In this thesis we study the asymptotic behaviour of the solution of a discrete time renewal equation depending on a small perturbation parameter. In particular, we construct asymptotic expansions for the solution of the renewal equation and related quantities. The results are applied to studies of quasi-stationary phenomena for regenerative processes and asymptotics of ruin probabilities for a discrete time analogue of the Cramér-Lundberg risk model.

<p>Current frontiers in complex stochastic modeling of high-dimensional processes include major emphases on so-called functional data: problems in which the data are snapshots of curves and surfaces representing fundamentally important scientific quantities. This thesis explores new Bayesian methodologies for functional data analysis. </p><p>The first part of the thesis places emphasis on the role of factor models in functional data analysis. Data reduction becomes mandatory when dealing with such high-dimensional data, more so when data are available on a large number of individuals. In Chapter 2 we present a novel Bayesian framework which employs a latent factor construction to represent each variable by a low dimensional summary. Further, we explore the important issue of modeling and analyzing the relationship of functional data with other covariate and outcome variables simultaneously measured on the same subjects.</p><p>The second part of the thesis is concerned with the analysis of circadian data. The focus is on the identification of circadian genes that is, genes whose expression levels appear to be rhythmic through time with a period of approximately 24 hours. While addressing this goal, most of the current literature does not account for the potential dependence across genes. In Chapter 4, we propose a Bayesian approach which employs latent factors to accommodate dependence and verify patterns and relationships between genes, while representing the true gene expression trajectories in the Fourier domain allows for inference on period, phase, and amplitude of the signal.</p><p>The third part of the thesis is concerned with the statistical analysis of computer models (simulators). The heavy computational demand of these input-output maps calls for statistical techniques that quickly estimate the surface output at untried inputs given a few preliminary runs of the simulator at a set design points. In this regard, we propose a Bayesian methodology based on a non-stationary Gaussian process. Relying on a model-based assessment of uncertainty, we envision a sequential design technique which helps choosing input points where the simulator should be run to minimize the uncertainty in posterior surface estimation in an optimal way. The proposed non-stationary approach adapts well to output surfaces of unconstrained shape.</p> / Dissertation

The mechanism by which chiral selectivity takes place is complicated by the surface morphology, the possible involvement of the solvent, and the characteristics of the chiral molecules at the surface. My goal is to model and understand the factors which lead to significant discrimination in the case of three closely related chiral stationary phases: N-(1-phenylethyl)-N’-[3-(triethoxysilyl)propyl]-urea (PEPU), [(3,5-dinitrobenzoyl)-amino]-N-[3-(triethoxysilyl)propyl]-2-phenylacetamide (DNB-phenyglycine), and [(3,5-dinitrobenzoyl)amino]-N-[3-(triethoxysilyl)propyl]-4-methylpentanamide (DNB-leucine). Ab initio calculations are used to develop molecular models of these chiral selectors. These models are employed in molecular dynamics (MD) simulations, which provide the theoretical framework for modelling chiral interfaces in different solvent mixtures. The MD simulations of PEPU interfaces show that, in alcohol/water mixtures, the alcohols form domains at the interface with the hydrophobic portions of the molecule tending to orient towards the surface. This disrupts the water hydrogen bonding networks at the interface and leads to the exclusion of water from the surface region relative to the bulk. The MD simulations of DNB-phenylglycine and DNB-leucine selectors in hexane/2-propanol mixtures demonstrate that the interfaces are distinct both in terms of the selector orientations at the surface and in the number of hydrogen bonds formed with 2-propanol. This occurs despite the structural similarity between these two selectors. The interfaces are also prepared experimentally by attaching the chiral selectors onto oxidized Si(111) samples and AFM tips. In particular, for DNB-phenylglycine and DNB-leucine samples, two synthetic routes have been explored. Using AFM, the morphologies of the resulting chiral interfaces are obtained. X-ray photoelectron spectroscopy and refraction-absorption infrared spectroscopy provide information regarding the relative distribution of the compounds on the surface. Using chemical force microscopy (CFM) measurements, chiral self-selectivity is examined in various solvent mixtures. For PEPU interfaces, the extent of hydrogen bonding at the surface is the dominant contributor to the measured forces. In the case of DNB-phenylglycine and DNB-leucine, CFM measurements of the chiral self-selectivity in 2-propanol demonstrate that chiral discrimination is present in both systems, but larger forces are observed for DNB-phenylglycine, consistent with the molecular dynamics study that shows much weaker solvent interactions with this species. / Thesis (Ph.D, Chemistry) -- Queen's University, 2008-08-14 11:26:37.436

The majority of this thesis is concerned with the development of efficient and robust numerical methods based on adaptive algebraic multigrid to compute the stationary distribution of Markov chains. It is shown that classical algebraic multigrid techniques can be applied in an exact interpolation scheme framework to compute the stationary distribution of irreducible, homogeneous Markov chains. A quantitative analysis shows that algebraically smooth multiplicative error is locally constant along strong connections in a scaled system operator, which suggests that classical algebraic multigrid coarsening and interpolation can be applied to the class of nonsymmetric irreducible singular M-matrices with zero column sums. Acceleration schemes based on fine-level iterant recombination, and over-correction of the coarse-grid correction are developed to improve the rate of convergence and scalability of simple adaptive aggregation multigrid methods for Markov chains. Numerical tests over a wide range of challenging nonsymmetric test problems demonstrate the effectiveness of the proposed multilevel method and the acceleration schemes. This thesis also investigates the application of adaptive algebraic multigrid techniques for computing the canonical decomposition of higher-order tensors. The canonical decomposition is formulated as a least squares optimization problem, for which local minimizers are computed by solving the first-order optimality equations. The proposed multilevel method consists of two phases: an adaptive setup phase that uses a multiplicative correction scheme in conjunction with bootstrap algebraic multigrid interpolation to build the necessary operators on each level, and a solve phase that uses additive correction cycles based on the full approximation scheme to efficiently obtain an accurate solution. The alternating least squares method, which is a standard one-level iterative method for computing the canonical decomposition, is used as the relaxation scheme. Numerical tests show that for certain test problems arising from the discretization of high-dimensional partial differential equations on regular lattices the proposed multilevel method significantly outperforms the standard alternating least squares method when a high level of accuracy is required.

Liquid chromatography is an important separation technique for a vast number of analytes. This thesis mainly focuses on the development of stationary phases for liquid chromatography based on zwitterionic sulfobetaine polymers. In the thesis, various ways to prepare zwitterionic polymers in an aqueous environment using reversible addition fragmentation chain transfer (RAFT) polymerization are described. Both telomers, i.e. short soluble polymer chains containing a functional terminal group, as well as graft polymers on various supports have been synthesized. The RAFT polymerization technique provides an increased degree of control of the final polymers, which may aid in the preparation of more specifically tailored separation materials. Sulfobetaine polymers carry both a positive and a negative charge within a single entity, which results in interesting solution properties as well as highly biocompatible features. These unique features make them especially suited for separation of highly polar and/or charged compounds. An example of the successful separation of short peptides using a stationary phase synthesized with the RAFT technique is given. The unusual properties of sulfobetaine-type polymers are believed to be associated with the structure of water close to the polymer. A study of water structure in some silica based stationary phase grafted with zwitterionic sulfobetaine polymers was conducted. The impact of water structure on retention characteristics was investigated.

The majority of this thesis is concerned with the development of efficient and robust numerical methods based on adaptive algebraic multigrid to compute the stationary distribution of Markov chains. It is shown that classical algebraic multigrid techniques can be applied in an exact interpolation scheme framework to compute the stationary distribution of irreducible, homogeneous Markov chains. A quantitative analysis shows that algebraically smooth multiplicative error is locally constant along strong connections in a scaled system operator, which suggests that classical algebraic multigrid coarsening and interpolation can be applied to the class of nonsymmetric irreducible singular M-matrices with zero column sums. Acceleration schemes based on fine-level iterant recombination, and over-correction of the coarse-grid correction are developed to improve the rate of convergence and scalability of simple adaptive aggregation multigrid methods for Markov chains. Numerical tests over a wide range of challenging nonsymmetric test problems demonstrate the effectiveness of the proposed multilevel method and the acceleration schemes. This thesis also investigates the application of adaptive algebraic multigrid techniques for computing the canonical decomposition of higher-order tensors. The canonical decomposition is formulated as a least squares optimization problem, for which local minimizers are computed by solving the first-order optimality equations. The proposed multilevel method consists of two phases: an adaptive setup phase that uses a multiplicative correction scheme in conjunction with bootstrap algebraic multigrid interpolation to build the necessary operators on each level, and a solve phase that uses additive correction cycles based on the full approximation scheme to efficiently obtain an accurate solution. The alternating least squares method, which is a standard one-level iterative method for computing the canonical decomposition, is used as the relaxation scheme. Numerical tests show that for certain test problems arising from the discretization of high-dimensional partial differential equations on regular lattices the proposed multilevel method significantly outperforms the standard alternating least squares method when a high level of accuracy is required.

Geodesy deals with the accurate analysis of spatial and temporal variations in the geometry and physics of the Earth at local and global scales. In geodesy, least-squares collocation (LSC) is a bridge between the physical and statistical understanding of different functionals of the gravitational field of the Earth. This thesis specifically focuses on the [incorrect] implicit LSC assumptions of isotropy and homogeneity that create limitations on the application of LSC in non-stationary gravity field modeling. In particular, the work seeks to derive expressions for local and global analytical covariance functions that account for the anisotropy and heterogeneity of the Earth's gravity field. / Standard LSC assumes 2D stationarity and 3D isotropy, and relies on a covariance function to account for spatial dependence in the observed data. However, the assumption that the spatial dependence is constant throughout the region of interest may sometimes be violated. Assuming a stationary covariance structure can result in over-smoothing, e.g., of the gravity field in mountains and under-smoothing in great plains. The kernel convolution method from spatial statistics is introduced for non-stationary covariance structures, and its advantage in dealing with non-stationarity in geodetic data is demonstrated. / Tests of the new non-stationary solutions were performed over the Darling Fault, Western Australia, where the anomalous gravity field is anisotropic and non-stationary. Stationary and non-stationary covariance functions are compared in 2D LSC to the empirical example of gravity anomaly interpolation. The results with non-stationary covariance functions are better than standard LSC in terms of formal errors and cross-validation. Both non-stationarity of mean and covariance are considered in planar geoid determination by LSC to test how differently non-stationarity of mean and covariance affects the LSC result compared with GPS-levelling points in this area. Non-stationarity of the mean was not very considerable in this case, but non-stationary covariances were very effective when optimising the gravimetric quasigeoid to agree with the geometric quasigeoid. / In addition, the importance of the choice of the parameters of the non-stationary covariance functions within a Bayesian framework and the improvement of the new method for different functionals on the globe are pointed out.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 84-86). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1996. / Typescript. Vita. Includes bibliographical references (leaves 84-86). Also available on the Internet.

Neste trabalho um modelo é apresentado que permite-nos prever quantitativamente o estado estacionário atingido por um plasma não-neutro durante um processo de relaxação nãocolisional. Como uma aplicação específica, a teoria é usada para estudar a relaxação de feixes de partículas carregadas na qual, mostra-se que, um feixe inicialmente casado relaxa para a distribuição de Lynden-Bell. No entanto, quando existe um descasamento inicial, o mesmo oscila e ressonâncias paramétricas conduzem-no a uma separação de fases: caroço e halo. A abordagem desenvolvida vale tanto para a densidade como para a distribuição de velocidade no estado final estacionário. / In this work a theoretical framework is presented which allows us to quantitatively predict the final stationary state achieved by a non-neutral plasma during a process of collisionless relaxation. As a specific application, the theory is used to study relaxation of charged-particle beams. It is shown that a fully matched beam relaxes to the Lynden-Bell distribution. However, when a mismatch is present and the beam oscillates, parametric resonances lead to a corehalo phase separation. The approach developed accounts for both the density and the velocity distributions in the final stationary state.