The analysis of joint distribution function with bivariate event time data is a challenging problem both theoretically and numerically. This thesis develops a tensor splinebased nonparametric maximum likelihood estimation method to estimate the joint distribution function with bivariate current status data. The tensor I-splines are developed to replace the traditional tensor B-splines in approximating joint distribution function in order to simplify the restricted maximum likelihood estimation problem in computing. The generalized gradient projection algorithm is used to compute the restricted optimization problem. We show that the proposed tensor spline-based nonparametric estimator is consistent and that the rate of convergence is obtained. Simulation studies with moderate sample sizes show that the finite-sample performance of the proposed estimator is generally satisfactory.

Background: Normal cardiac excitability depends upon the coordinated activity of ion channels and transporters. Mutations in genes encoding ion channels affecting their biophysical properties have been known for over twenty years as a root cause of potentially fatal human electrical rhythm disturbance (arrhythmias). More recently, defects in ion channel associated protein (e.g. adapter, regulatory, cytoskeletal proteins) have been shown to cause arrhythmia. Mathematical modeling is ideally suited to integrate large volumes of cellular and in vivo data from human patients and animal disease models with the over goal of determining cellular mechanisms for these atypical human cardiac diseases that involve complex defects in ion channel membrane targeting and/or regulation. Methods and Results: Computational models of ventricular, atrial, and sinoatrial cells were used to determine the mechanism for increased susceptibility to arrhythmias and sudden death in human patients with inherited defects in ankyrin-based targeting pathways. The loss of ankyrin-B was first incorporated into detailed models of the ventricular myocyte to identify the cellular mechanism for arrhythmias in human patients with loos-of-function mutations in ANK2 (encodes ankyrin-B). Mathematical modeling was used to identify the cellular pathway responsible for abnormal Ca2+ handling and cardiac arrhythmias in ventricular cells. A multi-scalar computational model of ankyrin-B deficiency in atrial and sinoatrial cells and tissue was then developed to determine the mechanism for the increased susceptibility to atrial fibrillation in these human patients. Finally, a state-based Markov model of the voltage-gated Na+ channel was incorporated into a ventricular cell model and parameter estimation was performed to determine the mechanism for a new class of human arrhythmia variants that confer susceptibility to arrhythmia by interfering with a regulatory complex comprised of a second member of the ankyrin family, ankyrin-G. Conclusions: Ca2+ accumulation was observed at baseline in the ankyrin-B deficient ventricular model, with pro-arrhythmic spontaneous release and afterdepolarizations in the presence of simulated â-adrenergic stimulation, consistent with the finding of catecholaminergic-induced arrhythmias in human patients. The simulations demonstrated that loss of membrane Na+/Ca2+ exchanger and Na+-K+-ATPase contributed to Ca2+ overload and afterdepolarizations, with loss of Na+/Ca2+ exchanger as the dominant mechanism. In the atrial model of ankyrin-B deficiency, the loss of the L-type Ca2+ channel targeting was identified as the dominant mechanism for the initiation of atrial fibrillation. Finally, the simulations showed that human variants affecting ankyrin-G dependent regulation of NaV1.5 results in arrhythmia by mimicking the phosphorylation of the channel. Most importantly, mathematical modeling has been used to the molecular mechanism underlying human arrhythmia syndromes.

This thesis studies the problem of production and inventory planning for an organization facing uncertainty in demand. Specifically, we examine the problem of assigning workers to tasks, seeking to maximize profits, while taking in consideration learning through experience and stochasticity in demand. As quantitative descriptions of human learning are nonlinear, we employ a reformulation technique that uses binary and continuous variables and linear constraints. Similarly, as demand is not assumed to be known with certainty, we embed this mixed integer representation of how experience translates to productivity in a stochastic workforce assignment model. We further present a matheuristic solution technique and a Markov decision process formulation with a one-step lookahead that allows for the problem to be solved in stages in time as demand information becomes available. With an extensive computational study, we demonstrate the advantages of the matheuristic approach over an off-the-shelf solver and derive managerial insights about task assignment, workforce capacity development, and inventory management. We show that cross training increases as demand uncertainty increases, worker practice increases as inventory holding costs increase, and workers with less initial experience receive more practice than workers with higher initial experience. We further observe that the proposed lookahead MDP model outperforms similar myopic models by producing both increased profit and decreased lost sales and is especially valuable when expecting high demand variation. By recognizing individual differences in learning and modeling the improvement in productivity through experience, results show that the ability to manage workforce capacity can be an effective substitute for inventory. Additionally, we observe that optimal solutions favor the use of inventory for more valuable products and rely on higher productivity for less valuable ones. Further analysis suggests that slower learners tend to specialize more and teams with slower average learning rate tend to produce more inventory.

A knot can be thought of as a knotted piece of string with the ends glued together. A tangle is formed by intersecting a knot with a 3-dimensional ball. The portion of the knot in the interior of the ball along with the fixed intersection points on the surface of the ball form the tangle. Tangles can be used to model protein- DNA binding, so another way to think of a tangle is in terms of segments of DNA (the strings) bounded by the protein complex (the 3-dimensional ball). In this thesis, we look at an algorithm used to list tangles. We also classify tangles into families.

In the field of cognitive neuroscience, there is a need for theory-based approaches to fMRI data analysis. The dynamic neural field model-based approach has been developing to meet this demand. This dissertation describes my contributions to this approach. The methods and tools were demonstrated through a case study experiment on response selection and inhibition. The experiment was analyzed via both the standard behavioral approach and the new model-based method, and the two methods were compared head to head. The methods were quantitatively comparable at the individual-level of the analysis. At the group level, the model-based method reveals distinct functional networks localized in the brain. This validates the dynamic neural field model-based approach in general as well as my recent contributions.

New numerical algorithms are devised for high-order, efficient quadrature in domains arising from the intersection of a hyperrectangle and a manifold implicitly defined by level sets. By casting the manifold locally as the graph of a function (implicitly evaluated through a recurrence relation for the zero level set), a recursion stack is set up in which the interface and integrand information of a single dimension after another will be treated. Efficient means for the resulting dimension reduction process are developed, including maps for identifying lower-dimensional hyperrectangle facets, algorithms for minimal coordinate-flip vertex traversal, which, together with our multilinear-form-based derivative approximation algorithms, are used for checking a proposed integration direction on a facet, as well as algorithms for detecting interface-free sub-hyperrectangles. The multidimensional quadrature nodes generated by this method are inside their respective domains (hence, the method does not require any extension of the integrand) and the quadrature weights inherit any positivity of the underlying single-dimensional quadrature method, if present. The accuracy and efficiency of the method are demonstrated through convergence and timing studies for test cases in spaces of up to seven dimensions. The strengths and weaknesses of the method in high dimensional spaces are discussed. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2017. / July 17, 2017. / approximate integration, C, C++ implementation, cubature quadrature, implicit manifold, level set, recursive dimension reduction algorithm / Includes bibliographical references. / Mark Sussman, Professor Directing Dissertation; Tomasz Plewa, University Representative; Nick Moore, Committee Member; Giray Okten, Committee Member.

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We present a new algorithm for computing the Perron root of a nonnegative irreducible matrix. The algorithm is formulated by combining a reciprocal of the well known Collatz's formula with a special inverse iteration algorithm discussed in [10, Linear Algebra Appl., 15 (1976), pp 235-242]. Numerical experiments demonstrate that our algorithm is able to compute the Perron root accurately and faster than other well known algorithms; in particular, when the size of the matrix is large. The proof of convergence of our algorithm is also presented.

Time reversal is a general technique in wave propagation in inhomogeneous media when a signal is recorded at points of a device called time reversal mirror, gets time reversed and radiated back in the medium. The resulting field has a property of refocusing. Time reversal in acoustics has been extensively studied both experimentally and theoretically. In this thesis we consider the problem of time reversal of electromagnetic waves in inhomogeneous layered media. We use Markov process model for the medium parameters which allows us to exploit diffusion approximation theorem. We show that the field generated by the time reversal mirror focuses at a point of initial source inside of the medium. The size of the focusing spot is of the kind that it is smaller than the one that would be obtained if the medium were homogeneous meaning that the super resolution phenomenon is observed.

This study analyzes two robust failure detection algorithms and applies the algorithms to three power system models. An optimal test signal to distinguish between a failure model and a normal model is calculated using the two algorithms. Advantages and disadvantages of each algorithm, Direct Optimization (DO) and Constrained Control (CC), are discussed. DO uses complex software (Sparse Optimal Control Software by The Boeing Corporation) to solve the necessary and boundary conditions of the optimization problem directly. CC utilizes free software (SciLab by Inria, Enpc.) to solve a two-point boundary value problem based on the necessary and boundary conditions of the optimization problem. Both algorithms yield similar signals, but DO is faster and more accurate yet requires expensive software. CC is not as robust, but can be run on free software and does not need as much fine tuning as the DO algorithm. Examples presented are two DC motor models and a linearized gas turbine model.