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Conjecturing In Dynamic Geometry: A Model For Conjecture-Generation Through Maintaining Dragging, Anna Baccaglini-Frank

Doctoral Dissertations

The purpose of this research is to study aspects of the impact of Dynamic Geometry Systems (DGS) in the process of producing conjectures in Euclidean geometry. Previous research has identified and classified a set of dragging schemes spontaneously used by students. Building on these findings, the study focuses on cognitive processes that arise in correspondence to particular dragging modalities in Cabri. Specifically, we have conceived a model describing what seems to occur during a process of conjecture-generation that involves the use of a particular dragging modality, described in the literature as dummy locus dragging. In order to accomplish this goal, …

The Impact Of A Mathematics Research Experience On Teachers' Conceptions Of Student Learning, Todd Abel

Doctoral Dissertations

Many mathematics teacher professional development programs have either incorporated or been organized around a goal of providing "research-like" (Cuoco, 2001) experiences. That is, teachers participate in a project that somehow simulates the mathematics research process. Though some research studies have shown positive outcomes from such programs, researchers have cautioned against assuming universally positive benefits without sufficient evidence (Proulx and Bednarz, 2001). Teacher conceptions of student learning play an important role in lesson development and preparation for classroom work (Penso & Shoham, 2003). Similarities between the processes of mathematics research and student learning (Dreyfus, 1991) beg the question of whether experience …

Admissible Orders On Quotients Of The Free Associative Algebra, Jeremiah William Johnson

Doctoral Dissertations

An admissible order on a multiplicative basis of a noncommutative algebra A is a term order satisfying additional conditions that allow for the construction of Grobner bases for A -modules. When A is commutative, a finite reduced Grobner basis for an A -module can always be obtained, but when A is not commutative this is not the case; in fact in many cases a Grobner basis theory for A may not even exist.

E. Hinson has used position-dependent weights, encoded in so-called admissible arrays, to partially order words in the free associative algebra in a way which produces a length-dominant …

Mf Algebras And A Bishop -Stone -Weierstrass Theorem Result, Qihui Li

Doctoral Dissertations

This dissertation consists of two parts. In the first part, we obtain many new results about MF algebras. First, we continue the work on D. Voiculescu's topological free entropy dimension deltatop (x1, ..., xn) for an n-tuple x&ar; = (x1, ..., xn) of elements in a unital C*-algebra. We also introduce a new invariant that is a C*-algebra analog of the invariant K3 introduced for von Neumann algebras. Second, we discuss a full amalgamated free product of unital MF (and residually finite-dimensional) algebras with amalgamation over a finite-dimensional C*-subalgebra. Necessary and sufficient conditions are given in this situation. In the …

Von Neumann Algebras, Affiliated Operators And Representations Of The Heisenberg Relation, Zhe Liu

Doctoral Dissertations

Von Neumann algebras are self-adjoint, strong-operator closed subalgebras (containing the identity operator) of the algebra of all bounded operators on a Hilbert space. Factors are von Neumann algebras whose centers consist of scalar multiples of the identity operator. In this thesis, we study unbounded operators affiliated with finite von Neumann algebras, in particular, factors of Type II1. Such unbounded operators permit all the formal algebraic manipulations used by the founders of quantum mechanics in their mathematical formulation, and surprisingly, they form an algebra. The operators affiliated with an infinite von Neumann algebra never form such an algebra. The Heisenberg commutation …

Covering Maps In The Uniform Category, Brendon L. Labuz

Doctoral Dissertations

First, generalized uniform covering maps are classified in terms of subgroups of the uniform fundamental group. Hausdorff, locally uniform joinable, and chain connected covering spaces of a uniformly locally joinable chain connected space are classifieded in terms of closed subgroups of its uniform fundamental group. If the space is also semilocally simply uniform joinable, uniform covering spaces are classified in terms of all subgroups of its uniform fundamental group. Next it is shown that the inverse limit of a strong Mittag-Leer inverse system of Hausdorff uniform covering spaces is a generalized uniform covering space. The question of the converse is …

Nonlinear Dissipative Wave Equations With Space-Time Dependent Potentials, Maisa Khader

Doctoral Dissertations

We study the long time behavior of solutions of the wave equations with absorption abs (u(t, x))[superscript p]⁻¹u(t, x) and variable damping a(t, x)u[subscript t](t, x), where p belongs to (1, n + 2/n - 2) and a(t, x) ~ a₀(1 + abs(x))⁻[superscript alpha](1 + t)⁻[superscript beta] for large abs x and t, a₀ > 0, for alpha belongs to (-infinity, 1), beta belongs to (-1, 1). We established decay estimates for the energy, L² and L[superscript p]⁺¹ norm of the solutions. 1. For alpha belongs to [0, 1), beta belongs to (-1, 1) and alpha + beta belongs to (0, …

Numerical Methods For Fully Nonlinear Second Order Partial Differential Equations, Michael Joseph Neilan

Doctoral Dissertations

This dissertation concerns the numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). The numerical methods and analysis are based on a new concept of weak solutions called moment solutions, which unlike viscosity solutions, are defined by a constructive method called the vanishing moment method. The main idea of the vanishing moment method is to approximate fully nonlinear second order PDEs by a family of fourth order quasi-linear PDEs. Because the method is constructive, we can develop a wealth of convergent numerical discretization methods to approximate fully nonlinear second order PDEs. We first study the numerical …

Invariant Frechet Algebras On Bounded Symmetric Domains, Oleg Eroshkin

Doctoral Dissertations

Let D be a bounded domain in the complex vector space Cn . We say that D is symmetric iff, given any two points p, q ∈ D, there is a biholomorphism &phis;, which interchanges p and q. These domains were classified abstractly by Elie Cartan in his general study of symmetric spaces, and were canonically realized in Cn by Harish-Chandra. They include polydisks and Siegel domains.

Let D be a bounded symmetric domain in Cn , and G be the largest connected group of biholomorphic automorphisms of D. The algebra C( D) of all continuous (not necessarily bounded) complex-valued …

Wavelet Regression With Long Memory Infinite Moving Average Errors, Juan Liu

Doctoral Dissertations

For more than a decade there has been great interest in wavelets and wavelet-based methods. Among the most successful applications of wavelets is nonparametric statistical estimation, following the pioneering work of Donoho and Johnstone (1994, 1995) and Donoho et al. (1995). In this thesis, we consider the wavelet-based estimators of the mean regression function with long memory infinite moving average errors, and investigate the rates of convergence of estimators based on thresholding of empirical wavelet coefficients. We show that these estimators achieve nearly optimal minimax convergence rates within a logarithmic term over a large class of non-smooth functions that involve …

Kadison -Singer Algebras With Applications To Von Neumann Algebras, Mohan Ravichandran

Doctoral Dissertations

I develop the theory of Kadison-Singer algebras, introduced recently by Ge and Yuan. I prove basic structure theorems, construct several new examples and explore connections to other areas of operator algebras. In chapter 1, I survey those aspects of the theory of non-selfadjoint algebras that are relevant to this work. In chapter 2, I define Kadison-Singer algebras and give different proofs of results of Ge-Yuan, which will be further extended in the last chapter. In chapter 3, I analyse in detail a class of elementary Kadison-Singer algebras that contain Hinfinity and describe their lattices of projections. In chapter 4, I …

Mathematics Of Double-Walled Carbon Nanotube Model: Asymptotic Spectral And Stability Analysis, Miriam Rojas-Arenaza

Doctoral Dissertations

This dissertation is devoted to analytical study of a contemporary model of a double-walled carbon nano-tube. Carbon nano-tubes have been considered outstanding candidates to innovate and to promote emerging technologies, due to their remarkable chemical, mechanical, and physical properties. For these technologies, there is a need to develop mathematical models that capture the nature of the responses of these structures under a variety of physical conditions. Developing these models is challenging because the behavior lies on the borderline between classical and quantum systems. The main goal of the present dissertation is to prove mathematically rigorous results concerning the vibrational behavior …

The Process Of Making Meaning: The Interplay Between Teachers' Knowledge Of Mathematical Proofs And Their Classroom Practices, Megan Paddack

Doctoral Dissertations

The purpose of this study was to investigate and describe how middle school mathematics teachers make meaning of proofs and the process of proving in the context of their classroom practices. A framework of making meaning, created by the researcher, guided the data collection and analysis phases of the study. This framework describes the five central aspects of the process of making meaning: knowledge, beliefs, utilization of knowledge, interconnections of practice and knowledge, and making sense of past knowledge and current experiences. The utilization of a qualitative research methodology that combined ethnographic fieldwork and discourse analysis allowed the researcher to …

Kadison -Singer Algebras, Wei Yuan

Doctoral Dissertations

In this dissertation, we defined a new class of non selfadjoint operator algebras---Kadison-Singer algebras or KS-algebras for simplicity. These algebras combine triangularity, reflexivity and von Neumann algebra property into one consideration. Generally speaking, KS-algebras are reflexive, maximal triangular with respect to its "diagonal subalgebra". Many selfadjoint features are preserved in them and concepts can be borrowed directly from the theory of von Neumann algebras. In fact, a more direct connection of KS-algebras and von Neumann algebras is through the lattice of invariant projections of a KS-algebra. The lattice is reflexive and "minimally generating" in the sense that it generates the …

Unitarily Invariant Norms And Tensor Products Of Maximal Injective Von Neumann Subalgebras, Junsheng Fang

Doctoral Dissertations

This dissertation consists of four contributions to the study of von Neumann algebras. In the first part, we set up a representation theorem for unitarily invariant norms on finite factor von Neumann algebras. In the second part, we set up a representation theorem for unitarily invariant norms related to infinite factor von Neumann algebras. In the third part, we introduce a notion of completely singular von Neumann subalgebras and characterize the class of completely singular von Neumann subalgebras. In the fourth part, we study a longstanding open question on the tensor product of maximal injective von Neumann subalgebras.

The first …

Planar Algebra Of Families Of Subfactors, Paramita Das

Doctoral Dissertations

The standard invariant is an object that classifies a fairly general class of subfactors and was reformulated by Vaughan Jones as 'planar algebras'. We compute the planar algebra of two families of subfactors arising out of actions of finitely generated groups on a II 1 factor in an attempt to see whether this class can be enlarged. This is a joint work with Dietmar Bisch and Shamindra Ghosh.

Free Entropy Dimensions And Approximate Liftings, Weihua Li

Doctoral Dissertations

In the first chapter of the dissertation, we give a very elementary proof of a more detailed version of one of D. Voiculescu's results, which was a key ingredient in Voiculescu's proof that his free entropy is additive when the variables are free.

In the second chapter of the dissertation, based on the notion of upper free orbit-dimension introduced by D. Hadwin and J. Shen, we introduce a new invariant on finite von Neumann algebras that do not necessarily act on separable Hilbert space. We show that this invariant is independent of the generating set, and we obtain a number …

The Relationship Among Working Memory, Mathematics Anxiety, And Mathematics Achievement In Developmental Mathematics Courses In Community College, Janet Marie Spybrook

Doctoral Dissertations

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Understanding Abstract Algebra Concepts, Anna S. Titova

Doctoral Dissertations

This study discusses various theoretical perspectives on abstract concept formation. Students' reasoning about abstract objects is described based on theoretical proposition that abstraction is a shift from abstract to concrete. Existing literature suggested a theoretical framework for the study. The framework describes process of abstraction through its elements: assembling, theoretical generalization into abstract entity, and articulation. The elements of the theoretical framework are identified from students' interpretations of and manipulations with elementary abstract algebra concepts including the concepts of binary operation, identity and inverse element, group, subgroup, cyclic group. To accomplish this, students participating in the abstract algebra class were …

Morita Equivalence For Group -Theoretical Categories, Deepak Naidu

Doctoral Dissertations

We give necessary and sufficient conditions for two pointed categories to be dual to each other with respect to a module category. Whenever the dual of a pointed category with respect to a module category is pointed, we give explicit formulas for the Grothendieck ring and for the associator of the dual. This leads to the definition of categorical Morita equivalence on the set of all finite groups and on the set of all pairs ( G, o), where G is a finite group and o ∈ H3(G, kx). A group-theoretical and cohomological interpretation of this relation is given. As …

Fighting The Freeloader Effect : Cooperation, Attitude, And Achievement In A Jesuit Secondary Math Classroom, Kevin Quattrin

Doctoral Dissertations

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Modeling Core Generation And Transport Phenomena Of Microcapsules, Xiaoshuo Wu

Doctoral Dissertations

Microcapsules play an important role in chemical, biomedical, and medical research. Of importance is their potential use as bio-sensors, as microreactors, and for drug delivery. In this dissertation, Computational Fluid Dynamics methodology is used to model the generation of droplets or cores (using alginate and oil) and to determine from the model process parameters needed for generating cores of homogeneous size for the manufacturing of microcapsules. Also, material diffusion through a microcapsule for controlled release is modeled based on finite difference methodology. The model was applied to experimental data on dextran release from microcapsules in order to estimate the diffusion …

An Analysis Of 2005 Naep 8th Grade Mathematics Achievement Items By Content Strand, Problem Type And Language Complexity, Yvette Marie Fagan

Doctoral Dissertations

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Planar Algebras: A Category Theoretic Point Of View, Shamindra Kumar Ghosh

Doctoral Dissertations

We define Jones's planar algebra as a map of multicategories and construct a planar algebra starting from a 1-cell in a pivotal strict 2-category. We introduce the concept of affine representations of a planar algebra and prove some finiteness results for the affine representations of finite depth planar algebras. We also show that the radius of convergence of the dimension of an affine representation of the planar algebra associated to a finite depth subfactor is at least as big as the inverse-square of the modulus.

An Investigation Of Pre-Service Teachers' And Professional Mathematicians' Perceptions Of Mathematical Proof At The Secondary School Level, David M. Gray

Doctoral Dissertations

The National Council of Teachers of Mathematics [NCTM] states that by the time students graduate high school, they should learn to present "arguments consisting of logically rigorous deductions of conclusions from hypotheses" (NCTM, 2000, p. 56) in "written forms that would be acceptable to professional mathematicians" (NCTM, 2000, p. 58). Research studies indicate, however, that students and teachers have difficulty with many aspects of mathematical proof, including its nature and meaning. In addition, there appears to be a disconnect between school teachers' and university mathematicians' expectations for their respective students regarding mathematical proof. This study examined the perceptions of thirteen …

Cupolets: Chaotic Unstable Periodic Orbits Theory And Applications, Kourosh Zarringhalam

Doctoral Dissertations

Recent theoretical work suggests that periodic orbits of chaotic systems are a rich source of qualitative information about the dynamical system. The presence of unstable periodic orbits located densely on the attractor is a typical characteristic of chaotic systems. This abundance of unstable periodic orbits can be utilized in a wide variety of theoretical and practical applications [19]. In particular, chaotic communication techniques and methods of controlling chaos depend on this property of chaotic attractors [12, 13].

In the first part of this thesis, a control scheme for stabilizing the unstable periodic orbits of chaotic systems is presented and the …

Exploring Lesson Study As A Form Of Professional Development For Enriching Teacher Knowledge And Classroom Practices, Melissa K. Mitcheltree

Doctoral Dissertations

This study tracked the development of teacher knowledge through a professional development experience called lesson study. Lesson study is a Japanese professional development process in which a group of teachers develop a series of lessons using the following stages: planning lessons, teaching/observing the lessons, reflecting on lessons taught as well as sharing and discussing the lessons with their colleagues (Lewis, 2002). The aim of this study was to explore how the lesson study process influenced teacher knowledge of mathematics content and pedagogical content.

Study participants were four secondary mathematics teachers from a rural high school in the Northeast. All participants …

Developing A B -Tagging Algorithm Using Soft Muons At Level-3 For The Dø Detector At Fermilab, Mayukh Das

Doctoral Dissertations

The current data-taking phase of the DØ detector at Fermilab, called Run II, is designed to aid the search for the Higgs Boson. The neutral Higgs is postulated to have a mass of 117 GeV. One of the channels promising the presence of this hypothetical particle is through the decay of b-quark into a muon. The process of identifying a b-quark in a jet using muon as a reference is b-tagging with a muon tag.

At the current data taking and analysis rate, it will take long to reach the process of identifying valid events. The triggering mechanism of the …

Differences In Teacher Qualifications And The Relationship To Middle School Student Achievement In Mathematics, Carrie R. Ferguson

Doctoral Dissertations

The purpose of the study was to compare mathematics teacher preparation and experience to the achievement of middle school students (grades 6–8) on the mathematics section of the Texas Assessment of Knowledge and Skills (TAKS). Using a theoretical framework, the research examined the extent to which the teacher qualification variables were related to the student achievement. Measures of teacher qualifications included four independent variables: (a) number of mathematics content hours, (b) level of teaching certification, (c) number of years teaching experience, and (d) number of years teaching middle school/junior high mathematics. Participants in this causal-comparative study included 97 teachers and …

Evaluating The Middle School Mathematics Teacher Preparation Program At Riyadh Teachers' College, Maher M. Alghanem

Doctoral Dissertations

Continuous evaluation and feedback are essential to improving the quality of teacher preparation programs. The purpose of this study was to evaluate the perceptions of various aspects of the preparation of preservice middle school mathematics teachers at Riyadh Teachers' College, Saudi Arabia, from the perspective of both preservice middle school mathematics teachers and mathematics department faculty members. The focus of this study was on the mathematics department's theoretical framework, its governance and resources, faculty qualifications and performance, courses and field experiences, approaches of teaching, and assessment forms. Two research questionnaires were developed for the purpose of collecting research data: one …