We define morphism (E, M)-structures in an abstract category, develop their basic properties and present some examples. We also consider the existence of such factorization structures, and find conditions under which they can be extended to factorization structures for certain classes of sources. There is a Galois correspondence between the collection of all subclasses of X-morphisms and the collection of all subclasses of X-objects. A-epimorphisms diagonalize over A-regular morphisms. Given an (E, M)-factorization structure on a finitely complete category, E-separated objects are those for which diagonal morphisms lie in M. Other characterizations of E-separated objects are given. We give a bijective correspondence between the class of all (E, M)factorization structures with M contained in the class of all X-embeddings and the class of all strong limit operators. We study M-preserving morphisms, M-perfect morphisms and M-compact objects in a morphism (E, M)-hereditary construct, and prove some of their properties which are analogous to the topological ones. / Mathematical Sciences / M. Sc. (Mathematics)

The entire thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file; a non-technical public abstract appears in the public.pdf file. Title from PDF of title page (University of Missouri--Columbia, viewed on December 30, 2009). Thesis advisor: Dr. William Banks. Includes bibliographical references.

In this study, we address the factorization problem in the Hardy H('p) spaces, and provide a fast algorithm for its implementation with applications to some important engineering problems. The Thesis is presented in three autonomous papers. / In the first paper we lay down the technical foundation of the new approach in the scalar case. First, the factorization problem is formulated in the H('p) spaces. A formulation with sufficient generality to encompass practically all such engineering problems. Necessary and sufficient conditions for the existence of the spectral factors are derived, and a characterization of the class of functions admitting a canonical factorization is obtained. The reduction method is applied to certain Toeplitz equations in H('2) space to generate a sequence of approximate spectral factors. When the Laguerre basis is used in the reduction method the Toeplitz equation turns out to a Toeplitz set of linear equations. We also provide an error bound and an estimate for the speed of convergence. / In the second paper the matrix version of all the scalar results is provided and enriched with discussions and extension. In particular, we have shown that the factorization problem is associated with the solutions of certain Toeplitz equations in / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / spaces. The classical Gohberg-Krein factorization is re-examined within the framework developed here, and the connections between the outer-factorization, the canonical factorization, and inversion of certain Toeplitz operators have also been unveiled. / In Part III we generalize the Davis and Barry formula for the feedback gain in the LQR problems. The new setting, equipped with the spectral factorization method, provides fast and efficient algorithms for solving a wide class of LQR problems, rational matrix factorization, and positive polynomials factorization. Our parallel results for the discrete time case are given in brief together with many interesting computational properties.

Thesis (M.A.)--The University of North Carolina at Greensboro, 2010. / Directed by Paul Duvall; submitted to the Dept. of Mathematics and Statistics. Title from PDF t.p. (viewed Jul. 16, 2010). Includes bibliographical references (p. 39-40).

Thesis (M.A.)--University of the Western Cape, 2008. / Includes bibliographical references (leaves 81- 82).

We define morphism (E, M)-structures in an abstract category, develop their basic properties and present some examples. We also consider the existence of such factorization structures, and find conditions under which they can be extended to factorization structures for certain classes of sources. There is a Galois correspondence between the collection of all subclasses of X-morphisms and the collection of all subclasses of X-objects. A-epimorphisms diagonalize over A-regular morphisms. Given an (E, M)-factorization structure on a finitely complete category, E-separated objects are those for which diagonal morphisms lie in M. Other characterizations of E-separated objects are given. We give a bijective correspondence between the class of all (E, M)factorization structures with M contained in the class of all X-embeddings and the class of all strong limit operators. We study M-preserving morphisms, M-perfect morphisms and M-compact objects in a morphism (E, M)-hereditary construct, and prove some of their properties which are analogous to the topological ones. / Mathematical Sciences / M. Sc. (Mathematics)

In this dissertation, we examine atomicity in rings with zero divisions. We begin by examining the relationship between a ring’s level of atomicity and the highest level of irreducibility shared by the ring’s irreducible elements. Later, we chose one of the higher forms of atomicity and identify ways of building large classes of examples of rings that rise to this level of atomicity but no higher. Characteristics of the various types of irreducible elements will also be examined. Next, we extend our view to include polynomial extensions of rings with zero divisors. In particular, we focus on properties of the three forms of maximal common divisors and how a ring’s classification as an MCD, SMCD, or VSMCD ring affects its atomicity. To conclude, we identify some unsolved problems relating to the topics discussed in this dissertation.

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by Chiu Chak Lam. / Thesis submitted in: June, 1997. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 67-69). / Abstract also in Chinese. / Chapter 1 --- Basic Knowledge of Elliptic Curve --- p.2 / Chapter 1.1 --- Elliptic Curve Group Law --- p.2 / Chapter 1.2 --- Discriminant and j-invariant --- p.7 / Chapter 1.3 --- Elliptic Curve over C --- p.10 / Chapter 1.4 --- Complex Multiplication --- p.15 / Chapter 2 --- Order of Elliptic Curve Group Over Finite Fields and the Endo- morphism Ring --- p.18 / Chapter 2.1 --- Hasse's Theorem --- p.18 / Chapter 2.2 --- The Torsion Group --- p.23 / Chapter 2.3 --- The Weil Conjectures --- p.33 / Chapter 3 --- Computing the Order of an Elliptic Curve over a Finite Field --- p.35 / Chapter 3.1 --- Schoof's Algorithm --- p.35 / Chapter 3.2 --- Computation Formula --- p.38 / Chapter 3.3 --- Recent Works --- p.42 / Chapter 4 --- Primality Test Using Elliptic Curve --- p.43 / Chapter 4.1 --- Goldwasser-Kilian Test --- p.43 / Chapter 4.2 --- Atkin's Test --- p.44 / Chapter 4.3 --- Binary Quadratic Form --- p.49 / Chapter 4.4 --- Practical Consideration --- p.51 / Chapter 5 --- Elliptic Curve Factorization Method --- p.54 / Chapter 5.1 --- Lenstra's method --- p.54 / Chapter 5.2 --- Worked Example --- p.56 / Chapter 5.3 --- Practical Considerations --- p.56 / Chapter 6 --- Elliptic Curve Public Key Cryptosystem --- p.59 / Chapter 6.1 --- Outline of the Cryptosystem --- p.59 / Chapter 6.2 --- Index Calculus Method --- p.61 / Chapter 6.3 --- Weil Pairing Attack --- p.63