Thesis (Ph. D.)--University of Wisconsin--Madison, 1974. / Typescript. Vita. Description based on print version record. Includes bibliographical references (leaves 82-83).

Diss.--Genève.

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 44)

Diss.--Genève.

Thesis (M.Sc.)(Mathematics)--University of Pretoria, 2006. / Includes summary. Includes bibliographical references. Available on the Internet via the World Wide Web.

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.

In 1967, J. A. Erdős showed, using a matrix theory approach that the semigroup Sing[sub]n of singular endomorphisms of an n-dimensional vector space is generated by the set E of idempotent endomorphisms of rank n - 1. This thesis gives an alternative proof using a linear algebra and semigroup theory approach. It is also shown that not all the elements of E are needed to generate Sing[sub]n. Necessary conditions for a subset of E to generate found; these conditions are shown to be sufficient if the vector space is defined over a finite field. In this case, the minimum order of all subsets of E that generate Sing[sub]n is found. The problem of determining the number of subsets of E that generate Sing[sub]n and have this minimum order is considered; it is completely solved when the vector space is two-dimensional. From the proof given by Erdős, it could be deduced that any element of Sing[sub]n could be expressed as the product of, at most, 2n elements of E. It is shown here that this bound may be reduced to n, and that this is best possible. It is also shown that, if E+ is the set of all idempotent of Singn, then (E+)n−1 is strictly contained in Sing[sub]n. Finally, it is shown that Erdős's result cannot be extended to the semigroup Sing of continuous singular endomorphisms of a separable Hilbert space. The sub semigroup of Sing generated by the idempotent of Sing is determined and is, clearly, strictly contained in Sing.

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

In this work torque vectoring methods are used to compensate for a low performance steer-by-wire system. Currently a number of vehicle manufacturers are considering introducing steer-by-wire into their range of vehicles. Some of the key concerns for the manufacturers are safety and cost. The safety can be subdivided in the integrity of the steering system and the effect on handling. The focus of this study is the use of low cost steering actuators on a vehicle and identify its effects on the vehicle's handling response. The test vehicle is dune buggy modified to accommodate the low performance steer-by-wire system without a direct mechanical link between the steering wheel and the wheels and equipped with various sensors to data recording. In order to investigate the influence of torque vectoring system on the steer-by-wire, an eight degrees of freedom vehicle model in Matlab/Simulink has been developed. The eight degrees of freedom are longitudinal and lateral translations, yaw and roll motion and rotation of each wheel. The Matlab/Simulink model also includes the dynamics of the actuators, which is validated against the experimental data. The actuator was shown to have a bandwidth of less than 0.3 Hz. The eight degrees of freedom model's response was validated against experimental data for both steady state and transient response up to 0.5 g. The tyre forces and moments are implemented by using the Dugoff tyre model, which has been validated against experimentally measured data. The torque vectoring system uses the cascade approach based on a reference model, which uses a two degrees of freedom (bicycle model) to generate the reference signal for control purposes. The upper level yaw controller is based on the optimal control theory and uses the LQR (Linear-quadratic regulator) approach. The lower level wheel slip controller is based on a slidingmode structure and prevents tyre force saturation. The simulation results show that the vehicle augmented with the torque vectoring system outperforms the low performance steer-by-wire vehicle and also the vehicle with conventional steering arrangement.

In this thesis we prove the existence of vector-valued automorphic forms for an arbitrary Fuchsian group and an arbitrary finite dimensional complex representation of this group. For small enough values of the weight as well as for large enough values, we provide explicit formulas for the spaces of these vector-valued automorphic forms (holomorphic and cuspidal). To achieve these results, we realize vector-valued automorphic forms as global sections of a certain family of holomorphic vector bundles on a certain Riemann surface associated to the Fuchsian group. The dimension formulas are then provided by the Riemann-Roch theorem. In the cases of 1 and 2-dimensional representations, we give some applications to the theories of generalized automorphic forms and equivariant functions.