No description available.

No description available.

A fundamental task of algebraists is to classify algebraic structures. For example, the classification of finite groups has been widely studied and has benefited from the use of computational tools. Advances in computer power have allowed researchers to attack problems never possible before. In this dissertation, algorithms for <I>noncommutative</I> algebra, when <I>ab</I> is not necessarily equal to <I>ba</I>, are examined with practical implementations in mind. Different encodings of <I>associative algebras</I> and <I>modules</I> are also considered. To effectively analyze these algorithms and encodings, the <I>encoding neutral</I> analysis framework is introduced. This framework builds on the ideas used in the arithmetic complexity framework defined by Winograd. Results in this dissertation fall into three categories: analysis of algorithms, experimental results, and novel algorithms. Known algorithms for calculating the Jacobson radical and Wedderburn decomposition of associative algebras are reviewed and analyzed. The algorithms are compared experimentally and a recommendation for algorithms to use in computer algebra systems is given based on the results. A new algorithm for constructing the Drinfel'd double of finite dimensional Hopf algebras is presented. The performance of the algorithm is analyzed and experiments are performed to demonstrate its practicality. The performance of the algorithm is elaborated upon for the special case of group algebras and shown to be very efficient. The <tt>MeatAxe</tt> algorithm for determining whether a module contains a proper submodule is reviewed. Implementation issues for the <tt>MeatAxe</tt> in an encoding neutral environment are discussed. A new algorithm for constructing endomorphism rings of modules defined over path algebras is presented. This algorithm is shown to have better performance than previously used algorithms. Finally, a linear time algorithm, to determine whether a quotient of a path algebra, with a known Gröbner basis, is finite or infinite dimensional is described. This algorithm is based on the Aho-Corasick pattern matching automata. The resulting automata is used to efficiently determine the dimension of the algebra, enumerate a basis for the algebra, and reduce elements to normal forms. / Ph. D.

This work focuses on commutative algebra, its combinatorial and computational aspects, and its interactions with statistics. The main objects of interest are projective varieties in Pn, algebraic properties of their coordinate rings, and the combinatorial invariants, such as Hilbert series and Gröbner fans, of their defining ideals. Specifically, the ideals in this work are all toric ideals, and they come in three flavors: they are defining ideals of a family of classical varieties called rational normal scrolls, cut ideals that can be associated to a graph, and phylogenetic ideals arising in a new and increasingly popular area of algebraic statistics.

The purpose of this study was to determine whether there were any differences in basic skills maintenance between Algebra I students who used calculators during classroom mathematics instruction and Algebra I students who did not use calculators during classroom mathematics instruction. Another purpose of this study was to determine whether there were any differences in algebra achievement between Algebra I students who used calculators during classroom mathematics instruction and Algebra I students who did not use calculators during classroom mathematics instruction. This study also investigated the effects of the use of the calculator in Algebra I classes on students' attitudes toward mathematics.

El objetivo principal de esta tesis es calcular la K-Teoría de las C∗-Álgebras y con aplicación al cálculo de la K-Teoría del Álgebra de Cuntz y Álgebra de Toeplitz mediante la K-teoría de las C∗-Álgebras de grafos dirigidos. / Tesis

Thesis (M.Sc. (Mathematics) -- University of Limpopo, 2010 / Summary not available

We consider and solve some open conjectures on the asymptotic behavior of the number of different numbers of the nonassociate factorizations of prescribed minimal length for specific finite factorization domains. The asymptotic behavior will be classified for Cohen-Kaplansky domains in Chapter 1 and for domains of the form R=K+XF[X] for finite fields K and F in Chapter 2. A corollary of the main result in Chapter 3 will determine the asymptotic behavior for Krull domains with finite divisor class group.

Thesis (Ed.D.)--University of West Florida, 2007. / Title from title page of source document. Document formatted into pages; contains 87 pages. Includes bibliographical references.

Thesis (Ph. D.)--Open University. BLDSC no. DX212669.