Problems involving partial or ordinary differential equations arise in various fields of science. Therefore, the task of obtaining exact solutions of differential equations is of primary importance, and attracts high attention. The main purpose of the current thesis is the development of a Maple-based, symbolic software package for symmetry reduction of differential equations and computation of symmetry-invariant solutions. The package developed in the current thesis is compatible with and can be viewed as an extension of the package GeM for symbolic symmetry analysis, developed by Prof. Alexei Cheviakov. The reduction procedure is based on the Lie's classical symmetry reduction method involving canonical coordinates. The developed package is applicable for obtaining solutions arising from extension of Lie's method, in particular, nonlocal and approximate symmetries. The developed software is applied to a number of PDE problems to obtain exact invariant solutions. The considered equations include the one-dimensional nonlinear heat equation, the potential Burgers' equation, as well as equations arising in nonlinear elastostatics and elastodynamics.

Problems involving partial or ordinary differential equations arise in various fields of science. Therefore, the task of obtaining exact solutions of differential equations is of primary importance, and attracts high attention. The main purpose of the current thesis is the development of a Maple-based, symbolic software package for symmetry reduction of differential equations and computation of symmetry-invariant solutions. The package developed in the current thesis is compatible with and can be viewed as an extension of the package GeM for symbolic symmetry analysis, developed by Prof. Alexei Cheviakov. The reduction procedure is based on the Lie's classical symmetry reduction method involving canonical coordinates. The developed package is applicable for obtaining solutions arising from extension of Lie's method, in particular, nonlocal and approximate symmetries. The developed software is applied to a number of PDE problems to obtain exact invariant solutions. The considered equations include the one-dimensional nonlinear heat equation, the potential Burgers' equation, as well as equations arising in nonlinear elastostatics and elastodynamics.

An exponential growth in the offshore industry has resulted in a corresponding increase in demand for quick, accurate, and implementable designs. With the increase in size of the structure relative to the wave amplitude, analysis should be performed using the diffraction theory. The steady wave drift force on a submerged body is a second-order quantity. With a potential flow assumption, the force arises from the diffraction and radiation of the waves from the interaction with the body. For a fixed body in waves the steady force is contributed from the wave diffraction effect alone. Numerical solutions for are generally needed for the computation of the steady drift force on submerged structures. In this study the steady wave drift forces on several fixed bodies of basic shapes are derived in closed form. The thesis addresses the steady drift forces on the following basic structures: a box, a vertical circular cylinder, a submerged horizontal cylinder, a bottom-seated horizontal half cylinder, a bottom-seated hemisphere, a submerged sphere, and an ellipsoid. The results developed demonstrate the importance of various independent non-dimensional parameters. To achieve speed and accuracy of the analytical/numerical solutions for the second order forces on the basic bodies with symmetry has been presented. Mathematical formulation of the boundary value problem and its second order solution have been described using the different coordinates depending on the symmetry and nature of the object. Charts and formulas have been developed to provide solution for the second order wave forces on different basic structures like cylinder, sphere and ellipsoids. This study is helpful for a first pass estimate of the steady drift force where the translational and rotational contributions are neglected. The illustrative examples provide a sense of the accuracy and an approach to bound the results of complex geometries by approximating them as simpler geometries.

The symmetry energy and the isoscaling properties of the fragments produced in multifragmentation of 40Ar, 40Ca + 58Fe, 58Ni reactions at 25, 33, 45 and 53 MeV/nucleon were investigated within the framework of a statistical multifragmentation model. The isoscaling parameter, α from the hot primary and cold secondary fragment yield distributions, was studied as a function of the excitation energy, isospin (neutronto- proton asymmetry), and fragment symmetry energy. Through changing the symmetry energy in the statistical multifragmentation model to describe the experimental data, it is observed that the isoscaling parameter α decreases with increasing excitation energy and decreases with decreasing symmetry energy. The parameter α is also observed to increase with increasing difference in the isospin of the fragmenting system. The sequential decay of the primary fragments into secondary fragments show very little influence on the isoscaling parameter when studied as a function of excitation energy. However, the symmetry energy has a strong influence on the isospin properties of the hot fragments. The results indicate that the isospin properties of the fragments produced at high excitation energy and reduced density in multifragmentation reactions are sensitive to the symmetry energy, indicating that the properties of hot nuclei at excitation energies, densities, and isospin away from normal ground state nuclei are significantly different than those of normal (cold) nuclei at saturation density.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2001. / Includes bibliographical references. Also available on the Internet.

Thesis (M.Sc.(Natural and Agricultural Sciences)) -- University of Pretoria, 2009. / Summary in English. Includes bibliographical references (leaves 55-56).

Thesis (Ph. D.)--University of Oregon, 2003. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 69-71). Also available for download via the World Wide Web; free to University of Oregon users.

Cityscapes is a 13-minute work for full orchestra. It serves not only as a summary of my experiences while in Austin, Texas, but also as a representation of my musical style and creative influences. The composition consists of three movements, the first and second of which are elided into one another. The formal structure of Cityscapes is governed by the meshing and combining of recurring motivic figures, the framing of themes and sections based on two distinct scalar patterns, and the symmetry and intervallic design extrapolated from these two patterns. / text

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