In this thesis, we establish the asymptotic analysis of the singularly perturbed reaction diffusion equation [cataloger unable to transcribe mathematical equations].... Our results establish the specific dependency on the coefficients of this equation and the size of the parameter [delta] with respect to [epsilon]. The analyses include equation subject to Dirichlet and Neumann boundary conditions. In both cases, the solutions u[superscript epsilon] converge locally uniformally to the equilibria of the reaction term f. We characterize the limiting behavior of the solutions through the viscosity solution of a variational inequality. To construct the coefficients defining the variational inequality, we apply concepts developed for the homogenization of elliptic operators. In chapter two, we derive the convergence results in the Neumann case. The third chapter is dedicated to the analysis of the Dirichlet case. / text

Thesis (Ph.D)--University of Wollongong, 2004. / Typescript. Includes bibliographical references: leaf 163-173.

Thesis (M.Sc.) - Carleton University, 2006. / Includes bibliographical references (p. 88-93). Also available in electronic format on the Internet.

Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1993. / Vita. Abstract. Includes bibliographical references (leaves 98-102). Also available via the Internet.

Thesis (doctoral)--Justus Liebig-Universität Giessen, 2005. / "Teil 1 der Arbeit mit dem Literaturverzeichnis ist in Heft 255 erschienen."

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

No description available.

In this dissertation, we discuss spatiotemporal patterns in the wake of traveling waves in a microelectrode array (MEA) recording of a human epileptic seizure. In chapter two, we describe a method for estimating the direction of planar waves found in the last third of the seizure. We categorize the different phenomena that occur during those waves when projected along a one-dimensional slice in the domain. In chapter three, we summarize known examples of patterns in the wake of traveling wave solutions to reaction-diffusion systems. A brief review of results regarding the spectral stability of traveling wave solutions to reaction-diffusion equations is provided in chapter four. We review the essential spectrum and absolute spectrum, and summarize results about glued front-and-back pulse solutions. Using a reaction-diffusion model with Morris-Lecar dynamics, we present numerical experiments on a one-dimensional domain that exhibit spatiotemporal patterns in the wake of traveling waves. These patterns are precipitated by “backfiring” waves emitted from the primary wave in the opposite direction of initial travel, and qualitatively reproduce many of the features found in the last third of the seizure. A review of the model is given in chapter five. and a description of the phenomena found over an exhaustive set in a relevant parameter space of the model is given in chapter six. We compute branches of solutions in the parameter plane using numerical continuation in chapter seven. We describe the different types of solutions found along these branches. We present results on a curve of solutions where two branches of homoclinic orbits to equilibria in the moving coordinate frame meet at a heteroclinic loop, or T-point. We analyze the linear stability of solutions along this branch and draw comparisons to a known model that exhibits backfiring behavior. In chapter eight, we discuss seizure behavior in two spatial dimensions and present numerical experiments of the Morris-Lecar model in two dimensions. We describe results from backfiring waves initiated by a single point source and by two point sources in a two-dimensional domain. We show examples of simulations generated by two point sources that mimic the patterns in the empirical data.

No description available.

Members of the class Aves possess integumentary structures which distinguish them from other vertebrate lineages. The characteristic integumentary structure that defines the Aves from other vertebrates are the feathers, whose functions include insulation, camouflage, visual display, gliding, and powered flight. The recent discoveries of theropod dinosaur fossils displaying feather-like structures have led to interest in the morphological innovations of the feathers, which are associated with the evolution of flight in Aves. Most modern birds, display a highly ordered, hexagonal arrangement of feather follicles, which aids in the streamlining of the body to increase aerodynamic efficiency. Using the chicken embryo as a developmental model, I address the cellular and molecular processes involved in the initiation and formation of a high fidelity periodic pattern of feather primordia. From my studies, I propose a model in which the induction of individual feather primordia begins with the activation of FGF20 expression. This gene encodes a protein that serves as a chemoattractant. Aggregation of cells towards sources of FGF20 stimulates and reinforces FGF20 expression and also induces the expression of BMP4. Via a reaction-diffusion-like mechanism, BMP4 acts to limit the growth of the cell aggregate and promotes lateral inhibition to prevent fusions between neighbouring feather primordia through transcriptional regulation of FGF20. In order to achieve a high fidelity periodic pattern of feather primordia, three components are required; 1) a competent epidermis displaying β-Catenin and EDAR expression, 2) wave-like propagation of EDA expression, which acts synergistically with β-Catenin expression to activate FGF20 expression at the β-Catenin/EDA junction, 3) and a dermis of sufficient cell density. The spatiotemporal wave-like propagation of EDA expression, specifically, promotes the sequential induction of new feather primordium rows and is associated with the formation of a high fidelity periodic pattern. The importance of these three components appears to be evolutionarily conserved among the Aves and differences in the periodic pattern of feather primordia between species can be explained by how the three components are expressed or regulated in individual species. Independent losses of flight in ratites, such as ostriches and emus, are associated with the loss of feather pattern fidelity. In emus, this loss of pattern fidelity results from the delayed formation of a dermis of sufficient cell density, which prevents the induction of feather primordium formation within the dorsal tract, despite the presence of a fully primed and competent epidermis. These studies demonstrate how the precise feather pattern arises during embryonic development in birds, and how feather patterns can be modified through differential regulation of the molecular and cellular toolkit involved in feather primordium induction in different bird species.