Bessel beams are optical fields which falls into the category of non-diffracting beams. Vector Bessel beams are vector beams possessing cylindrical symmetry. Cylindrically symmetric beams tend to have a tight focal point during propagation. The tight focal beam nature of vector Bessel beams makes them a good potential in various facets of science such as biological optical trapping, wireless communications, remote sensing, microscopy etc. In this research work, vector Bessel beams were generated using the phase of an Axicon that was encoded into a spatial light modulator. Firstly, scalar Bessel beams which possess linear polarization were generated and converted to circularly polarized vector beams by the use of a q-plate. The orbital angular momentum (OAM) modes that are embedded in the vortex beams were detected using modal decomposition technique. This was implemented for both the scalar and vector case using a quarter wave plate. The measure of the degree of non-separability of the vector Bessel beams using tomographic quantum tools was also implemented where the density matrix was reconstructed. The concurrence and fidelity which explore the measure of vectorness of both scalar and vector Bessel beams were calculated from the density matrix. The obtained results show that the spatial modes and polarization are coupled in the vector case as expected.

The d-MUSIC algorithm estimates the direction-of-arrival of two closely spaced sources using a single array snapshot. To make the problem full rank, d-MUSIC utilizes additional information, specifically the derivative of the input snapshot vector. The combined vector set yields a rank two signal space projector that can be used to estimate the source directions. To construct this projector, an estimate for the center of the target cluster is required. In many radar low angle tracking problems involving distant aircraft, the center of the target plus multipath cluster is known a priori (flat earth approximation). Otherwise, d-MUSIC estimates the source bearings for a grid of center angles and selects the grid point where the signal space of the solution is most consistent with the input vector. Following the approach of Stoica and Nehorai [10], a theoretical estimate for the d-MUSIC error variance is derived and compared to the Cramér-Rao bound for the case of a known cluster centroid (typical air traffic control problem). The algorithm nearly attains the Cramér-Rao bound, displaying a low sensitivity to signal correlation. A number of Monte Carlo tests are also performed to compare the performance of MUSIC to the two d-MUSIC algorithms (cluster center known or unknown). These tests demonstrate that both versions of d-MUSIC is highly resilient to signal correlation whereas MUSIC is not. The algorithm is field tested using data from a X-band radar tracking a low flying helicopter. The receive array is a 6 channel vertical linear array of horns with an array aperture of nearly 19 wavelengths. As the flat earth approximation is not appropriate to this experiment the grid search version of d-MUSIC is employed (unknown cluster center). The array is calibrated using the method of Wylie et al. [30] to restore the Toeplitz structure of the covariance matrix. With a spacing of 16% to 35% of a beamwidth between the direct and multipath signals, the d-MUSIC rms error for the source spacing is 9.6% of a beamwidth for the 4 data collections while MUSIC resolved the two signals for 2 of the 4 cases with a rms error of 18.1%. / Graduate

Matricial norms, minimal matricial norms, vectorial norms and vectorial norms subordinate to matricial norms, which are respectively generalizations of matrix norms, minimal matrix norms, vector norms and vectorial norms subordinate to matrix norms, are defined and their various applications and properties are discussed. / Science, Faculty of / Mathematics, Department of / Graduate

The discovery of a new truth in the older fields of mathematics is a rare event. Here an investigator may hope at best to secure greater elegance in method or notation, or to extend known results by some process of generalization. It is our purpose to make a study of conic sections in the spirit of the above remark, using the symbolism developed by Josiah Williard Gibbs.

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Shea Yuen Cheuk. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 91-94). / Abstracts in English and Chinese. / Introduction --- p.5 / Chapter 1 --- Prelimaries --- p.9 / Chapter 1.1 --- Basic Notations --- p.9 / Chapter 1.2 --- Multiresolution Analysis --- p.12 / Chapter 1.3 --- Orthonormal Wavelets --- p.17 / Chapter 1.4 --- Theory of Frames --- p.24 / Chapter 2 --- Construction of Orthonormal Wavelets --- p.33 / Chapter 2.1 --- Compactly Supported Smooth Orthonormal Wavelet in R --- p.33 / Chapter 2.2 --- Compactly Supported Smooth Orthonormal Wavelet in R2 --- p.40 / Chapter 3 --- Wavelet Frames --- p.51 / Chapter 3.1 --- Basic Properties --- p.51 / Chapter 3.2 --- Dual Wavelet Frame --- p.56 / Chapter 3.3 --- Canonical Dual Frame --- p.66 / Chapter 3.4 --- Oversampling --- p.69 / Chapter 4 --- MRA-Based Wavelet Frames --- p.74 / Chapter 4.1 --- Definitions --- p.74 / Chapter 4.2 --- Tight Frames Constructed by MRA --- p.77 / Chapter 4.3 --- Approximation Order and Vanishing Moments for Wavelet Frames --- p.82 / Chapter 4.4 --- Construction of MRA-Based Wavelet Frames --- p.85 / Bibliography --- p.91

Thesis (Ph. D.)--University of Chicago, 1921. / Published also without thesis note.

Thesis (Ph. D.)--University of Chicago, 1921. / Published also without thesis note.