An integral transform which reproduces a transformable input function after a finite number $N$ of successive applications is known as a {\it cyclic} transform. Of course, such a transform will reproduce an arbitrary transformable input after $N$ applications, but it also admits eigenfunction inputs which will be reproduced after a...
Interfacial phenomena play a considerable role in many physical, chemical and biological systems. Some of these systems exhibit interaction of several interfaces. This thesis contains a study of two types of systems characterized by interacting interfaces. First, a theory of the formation of nanoscale porous structures in oxides of metals...
Problems involving irregularly shaped domains or embedded interfaces occur in a wide range of mathematical models. These include models in computational biology, fluid mechanics, and solidification. Numerical solutions for this type of problem can be difficult to obtain, and therefore a variety of methods have been devised to solve them....
This work explores the development and implementation of a biofilm model for multiple solid species and multiple dissolved components. The model makes use of several different advanced numerical methods in order to achieve the goal of describing the behavior of biofilms growing in wastewater treatment reactors. Using the model we...
Motivated by the rich variety of complex patterns observed on the surface of fluid layers that are vibrated at multiple frequencies, we investigate the effect of such resonant forcing on systems undergoing a Hopf bifurcation to spatially homogeneous oscillations. We use an extension of the complex Ginzburg-Landau equation (CGLE) that...
We consider two problems in low Reynolds-number, interfacial fluid mechanics: the rupture of thin liquid films on chemically patterned solid substrates, and the engulfment of foreign particles by a solidification front progressing through a binary alloy.
First we investigate the stability and rupture of thin liquid films on patterned sub-...
Rare events are studied in an increasing number of areas, ranging from lightwave and optical communication systems, to industrial routing problems, to rogue ocean waves, to financial asset pricing, to the rare failure of something as common as Gaussian elimination, as well as numerous others. In optical communication systems, a...
Across a broad spectrum of mathematical models, problems arise that have irregular domains or contain embedded interfaces. The complexity due to the introduction of these interfaces makes it more difficult to develop efficient and accurate numerical methods for their solutions.
This thesis discusses modifications to the eXtended Finite Element Method...
Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to chemical stimuli that can be either supplied externally or produced locally.
Understanding of the fundamental mechanisms that govern angiogenesis suggests a powerful therapeutic approach that will allow to combat...
The self-assembly of quantum dots (QDs) in thin solid films is an important area of nanotechnology with many relevant applications. In the present thesis, three problems related to the growth and self-assembly of QDs are investigated.
In Chapter 1, a new instability mechanism for the formation of QDs associated with...