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Interviews were conducted with 20 parents in the US who have adopted one or more children from China. The study focuses on the motivation to adopt, decision making regarding adoption and the process in relation to rational choice theory. The interviews also inquired about their required adoption trip to China and the post-adoption adjustment phase including bonding and developmental delays, as well as about why families chose to adopt from China, how they learned about the adoption agency they used and whether or not they knew families that had adopted internationally and more specifically from China. This information provided insight into the way that families obtained information that helped them reach important decisions throughout the adoption process. / Master of Science

Dynamical systems are mathematical models characterized by a set of differential or difference equations. Due to the increasing demand for more accuracy, the number of equations involved may reach the order of thousands and even millions. With so many equations, it often becomes computationally cumbersome to work with these large-scale dynamical systems. Model reduction aims to replace the original system with a reduced system of significantly smaller dimension which will still describe the important dynamics of the large-scale model. Interpolation is one method used to obtain the reduced order model. This requires that the reduced order model interpolates the full order model at selected interpolation points. Reduced order models are obtained through the Krylov reduction process, which involves solving a sequence of linear systems. The Iterative Rational Krylov Algorithm (IRKA) iterates this Krylov reduction process to obtain an optimal Η₂ reduced model. Especially in the large-scale setting, these linear systems often require employing inexact solves. The aim of this thesis is to investigate the impact of inexact solves on interpolatory model reduction. We considered preconditioning the linear systems, varying the stopping tolerances, employing GMRES and BiCG as the inexact solvers, and using different initial shift selections. For just one step of Krylov reduction, we verified theoretical properties of the interpolation error. Also, we found a linear improvement in the subspace angles between the inexact and exact subspaces provided that a good shift selection was used. For a poor shift selection, these angles often remained of the same order regardless of how accurately the linear systems were solved. These patterns were reflected in Η₂ and Η∞ errors between the inexact and exact subspaces, since these errors improved linearly with a good shift selection and were typically of the same order with a poor shift. We found that the shift selection also influenced the overall model reduction error between the full model and inexact model as these error norms were often several orders larger when a poor shift selection was used. For a given shift selection, the overall model reduction error typically remained of the same order for tolerances smaller than 1 x 10^-3, which suggests that larger tolerances for the inexact solver may be used without necessarily augmenting the model reduction error. With preconditioned linear systems as well as BiCG, we found smaller errors between the inexact and exact models while the order of the overall model reduction error remained the same. With IRKA, we observed similar patterns as with just one step of Krylov reduction. However, we also found additional benefits associated with using an initial guess in the inexact solve and by varying the tolerance of the inexact solve. / Master of Science

One of the marvelous phenomena of coordinate geometry is the equivalence of Desargues' Theorem to the presence of an underlying division ring in a projective plane. Supplementing this correspondence is the general theory of intersection theorems, which, restricted to desarguian projective planes P, corresponds precisely to the theory of integral rational identities, restricted to division rings D. The first chapter of this paper introduces projective planes, develops the concept of an intersection theorem, and expounds upon the Theorem of Desargues; the discussion culminates with a proof of the desarguian phenomenon in the second chapter. The third chapter characterizes the automorphisms of P and introduces the theory of polynomial identities; the fourth chapter expands this discussion to rational identities and cements the ``dictionary''. The last section describes a measure of complexity for these intersection theorems, and the paper concludes with a curious spawn of the correspondence. / Master of Science

Ferroelectric materials have been an area of keen interest for researchers due to their useful electro-mechanical coupling properties for a range of modern applications, such as sensing, precision actuation, or energy harvesting. The distribution of the piezoelectric coefficients, which corresponds to the piezoelectric properties, in traditional crystalline ferroelectric materials are determined by their inherent crystalline structure. This restriction limits the tunability of their piezoelectric properties. In the present work, ferroelectric lattice materials capable of a wide range of rationally designed piezoelectric coefficients are achieved through lattice micro-architecture design. The piezoelectric coefficients of several lattice designs are analyzed and predicted using an analytical volume-averaging approach. Finite element models were used to verify the analytical predictions and strong agreement between the two sets of results were found. Select lattice designs were additively manufactured using projection microstereolithography from a PZT-polymer composite and their piezoelectric coefficients experimentally verified and also found to be in agreement with the analytical and numerical predictions. The results show that the use of lattice micro-architecture successfully decouples the dependency of the piezoelectric properties on the material's crystalline structure, giving the user a means to tune the piezoelectric properties of the lattice materials. Real-world application of a ferroelectric lattice structure is demonstrated through application as a multi-directional stress sensor. / Master of Science / Ferroelectric materials have been an area of keen interest for researchers due to their useful electro-mechanical coupling properties for a range of modern applications, such as sensing, precision actuation, or energy harvesting. However, the piezoelectric properties of traditional materials are not easily augmented due to their dependency on material crystalline structure. In the present work, material architecture is investigated as a means for designing new piezoelectric materials with tunable sets of piezoelectric properties. Analytical predictions of the properties are first obtained and then verified using finite element models and experimental data from additively manufactured samples. The results indicate that the piezoelectric properties of a material can in fact be tuned by varying material architecture. Following this, real-world application of a ferroelectric lattice structure is demonstrated through application as a multi-directional stress sensor.

Dynamical systems are widely used to model physical phenomena and, in many cases, these physical phenomena are parameter dependent. In this thesis we investigate three prominent problems related to the simulation of parametric dynamical systems and develop the analysis and computational framework to solve each of them. In many cases we have access to data resulting from simulations of a parametric dynamical system for which an explicit description may not be available. We introduce the parametric AAA (p-AAA) algorithm that builds a rational approximation of the underlying parametric dynamical system from its input/output measurements, in the form of transfer function evaluations. Our algorithm generalizes the AAA algorithm, a popular method for the rational approximation of nonparametric systems, to the parametric case. We develop p-AAA for both scalar and matrix-valued data and study the impact of parameter scaling. Even though we present p-AAA with parametric dynamical systems in mind, the ideas can be applied to parametric stationary problems as well, and we include such examples. The solution of a dynamical system can often be expressed in terms of an eigenvalue problem (EVP). In many cases, the resulting EVP is nonlinear and depends on a parameter. A common approach to solving (nonparametric) nonlinear EVPs is to approximate them with a rational EVP and then to linearize this approximation. An existing algorithm can then be applied to find the eigenvalues of this linearization. The AAA algorithm has been successfully applied to this scheme for the nonparametric case. We generalize this approach by using our p-AAA algorithm to find a rational approximation of parametric nonlinear EVPs. We define a corresponding linearization that fits the format of the compact rational Krylov (CORK) algorithm for the approximation of eigenvalues. The simulation of dynamical systems may be costly, since the need for accuracy may yield a system of very large dimension. This cost is magnified in the case of parametric dynamical systems, since one may be interested in simulations for many parameter values. Interpolatory model order reduction (MOR) tackles this problem by creating a surrogate model that interpolates the original, is of much smaller dimension, and captures the dynamics of the quantities of interest well. We generalize interpolatory projection MOR methods from parametric linear to parametric bilinear systems. We provide necessary subspace conditions to guarantee interpolation of the subsystems and their first and second derivatives, including the parameter gradients and Hessians. Throughout the dissertation, the analysis is illustrated via various benchmark numerical examples. / Doctor of Philosophy / Simulation of mathematical models plays an important role in the development of science. There is a wide range of models and approaches that depend on the information available and the goal of the problem. In this dissertation we focus on three problems whose solution depends on parameters and for which we have either data resulting from simulations of the model or a explicit structure that describes the model. First, for the case when only data are available, we develop an algorithm that builds a data-driven approximation that is then easy to reevaluate. Second, we embed our algorithm in an already developed framework for the solution of a specific kind of model structure: nonlinear eigenvalue problems. Third, given a model with a specific nonlinear structure, we develop a method to build a model with the same structure, smaller dimension (for faster computation), and that provides an accurate approximation of the original model.