Thesis (Ph. D.)--Illinois State University, 1998. / Title from title page screen, viewed July 5, 2006. Dissertation Committee: Jane O. Swafford (chair), John A. Dossey, Cheryl Hawker, Cynthia W. Langrall. Includes bibliographical references (leaves 142-158) and abstract. Also available in print.

This thesis concerns representation theory of the symmetric groups and related algebras. In recent years, the study of the “quiver Hecke algebras”, constructed independently by Khovanov and Lauda and by Rouquier, has become extremely popular. In this thesis, our motivation for studying these graded algebras largely stems from a result of Brundan and Kleshchev – they proved that (over a field) the KLR algebras have cyclotomic quotients which are isomorphic to the Ariki–Koike algebras, which generalise the Hecke algebras of type A, and thus the group algebras of the symmetric groups. This has allowed the study of the graded representation theory of these algebras. In particular, the Specht modules for the Ariki–Koike algebras can be graded; in this thesis we investigate graded Specht modules in the KLR setting. First, we conduct a lengthy investigation of the (graded) homomorphism spaces between Specht modules. We generalise the rowand column removal results of Lyle and Mathas, producing graded analogues which apply to KLR algebras of arbitrary level. These results are obtained by studying a class of homomorphisms we call dominated. Our study provides us with a new result regarding the indecomposability of Specht modules for the Ariki–Koike algebras. Next, we use homomorphisms to produce some decomposability results pertaining to the Hecke algebra of type A in quantum characteristic two. In the remainder of the thesis, we use homogeneous homomorphisms to study some graded decomposition numbers for the Hecke algebra of type A. We investigate graded decomposition numbers for Specht modules corresponding to two-part partitions. Our investigation also leads to the discovery of some exact sequences of homomorphisms between Specht modules.

Title from title page of PDF (University of Missouri--St. Louis, viewed February 9, 2010). Includes bibliographical references (p.174-195).

This thesis consists of three parts. In the first part we look at Hopf algebras. We classify pointed rank one Hopf algebras over fields of prime characteristic which are generated as algebras by the first term of the coradical filtration. These Hopf algebras were classified by Radford and Krop for fields of characteristic zero. We obtain three types of Hopf algebras presented by generators and relations. The third type is new and has not previously appeared in literature. The second part of this thesis deals with Auslander-Reiten theory of finitedimensional algebras over fields. We consider G-transitive algebras and develop necessary conditions for them to have Auslander-Reiten components with Euclidean tree class. Thereby a result in [F3, 4.6] is corrected and generalized. We apply these results to G-transitive blocks of the universal enveloping algebras of restricted p-Lie algebras. Finally we deduce a condition for a smash product of a local basic algebra Λ with a commutative semi-simple group algebra to have components with Euclidean tree class, in terms of the components of the Auslander-Reiten quiver of Λ. In the last part we introduce and analyze Auslander-Reiten components for the bounded derived category of a finite-dimensional algebra. We classify derived categories whose Auslander-Reiten quiver has either a finite stable component or a stable component with finite Dynkin tree class or a bounded stable component. Their Auslander-Reiten quiver is determined. We use these results to show that certain algebras are piecewise hereditary. Also a necessary condition for the existence of components of Euclidean tree class is deduced. We determine components that contain shift periodic complexes.

Thesis (Ph.D)--Mississippi State University. Department of Instructional Systems, Leadership and Workforce Development. / Title from title screen. Includes bibliographical references.

The aim of this research was to investigate how I engaged in constructivist teaching (CT) when helping a group of low-performing Grade 7 students to develop new meanings of notation as they started to learn formal algebra. Data was collected over a period of one scholastic year, in which I explored the teacher-student dynamics during my mathematics lessons, where students learnt new representations and interpretations of notation with the help of the computer software Grid Algebra. Analysing video recordings of my lessons, I observed myself continuously changing my teaching purpose as I negotiated between the mathematics I intended to teach and the mathematics being constructed by my students. These shifts of focus and purpose were used to develop a conceptual framework called Mathematics-Negotiation-Learner (M-N-L). Besides serving as a CT model, the M-N-L framework was found useful to determine the extent to which I managed to engage in CT during the lessons and also to identify moments where I lost my sensitivity to students constructions of knowledge. The effectiveness of my CT was investigated by focusing on students learning, for which reason I developed the analytical framework called CAPS (Concept-Action-Picture-Symbol). The CAPS framework helped me to analyse how students developed notions about properties of operational notation, the structure and order of operations in numerical and algebraic expressions, and the relational property of the equals sign. Grid Algebra was found to be a useful tool in helping students to enrich their repertoire of representations and to develop new interpretations of notation through what I defined as informal- and formal-algebraic activities. All students managed to transfer these representations and interpretations of notation to pen-and-paper problems, where they successfully worked out traditionally set substitution-and-evaluation tasks.

This diploma thesis deals with the comparison of historical solutions of word problems with student solutions. Its aim was to describe how students solve historical word problems, while looking for analogies between student and historical solutions. This intention led me to a better understanding of student solutions. The theoretical part of the thesis describes important concepts for algebraic word problems, such as a variable, algebraic expression or algebraic word problem. In the historical part I describe chronologically the development of algebra from antiquity through the Middle Ages and the Renaissance to the Baroque. In each period, I mention important mathematicians of the time and present several solved word problems. In most cases, I analyze these solutions from the perspective of today's mathematics. The theoretical part describes the research that took place at the eight-year grammar school. As part of the research, I gave students 6 historical tasks across historical periods and then analyzed the ways in which students solved problems. I found that for most of the tasks, there were solutions similar to the historical solutions among the student's solutions. Some historical methods appeared very often. An example is the use of addition instead of multiplication, or division, as used by the...