Mathematics

Sound propagation in a waveguide is greatly dependent on the acoustic properties of the boundaries. The effect of these properties can be described by a bottom reflection coefficient RB, and surface reflection coefficient RS. Two methods for estimating reflection coefficients are used in this research. The first, the ratio method, is based on the variations of the Green's function with depth utilizing the ratio of the wavenumber spectra at two depths. The second, the pole method, is based on the wavenumbers of the modal peaks in the spectrum at a particular depth. A method to invert for sound speed and density is also examined. Estimates of RB and RS based on synthetic data by the ratio method were very close to their predicted values, especially for higher frequencies and longer apertures. The pole method returned less precise estimates though with longer apertures, the estimates were better. Using experimental data, results of the pole method as well a geoacoustic inversion technique based on them were mixed. The ratio method was used to estimate RS based on the actual data and returned results close to the predicted phase of p.

Number perception, its neural basis and its relationship to how numerical stimuli are presented have been challenging research topics in cognitive neuroscience for many years. A primary question that has been addressed is whether the perception of the quantity of a visually presented number stimulus is dissociable from its early visual perception. The present study examined the possible influence of visual quality judgment on quantity judgments of numbers. To address this issue, volunteer adult subjects performed a mental number comparison task in which two-digit stimulus numbers (Arabic number format), among the numbers between 31 and 99 were mentally compared to a memorized reference number, 65. Reaction times (RTs) and neurophysiological (i.e. electroencephalographic (EEG) data) responses were acquired simultaneously during performance of the two-digit number comparison task. In this particular quantity comparison task, the number stimuli were classified into three distance factors. That is, numbers were a close, medium or far distance from the reference number (i.e., 65). In order to evaluate the relationship between numerical stimulus quantity and quality, the number stimuli were embedded in varying degrees of a typical visual noise form, known as "salt and pepper noise" (e.g., the visual noise one perceives when viewing a photograph taken with a dusty camera lens). In this manner, the visual noise permitted visual quality to be manipulated across three levels: no noise, medium noise (approximately 60% degraded visual quality from nonoise), and dense noise (75% degraded visual quality from no-noise).

The main objective of the thesis is to carry out a rigorous hydrodynamic analysis of ocean current turbines and determine power for a range of flow and geometric parameters. For the purpose, a computational tool based on the vortex lattice method (VLM) is developed. Velocity of the flow on the turbine blades, in relation to the freestream velocity, is determined through induction factors. The geometry of trailing vortices is taken to be helicoidal. The VLM code is validated by comparing its results with other theoretical and experimental data corresponding to flows about finite-aspect ratio foils, swept wings and a marine current turbine. The validated code is then used to study the performance of the prototype gulfstream turbine for a range of parameters. Power and thrust coefficients are calculated for a range of tip speed ratios and pitch angles. Of all the cases studied, the one corresponding to tip speed ratio of 8 and uniform pitch angle 20 produced the maximum power of 41.3 [kW] in a current of 1.73 [m/s]. The corresponding power coefficient is 0.45 which is slightly less than the Betz limit power coefficient of 0.5926. The VLM computational tool developed for the research is found to be quite efficient in that it takes only a fraction of a minute on a regular laptop PC to complete a run. The tool can therefore be efficiently used or integrated into software for design optimization.

The purpose of this research was to identify if 1) there is a difference in student achievement between students who use the GeoLeg manipulative tool and students who use a traditional compass, protractor, and ruler on the same geometry unit; 2) there is a difference in student achievement between the genders between those who use the GeoLeg manipulative tool and those students who do not; and 3) there is a relationship between identified learning styles and student achievement on a geometry unit posttest after using the GeoLeg manipulative tool. There were 317 students in the study. The research found that students using the GeoLeg manipulative tool produced significantly better student performance on a posttest in this particular school setting. Although these results cannot be generalized to other school sites, it is plausible that these results could generalize to school sites whose demographics are similar. The research findings revealed that there was no statistically significant difference between male and female students within the treatment group. The significant finding is that the GeoLeg manipulative tool appears to work equally well with both genders. None of the learning styles, as identified by the Honey and Mumford Learning Styles Questionnaire, were correlated with student posttest score achievement on the tested geometry unit. In addition, there was no evidence to suggest that a student's learning style moderates the effectiveness of the use of the GeoLeg manipulative tool. There is no evidence to suggest that the effectiveness of the GeoLeg manipulative tool is any different depending upon the student's gender or learning style. The results of this research provide strong support for the use of the GeoLeg manipulative tool for improving student performance. Further research is needed to confirm these results in similar and different populations.

Epidemic models help us predict the outcome of an epidemic. I will discuss and compare two simple epidemic models: a deterministic model implemented by a simple differential equation, and a stochastic model, which is more realistic, but harder to analyze. In both models we assume, for simplicity, that each individual goes through only two stages: healthy (susceptible) and sick (infective). Such models, called SI epidemic models, describe infections with no immunity. We will show that, when the population gets large, the more realistic stochastic model approaches the simple deterministic model on the average, which will allow us to see that the deterministic model is used for a good reason.

Famous mathematician Paul Erdèos conjectured the existence of infinite sequences of symbols where no two adjacent subsequences are permutations of one another. It can easily be checked that no such sequence can be constructed using only three symbols, but as few as four symbols are sufficient. Here, we expand this concept to include sequences that may contain 'do not know'' characters, called holes. These holes make the undesired subsequences more common. We explore both finite and infinite sequences. For infinite sequences, we use iterating morphisms to construct the non-repetitive sequences with either a finite number of holes or infinitely many holes. We also discuss the problem of using the minimum number of different symbols.