Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematical Sciences." Includes bibliographical references (p. 52-53).

This thesis is mainly concerned with problems in the areas of the Calculus of Variations and Partial Differential Equations (PDEs). The properties of the functional to minimize with respect to the given topology play an important role in the existence of minimizers of integral problems. We will introduce the important concepts of quasiconvexity and polyconvexity. Inspired by finite element methods from Numerical Analysis, we introduce a perturbed problem which has some surprising uniqueness properties.

This paper investigates some of the ideas and algorithms developed for exploiting the structure of quasiseparable matrices. The case of purely scalar generators is considered initially. The process by which a quasiseparable matrix is represented as the product of matrices comprised of its generators is explained. This is done clearly in the scalar case, but may be extended to block generators. The complete factoring approach is then considered. This consists of two stages: inner-outer factorization followed by inner-coprime factorization. Finally, the stability of the algorithm is investigated. The algorithm is used to factor various quasiseparable matrices R created first using minimal generators, and subsequently using non-minimal generators. The result is that stability of the algorithm is compromised when non-minimal generators are present.

This study investigated the source attribution of ship emissions to atmospheric particulate matter with a median aerodynamic diameter less than, or equal to 2.5 micron (PM2.5) in the port city of Halifax, Nova Scotia, Canada. The USEPA PMF model successfully determined the following sources with the average mass (percentage) contribution: Sea salt 0.147 µg m-3 (5.3%), Surface dust 0.23 µg m-3 (8.3%), LRT Secondary (ammonium sulfate) 0.085 µg m-3 (3.1%), LRT Secondary (nitrate and sulfate) 0.107 µg m-3 (3.9%), Ship emissions 0.182 µg m-3 (6.6%), and Vehicles and re-suspended gypsum 2.015 µg m-3 (72.8%). A good correlation was achieved between PM2.5 total mass predicted and observed with R2 = 0.83, bias = -0.23, and RMSE = 0.09 µg m-3. In addition, a 2.5 times (60%) reduction in sulfate was estimated, when compared to 2006-2008 Government data in Halifax.

Baigiamajame darbe nagrinėjama paprastoji antros eilės diferencialinė lygtis. Jos sprendinių gavimui ir analizei naudojamas faktorizacijos metodas – ieškomosios funkcijos skaidymas dauginamaisiais bei kiti tradiciniai paprastųjų diferencialinių lygčių sprendimo metodai: nepriklausomo kintamojo keitimo metodas, konstantų varijavimo metodas. / In this work is analyzed second-order differential equation. I use factorization method and other traditional ordinary differential equations approaches as an example: independent variable exchange method, variation of constants method and direct integration, to find solutions of the equation.

Thesis (M.S.)--Brigham Young University. Dept. of Statistics, 2006. / Includes bibliographical references (p. 96-98).

Thesis--Darmstadt. / Includes bibliographical references (p. 109-112).

Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2007. / In this thesis, we surveyed the most important methods for factorization of polynomials over a global field, focusing on their strengths and showing their most striking disadvantages. The algorithms we have selected are all modular algorithms. They rely on the Hensel factorization technique, which can be applied to all global fields giving an output in a local field that can be computed to a large enough precision. The crucial phase of the reconstruction of the irreducible global factors from the local ones, determines the difference between these algorithms. For different fields and cases, different techniques have been used such as residue class computations, ideal calculus, lattice techniques. The tendency to combine ideas from different methods has been of interest as it improves the running time. This appears for instance in the latest method due to van Hoeij, concerning the factorization over a number field. The ideas here can be used over a global function field in the form given by Belabas et al. using the logarithmic derivative instead of Newton sums. Complexity analysis was not our objective, nevertheless it was important to mention certain results as part of the properties of these algorithms.

Chapter 1 reviews better-known factorization theorems of a square matrix. For example, a square matrix over a field can be expressed as a product of two symmetric matrices; thus square matrices over real numbers can be factorized into two diagonalizable matrices. Factorizing matrices over complex num hers into Hermitian matrices is discussed. The chapter concludes with theorems that enable one to prescribe the eigenvalues of the factors of a square matrix, with some degree of freedom. Chapter 2 proves that a square matrix over arbitrary fields (with one exception) can be expressed as a product of two diagona lizab le matrices. The next two chapters consider decomposition of singular matrices into Idempotent matrices, and of nonsingutar matrices into Involutions. Chapter 5 studies factorization of a comp 1 ex matrix into Positive-( semi )definite matrices, emphasizing the least number of such factors required / Mathematical Sciences / M.Sc. (MATHEMATICS)

Representing and handling high-dimensional data has been increasingly ubiquitous in many real world-applications, such as computer vision, machine learning, and data mining. High-dimensional data usually have intrinsic low-dimensional structures, which are suitable for subsequent data processing. As a consequent, it has been a common demand to find low-dimensional data representations in many machine learning and data mining problems. Factorization methods have been impressive in recovering intrinsic low-dimensional structures of the data. When seeking low-dimensional representation of the data, traditional methods mainly face two challenges: 1) how to discover the most variational features/information from the data; 2) how to measure accurate nonlinear relationships of the data. As a solution to these challenges, traditional methods usually make use of a two-step approach by performing feature selection and manifold construction followed by further data processing, which omits the dependence between these learning tasks and produce inaccurate data representation. To resolve these problems, we propose to integrate feature learning and graph learning with factorization model, which allows the goals of learning features, constructing manifold, and seeking new data representation to mutually enhance and lead to powerful data representation capability. Moreover, it has been increasingly common that 2-dimensional (2D) data often have high dimensions of features, where each example of 2D data is a matrix with its elements being features. For such data, traditional data usually convert them to 1-dimensional vectorial data before data processing, which severely damages inherent structures of such data. We propose to directly use 2D data for seeking new representation, which enables the model to preserve inherent 2D structures of the data. We propose to seek projection directions to find the subspaces, in which spatial information is maximumly preserved. Also, manifold and new data representation are learned in these subspaces, such that the manifold are clean and the new representation is discriminative. Consequently, seeking projections, learning manifold and constructing new representation mutually enhance and lead to powerful data representation technique.