The aim of this article is to give a pedagogical introduction to the exact equilibrium and nonequilibrium properties of free fermionic quantum spin chains. In a first part we present in full details the canonical diagonalisation procedure and review quickly the equilibrium dynamical properties. The phase diagram is analysed and possible phase transitions are discussed. The two next chapters are concerned with the effect of aperiodicity and quenched disorder on the critical properties of the quantum chain. The remaining part is devoted to the nonequilibrium dynamical behaviour of such quantum chains relaxing from a nonequilibrium pure initial state. In particular,<br />a special attention is made on the relaxation of transverse magnetization. Two-time linear response functions and correlation functions are also considered, giving insights on the nature of the final nonequilibrium stationnary state. The possibility of aging is also discussed.

Nutrient pollution has been identified as a significant threat to U.S. coastal and estuarine water quality. Though coastal and estuarine waters need nutrients to maintain a healthy, productive ecosystem, excess nutrients can lead to eutrophication. There are significant potential negative consequences associated with eutrophication, including loss of habitat, loss of economic activity, and direct threats to human health. Hillsborough Bay experienced eutrophication in the 1960s and 1970s due to a rapidly growing population and associated increases in nutrient pollution. These eutrophic conditions led to more frequent phytoplankton and macroalgae blooms and declines in seagrasses. To address these problems, a series of actions were taken including legislation limiting nutrient concentrations from domestic wastewater treatment plants, development of water quality and nutrient loading targets, and establishment of seagrass restoration and protection goals. Since the 1970s, water quality improvements and increasing seagrass acreages have been documented throughout Tampa Bay. In the current study, a series of analyses and tools are developed to obtain a more in depth understanding of water quality in Hillsborough Bay. The first tool is a linked hydrodynamic and water quality model (Environmental Fluid Dynamics Code) of Hillsborough Bay which can be employed to predict water quality responses to proposed management actions. In the second part of the study, a series of water quality indices were evaluated. The most appropriate index for determining overall water quality in Hillsborough Bay was identified. Chlorophyll a is one of the constituents in the water quality index and is currently used to evaluate annual water quality conditions in Hillsborough Bay. Therefore, the statistical distribution that describes chlorophyll a concentrations in Hillsborough Bay was identified and robust confidence intervals were developed to better understand the uncertainty associated with chlorophyll a measurements. Previous work linked chlorophyll a concentrations in Hillsborough Bay to explanatory variables based on monthly estimates. These relationships were used to develop water quality targets for the system. In this study, the previously developed relationship was revisited, resulting in an improved statistical model that is more robust. This improved model can also be used to evaluate the previously proposed targets and to better predict future changes due to climate change, sea level rise, and management actions. Lastly, a new method was developed to estimate atmospheric temperature in the contiguous United States.

The objective of this study is to compare Bayesian and parametric approaches to determine the best for estimating reliability in complex systems. Determining reliability is particularly important in business and medical contexts. As expected, the Bayesian method showed the best results in assessing the reliability of systems. In the first study, the Bayesian reliability function under the Higgins-Tsokos loss function using Jeffreys as its prior performs similarly as when the Bayesian reliability function is based on the squared-error loss. In addition, the Higgins-Tsokos loss function was found to be as robust as the squared-error loss function and slightly more efficient. In the second study, we illustrated that--through the power law intensity function--Bayesian analysis is applicable in the power law process. The power law intensity function is the key entity of the power law process (also called the Weibull process or the non-homogeneous Poisson process). It gives the rate of change of a system's reliability as a function of time. First, using real data, we demonstrated that one of our two parameters behaves as a random variable. With the generated estimates, we obtained a probability density function that characterizes the behavior of this random variable. Using this information, under the commonly used squared-error loss function and with a proposed adjusted estimate for the second parameter, we obtained a Bayesian reliability estimate of the failure probability distribution that is characterized by the power law process. Then, using a Monte Carlo simulation, we showed the superiority of the Bayesian estimate compared with the maximum likelihood estimate and also the better performance of the proposed estimate compared with its maximum likelihood counterpart. In the next study, a Bayesian sensitivity analysis was performed via Monte Carlo simulation, using the same parameter as in the previous study and under the commonly used squared-error loss function, using mean square error comparison. The analysis was extended to the second parameter as a function of the first, based on the relationship between their maximum likelihood estimates. The simulation procedure demonstrated that the Bayesian estimates are superior to the maximum likelihood estimates and that the selection of the prior distribution was sensitive. Secondly, we found that the proposed adjusted estimate for the second parameter has better performance under a noninformative prior. In the fourth study, a Bayesian approach was applied to real data from breast cancer research. The purpose of the study was to investigate the applicability of a Bayesian analysis to survival time of breast cancer data and to justify the applicability of the Bayesian approach to this domain. The estimation of one parameter, the survival function, and hazard function were analyzed. The simulation analysis showed that the Bayesian estimate of the parameter performed better compared with the estimated value under the Wheeler procedure. The excellent performance of the Bayesian estimate is reflected even for small sample sizes. The Bayesian survival function was also found to be more efficient than its parametric counterpart. In the last study, a Bayesian analysis was carried out to investigate the sensitivity to the choice of the loss function. One of the parameters of the distribution that characterized the survival times for breast cancer data was estimated applying a Bayesian approach and under two different loss functions. Also, the estimates of the survival function were determined under the same setting. The simulation analysis showed that the choice of the squared-error loss function is robust in estimating the parameter and the survival function.

Numerous ways to meta-analyze single-case data have been proposed in the literature, however, consensus on the most appropriate method has not been reached. One method that has been proposed involves multilevel modeling. This study used Monte Carlo methods to examine the appropriateness of Van den Noortgate and Onghena's (2008) raw data multilevel modeling approach to the meta-analysis of single-case data. Specifically, the study examined the fixed effects (i.e., the overall average baseline level and the overall average treatment effect) and the variance components (e.g., the between person within study variance in the average baseline level, the between study variance in the overall average baseline level, the between person within study variance in the average treatment effect) in a three level multilevel model (repeated observations nested within individuals nested within studies). More specifically, bias of point estimates, confidence interval coverage rates, and interval widths were examined as a function of specific design and data factors. Factors investigated included (a) number of primary studies per meta-analysis, (b) modal number of participants per primary study, (c) modal series length per primary study, (d) level of autocorrelation, and (3) variances of the error terms. The results of this study suggest that the degree to which the findings of this study are supportive of using Van den Noortgate and Onghena's (2008) raw data multilevel modeling approach to meta-analyzing single-case data depends on the particular effect of interest. Estimates of the fixed effects tended to be unbiased and produced confidence intervals that tended to overcover but came close to the nominal level as level-3 sample size increased. Conversely, estimates of the variance components tended to be biased and the confidence intervals for those estimates were inaccurate.

Pandemic influenza outbreaks have historically entailed significant societal and economic disruptions. Today, our quality of life is threatened by our inadequate preparedness for the imminent pandemic. The key challenges we are facing stem from a significant uncertainty in virus epidemiology, limited response resources, inadequate international collaboration, and the lack of appropriate science-based decision support tools. The existing literature falls short of comprehensive models for global pandemic spread and mitigation which incorporate the heterogeneity of the world regions and realistic travel networks. In addition, there exist virtually no studies which quantify the impact of resource sharing strategies among multiple countries. This dissertation presents three related models that contribute to filling the existing vacuum. The first model develops optimal capacity management strategies for multi-region pandemic surveillance. The second model estimates the pandemic propagation time from the onset to a likely pandemic export region, such as a major transportation hub. The model builds on a large-scale agent-based simulation and geographic information systems (GIS). The model is tested on a hypothetical outbreak in Mexico involving 155 regions and over 100 million people. The third model develops an empirical relationship to quantify the impact of various U.S. - Mexico antiviral sharing strategies under several pandemic detection and response scenarios.

Epithelial ovarian cancer (EOC) is a leading cause of morbidity and mortality among women in the United States, and the etiology is incompletely understood. Common, low penetrant genetic variants such as single nucleotide polymorphisms (SNPs) likely contribute to a significant proportion of EOC. We examined whether SNPs in two understudied yet biologically important types of genes, mitochondrial-related and miRNA-related genes, may contribute to EOC susceptibility using data from a large, homogeneous study population of 1,815 EOC cases and 1,900 controls (frequency-matched on age-group and race/ethnicity) genotyped through stage 1 of an ongoing genome-wide association study. Inter-individual variation in genes involved in mitochondrial biogenesis was strongly associated with EOC risk (empirical P=0.050), especially for genes NRF1, PPARGC1A, MTERF, ESRRA, and CAMK2D. SNPs in several genes involved in the biogenesis of miRNAs (LIN28, LIN28B, AGO2, DICER, and DROSHA) also demonstrated associations with EOC risk; a joint meta-analysis and in vitro investigations reinforced evidence for a protective role of LIN28B rs12194974 (combined OR= 0.90, 95% CI: 0.82-0.98), a G>A SNP predicted to reside in a transcription factor binding site in the highly conserved LIN28B promoter. Our findings provide valuable insight into the pathogenesis of EOC, and support the consideration of variants in these genes as candidates when building risk prediction models. Most importantly, this work has provided a strong foundation for further lines of research that may aid in reducing the burden of this disease.

Stochastic hybrid dynamic systems that incorporate both continuous and discrete dynamics have been an area of great interest over the recent years. In view of applications, stochastic hybrid dynamic systems have been employed to diverse fields of studies, such as communication networks, air traffic management, and insurance risk models. The aim of the present study is to investigate properties of some classes of stochastic hybrid dynamic systems. The class of stochastic hybrid dynamic systems investigated has random jumps driven by a non-homogeneous Poisson process and deterministic jumps triggered by hitting the boundary. Its real-valued continuous dynamic between jumps is described by stochastic differential equations of the It\^o-Doob type. Existing results of piecewise deterministic models are extended to obtain the infinitesimal generator of the stochastic hybrid dynamic systems through a martingale approach. Based on results of the infinitesimal generator, some stochastic stability results are derived. The infinitesimal generator and stochastic stability results can be used to compute the higher moments of the solution process and find a bound of the solution. Next, the study focuses on a class of multidimensional stochastic hybrid dynamic systems. The continuous dynamic of the systems under investigation is described by a linear non-homogeneous systems of It\^o-Doob type of stochastic differential equations with switching coefficients. The switching takes place at random jump times which are governed by a non-homogeneous Poisson process. Closed form solutions of the stochastic hybrid dynamic systems are obtained. Two important special cases for the above systems are the geometric Brownian motion process with jumps and the Ornstein-Uhlenbeck process with jumps. Based on the closed form solutions, the probability distributions of the solution processes for these two special cases are derived. The derivation employs the use of the modal matrix and transformations. In addition, the parameter estimation problem for the one-dimensional cases of the geometric Brownian motion and Ornstein-Uhlenbeck processes with jumps are investigated. Through some existing and modified methods, the estimation procedure is presented by first estimating the parameters of the discrete dynamic and subsequently examining the continuous dynamic piecewisely. Finally, some simulated stochastic hybrid dynamic processes are presented to illustrate the aforementioned parameter-estimation methods. One simulated insurance example is given to demonstrate the use of the estimation and simulation techniques to obtain some desired quantities.

The objective of the present study is to address some important questions related to prostate cancer treatments and survivorship among White and African American men. It is commonly understood that the risk of developing prostate cancer is higher in African American men than the other races. However, using parametric analysis, this study demonstrates that this perception is a "myth" not a "reality". The study further identifies the existence of racial/ethnic disparities by comparing the average mean tumor size, the median of survival time, and the survival function between White and African American men. These results underline the necessity of understanding the role of racial background in working towards improved clinical targeting, and thereby, improving clinical outcomes. Furthermore, parametric survival analysis was performed to estimate the survivorship of white men undergoing different treatments at each stage of prostate cancer. Additionally, to better understand the risk factors (age, tumor size, the interaction between age and tumor size) associated with survival time, an accelerated failure time model was developed that could accurately predict the rates of survivorship of white men at each stage of prostate cancer in accordance with whatever treatment they had received. Finally, the results of parametric survival analysis and the accelerated failure time model are compared among white men undergoing similar treatment at each stage of the disease.

Growth rate of prostate cancer tumor is an important aspect of understanding the natural history of prostate cancer. Using real prostate cancer data from the SEER database with tumor size as a response variable, we have clustered the cancerous tumor sizes into age groups to enhance its analytical behavior. The rate of change of the response variable as a function of age is given for each cluster. Residual analysis attests to the quality of the analytical model and the subject estimates. In addition, we have identified the probability distribution that characterize the behavior of the response variable and proceeded with basic parametric analysis. There are several remarkable treatment options available for prostate cancer patients. In this present study, we have considered the three commonly used treatment for prostate cancer: radiation therapy, surgery, and combination of surgery and radiation therapy. The study uses data from the SEER database to evaluate and rank the effectiveness of these treatment options using survival analysis in conjunction with basic parametric analysis. The evaluation is based on the stage of the prostate cancer classification. Improvement in prostate cancer disease can be measured by improvement in its mortality. Also, mortality projection is crucial for policy makers and the financial stability of insurance business. Our research applies a parametric model proposed by Renshaw et al. (1996) to project the force of mortality for prostate cancer. The proposed modeling structure can pick up both age and year effects.

The efforts to improve health care delivery usually involve studies and analysis of patient populations and healthcare systems. In this dissertation, I present the research conducted in the following areas: identifying patient groups, improving treatments for specific conditions by using statistical as well as data mining techniques, and developing new operation research models to increase system efficiency from the health institutes’ perspective. The results provide better understanding of high risk patient groups, more accuracy in detecting disease’ correlations and practical scheduling tools that consider uncertain operation durations and real-life constraints.