Novel Wavelet-Based Statistical Methods with Applications in Classification, Shrinkage, and Nano-Scale Image Analysis

Given the recent popularity and clear evidence of wide applicability of wavelets, this thesis is devoted to several statistical applications of Wavelet transforms. Statistical multiscale modeling has, in the most recent decade, become a well-established area in both theoretical and applied statistics, with impact on developments in statistical methodology. Wavelet-based methods are important in statistics in areas such as regression, density and function estimation, factor analysis, modeling and forecasting in time series analysis, assessing self-similarity and fractality in data, and spatial statistics. In this thesis we show applicability of the wavelets by considering three problems: First, we consider a binary wavelet-based linear classifier. Both consistency results and implemental issues are addressed. We show that under mild assumptions wavelet-based classification rule is both weakly and strongly universally consistent. The proposed method is illustrated on synthetic data sets in which the truth is known and on applied classification problems from the industrial and bioengineering fields. Second, we develop wavelet shrinkage methodology based on testing multiple hypotheses in the wavelet domain. The shrinkage/thresholding approach by implicit or explicit simultaneous testing of many hypotheses had been considered by many researchers and goes back to the early 1990's. We propose two new approaches to wavelet shrinkage/thresholding based on local False Discovery Rate (FDR), Bayes factors and ordering of posterior probabilities. Finally, we propose a novel method for the analysis of straight-line alignment of features in the images based on Hough and Wavelet transforms. The new method is designed to work specifically with Transmission Electron Microscope (TEM) images taken at nanoscale to detect linear structure formed by the atomic lattice.Ph.D.Committee Chair: Vidakovic, Brani; Committee Member: Hayter, Anthony; Committee Member: Heil, Chris; Committee Member: Huo, Xiaoming; Committee Member: Wang, Yan

Fractions speak louder than words: Investigating preservice primary teachers’ knowledge and understanding for teaching fractions with representations

Mathematics presents specific challenges for primary preservice teachers and fractions is among the most problematic of topics. This thesis investigates preservice primary teachers’ understanding and use of fractions and fraction representations. Preservice teachers have particular difficulty explaining the rationale behind fraction operations, often only demonstrating superficial knowledge of symbolic procedures. This level of knowledge is insufficient for teaching and, thus, initial teacher education presents a crucial opportunity to deepen teachers’ knowledge before they begin their teaching careers. The study addresses the crucial need for further research into the initial teacher education of preservice teachers at a time where there is a national agenda for improving education in Australia. However, despite the potential to redress preservice teachers’ knowledge of fractions, there is a dearth of studies elucidating how fraction knowledge develops over a program in initial teacher education, particularly in an Australian context. To address this gap, the current study aimed to investigate the development of preservice primary teachers’ knowledge about teaching fractions during a Graduate Diploma of Education (GradDipEd) program with a focus on their understanding and use of fraction representations. To focus the study, the following research questions were posed: RQ1. How do preservice teachers’ understandings of fractions and fraction representations develop over a teacher education program? RQ2. How and why do preservice teachers use fraction representations for learning and teaching tasks over the course of a teacher education program

The narrative of dream reports

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Two questions are addressed: 1) whether a dream is meaningful as a whole, or whether the scenes are separate and unconnected, and 2) whether dream images are an epiphenomenon of a functional physiologicaL process of REM sleep, or whether they are akin to waking thought. Theories of REM sleep as a period of information-processing are reviewed. This is Linked with work on the relationship between dreaming and creativity, and between memory and imagery. Because of the persuasive evidence that REM sleep is implicated in the consolidation of memories there is a review of recent work on neural associative network models of memory. Two theories of dreams based on these models are described, and predictions with regard to the above two questions are made. Psychological evidence of relevance to the neural network theories is extensively reviewed. These predictions are compared with those of the recent application of structuralism to the study of dreams, which is an extension from its usual field of mythology and anthropology. The different theories are tested against four nights of dreams recorded in a sleep Lab. The analysis shows that not only do dreams concretise waking concerns as metaphors but that these concerns are depicted in oppositional terms, such as, for example, inside/outside or revolving/static. These oppositions are then permuted from one dream to the next until a resolution of the initial concern is achieved at the end of the night. An account of the use of the single case-study methodology in psychology is given, in addition to a replication of the analysis of one night's dreams by five independent judges. There is an examination of objections to the structuralist methodology, and of objections to the paradigm of multiple dream awakenings. The conclusion is drawn that dreams involve the unconscious dialectical step-by-step resolution of conflicts which to a great extent are consciously known to the subject. The similarity of dreams to day-dreams is explored, with the conclusion that the content of dreams is better explained by an account of metaphors we use when awake and by our daily concerns, than by reference to the physiology of REM sleep. It is emphasised that dreams can be meaningful even if they do not have a function.Ann Murray Award Fun

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition