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A Fourier transform for a finite group G is an isomorphism from the complex group algebra CG to a direct product of complex matrix algebras, which are determined beforehand by the structure of G. Given such an isomorphism, naive application of that isomorphism to an arbitrary element of CG takes time proportional to |G|2. A fast Fourier transform for some (family of) groups is an algorithm which computes the Fourier transform of a group G of the family in less than O(|G|2) time, generally O(|G| log |G|) or O(|G|(log |G|)2). I describe the construction of a fast Fourier transform for the special linear groups SL(q) with q = 2n.

We develop noncommutative field theory, starting from a very basic background and explore recent and important results in classical noncommutative field theory. The background section is of interest because it presents mathematical and physical interpretations of differential geometry together in a coherent way, not seen in most of the literature. We present several interesting examples that resulted from recent research in the field.

An experimental search based on previous published theoretical work was made for de Haas-van Alphen-like quantum oscillations in field emission current. The study was motivated by the possible applicability of de Haas-van Alphen measurements to the study of Fermi surfaces near real surfaces, Field emitters were fabricated from bismuth single crystals grown from the melt by a modified Bridgeman technique. Field emission current was measured with the field emitter cooled by contact with a liquid helium bath. Most measurements were made at 4.2° K, although a few measurements were made at 2.02°K; Fowler-Nordheim plots of the experimental current-voltage data were linear over several orders of magnitude. The field emission current was measured as a function of magnetic field strength to twenty kilogauss and as a function of direction, with respect to the emitter axis, for a steady field of ten kilogauss. The results of measurements on four field emitter crystals are reported in this thesis. In most of this work, de Haas-van Alphen-like quantum oscillations were not observed. In one set of data, however, de Haas-van Alphen-like oscillations having a period of 0.50 x 10⁻⁵ G⁻¹ appear to be present. The published theoretical work was reevaluated. As a result of a large effective Dingle temperature for field emitters, and because the effective masses of carriers in bismuth are anisotropic, the de Haas-van Alphen effect in field emission current could be several times smaller than has been suggested in the literature. The sensitivity of the experiments reported in this thesis was not sufficient to allow unambiguous identification of quantum oscillations at the reduced level. An outline of an experimental procedure suitable for observation of the de Haas-van Alphen-like effect in field emission current at the reduced level is provided in the final chapter.

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The cerebral cortex contains many neurons. The neuron is part of the nervous system and it receives and transmits the electrical signals. These signals are significant to a human's behaviour. Since the neurons are charged, these charges produce electrical fields, so these neural signals can be measured by using scalp electrodes in electroencephalography (EEG). As long as the brain is not dead, the spontaneous activities of neurons will produce a series of EEG signals. There are many models that have been developed for simulating the cortical signal, and mostly each model is focused towards a different purpose or application. Often, a different computer code has to be written for each different application, and this can be inefficient. Therefore, this project aims to develop a software system for simulating cortical signals where the model used for the system can be changed easily. Furthermore, the system is requested to be versatile and easy-to-use for many applications. The developed system is written in MATLAB in response to a user requirement and mostly applies to any model which uses a mean-field approach. Only the specific inputs need to be modified for changing the model. This thesis details how this system is developed. The main limitation of the system is computational resources, much the same as other cortical modelling. However, all the user requirements had been satisfied. The system can simulate the response of the neurons for any condition and generate simulated EEG data to the user. The user can analyze the cortical activities using the standard signal processing techniques such as a power spectrum. This software is very helpful for the research of sleep and anaesthesia.

Thesis (M.S.)--Virginia Polytechnic Institute and State University, 1988. / Vita. Abstract. Includes bibliographical references (leaves 82-84). Also available via the Internet.

The human visual system has the ability o utilize motion information to infer the shapes of surfaces. More specifically, we are able to derive descriptions of rigidly rotating smooth surfaces entirely from the orthographic projection of the motions of their surface markings. A computational analysis of this ability is proposed based on "shape from motion" proposition. This proposition states that given the first spatial derivatives of the orthographically projected velocity and the acceleration fields of a rigidly rotating regular surface, then the angular velocity and the surface normal at each visible point on that surface are uniquely determined up to a reflection.

Thesis (Ph.D.)--University of Delaware, 2006. / Principal faculty advisor: Dennis W. Prather, Dept. of Electrical and Computer Engineering. Includes bibliographical references.

Thesis (M.S.)--Ohio State University. / Bibliography: leaves 62-64. Available online via OhioLINK's ETD Center