The purpose of this thesis is to examine general properties, convergence, and limit points of sequences of real numbers.

Thesis (Ph. D.)--Oregon State University, 1963. / Typescript. Includes bibliographical references (leaves 180-182). Also available on the World Wide Web.

The subject of this work is functional equations with direction towards linear functional equations. The .rst part describes function sets where iterates of the functions converge to a .xed point. In the second part the convergence property is used to provide solutions to linear functional equations by de.ning solutions as in.nite sums. Furthermore, this work contains some transforms to linear form, examples of functions that belong to di¤erent classes and corresponding linear functional equations. We use Mathematica to generate solutions and solve itera- tively equations.

The directionality and consistency of evolution has long been a subject of contention among evolutionary biologists since the days of Darwin. However, it is unknown how much can be quantified and how much results from more complex variables. It is also unknown whether evolution is consistent or whether it occurs differently in each system. My study focuses on predation and habitat as ecological gradients that may create convergent evolution in livebearing fishes. In Chapter I, I focus on predation as a mechanism for driving convergent evolution in Gambusia affinis. A suite of 7 microsatellite markers was used in order to determine independence of morphological evolution. Mantel tests were used to compare genetic, phenotypic, geographical and environmental distances among the six focal populations. These tests showed that there was a significant correlation between genetic and geographic distance but no significant correlation between genetic and phenotypic distances, which may indicate that phenotypic divergence has arisen independently in multiple instances. The second chapter focuses on a unique form of convergence that arose during speciation of three livebearing fishes, which we termed "convergent speciation." I focus on habitat type as a selective pressure in the lake system of Lake Catemaco, Mexico and the surrounding rivers. Lake Catemaco has been isolated from the surrounding rivers for approximately 1.2 million years and during that time several endemic species have evolved in the lake. This provides an excellent study system for studying convergent divergence. To test the theory of convergent speciation in this system, a MANOVA was conducted. The effect of habitat was an important source of variance in the system, indicating that habitat is a likely driving force responsible for convergent speciation in the system. Using discriminant functions I was able to correctly predict the habitat of fish of six different species between 68% and 71% of the time. This may indicate that evolutionary response to habitat is consistent across taxa (i.e., convergent divergence is taking place).

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An increase in the amount of lightning and rainfall over the Houston area, compared to the surrounding rural areas, has been well documented in previous studies. The placement of a Shared Mobile Atmospheric Research and Teaching Radar (SMART-R) in the Houston area during the summer season of 2005 presented a unique opportunity to investigate the role of boundary-layer convergence in modulating convective frequency and intensity and, thereby, likely causing the rainfall and lightning anomalies. The role of the urban heat island (UHI) and the sea-breeze, as a source of low-level convergence leading to enhanced convection over Houston, was examined. Hourly average dual-Doppler wind and convergence maps were created on 1 X 1 km grids for an eleven-week period. By using these images along with average lightning, rainfall, and reflectivity for a large Houston-centered domain, it was possible to discern a correlation between low-level convergence and convection. Also, past findings of enhancement in lightning and rainfall over Houston and downwind of Houston were validated. High convergence levels for the Houston area in the mid-morning were followed closely by a peak in convection in the early afternoon. The enhancement of rainfall and lightning over and downwind of downtown was found to be primarily from a large increase in frequency of deep convective events when compared to the surrounding domain. Also, it was found that UHI, rather than sea-breeze, was likely the primary causative mechanism in the development of convection over the Houston area because of the lack of deep convection in areas equally affected by the sea-breeze and the timing of the convection compared to time of peak sea-breeze. An area of weaker enhancement south of Houston, not discussed in previous studies, was found to be present, possibly from the interactions between the bay-breeze off of the Galveston Bay and the seabreeze.

Die ökonomische Wachstumsliteratur ging lange Zeit davon aus, dass die absolute Konvergenz zwischen Ländern (und Regionen) ein Naturgesetz ist. Mittlerweile weiß man, dass die Länder eher einer Clubkonvergenz, die Pro-Kopf- Einkommen also eher einer Polarisierung unterliegen. Neben einer theoretischen Beschreibung der beiden Ansätze erläutert der Beitrag einige ausgewählte Erklärungsansätze für das Phänomen der Clubkonvergenz. Außerdem wird die empirische Einkommensverteilung in den Regionen der EU dargestellt. / In its beginnings, growth literature viewed absolute convergence between countries (and regions) as a kind of natural law. In the meantime, however, it is known that countries follow rather club convergence, which means that per capita incomes polarise. The two concepts will first be defined. Thereafter, selected approaches seeking an explanation for this phenomenon are presented. The article closes with an analysis of the empirical distribution of incomes across the regions of the EU.

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

Thesis (doctoral)--Faculté des sciences de Lund, 1918. / Includes bibliographical references.

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