Thesis (M.S.)--Rochester Institute of Technology, 2009. / Typescript. Includes bibliographical references (leaves 108-110).

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (leaves 100-104). Also available in print.

Thesis (Ph. D.)--School of Industrial and Systems Engineering, Georgia Institute of Technology, 2004. Directed by Robert D. Foley. / Vita. Includes bibliographical references (leaves 85-86).

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Flag varieties of reductive Lie groups and their subvarieties play a central role in representation theory. In the early 1980s, V. Deodhar introduced a decomposition of the flag variety which was then used to study the Kazdan-Lusztig polynomials. A Deodhar-type decomposition of the product of the flag variety with itself, referred to as the double flag variety, was introduced in 2007 by B. Webster and M. Yakimov, and each piece of the decomposition was shown to be coisotropic with respect to a naturally defined Poisson structure on the double flag variety. The work of Webster and Yakimov was partially motivated by the theory of cluster algebras in which Poisson structures play an important role. The Deodhar decomposition of the flag variety is better understood in terms of a cell decomposition of Bott-Samelson varieties, which are resolutions of Schubert varieties inside the flag variety. In the thesis, double Bott-Samelson varieties were introduced and cell decompositions of a Bott-Samelson variety were constructed using shuffles. When the sequences of simple reflections defining the double Bott-Samelson variety are reduced, the Deodhar-type decomposition on the double flag variety defined by Webster and Yakimov was recovered. A naturally defined Poisson structure on the double Bott-Samelson variety was also studied in the thesis, and each cell in the cell decomposition was shown to be coisotropic. For the cells that are Poisson, coordinates on the cells were also constructed and were shown to be log-canonical for the Poisson structure. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Zero inflation occurs when the proportion of zeros of a model is greater than the proportion of zeros of the corresponding Poisson model. This situation is very common in count data. In order to model zero inflated count time series data, we propose the zero inflated autoregressive conditional Poisson (ZIACP) model by the extending the autoregressive conditional poisson (ACP) model of Ghahramani and Thavaneswaran (2009). The stationarity conditions and the autocorrelation functions of the ZIACP model are provided. Based on the expectation maximization (EM) algorithm an estimation method is developed. A simulation study shows that the estimation method is accurate and reliable as long as the sample size is reasonably high. Three real data examples, syphilis data Yang (2012), arson data Zhu (2012) and polio data Kitromilidou and Fokianos (2015) are studied to compare the performance of the proposed model with other competitive models in the literature. / February 2016

Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let g denote the complexification of the Lie algebra of U, g=u^C. Each u-compatible triangular decomposition g= n_- + h + n_+ determines a Poisson Lie group structure pi_U on U. The Evens-Lu construction produces a (U, pi_U)-homogeneous Poisson structure on X. By choosing the basepoint in X appropriately, X is presented as U/K where K is the fixed point set of an involution which stabilizes the triangular decomposition of g. With this presentation, a connection is established between the symplectic foliation of the Evens-Lu Poisson structure and the Birkhoff decomposition of U/K. This is done through reinterpretation of results of Pickrell. Each symplectic leaf admits a natural torus action. It is shown that these actions are Hamiltonian and the momentum maps are computed using triangular factorization. Finally, local formulas for the Evens-Lu Poisson structure are displayed in several examples.

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Prevalence is a valuable epidemiological measure about the burden of disease in a community for planning health services; however, true prevalence is typically underestimated and there exists no reliable method of confirming the estimate of this prevalence in question. This thesis studies imprecise priors for the development of a statistical reasoning framework regarding this epidemiological decision making problem. The concept of imprecise probabilities introduced by Walley (1991) is adopted for the construction of this inferential framework in order to model prior ignorance and quantify the degree of imprecision associated with the inferential process. The study is restricted to the standard and zero-truncated Poisson sampling models that give an exponential family with a canonical log-link function because of the mechanism involved with the estimation of population size. A three-parameter exponential family of posteriors which includes the normal and log-gamma as limiting cases is introduced by applying normal priors on the canonical parameter of the Poisson sampling models. The canonical parameters simplify dealing with families of priors as Bayesian updating corresponds to a translation of the family in the canonical hyperparameter space. The canonical link function creates a linear relationship between regression coefficients of explanatory variables and the canonical parameters of the sampling distribution. Thus, normal priors on the regression coefficients induce normal priors on the canonical parameters leading to a higher-dimensional exponential family of posteriors whose limiting cases are again normal or log-gamma. All of these implementations are synthesized to build the ipeglim package (Lee, 2013) that provides a convenient method for characterizing imprecise probabilities and visualizing their translation, soft-linearity, and focusing behaviours. A characterization strategy for imprecise priors is introduced for instances when there exists a state of complete ignorance. The learning process of an individual intentional unit, the agreement process between several intentional units, and situations concerning prior-data conflict are graphically illustrated. Finally, the methodology is applied for re-analyzing the data collected from the epidemiological disease surveillance of three specific cases – Cholera epidemic (Dahiya, 1973), Down’s syndrome (Zelterman, 1988), and the female users of methamphetamine and heroin (B ̈ ohning, 2009).

The magnetic spherical pendulum is a mechanical system consisting of a pendulum whereof the bob is electrically charged, moving under the influence of gravitation and the magnetic field induced by a magnetic monopole deposited at the origin. Physically not directly realizable, it turns out to be equivalent to a reduction of the Lagrange top. This work is essentially the logbook of our attempts at understanding the simplest contemporary approaches to the magnetic spherical pendulum.