If a partially hyperbolic diffeomorphism on a torus of dimension d greater than 3 has stable and unstable foliations which are quasi-isometric on the universal cover, and its center direction is one-dimensional, then the diffeomorphism is leaf conjugate to a linear toral automorphism. In other words, the hyperbolic structure of the diffeomorphism is exactly that of a linear, and thus simple to understand, example. In particular, every partially hyperbolic diffeomorphism on the 3-torus is leaf conjugate to a linear toral automorphism.

If a partially hyperbolic diffeomorphism on a torus of dimension d greater than 3 has stable and unstable foliations which are quasi-isometric on the universal cover, and its center direction is one-dimensional, then the diffeomorphism is leaf conjugate to a linear toral automorphism. In other words, the hyperbolic structure of the diffeomorphism is exactly that of a linear, and thus simple to understand, example. In particular, every partially hyperbolic diffeomorphism on the 3-torus is leaf conjugate to a linear toral automorphism.

Capture-recapture methods take their origins from animal abundance estimation, where they were used to estimate the unknown size of the animal population under study. In the late 1940s and again in the late 1960s and early 1970s these same capture-recapture methods were modified and applied to epidemiological list data. Since then through their continued use, in particular in the 1990s, these methods have become popular for the estimation of the completeness of disease registries and for the estimation of the unknown total size of human disease populations. In this thesis we investigate new methods for the analysis of epidemiological list data using capture-recapture methods. In particular we compare two standard methods used to estimate the unknown total population size, and examine new methods which incorporate list mismatch errors and model-selection uncertainty into the process for the estimation of the unknown total population size and its associated confidence interval. We study the use of modified tag loss methods from animal abundance estimation to allow for list mismatch errors in the epidemio-logical list data. We also explore the use of a weighted average method, the use of Bootstrap methods, and the use of a Bayesian model averaging method for incorporating model-selection uncertainty into the estimate of the unknown total population size and its associated confidence interval. In addition we use two previously unanalysed Diabetes studies to illustrate the methods examined and a well-known Spina Bifida Study for simulation purposes. This thesis finds that ignoring list mismatch errors will lead to biased estimates of the unknown total population size and that the list mismatch methods considered here result in a useful adjustment. The adjustment also approximately agrees with the results obtained using a complex matching algorithm. As for the incorporation of model-selection uncertainty, we find that confidence intervals which incorporate model-selection uncertainty are wider and more appropriate than confidence intervals that do not. Hence we recommend the use of tag loss methods to adjust for list mismatch errors and the use of methods that incorporate model-selection uncertainty into both point and interval estimates of the unknown total population size. / Subscription resource available via Digital Dissertations only.

Capture-recapture methods take their origins from animal abundance estimation, where they were used to estimate the unknown size of the animal population under study. In the late 1940s and again in the late 1960s and early 1970s these same capture-recapture methods were modified and applied to epidemiological list data. Since then through their continued use, in particular in the 1990s, these methods have become popular for the estimation of the completeness of disease registries and for the estimation of the unknown total size of human disease populations. In this thesis we investigate new methods for the analysis of epidemiological list data using capture-recapture methods. In particular we compare two standard methods used to estimate the unknown total population size, and examine new methods which incorporate list mismatch errors and model-selection uncertainty into the process for the estimation of the unknown total population size and its associated confidence interval. We study the use of modified tag loss methods from animal abundance estimation to allow for list mismatch errors in the epidemio-logical list data. We also explore the use of a weighted average method, the use of Bootstrap methods, and the use of a Bayesian model averaging method for incorporating model-selection uncertainty into the estimate of the unknown total population size and its associated confidence interval. In addition we use two previously unanalysed Diabetes studies to illustrate the methods examined and a well-known Spina Bifida Study for simulation purposes. This thesis finds that ignoring list mismatch errors will lead to biased estimates of the unknown total population size and that the list mismatch methods considered here result in a useful adjustment. The adjustment also approximately agrees with the results obtained using a complex matching algorithm. As for the incorporation of model-selection uncertainty, we find that confidence intervals which incorporate model-selection uncertainty are wider and more appropriate than confidence intervals that do not. Hence we recommend the use of tag loss methods to adjust for list mismatch errors and the use of methods that incorporate model-selection uncertainty into both point and interval estimates of the unknown total population size. / Subscription resource available via Digital Dissertations only.

Capture-recapture methods take their origins from animal abundance estimation, where they were used to estimate the unknown size of the animal population under study. In the late 1940s and again in the late 1960s and early 1970s these same capture-recapture methods were modified and applied to epidemiological list data. Since then through their continued use, in particular in the 1990s, these methods have become popular for the estimation of the completeness of disease registries and for the estimation of the unknown total size of human disease populations. In this thesis we investigate new methods for the analysis of epidemiological list data using capture-recapture methods. In particular we compare two standard methods used to estimate the unknown total population size, and examine new methods which incorporate list mismatch errors and model-selection uncertainty into the process for the estimation of the unknown total population size and its associated confidence interval. We study the use of modified tag loss methods from animal abundance estimation to allow for list mismatch errors in the epidemio-logical list data. We also explore the use of a weighted average method, the use of Bootstrap methods, and the use of a Bayesian model averaging method for incorporating model-selection uncertainty into the estimate of the unknown total population size and its associated confidence interval. In addition we use two previously unanalysed Diabetes studies to illustrate the methods examined and a well-known Spina Bifida Study for simulation purposes. This thesis finds that ignoring list mismatch errors will lead to biased estimates of the unknown total population size and that the list mismatch methods considered here result in a useful adjustment. The adjustment also approximately agrees with the results obtained using a complex matching algorithm. As for the incorporation of model-selection uncertainty, we find that confidence intervals which incorporate model-selection uncertainty are wider and more appropriate than confidence intervals that do not. Hence we recommend the use of tag loss methods to adjust for list mismatch errors and the use of methods that incorporate model-selection uncertainty into both point and interval estimates of the unknown total population size. / Subscription resource available via Digital Dissertations only.

Capture-recapture methods take their origins from animal abundance estimation, where they were used to estimate the unknown size of the animal population under study. In the late 1940s and again in the late 1960s and early 1970s these same capture-recapture methods were modified and applied to epidemiological list data. Since then through their continued use, in particular in the 1990s, these methods have become popular for the estimation of the completeness of disease registries and for the estimation of the unknown total size of human disease populations. In this thesis we investigate new methods for the analysis of epidemiological list data using capture-recapture methods. In particular we compare two standard methods used to estimate the unknown total population size, and examine new methods which incorporate list mismatch errors and model-selection uncertainty into the process for the estimation of the unknown total population size and its associated confidence interval. We study the use of modified tag loss methods from animal abundance estimation to allow for list mismatch errors in the epidemio-logical list data. We also explore the use of a weighted average method, the use of Bootstrap methods, and the use of a Bayesian model averaging method for incorporating model-selection uncertainty into the estimate of the unknown total population size and its associated confidence interval. In addition we use two previously unanalysed Diabetes studies to illustrate the methods examined and a well-known Spina Bifida Study for simulation purposes. This thesis finds that ignoring list mismatch errors will lead to biased estimates of the unknown total population size and that the list mismatch methods considered here result in a useful adjustment. The adjustment also approximately agrees with the results obtained using a complex matching algorithm. As for the incorporation of model-selection uncertainty, we find that confidence intervals which incorporate model-selection uncertainty are wider and more appropriate than confidence intervals that do not. Hence we recommend the use of tag loss methods to adjust for list mismatch errors and the use of methods that incorporate model-selection uncertainty into both point and interval estimates of the unknown total population size. / Subscription resource available via Digital Dissertations only.

Capture-recapture methods take their origins from animal abundance estimation, where they were used to estimate the unknown size of the animal population under study. In the late 1940s and again in the late 1960s and early 1970s these same capture-recapture methods were modified and applied to epidemiological list data. Since then through their continued use, in particular in the 1990s, these methods have become popular for the estimation of the completeness of disease registries and for the estimation of the unknown total size of human disease populations. In this thesis we investigate new methods for the analysis of epidemiological list data using capture-recapture methods. In particular we compare two standard methods used to estimate the unknown total population size, and examine new methods which incorporate list mismatch errors and model-selection uncertainty into the process for the estimation of the unknown total population size and its associated confidence interval. We study the use of modified tag loss methods from animal abundance estimation to allow for list mismatch errors in the epidemio-logical list data. We also explore the use of a weighted average method, the use of Bootstrap methods, and the use of a Bayesian model averaging method for incorporating model-selection uncertainty into the estimate of the unknown total population size and its associated confidence interval. In addition we use two previously unanalysed Diabetes studies to illustrate the methods examined and a well-known Spina Bifida Study for simulation purposes. This thesis finds that ignoring list mismatch errors will lead to biased estimates of the unknown total population size and that the list mismatch methods considered here result in a useful adjustment. The adjustment also approximately agrees with the results obtained using a complex matching algorithm. As for the incorporation of model-selection uncertainty, we find that confidence intervals which incorporate model-selection uncertainty are wider and more appropriate than confidence intervals that do not. Hence we recommend the use of tag loss methods to adjust for list mismatch errors and the use of methods that incorporate model-selection uncertainty into both point and interval estimates of the unknown total population size. / Subscription resource available via Digital Dissertations only.

Master of Science / Department of Mathematics / Pietro Poggi-Corradini / This thesis contains three chapters split into two parts. In the first chapter, the discrete p-modulus of families of walks is introduced and discussed from various perspectives. Initially, we prove many properties by mimicking the theory from the continuous case and use Arne Beurling's criterion for extremality to build insight and intuition regarding the modulus. After building an intuitive understanding of the p-modulus, we proceed to switch perspectives to that of convex analysis. From here, uniqueness and existence of extremal densities is shown and a better understanding of Beurling's criterion is developed before describing an algorithm that approximates the value of the p-modulus arbitrarily well. In the second chapter, an exclusively edge-based approach to the discrete transboundary modulus is described. Then an interesting application is discussed with some preliminary numerical results. The final chapter describes four different takes of the Susceptible-Infected (SI) epidemic model on graphs and shows them to be equivalent. After developing a deep understanding of the SI model, the epidemic hitting time is compared to a variety of different graph centralities to indicate successful alternative methods in identifying important agents in epidemic spreading. Numerical results from simulations on many real-world graphs are presented. They indicate the effective resistance, which coincides with the 2-modulus for connecting families, is the most closely correlated indicator of importance to that of the epidemic hitting time. In large part, this is suspected to be due to the global nature of both the effective resistance and the epidemic hitting time. Thanks to the equivalence between the epidemic hitting time and the expected distance on an randomly exponentially weighted graph, we uncover a deeper connection- the effective resistance is also a lower bound for the epidemic hitting time, showing an even deeper connection.

Doctor of Philosophy / Department of Mathematics / John S. Maginnis / In this work, we find the module structure of the cohomology of the group of four by four upper triangular matrices (with ones on the diagonal) with entries from the field on three elements modulo its center. Some of the relations amongst the generators for the cohomology ring are also given. This cohomology is found by considering a certain split extension. We show that the associated Lyndon-Hochschild-Serre spectral sequence collapses at the second page by illustrating a set of generators for the cohomology ring from generating elements of the second page. We also consider two other extensions using more traditional techniques. In the first we introduce some new results giving degree four and five differentials in spectral sequences associated to extensions of a general class of groups and apply these to both the extensions.

Doctor of Philosophy / Department of Mathematics / Ivan Blank / We study the obstacle problem with an elliptic operator in nondivergence form with principal coefficients in VMO. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary, and we show existence of blowup limits, a basic measure stability result, and a measure-theoretic version of the Caffarelli alternative proven in Caffarelli's 1977 paper ``The regularity of free boundaries in higher dimensions." Finally, we show that blowup limits are in general not unique at free boundary points.