An interaction theory for scattering by defects in arrays

Wave scattering by an array of bodies that is periodic except for a finite number of missing or irregular elements is considered. The field is decomposed into contributions from a set of canonical problems, which are solved using a modified array scanning method. The resulting interaction theory for defects is very efficient and can be used to construct the field in a large number of different situations. Numerical results are presented for several cases, and particular attention is paid to the amplitude with which surface waves are excited along the array. We also show how other approaches can be incorporated into the theory so as to increase the range of problems that can be solved

Localizations with noncompact transverse spaces and covert symmetry breaking in supergravity

The focus of my doctoral work has been answering the question: Can lower-dimensional effective gravitational theories be found in a higher-dimensional theory with a non-compact transverse space? To answer this question this thesis is divided into two parts. First, I explore supergravity solutions on warped product manifolds and argue that they correspond to solutions of a modified Laplacian. I pay special attention to Type III † solutions, or solutions characterized by the presence of non-constant transverse zero modes, and emphasize that these are the only higher dimensional solutions corresponding to localized sources that yield effectively lower-dimensional physics when the transverse space has infinite volume. Second, I derive the lower dimensional effective field theory about such backgrounds. I discover that these effective field theories have covert symmetry breaking, spontaneous breaking of gauge symmetry which only appears at quartic order. I show this explicitly for D = d + 1 scalar electrodynamics with any boundary condition that corresponds to a non-constant transverse zero mode. The mathematical prerequisite for both of these conclusions is Sturm–Liouville theory with precise manipulations of Green’s formula. To support this I derive the fundamental conclusions of Sturm–Liouville theory for a restricted class of operator, that is the Laplacian, which relaxes some requirements for permissible boundary conditions of Sturm–Liouville theory. The answer to my focal question is: Yes; however, the lower-dimensional theory has novel corrections which were previously unexplored, and further research is indicated.Open Acces

Numerical Simulation

Nowadays mathematical modeling and numerical simulations play an important role in life and natural science. Numerous researchers are working in developing different methods and techniques to help understand the behavior of very complex systems, from the brain activity with real importance in medicine to the turbulent flows with important applications in physics and engineering. This book presents an overview of some models, methods, and numerical computations that are useful for the applied research scientists and mathematicians, fluid tech engineers, and postgraduate students