OU - Dissertations

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https://hdl.handle.net/11244/10476

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Open Access
A Guided Reinvention of Ring, Integral Domain, and Field
(2012) Cook, John Paul; McKnight, Curtis C
Abstract algebra enjoys a prestigious position in mathematics and the undergraduate mathematics curriculum. A typical abstract algebra course aims to provide students with a glimpse into the elegance of mathematics by exposing them to structures that form its foundation--it arguably approximates the actual practice of mathematics better than any of the courses by which it is typically preceded. Regrettably, despite the importance and weight carried by the abstract algebra, the educational literature is replete with suggestions that undergraduate students do not appear to be grasping even the most fundamental ideas of the subject. Additionally, many students fail to make the connection between abstract algebra and the algebra they learned at the primary and secondary levels, perpetually blind to any interpretations of the subject beyond surface-level. These discrepancies have two problematic consequences. First, students who were otherwise enthusiastic and interested in mathematics experience a complete reversal and become indifferent and disengaged. Second, future mathematics teachers at the primary and secondary levels do not build upon their elementary understandings of algebra, leaving them unable to communicate traces of any deep and unifying ideas that govern the subject. To address this problem, it has been suggested that the traditional lecture method be eschewed in favor of a student-centered, discovery-based approach. There have been several responses to this call; most notable and relevant to this project is the work of Larsen (2004, 2009), who developed an instructional theory to support students' reinvention of group and group isomorphism. As no such innovative methods of instruction exist regarding ring field theory, this project details the development of an instructional theory supporting students' reinvention of fundamental structures from ring theory: ring, integral domain, and field. Rooted in the theory of Realistic Mathematics Education, this dissertation reports on a developmental research project conducted via multiple iterations of the constructivist teaching experiment, wherein the primary goal was to test and revise an instructional theory supporting the guided reinvention of ring, integral domain, and field. The findings include an empirically tested and revised instructional theory, as well as conceptual frameworks detailing the emergence and progressive formalization of the key features in a ring structure.
Open Access
Affine group actions on Euclidean space.
(2006) Sulman, Robert M.; Basmajian, Ara,
Upto affine conjugacy, we describe properly discontinuous rank two affine groups of Euclidean space (primarily dimensions two and three) in terms of "coordinates" of generators. In dimension two, a chosen generator is put into a normal form. The commuting condition simplifies a second generator, which can be identified with a point of the plane. Thus, R2 can be viewed as a parameter space of groups (since the first normalized generator is common to each group). A homomorphism Res:R2->R (the residue) singles out properly discontinuous groups G isomorphic to Z+Z. Affine conjugacy of two groups is characterized by their residues and an element of GL (2, Z). As a consequence, we show (i) There are uncountably many conjugacy classes of properly discontinuous rank two groups, and (ii) Each point of Ker(Res) is the limit point of every conjugacy class. An analog of the residue is used to determine three dimensional properly discontinuous rank two groups, although this description does not cover all normalized forms.
Open Access
The algebra of a computer integer arithmetic system.
(1977) Oldroyd, Lawrence Andrew.
Open Access
An Exploration of the Transition to Graduate School in Mathematics
(2011) Marsh, Sarah Leanne; Grasse, Kevin A||Murphy, Teri J
This purpose of this research was to explore the transition to graduate school in mathematics--the struggles students face, the expectations they must meet, and the strategies they use to deal with this new chapter in their academic experience. In this qualitative study, interview data regarding this transition--collected from 13 mathematics graduate students and eight mathematics faculty members on one university campus--is presented. Based on transcriptions of digital audio recordings of the interviews, interview data were thematically coded and analyzed to yield the themes below.
Open Access
Arithmetic in Quaternion Algebras and Quaternionic Modular Forms
(2019-05-10) Wiebe, Jordan; Martin, Kimball; Schmidt, Ralf; Roche, Alan; Pitale, Ameya; Cheng, Qi
This dissertation has two parts. In the first part, we revisit the correspondence between spaces of modular forms and orders in quaternion algebras addressed first by Eichler and completed by Hijikata, Pizer, and Shemanske, using an arbitrary definite quaternion algebra with arbitrary level. We present explicit bases for orders of arbitrary level N>1 in definite rational quaternion algebras. These orders have applications to computations of spaces of elliptic and quaternionic modular forms. In the second part, we investigate the behavior of quaternionic modular forms. In particular, we calculate quaternionic modular forms of weight 2, and illustrate a use of the orders constructed in the first part. We use these forms to explore the behavior of spaces of quaternionic cusp forms of weight 2 and level N, and make a number of conjectures concerning the behavior of zeros of such quaternionic modular forms. In particular, we use dimension formulas and the action of involutions on our space to predict certain zeros of quaternionic modular forms (which we call trivial zeros), and conjecture that the ratio of the number of zerofree forms of level < N to the number of forms with no trivial zeros tends to 1 as N goes to infinity. Finally, we analyze asymptotics of the growth rate of trivial zeros, and provide a histogram of the distribution of nontrivial zeros with respect to the degrees of factors associated to them. We also provide data on a variety of quaternionic modular forms in Appendix A.
Open Access
Bockstein Basis and Resolution Theorems in Extension Theory
(2009) Tonic, Vera; Rubin, Leonard R
Resolution refers to a map (a continuous function) between topological spaces, where the domain is in some way better than the range, and the fibers (point preimages) meet certain requirements. We will be interested in the relationship between covering dimension and cohomological dimension, so the resolution we obtain will be between a domain of finite covering dimension, and a range of finite cohomological dimension, with cell-like or G-acyclic fibers. Both domain and range will be compact metrizable spaces.
Open Access
Brauer groups of H-dimodule algebras and truncated power series Hopf algebras.
(1980) Tan, Richard Beng-tok,
The Brauer group of H-dimodule algebras, BD(R, H), consists of equivalence classes of H-Azumaya algebras; where H is a Hopf algebra over a commutative ring R, and an H-dimodule algebra A is defined to be H-Azumaya if certain maps (analogous to the usual map A (CRTIMES) A (--->) End(A)) are isomorphisms. It is shown that if R is a separably closed field of characteristic p and H is a truncated power series Hopf algebra then a necessary and sufficient condition for an H-Azumaya algebra A to be R-Azumaya (the usual Azumaya R-algebra) is that it be semisimple. An example is given to show that semisimplicity is necessary for this to be true.
Open Access
Building fences around the chromatic coefficients.
(1997) Strickland, Debra Ann Mullins.; Breen, Marilyn,
Although these bounding conditions do not allow us to completely predict all chromatic polynomials, they do serve to severely limit the form of polynomials considered to be candidates for the chromatic polynomial of some graph.
Open Access
The business calculus GMTA: An exploration of teaching experiences in an unfamiliar course.
(2007) Hands, Krista Brooke.; Murphy, Teri J.,
Abstract not available.
Open Access
Calculus Instructors' Resources, Orientations, and Goals in Teaching Low-Achieving Students
(2014-05-09) Lee, Misun; Stewart, Sepideh; Rubin, Leonard; Albert, John; Schmidt, Ralf; Terry, Robert
Teaching and learning calculus has been the subject of mathematics education research for many years. Although the literature is mainly concerned with students’ difficulties with calculus, research on mathematicians’ day to day activities is still scarce. Using Schoenfeld’s Resources, Orientations and Goals (ROGs) framework, this study examined four instructors’ ROGs in teaching calculus to low-achieving students. The findings revealed the in depth pedagogical experiences of mathematicians and their deliberation in helping students. The study suggested that building research based models and frameworks results in richer studies that would be more beneficial to both students and instructors.
Open Access
Calculus Instructors' Responses to Prior Knowledge Errors
(2009) Talley, Jana Renee; McKnight, Curtis||Murphy, Teri J.
This study investigates the responses to prior knowledge errors that Calculus I instructors make when assessing students. Prior knowledge is operationalized as any skill or understanding that a student needs to successfully navigate through a Calculus I course. A two part qualitative study consisting of student exams and instructor interviews was employed to examine how instructors approach prior knowledge mistakes when they are evaluating students. Analysis of these interviews revealed that calculus instructors agree that algebra and trigonometry are essential components of prior knowledge within a calculus course. Additionally, perceived inconsistencies in instructor grading were reconciled using a sensible system framework.
Open Access
Categorizing Students' Difficulties with Mathematical Proofs: Developing a Model of the Structure of Proof Construction
(2015-08-14) Yamamoto, Tetsuya; Stewart, Sepideh; Albert, John; Kyung-Bai, Lee; Miller, Andrew; Savic, Milos; Terry, Robert
Studies have shown that proof construction is a challenging task for students at all levels. The purposes of my study were to examine students' difficulties with proof construction and to explore a practical method to help them overcome their difficulties. I created a model of the structure of proof construction. The model was useful in explaining students' difficulties and producing metacognitive knowledge for proof construction in the form of algorithm.
Open Access
Classification of codimension one biquotient foliations in low dimensions
(2021-05-14) Lutz, Andrew; Shankar, Krishnan; DeVito, Jason; Lifschitz, Lucy; Malestein, Justin; Mendes, Ricardo; Weldon, Stephen
In this thesis we study compact simply connected C1BFs (manifolds which arise as quotients of cohomogeneity one manifolds). In particular, we study various elementary properties of C1BFs including their topological and curvature properties. Moreover, we give a classification of their structures in low dimensions and also show that all simply connected manifolds which admit non-negative curvature in low dimensions admit a C1BF structure.
Open Access
Classification of the normal subgroups of GL (R).
(1979) Moore, Jeanna Beth,
Open Access
Classification, Cobordism, and Curvature of Four-Dimensional Infra-Solvmanifolds
(2014) Thuong, Scott Van; Lee, Kyung-Bai; Jablonski, Michael; Shankar, Krishnan; Walschap, Gerard; Hahn, Sowon
Crystallographic groups of solvable Lie groups generalize the crystallographic groups of Euclidean space. The quotient of a solvable Lie group G by the action of a torsion-free crystallographic group of G is an infra-solvmanifold of G. Infra-solvmanifolds are aspherical manifolds that generalize both closed flat manifolds and closed almost flat manifolds (that is, infra-nilmanifolds). Here we complete the classification of 4-dimensional infra-solvmanifolds by classifying torsion-free crystallographic groups of certain 4-dimensional solvable Lie groups. The classification also includes those crystallographic groups with torsion. We prove that every 4-dimensional infra-solvmanifold is the boundary of a compact 5-dimensional manifold by constructing an involution on certain 4-dimensional infra-solvmanifolds which is either free, or has 2-dimensional fixed set. The Ricci signatures (that is, signatures of the Ricci transformation) of 4-dimensional Lie groups have been classified. A Ricci signature can be realized on an infra-solvmanifold M of G if M is a compact isometric quotient of G, where G has left invariant metric with prescribed Ricci signature. We classify which Ricci signatures can be realized on certain 4-dimensional infra-solvmanifolds.
Open Access
Collections of covers which imply compactness.
(1979) Pettis, Ted Ray,
Open Access
A comparison of solutions of a Boussinesq system and the Benjamin-Bona-Mahony equation.
(2000) Alazman, Abdulrahman A.; Albert, John,
We compare solutions of a regularized Boussinesq model system for water waves to solutions of the Benjamin-Bona-Mahony equation. Both the system and the equation are obtained as formal approximations to the Euler equations of motion in which terms of order epsilon2 are neglected, where epsilon is a small parameter related to the wave amplitude and the inverse wavelength. For each choice of initial free-surface profile in an appropriate Sobolev space, we show that the Boussinesq system has a unidirectional solution which exists on a time interval 0 ≤ t ≤ T of length T = O( 1/e ), and that on this time interval the solution agrees with the corresponding solution of the Benjamin-Bona-Mahony equation to within Cepsilon2t, where C is independent of epsilon and t. Therefore the solution of the system agrees with the solution of the equation to within the accuracy of either model.
Open Access
A comparison study between a traditional and experimental first year linear algebra program.
(2001) Dogan, Hamide.; McKnight, Curtis,
The traditional section was in lecture format whereas the experimental section was in mostly discovery format; Students in the experimental group discovered definitions of basic abstract concepts mostly through visual-based Mathematica notebook demonstrations, whereas the students in the traditional group were given the definitions.
Open Access
Comparison Theorems, Geomatric Inequalities, And Applications to p-harmonic Geometry
(2012) Li, Ye; Wei, Shihshu Walter
In this dissertation, we consider three aspects: comparison theorems on complete manifold which posses a pole, geometric inequalities on complete manifolds, and the applications of inequalities to p-harmonic geometry. More precisely, wefirst derive a comparison theorem of the matrix-valued Riccati equation with certain initial conditions, and then use this as a tool to obtain
Open Access
Comparison Theorems, Geomatric Inequalities, And Applications to p-harmonic Geometry
(2012) Li, Ye; Wei, Shihshu Walter
In this dissertation, we consider three aspects: comparison theorems on complete manifold which posses a pole, geometric inequalities on complete manifolds, and the applications of inequalities to p-harmonic geometry. More precisely, wefirst derive a comparison theorem of the matrix-valued Riccati equation with certain initial conditions, and then use this as a tool to obtain