Browsing by Author "Dr. Ernest Stitzinger, Committee Member"

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Algorithms for Computing Restricted Root Systems and Weyl Groups
(2006-05-04) Cicco, Tracey Martine Westbrook; Dr. Aloysius Helminck, Committee Chair; Dr. Ernest Stitzinger, Committee Member; Dr. Tom Lada, Committee Member; Dr. Amassa Fauntleroy, Committee Member
While the computational packages LiE, Gap4, Chevie, and Magma are sufficient for work with Lie Groups and their corresponding Lie Algebras, no such packages exist for computing the k-structure of a group or the structure of symmetric spaces. My goal is to examine the k-structure of groups and the structure of symmetric spaces and arrive at various algorithms for computing in these spaces.
Characteristics of Complexity within the Lattice of Compactifications
(2003-07-07) Mawhinney, Katherine Joyner; Dr. Ernest Stitzinger, Committee Member; Dr. William Swallow, Committee Member; Dr. Gary D. Faulkner, Committee Chair; Dr. Richard Chandler, Committee Member; Dr. Jo-Ann Cohen, Committee Member
The purpose of this research is to determine some topological characteristics that may be used to classify a Hausdorff compactification of a topological space as a complex compactification, within the lattice of compactifications. The Stone-Cech compactification is the supremum of the lattice, the Alexandroff one-point compactification the infimum. We look to characteristics that the Stone-Cech compactification holds and whether or not those properties are found in compactifications "close" to it. The idea of a complex compactification has not been strictly defined and there are numerous properties that could be used in a definition. Beginning with mappings with a finite number of nontrivial fibers,, we find that F-space is invariant. F-space can not be guaranteed for all finite-to-one mappings. The characteristic we call G-int under any finite-to-one irreducible mapping and the continuous image of a nowhere F space is nowhere F, a characteristic of compactifications that are simple. We also consider the mappings on the Stone-Cech compactification of the natural numbers that are simple mappings, proving that if a simple mapping is finite-to-one, then so is its generator and vice versa.
Effective Teaching and Uses of Instructional Representations in Secondary Geometry: A Comparison of a Novice and an Experienced Mathematics Teacher
(2006-12-19) Slaten, Kelli Marlene; Dr. Ernest Stitzinger, Committee Member; Dr. Sarah B. Berenson, Committee Co-Chair; Dr. Karen F. Hollebrands, Committee Co-Chair; Dr. Lee V. Stiff, Committee Member
The purpose of this qualitative case study is to investigate the uses of instructional representations of a novice secondary mathematics teacher and an experienced mathematics teacher in the content area of secondary geometry. Instructional representations are the tools teachers use to communicate their mathematical knowledge to students (Berenson & Nason, 2003). Classroom observations and semi-structured interviews were conducted and analyzed in order to find emergent patterns among the participants' uses of instructional representations. Patterns are reported from each participant's uses of instructional representations and from a cross-case analysis of both participants' uses of instructional representations. The Pirie-Kieren theory (Pirie & Kieren, 1994b) for students' growth of mathematical understanding serves as the framework for the study. The Pirie-Kieren theory describes eight potential levels of student understanding based on the cognitive changes that occur during the processes of learning mathematics. These eight levels were adapted for the purposes of this study and used to describe how the participants' uses of instructional representations allowed and fostered opportunities for students to engage within those levels of mathematical understanding. In order to facilitate students' growth of mathematical understanding, effective teachers have a well-developed knowledge base for teaching, including knowledge of multiple instructional representations and the connections between them (Lesh, Post, & Behr, 1987; Moseley & Brenner, 1997; NCTM, 2000; Rider, 2004; Wilson, Shulman, & Richert, 1987). The results of this study reveal the importance of examining how they use those representations in order to better understand how teacher knowledge contributes to effective teaching and student learning.
The Lattice of Equivalence Classes of Closed Sets and the Stone-Cech Compactification.
(2005-03-16) Seaton, Gerald Arthur; Dr. Gary Faulkner, Committee Chair; Dr. Richard Chandler, Committee Member; Dr. Kailash Misra, Committee Member; Dr. Ernest Stitzinger, Committee Member
βX X is the remainder of the Stone-Cech compactification of a locally compact space X. This paper introduces a lattice which we call L(X) that is constructed using equivalence classes of closed sets of X. We then determine that St(L(X)) (the set of ultrafilters on L(X)) is homeomorphic to βX X. We subsequently give some examples. Most notably, for X = H this now provides a lattice-theoretic approach for representing βH H. In addition, we expand and clarify some aspects of lattice theory related to our constructions. We introduce the term "upwardly nonlinear" as a way to describe lattices with a certain property related to the ultrafilters on it. We also investigate some of the lattice properties of L(X).
On Locally Invertible Encoders and Multidimensional Convolutional Codes
(2006-08-10) Lobo, Ruben Gerald; Dr. Mladen A. Vouk, Committee Co-Chair; Dr. Donald L. Bitzer, Committee Co-Chair; Dr. Brian L. Hughes, Committee Member; Dr. Alexandra Duel-Hallen, Committee Member; Dr. Ernest Stitzinger, Committee Member
Multidimensional (m-D) convolutional codes generalize the well known notion of a 1-D convolutional code defined over a univariate polynomial ring with coefficients in a finite field to multivariate polynomial rings. The more complicated structure of a multivariate polynomial ring when compared to a univariate one, however, makes the generalization nontrivial. While 1-D convolutional codes have been thoroughly understood and have wide applications in communication systems, the theory of m-D convolutional codes is still in its infancy, and these codes lack unified notation and practical implementation. This dissertation develops a sequence space approach for realizing m-D convolutional codes. While most of the existing research is focused on algebraic aspects, fundamental issues regarding practical implementation that are well developed and fairly straightforward in the 1-D case have remained undefined for m-D convolutional codes. In this dissertation we address some of these issues. We define a new notion of sequence space ordering and show that certain multivariate polynomial matrices which we call as locally invertible encoders, when transformed to the sequence space domain, have an invertible subsequence map between their input and output sequences. This subsequence map has a well defined structure that allows for the explicit construction of locally invertible encoders by performing elementary operations on the ground field without the use of any polynomial operations. We use the invertible subsequence map to introduce a novel method to encode and invert multidimensional sequences. We show that locally invertible encoders have good structural properties which make them a natural choice to generate multidimensional convolutional codes.
On the classification of orbits of minimal parabolic k-subgroups acting on symmetric k-varieties of SL(n,k)
(2008-04-25) Beun, Stacy L.; Dr. Tom Lada, Committee Member; Dr. Ernest Stitzinger, Committee Member; Dr. Aloysius Helminck, Committee Chair; Dr. Amassa Fauntleroy, Committee Member
Relationship Between Symmetric and Skew-Symmetric Bilinear Forms on V=kn and Involutions of SL(n,k) and SO(n,k,beta)
(2003-10-22) Dometrius, Christopher; Dr. Aloysius Helminck, Committee Chair; Dr. Naihuan Jing, Committee Member; Dr. Tom Lada, Committee Member; Dr. Ernest Stitzinger, Committee Member
In this paper, we show how viewing involutions on matrix groups as having been induced by a given non-degenerate symmetric or skew-symmetric bilinear form on the vector space of corresponding dimension can lead to a classification up to isomorphism of the resulting reductive symmetric space in the group setting. We establish a direct link between bilinear algebraic properties of the vector space V=kˆn for an arbitrary field k of characteristic not 2 and involutions of the matrix group G, where G is a subgroup of GL(n,k). Symmetric spaces are defined in terms of involutions, and the development of this classification theory which classifies the involutions also classifies the symmetric spaces coming from these involutions. We classify all involutions on SL(n,k) and develop important foundations for a full classification of the involutons of SO(n,k,beta) where beta is any non-degenerate symmetric bilinear form. We prove that all involutions of SO(n,k,beta) are inner when n is odd. Additionally, we provide criteria for the matrix which gives the conjugation that is the inner involution of SO(n,k,beta), which covers all involutions when n is odd and all involutions which can be written as conjugations when n is even, a fact which is proven in this thesis.
Root multiplicities of the indefinite type Kac-Moody algebras HC[subscript n]1
(2003-06-27) Williams, Vicky Lynn; Dr. Kailash Misra, Committee Chair; Dr. Ernest Stitzinger, Committee Member; Dr. Naihuan Jing, Committee Member; Dr. Jacqueline Hughes-Oliver, Committee Member
Victor Kac and Robert Moody independently introduced Kac-Moody algebras around 1968. These Lie algebras have numerous applications in physics and mathematics and thus have been the subject of much study over the last three decades. Kac-Moody algebras are classified as finite, affine, or indefinite type. A basic problem concerning these algebras is finding their root multiplicities. The root multiplicities of finite and affine type Kac-Moody algebras are well known. However, determining the root multiplicities of indefinite type Kac-Moody algebras is an open problem. In this thesis we determine the multiplicities of some roots of the indefinite type Kac-Moody algebras HC[subscript n]⁽¹⁾. A well known construction allows us to view HC[subscript n]⁽¹⁾ as the minimal graded Lie algebra with local part V direct sum g₀ direct sum V', where g₀ is the affine Kac-Moody algebra C[subscript n]⁽¹⁾. and V,V' are suitable g₀-modules. From this viewpoint, root spaces of HC[subscript n]⁽¹⁾ become weight spaces of certain C[subscript n]⁽¹⁾-modules. Using a multiplicity formula due to Kang we reduce our problem to finding weight multiplicities in certain irreducible highest weight C[subscript n]⁽¹⁾-modules. We then use crystal basis theory for the affine Kac-Moody algebras C[subscript n]⁽¹⁾ to find these weight multiplicities. With this strategy we calculate the multiplicities of some roots of HC[subscript n]⁽¹⁾. In particular, we determine the multiplicities of the level two roots -2(alpha₋₁)-k(delta) of HC[subscript n]⁽¹⁾ for 1 less than or equal to k less than or equal to 10. We also show that the multiplicities of the roots of HC[subscript n]⁽¹⁾ of the form -l(alpha₋₁) -k(delta) are n for l equal to k and 0 for l greater than k. In the process, we observe that Frenkel's conjectured bound for root multiplicities does not hold for the indefinite Kac-Moody algebras HC[subscript n]⁽¹⁾.