<p>The purpose of this thesis is to find sufficient conditions under which a non-commutative version of the polynomial ring in two variables exists. The non-commutative rings we construct are non-commutative symmetric algebras over a two-sided vector space. After reviewing the definition of a two-sided vector space and giving some examples, we briefly recall the theory of simple two-sided vector spaces. We then assume k is a field of characteristic zero and t is transcendental over k and we find sufficient conditions under which a simple k-central two-sided vector space V over k(t) has left and right dimension two. Given such a V, and letting ^*V and V^* denote the left and right duals we find conditions under which (V^i*,V^(i+1)*,V^(i+2)* ) has a simultaneous for all i, i an integer. This condition implies the non-commutative symmetric algebra over V can be constructed. We conclude by exhibiting a five-dimensional family of simple k-central two-sided vector spaces over k(t) of left and right dimension two who non-commutative symmetric algebras exist.</p>

A common problem in imaging science is to estimate some underlying true image given noisy measurements of image intensity. When image intensity is measured by the counting of incident photons emitted by the object of interest, the data-noise is accurately modeled by a Poisson distribution, which motivates the use of Poisson maximum likelihood estimation. When the underlying model equation is ill-posed, regularization must be employed. I will present a computational framework for solving such problems, including statistically motivated methods for choosing the regularization parameter. Numerical examples will be included.

Trend estimation is important in many fields, though arguably the most important applications appear in ecology. Trend is difficult to quantify; in fact, the term itself is not well-defined. Often, trend is quantified by estimating the slope coefficient in a regression model where the response variable is an index of population size, and time is the explanatory variable. Linear trend is often unrealistic for biological populations; in fact, many critical environmental changes occur abruptly as a result of very rapid changes in human activities. My PhD research has involved formulating methods with greater flexibility than those currently in use. Penalized spline regression provides a flexible technique for fitting a smooth curve. This method has proven useful in many areas including environmental monitoring; however, inference is more difficult than with ordinary linear regression because so many parameters are estimated. My research has focused on developing methods of trend detection and comparing these methods to other methods currently in use. Attention is given to comparing estimated Type I error rates and power across several trend detection methods. This was accomplished through an extensive simulation study. Monte Carlo simulations and randomization tests were employed to construct an empirical sampling distribution for the test statistic under the null hypothesis of no trend. These methods are superior over smoothing methods over other smoothing methods of trend detection with respect to achieving the designated Type I error rate. The likelihood ratio test using a mixed effects model had the most power for detecting linear trend while a test involving the first derivative was the most powerful for detecting nonlinear trend for small sample sizes.

Tree ensembles have proven to be a popular and powerful tool for predictive modeling tasks. The theory behind several of these methods (e.g. boosting) has received considerable attention. However, other tree ensemble techniques (e.g. bagging, random forests) have attracted limited theoretical treatment. Specifically, it has remained somewhat unclear as to why the simple act of randomizing the tree growing algorithm should lead to such dramatic improvements in performance. It has been suggested that a specific type of tree ensemble acts by forming a locally adaptive distance metric [Lin and Jeon, 2006]. We generalize this claim to include all tree ensembles methods and argue that this insight can help to explain the exceptional performance of tree ensemble methods. Finally, we illustrate the use of tree ensemble methods for an ecological niche modeling example involving the presence of malaria vectors in Africa.

<P>The noise contained in images collected by a charge coupled device (CCD) camera is predominantly of Poisson type. This motivates the use of the negative logarithm of the Poisson likelihood in place of the ubiquitous least squares t-to-data. However, if the underlying mathematical model is assumed to have the form z = Au, where A is a linear, compact operator, the problem of minimizing the negative log-Poisson likelihood function is ill-posed, and hence some form of regularization is required. In this work, it involves solving a variational problem of the form u def = arg min u0 `(Au; z) + J(u); where ` is the negative-log of a Poisson likelihood functional, and J is a regularization functional. The main result of this thesis is a theoretical analysis of this variational problem for four dierent regularization functionals. In addition, this work presents an ecient computational method for its solution, and the demonstration of the eectiveness of this approach in practice by applying the algorithm to simulated astronomical imaging data corrupted by the CCD camera noise model mentioned above.</P>

The first half of this dissertation focuses on computational methods for solving the constrained quadratic program (QP) within the support vector machine (SVM) classifier. One of the SVM formulations requires the solution of bound and equality constrained QPs. We begin by describing an augmented Lagrangian approach which incorporates the equality constraint into the objective function, resulting in a bound constrained QP. Furthermore, all constraints may be incorporated into the objective function to yield an unconstrained quadratic program, allowing us to apply the conjugate gradient (CG) method. Lastly, we adapt the scaled gradient projection method of [10] to the SVM QP and compare the performance of these methods with the state-of-the-art sequential minimal optimization algorithm and MATLAB's built in constrained QP solver, quadprog. The augmented Lagrangian method outperforms other state-of-the-art methods on three image test cases. The second half of this dissertation focuses on computational methods for large-scale Kalman filtering applications. The Kalman filter (KF) is a method for solving a dynamic, coupled system of equations. While these methods require only linear algebra, standard KF is often infeasible in large-scale implementations due to the storage requirements and inverse calculations of large, dense covariance matrices. We introduce the use of the CG and Lanczos methods into various forms of the Kalman filter for low-rank approximations of the covariance matrices, with low-storage requirements. We also use CG for efficient Gaussian sampling within the ensemble Kalman filter method. The CG-based KF methods perform similarly in root-mean-square error when compared to the standard KF methods, when the standard implementations are feasible, and outperform the limited-memory Broyden-Fletcher-Goldfarb-Shanno approximation method.

General sufficient conditions are established for maps between function algebras to be composition or weighted composition operators, which extend previous results regarding spectral conditions for maps between uniform algebras. Let X and Y be a locally compact Hausdorff spaces, where A \subset C(X) and B \subset C(Y) are function algebras, not necessarily with unit. Also let \partial A be the Shilov boundary of A, \delta A the Choquet boundary of A, and p(A) the set of p-points of A. A map T \colon A \to B is called weakly peripherally-multiplicative if the peripheral spectra of fg and TfTg have non-empty intersection for all f,g in A. (i.e. \sigma_{pi}( fg ) \cap \sigma_{pi}(TfTg ) \neq \emptyset for all f,g in A) The map is said to be almost peripherally-multiplicative if the peripheral spectrum of fg is contained in the peripheral spectrum of TfTg (or if the peripheral spectrum of TfTg is contained in the peripheral spectrum of fg) for all f,g in A. Let X be a locally compact Hausdorff space and A \subset C(X) be a dense subalgebra of a function algebra, not necessarily with unit, such that \delta A = p(A). We show that if T\colon A \to B is a surjective map onto a function algebra B\subset C(Y) that is almost peripherally-multiplicative, then there is a homeomorphism \psi\colon \delta B\to\delta A and a function \alpha on \delta B so that (Tf)(y)=\a(y)\,f(\psi(y)) for all f \in A and y \in\delta B, i.e. T is a weighted composition operator where the weight function is a signum function. We also show that if T is weakly peripherally-multiplicative, and either \sigma_{pi}(f)\subset \sigma_{pi}(Tf) for all f in A, or, alternatively, \sigma_{pi}(Tf) \subset \sigma_{pi}(f) for all f in A, then (Tf)(y)=f(\psi(y)) for all f \in A and y \in \delta B. In particular, if A and B are uniform algebras and T \colon A \to B is a weak peripherally-multiplicative operator, that has a limit, say b, at some a in A with a^2=1, then (Tf)(y)=b(y)\,a(\psi(y))\, f(\psi(y)) for every f in A and y in \delta B. Also, we show that if a weak peripherally-multiplicative map preserving peaking functions in the sense \mathcal{P}(B) \subset T[ \mathbb{T} \cdot \mathcal{P}(A)] or T[\mathcal{P}(A)] \subset \mathbb{T} \cdot \mathcal{P}(B) then T is a weighted composition operator with a signum weight function. Finally, for function algebras containing sufficiently many peak functions, including function algebras on metric spaces, it is shown that weak peripherally-multiplicative maps are necessarily composition operators.

<p>There are many operators associated with a domain Ω â âⁿ with smooth boundary âΩ. There are two closely related projections that are of particular interest. The <i>Bergman projection</i> <b><i>B</b></i> is the orthogonal projection of L²(Ω) onto the closed subspace L²(Ω)â©O(Ω), where O(Ω)is the space of all holomorphic functions on â¦. The <i>Szeg� projection</i> <b><i>S</b></i> is the orthogonal projection of L²(ââ¦) onto the space H²(Ω) of boundary values of elements of O(Ω). These projection operators have integral representations</p> <p><b><i>B</b></i>[f](z) = <big>â«</big>_â¦,f(w)<b><i>B</b></i>(z,w)dw, <b><i>S</b></i>[f](z) = <big>â«</big>_ââ¦,f(w)<b><i>S</b></i>(z,w)do(w).</p> <p>The distributions <b><i>B</b></i> and <b><i>S</b></i> are known respectively as the Bergman and Szeg� kernels. In an attempt to prove that <b><i>B</b></i> and <b><i>S</b></i> are bounded operators on L^p, 1 < p < â, many authors have obtained size estimates for the kernels B and S for <i>pseudoconvex</i> domains in âⁿ.</p> <p>In this thesis, we restrict our attention to the Szeg� kernel for a large class of domains of the form 1 Such a domain fails to be pseudoconvex precisely when b is not convex on all of R. In an influential paper, Nagel, Rosay, Stein, and Wainger obtain size estimates for both kernels and sharp mapping properties for their respective operators in the convex setting. Consequently, if b is a convex polynomial, the Szeg� kernel S is absolutely convergent off the diagonal only. Carracino proves that the Szeg� kernel has singularities on <i>and off</i> the diagonal for a specific non-smooth, <i>{non-convex</i> piecewise defined quadratic b. Her results are novel since very little is known for the Szeg� kernel for non-pseudoconvex domains 2. I take b to be an arbitrary even-degree polynomial with positive leading coefficient and identify the set in 3 on which the Szeg� kernel is absolutely convergent. For a polynomial b, we will see that the Szeg� kernel is smooth off the diagonal if and only if b is convex. These results provide an incremental step toward proving the projection S is bounded on 4, for a large class of non-pseudoconvex domains â¦.</p>

Includes bibliographical references. / A popular approach to formulating the initial-boundary value problem for fibre suspension flows is that in which fibre orientation is accounted for in an averaged sense, through the introduction of a second-order orientation tensor A. This variable, together with the velocity and pressure, then constitutes the set of unknown variables for the problem. The governing equations are balance of linear momentum, the incompressibility condition, an evolution equation for A, and a constitutive equation for the stress. The evolution equation contains a fourth-order orientation tensor A, and it is necessary to approximate A as a function of A, through a closure relation. The purpose of this these is to examine the well-posedness of the equations governing fibre fibre suspension flows, for various closure relations. It has previously been shown by GP Galdi and BD Reddy that, for the linear closure, the problem is wellposed provided that the particle number, a material constant, is less than a critical value. The work by Galdi and Reddy made of a model in which rotary diffusivity is a function of the flow. This thesis re-examines these issues in two different ways. First, the second law of thermodynamics is used to establish the constraints that the constitutive equations have to satisfy in order to be compatible with this law. This investigation is carried out for a variety of closure rules. The second contribution of the thesis concerns the existence and uniqueness of solutions to the governing equations, for the linear and quadratic closures; for a model in which the rotary diffusivity is treated as a constant, local and global existence of solutions are established, for sufficiently small data, and in the case of the linear closure, for admissible values of the particle number. The existence theory uses a Schauder fixed point approach.

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