A discrete time Markov Chain is used to model enrollment at a regional university. A preliminary analysis is conducted on the data set in order to determine the classes for the Markov chain model. The semester, yearly, and long term results of the model are examined thoroughly. A sensitivity analysis of the probability matrix entries is then conducted to determine the overall greatest influence on graduation rates.

The discrete Fourier transform has become an essential tool in the analysis of digital signals. Applications have become widespread since the discovery of the Fast Fourier Transform and the rise of personal computers. The field of digital signal processing is an exciting intersection of mathematics, statistics, and electrical engineering. In this study we aim to gain understanding of the mathematics behind algorithms that can extract chord information from recorded music. We investigate basic music theory, introduce and derive the discrete Fourier transform, and apply Fourier analysis to audio files to extract spectral data.

The concept of rendering an object invisible, once considered unfathomable, can now be deemed achievable using artificial metamaterials. The ability for these advanced structures to refract waves in the negative direction has sparked creativity for future applications. Manipulating electromagnetic waves of all frequencies around an object requires precise and unique parameters, which are calculated from various mathemat- ical laws and equations. We explore the possible interpretations of these parameters and how they are implemented towards the construction of a suitable metamaterial. If carried out correctly, the wave will exit the metamaterial exhibiting the same behavior as when it had entered. Thus, an outside observer will not be able to recognize any abnormal changes in wave frequency or direction. This paper will survey studies and technologies from the past 20 years to arrive at a concise mathematical examination of the possibilities and inherent issues under the umbrella of modern ”cloaking.”

p, li { white-space: pre-wrap; } Mushy layers are regions of intermixed liquid and solid which can arise during the solidification of binary alloys, generally consisting of dendritic solids with solute-rich liquid in the interstices. They occur due to an instability resulting from the buildup of rejected solute along the solidification front. Liquid ahead of the front becomes supercooled, so disturbances to the interface grow more rapidly than the interface itself. A simple experiment has a tank filled with a uniform solution at uniform temperature being placed upon a cold surface. Early on, a small solid layer forms at the bottom capped by a rapidly advancing mushy layer. Typical modeling efforts have made at least one of two assumptions, that the tank is of infinite depth or that the diffusion of solute is negligible. This dissertation investigates the finite-domain problem in the presence of solute diffusion, highlighting new interfacial dynamics and other behaviors that arise in this case.

Over the course of a college mathematics degree, students are inevitably exposed to elementary physics. The derivation of the equations of motion are the classic examples of applications of derivatives and integrals. These equations of motion are easy to understand, however they can be expressed in other ways that students aren't often exposed to. Using the Lagrangian and the Hamiltonian, we can capture the same governing dynamics of Newtonian mechanics with equations that emphasize physical quantities other than position, velocity, and acceleration like Newton's equations do. Building o of these alternate interpretations of mechanics and understanding gauge transformations, we begin to understand some of the mathematical physics relating to gauge theories. In general, gauge theories are eld theories that can have gauge transformations applied to them in such a way that the meaningful physical quantities remain invariant. This paper covers the buildup to gauge theories, some of their applications, and some computational approaches to understanding them.

Sports analytics problems have become increasingly prominent in the past decade. Modern image processing capabilities allow coaching staff to easily capture detailed game-time statistics on their players, opponents, team configurations, and plays. The challenge is to turn that data into meaningful insights for team managers and coaches. This project uses descriptive and predictive techniques on publicly available NBA basketball data to identify powerful combinations of players and predict how they will perform against other teams.

The intent of this paper is to present the reader with a simple comparison of two systems of vertical transportation. Vertical transportation is a a relatively new field and is the subject of much interest in today's world. As buildings get taller and real estate becomes more expensive, the need to find a quick, efficient system with a small footprint becomes important. By performing a simulation and subjecting the two systems under study to similar traffic conditions, one can determine the effectiveness of one system relative to the other. Additionally, we look at the effects of changing various system attributes to gain a better understanding of the primary drivers of average travel time.

<p>Mathematical modelling has become a powerful tool used to predict the spread of infectious diseases in populations. Successful analysis and modeling of historical infectious disease data can explain changes in the pattern of past epidemics and lead to a better understanding of epidemiological processes. The lessons learned can be used to predict future epidemics and help to improve public healthstrategies for control and eradication.</p> <p>This thesis is focused on the analysis and modelling of smallpox dynamics based on the weekly smallpox mortality records in London, England, 1664-1930. Statistical analysis of these records is presented. A timeline of significant historical events related to changes in variolation and vaccination uptake levels and demographics was established. These events were correlated with transitions observed in smallpox dynamics. Seasonality of the smallpox time series was investigated and the contact rate between susceptible and infectious individuals was found to be seasonally forced. Seasonal variations in smallpox transmission and changes in their seasonality over long time scale were estimated. The method of transition analysis, which is used to predict qualitative changes in epidemiological patterns, was used to explain the transitions observed in the smallpox time series. We found that the standard SIR model exhibits dynamics similar to the more realistic Gamma distributed SEIR model if the mean serial interval is chosen the same, so we used the standard SIR model for our analysis. We conclude that transitions observed in the temporal pattern of smallpox dynamics can be explained by the changes in birth, immigration and intervention uptake levels.</p> / Doctor of Philosophy (PhD)

Poker is a complex game to analyze. In this project we will use the mathematics of game theory to solve some simplified variations of the game. Probability is the building block behind game theory. We must understand a few concepts from probability such as distributions, expected value, variance, and enumeration methods to aid us in studying game theory. We will solve and analyze games through game theory by using different decision methods, decision trees, and the process of domination and simplification. Poker models, with and without cards, will be provided to illustrate optimal strategies. Extensions to those models will be presented, and we will show that optimal strategies still exist. Finally, we will close this paper with an original work to an extension that can be used as a medium to creating more extensions and, or, different games to explore.

Clustering is a mathematical method of data analysis which identifies trends in data by efficiently separating data into a specified number of clusters so is incredibly useful and widely applicable for questions of interrelatedness of data. Two methods of clustering are considered here. K-means clustering defines clusters in relation to the centroid, or center, of a cluster. Spectral clustering establishes connections between all of the data points to be clustered, then eliminates those connections that link dissimilar points. This is represented as an eigenvector problem where the solution is given by the eigenvectors of the Normalized Graph Laplacian. Spectral clustering establishes groups so that the similarity between points of the same cluster is stronger than similarity between different clusters. K-means and spectral clustering are used to analyze adolescent data from the 2009 California Health Interview Survey. Differences were observed between the results of the clustering methods on 3294 individuals and 22 health-related attributes. K-means clustered the adolescents by exercise, poverty, and variables related to psychological health while spectral clustering groups were informed by smoking, alcohol use, low exercise, psychological distress, low parental involvement, and poverty. We posit some guesses as to this difference, observe characteristics of the clustering methods, and comment on the viability of spectral clustering on healthcare data.