antennas have been used in telecommunication systems for years, and are generally exploited for their compact nature, allowing for the same performance of a larger antenna, in a smaller space. They have also been shown to be capable of a wider transmission bandwidth (BW) than a standard antenna and with MRI applications they have been shown to provide a small SNR increase in proton imaging. It is hypothesized that a surface coil with a Koch snowflake fractal geometry can provide increased SNR for a sodium MRI scan, compared to that of a standard circular geometry coil, by producing a more homogeneous magnetic field in both space and frequency. To test the hypothesis two coils, one circular and the other a Koch snowflake fractal, were simulated. The simulated magnetic fields were compared on their homogeneity and magnitude before the two coils were constructed and implemented with a variety of sodium MRI scans. B1+ maps were acquired to measure RF field homogeneity, and SNR was determined for both coil geometries. The coils were also tested for their homogeneity over varied transmit BWs by comparing images with various field of view (FOV) sizes. Finally the coils were compared for clinical viability in a test of healthy human knee imaging. The circular coil had a more homogeneous B1+ field than the fractal at depths between 10-40mm, and had a higher SNR in its produced images. The circular coil acquired more signal in vivo which provided a higher detail image, but the fractal coil's SNR was higher due to reduced noise. The fractal coil performed better over a wider BW which indicates that further research should be conducted into the applications of fractal coils in multi-nuclear MRI scans. / Thesis / Master of Applied Science (MASc)

The use of fractals in fields such as molecular biology, epidemiology, landscape, ecology, geology, physics, etc., is becoming more common. In order to use fractals to model many phenomena, the researcher requires the knowledge of the fractal, or Hausdorff-Besicovitch, dimension. However, no statistical properties of the usual estimator, the entropy estimator, are known. In addition, the entropy estimator is biased high when an inefficient net is used. This dissertation develops a new estimator, the relative entropy estimator, which is asymptotically unbiased and is consistent. The estimator is asymptotically normal, and asymptotic confidence intervals are presented. An estimate of the variance of the estimator is given which does not depend on the dimension, or its estimate, using an occupancy model. The exact distribution of the estimator is also derived. Applications of the theory to various fields are presented. For example, I find that from the point of view of dimension, the logarithms of stock prices behave consistently with the classical Brownian function. Also, the relative entropy estimator gives a more realistic estimate of the dimension of surface terrain than an ad hoc estimate found in the literature. The Hausdorff dimensions of nursery-grown tree roots were estimated, and it was found that the dimension is related to the probability of the tree’s survival when the tree is planted in the wild. The dimensions of Julia sets and of the Hénon attractor were also investigated. A computer program for calculating the estimates is included. / Ph. D.

The term quantization refers to the process of estimating a given probability by a discrete probability supported on a finite set. The quantization dimension Dr of a probability is related to the asymptotic rate at which the expected distance (raised to the rth power) to the support of the quantized version of the probability goes to zero as the size of the support is allowed to go to infinity. This assumes that the quantized versions are in some sense ``optimal'' in that the expected distances have been minimized. In this dissertation we give a short history of quantization as well as some basic facts. We develop a generalized framework for the quantization dimension which extends the current theory to include a wider range of probability measures. This framework uses the theory of thermodynamic formalism and the multifractal spectrum. It is shown that at least in certain cases the quantization dimension function D(r)=Dr is a transform of the temperature function b(q), which is already known to be the Legendre transform of the multifractal spectrum f(a). Hence, these ideas are all closely related and it would be expected that progress in one area could lead to new results in another. It would also be expected that the results in this dissertation would extend to all probabilities for which a quantization dimension function exists. The cases considered here include probabilities generated by conformal iterated function systems (and include self-similar probabilities) and also probabilities generated by graph directed systems, which further generalize the idea of an iterated function system.

We consider random fractals generated by random recursive constructions, prove zero-one laws concerning their dimensions and find their packing and Minkowski dimensions. Also we investigate the packing measure in corresponding dimension. For a class of random distribution functions we prove that their packing and Hausdorff dimensions coincide.