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Please read the abstract in the section front of this document. / Thesis (PhD)--University of Pretoria, 2010. / Mathematics and Applied Mathematics / unrestricted

This thesis is broadly concerned with two problems: investigating the ergodic properties of boundary actions, and investigating various properties of Schreier graphs. Our main result concerning the former problem is that, in a variety of situations, the action of an invariant random subgroup of a group G on a boundary of G (e.g. the hyperbolic boundary, or the Poisson boundary) is conservative (there are no wandering sets). This addresses a question asked by Grigorchuk, Kaimanovich, and Nagnibeda and establishes a connection between invariant random subgroups and normal subgroups. We approach the latter problem from a number of directions (in particular, both in the presence and the absence of a probability measure), with an emphasis on what we term Schreier structures (edge-labelings of a given graph which turn it into a Schreier coset graph). One of our main results is that, under mild assumptions, there exists a rich space of invariant Schreier structures over a given unimodular graph structure, in that this space contains uncountably many ergodic measures, many of which we are able to describe explicitly.

Eastern Newts (Notophthalmus viridescens) are a ubiquitous member of eastern North America’s caudate fauna. Unlike the typical amphibian, their life cycle is split into three phases instead of two, commonly called a triphasic life cycle. The larvae of N. viridescens are fully aquatic, eventually metamorphosing to become terrestrial juveniles, called efts. Upon sexual maturity, the eft will metamorphose into a semi-aquatic adult where its external morphology is typical of an aquatic salamander. Since there are apparent differences in their ecological niche, there are different forces acting on their skeletons. We hypothesize that due to differences in buoyancy, torsion, and locomotion, differences are expected in the morphology of the axial skeleton. Using image data generated on a SkyScan 1273 micro-computed tomography (µCT) scanner, 3D shape analyses will be used to quantify shape differences between vertebrae and test the hypothesis. Three dimensional digital models of each vertebrae of interest will be rendered from the scans in Dragonfly (Object Research Systems). Each 3D model is then loaded into SlicerMorph (3D Slicer), where landmarks are placed upon homologous structures on each vertebra. A Generalized Procrustes Analysis (GPA) followed by a Principal Component Analysis (PCA) is conducted for each vertebra to test for potential shape differences between each life stage. GPA and PCA analysis will be conducted on 10 terrestrial juveniles, 10 semi-aquatic adults, 5 aquatic juveniles, and 5 paedomorphic adults. The 5 aquatic juveniles and 5 paedomorphic adults, eastern newts that remain in the water through their entire lives, will validate if the semi-aquatic adult is truly adapting towards an aquatic lifestyle. If GPA and PCA indicate statistical shape differences between certain vertebrae, those vertebrae will be run through the Automated Landmarking through Pointcloud Alignment and Correspondence Analysis (ALPACA) module of SlicerMorph to produce heatmap data on the 3D models showing where exactly the shape changes are occurring in the vertebra.

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No description available.

Geometric trimming of a surface involves removal of unwanted portions of the surface and providing a new mathematical description for the trimmed patch. This entails creating a new geometry for the trimmed patch, which closely approximates the corresponding portion on the original patch. The procedure is shown to involve obtaining data points on the B-spline surface that lie within the region specified by the parameter values for the trimming curve and describing a new surface which interpolates this new set of data points. This research looks at optimizing the procedure described above by basing the choice of parameter values for the trimming curve, at points where curvature optima occur over the surface. A visualization tool kit has been developed using OpenGL, as a means to discern the difference between the two surfaces. In order to quantify and aid in minimizing the error (difference) in approximating the original surface with the trimmed patch, an error measurement tool developed in MATLAB has been employed. / Master of Science

Most of the work accomplished on surface blending is based on visual trimming. In the process of visual trimming, the unwanted portion of a surface is only hidden but not removed. Geometric trimming provides a complete mathematical description of the wanted portion of the trimming surface, and generates a new mathematical surface or sets of surface patches. The new surface is intended to resemble closely the corresponding portion of the original surface. A robust procedure is developed to geometrically trim the intersecting surfaces and blend the trimmed surface patches into one new surface. This research generates a filleting algorithm for surface blending of an aircraft fuselage shape and a wing shape at a closed trimming intersection curve, and verifies the properties of the newly created surface. In order to distinguish how well the new surface approximates the original, an error comparison tool developed in MATLAB has been employed. / Master of Science