Master of Science / Department of Mathematics / Hrant Hakobyan / When studying geometrical objects less regular than ordinary ones, fractal analysis becomes a valuable tool. Over the last 30 years, this small branch of mathematics has developed extensively. Fractals can be de fined as those sets which have non-integral Hausdor ff dimension. In this thesis, we take a look at some basic measure theory needed to introduce certain de finitions of fractal dimensions, which can be used to measure a set's fractal degree. We introduce Minkowski dimension and Hausdor ff dimension as well as explore some examples where they coincide. Then we look at the dimension of a measure and some very useful applications. We conclude with a well known result of Bedford and McMullen about the Hausdor ff dimension of self-a ffine sets.

Doctor of Philosophy / Department of Mathematics / Andrew G. Bennett / The goal of this work is to provide instructors with detailed information about their classes at each assignment during the term. The information is both on an individual level and at the aggregate level. We used the large number of grades, which are available online these days, along with data-mining techniques to build our models. This enabled us to profile each student so that we might individualize our approach. From these profiles, we began to investigate what can be done in order to get students to do better, or at least be less frustrated. Regardless, the interactions with our undergraduates will improve as our knowledge about them increases. We start with a categorization of Studio College Algebra students into groups, or clusters, at some point in time during the semester. In our case, we used the grouping just after the first exam, as described by Dr. Rachel Manspeaker in her PhD. dissertation. From this we built a naive Bayesian model which extends these student clusters from one point in the semester, to a classification at every assignment, attendance score, and exam in the course. A hidden Markov model was then constructed with the transition probabilities being derived from the Bayesian model. With this HMM, we were able to compute the most likely path that students take through the various categories over the semester. We observed that a majority of students settle into a group within the first two weeks of the term.

Doctor of Philosophy / Department of Electrical and Computer Engineering / Caterina Maria Scoglio / We have analytically discovered the existence of two global epidemic invasion thresholds in a directed meta-population network model of the United States cattle industry. The first threshold describes the outbreak of disease first within the core of the livestock system while the second threshold describes the invasion of the epidemic into a second class of locations where the disease would pose a risk for contamination of meat production. Both thresholds have been verified through extensive numerical simulations. We have further derived the relationship between the pair of thresholds and discovered a unique dependence on the network topology through the fractional compositions and the in-degree distributions of the transit and sink nodes. We then addressed a major challenge for epidemiologists and their efforts to model disease outbreaks in cattle. There is a critical shortfall in the availability of large-scale livestock movement data for the United States. We meet this challenge by developing a method to estimate cattle movement parameters from publicly available data. Across 10 Central States of the US, we formulated a large, convex optimization problem to predict the cattle movement parameters which, having minimal assumptions, provide the best fit to the US Department of Agriculture's Census database and follow constraints defined by scientists and cattle experts. Our estimated parameters can produce distributions of cattle shipments by head which compare well with shipment distributions also provided by the US Department of Agriculture. This dissertation concludes with a brief incorporation of the analytic models and the parameter estimation. We approximated the critical movement rates defined by the global invasion thresholds and compared them with the average estimated cattle movement rates to find a significant opportunity for epidemics to spread through US cattle populations.

Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd Easton / Integer programming (IP) is a class of mathematical models useful for modeling and optimizing many theoretical and industrial problems. Unfortunately, IPs are NP-complete, and many integer programs cannot currently be solved. Valid inequalities and their respective cuts are commonly used to reduce the effort required to solve IPs. This thesis poses the questions, do valid equality cuts exist and can they be useful for solving IPs? Several theoretical results related to valid equalities are presented in this thesis. It is shown that equality cuts exist if and only if the convex hull is not full dimensional. Furthermore, the addition of an equality cut can arbitrarily reduce the dimension of the linear relaxation. In addition to the theory on equality cuts, the idea of infeasibility conditions are presented. Infeasibility conditions introduce a set of valid inequalities whose intersection is the empty set. infeasibility conditions can be used to rapidly terminate a branch and cut algorithm. Applying the idea of equality cuts to the multi-demand multidimensional knapsack problem resulted in a new class of cutting planes named anticover cover equality (ACE) cuts. A simple algorithm, FACEBT, is presented for finding ACE cuts in a branching tree with complexity O(m n log n). A brief computational study shows that using ACE cuts exist frequently in the MDMKP instances studied. Every instance had at least one equality cut, while one instance had over 500,000. Additionally, computationally challenging instances saw an 11% improvement in computational effort. Therefore, equality cuts are a new topic of research in IP that is beneficial for solving some IP instances.

Master of Science / Department of Mathematics / Nathan Albin / A new methodology is introduced for use in discrete periodic extension of non-periodic functions. The methodology is based on a band-limited step function, and utilizes the computational efficiency of FC-Gram (Fourier Continuation based on orthonormal Gram polynomial basis on the extension stage) extension database. The discrete periodic extension is a technique for augmenting a set of uniformly-spaced samples of a smooth function with auxiliary values in an extension region. If a suitable extension is constructed, the interpolating trigonometric polynomial found via an FFT(Fast Fourier Transform) will accurately approximate the original function in its original interval. The discrete periodic extension is a key construction in the FC-Gram algorithm which is successfully implemented in several recent efficient and high-order PDEs solvers. This thesis focuses on a new flexible discrete periodic extension procedure that performs at least as well as the FC-Gram method, but with somewhat simpler implementation and significantly decreased setup time.

Doctor of Philosophy / Department of Computing and Information Sciences / William Henry Hsu / Topic models are probabilistic models for discovering topical themes in collections of documents. In real world applications, these models provide us with the means of organizing what would otherwise be unstructured collections. They can help us cluster a huge collection into different topics or find a subset of the collection that resembles the topical theme found in an article at hand. The first wave of topic models developed were able to discover the prevailing topics in a big collection of documents spanning a period of time. It was later realized that these time-invariant models were not capable of modeling 1) the time varying number of topics they discover and 2) the time changing structure of these topics. Few models were developed to address this two deficiencies. The online-hierarchical Dirichlet process models the documents with a time varying number of topics. It varies the structure of the topics over time as well. However, it relies on document order, not timestamps to evolve the model over time. The continuous-time dynamic topic model evolves topic structure in continuous-time. However, it uses a fixed number of topics over time. In this dissertation, I present a model, the continuous-time infinite dynamic topic model, that combines the advantages of these two models 1) the online-hierarchical Dirichlet process, and 2) the continuous-time dynamic topic model. More specifically, the model I present is a probabilistic topic model that does the following: 1) it changes the number of topics over continuous time, and 2) it changes the topic structure over continuous-time. I compared the model I developed with the two other models with different setting values. The results obtained were favorable to my model and showed the need for having a model that has a continuous-time varying number of topics and topic structure.