Thesis (Ph. D.)--University of Oregon, 2001. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 124-127). Also available for download via the World Wide Web; free to University of Oregon users.

The purpose of this paper is to generalize regression models for repeated categorical data based on maximizing a conditional likelihood. Some existing methods, such as those proposed by Duncan (1985), Fischer (1989), and Agresti (1993, and 1997) are special cases of this latent variable approach, used to account for dependencies in clustered observations. The generalization concerns the incorporation of rather general data structures such as subject-specific time-dependent covariates, a variable number of observations per subject and time periods of arbitrary length in order to evaluate treatment effects on a categorical response variable via a linear parameterization. The response may be polytomous, ordinal or dichotomous. The main tool is the log-linear representation of appropriately parameterized Rasch-type models, which can be fitted using standard software, e.g., R. The proposed method is applied to data from a psychiatric study on the evaluation of psychobiological variables in the therapy of depression. The effects of plasma levels of the antidepressant drug Clomipramine and neuroendocrinological variables on the presence or absence of anxiety symptoms in 45 female patients are analyzed. The individual measurements of the time dependent variables were recorded on 2 to 11 occasions. The findings show that certain combinations of the variables investigated are favorable for the treatment outcome. (author´s abstract) / Series: Research Report Series / Department of Statistics and Mathematics

Tessellations have a rich mathematical structure and are especially appropriate as a context for teaching geometry in the middle grades. Few studies have researched how children conceptualize and learn tessellations in spite of their international use in educational contexts. This exploratory study looks at how fourth grade students conceptualize tessellations before instruction. The analysis is done from a Piagetian, cognitive viewpoint and from an aesthetic viewpoint. It is argued that the aesthetic viewpoint is crucial and foundational to children's mathematical understanding, just as it is for mathematicians. A series of clinical interviews was conducted with six fourth grade children. The results identified common themes of children's understanding, strategies, reasoning, and aesthetic criteria for tessellations. Children's ontology varied between object and process conceptions of tessellations. Children struggled especially with the infinite space of mathematical tessellations. Children's aesthetics, including symmetry, influenced their choices in creating tessellations and are shown to have played a cognitive role in children's mathematical exploration of tessellation structures. Mathematics influences students' aesthetic appreciation of tessellations and, more importantly, aesthetics drives the study of the mathematical structure of tessellations. Children's aesthetic criteria were the same as mathematicians', but with much different emphases. Other results are discussed, including the mathematical content elicited by the tasks, the influence of the tools used to create tessellations, the children's epistemology of their tessellations, and the role symmetry played in giving children confidence. Recommendations for future research and possible implications for curriculum and instruction are noted. / text

We review the important features of Little Higgs models, illuminating the mechanisms that generate a naturally light Higgs field while avoiding excessive fine-tuning. The full spectrum of the Littlest Higgs model is analyzed in light of the recent discovery of the Higgs boson mass. We find viable regions in parameter space that are consistent with the standard model. / text

We describe combinatorial techniques for studying log Calabi-Yau surfaces. These can be viewed as generalizing the techniques for studying toric varieties in terms of their character and cocharacter lattices. These lattices are replaced by certain integral linear manifolds described in [GHK11], and monomials on toric varieties are replaced with the canonical theta functions defined in [GHK11] using ideas from mirror symmetry. We classify deformation classes of log Calabi-Yau surfaces in terms of the geometry of these integral linear manifolds. We then describe the tropicalizations of theta functions and use them to generalize the dual pairing between the character and cocharacter lattices. We use this to describe generalizations of dual cones, Newton and polar polytopes, Minkowski sums, and finite Fourier series expansions. We hope that these techniques will generalize to higher rank cluster varieties. / text

Many physical systems break symmetry in their evolution. Biophysical systems, such as cells, developing organisms, and even entire populations, are no exception. Developing neurons represent a striking example of a biophysical system that breaks symmetry: neurons cultured in vitro begin as cell bodies with several tendrils (“neurites”) growing outward. A few days later, these same neurons invariably have the same new morphology: exactly one of the neurites (the “axon”) has grown hundreds of microns in length, while the others (the “dendrites”) are much shorter and are more branched. Previous work has shown that any of the neurites can become the axon, and so neurons must break symmetry during their development. The mechanisms underlying neuronal symmetry breaking and axon specification have recently attracted attention, with multiple groups proposing biophysical models to explain the phenomena. In this thesis, we perform the first analytical comparisons of these models by conducting multiple phenotypic and morphological studies of neurite growth in developing neurons. Studying neurite dynamics is technically challenging because neurites have unpredictable morphologies. In Chapter 4, we study neurite competition and neuronal symmetry breaking in hundreds of neurons by optically patterning micron-wide stripes to which the neurons adhere, and on which they grow exactly two neurites. We then use our measurements to test the accuracy of the models in the simple case when a neuron has exactly two neurites. In Chapter 5, we no longer constrain neuronal morphology. One characteristic of symmetry breaking systems is how the system’s complexity affects the symmetry breaking. We find that a majority of the models predict that neurons with more neurites break symmetry much slower than neurons with fewer neurites. Experimentally, we find that neurons with different neurite counts break symmetry at the same rate, consistent with previous observations. We then determine why the models disagree in their predictions, and rectify the models using our own experimental data. In particular, we find that neurons with higher neurite counts have higher concentrations of key proteins involved in symmetry breaking, so that neurons, regardless of neurite count, can break symmetry at the same rate. / Physics

Can symmetry be utilized as a design principle to constrain evolutionary search, making it more effective? This dissertation aims to show that this is indeed the case, in two ways. First, an approach called ENSO is developed to evolve modular neural network controllers for simulated multilegged robots. Inspired by how symmetric organisms have evolved in nature, ENSO utilizes group theory to break symmetry systematically, constraining evolution to explore promising regions of the search space. As a result, it evolves effective controllers even when the appropriate symmetry constraints are difficult to design by hand. The controllers perform equally well when transferred from simulation to a physical robot. Second, the same principle is used to evolve minimal-size sorting networks. In this different domain, a different instantiation of the same principle is effective: building the desired symmetry step-by-step. This approach is more scalable than previous methods and finds smaller networks, thereby demonstrating that the principle is general. Thus, evolutionary search that utilizes symmetry constraints is shown to be effective in a range of challenging applications. / text

published_or_final_version / Applied Statistics / Master / Master of Social Sciences

No abstract available. / Thesis (M.Sc.)-University of Natal, 1997.

We consider the study of symmetry analysis of difference equations. The original work done by Lie about a century ago is known to be one of the best methods of solving differential equations. Lie's theory of difference equations on the contrary, was only first explored about twenty years ago. In 1984, Maeda [42] constructed the similarity methods for difference equations. Some work has been done in the field of symmetries of difference equations for the past years. Given an ordinary or partial differential equation (PDE), one can apply Lie algebra techniques to analyze the problem. It is commonly known that the number of independent variables can be reduced after the symmetries of the equation are obtained. One can determine the optimal system of the equation in order to get a reduction of the independent variables. In addition, using the method, one can obtain new solutions from known ones. This feature is interesting because some differential equations have apparently useless trivial solutions, but applying Lie symmetries to them, more interesting solutions are obtained. The question arises when it happens that our equation contains a discrete quantity. In other words, we aim at investigating steps to be performed when we have a difference equation. Doing so, we find symmetries of difference equations and use them to linearize and reduce the order of difference equations. In this work, we analyze the work done by some researchers in the field and apply their results to some examples. This work will focus on the topical review of symmetries of difference equations and going through that will enable us to make some contribution to the field in the near future. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.