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Numerical Methods For Stochastic Stokes And Navier-Stokes Equations, Liet Vo

Doctoral Dissertations

This dissertation consists of three main parts with each part focusing on numerical approximations of the stochastic Stokes and Navier-Stokes equations.

Part One concerns the mixed finite element methods and Chorin projection methods for solving the stochastic Stokes equations with general multiplicative noise. We propose a modified mixed finite element method for solving the Stokes equations and show that the numerical solutions converge optimally to the PDE solutions. The convergence is under energy norms (strong convergence) for the velocity and in a time-averaged norm (weak convergence) for the pressure. In addition, after establishing the error estimates in second moment, high …

Examining Small-Group Discourse Through The Lens Of Students’ Beliefs About Mathematics And The Instructional Triangle, Jennifer Ericson

Doctoral Dissertations

Standards-based mathematics instruction [SBMI], with opportunities to engage in high-quality mathematical discourse about meaningful tasks, leads to increased student achievement. However, not all students participate in mathematical discourse with high-quality. Esmonde & Langer-Osuna (2013) found that students participate in discourse with various degrees of engagement based upon individual beliefs. Boaler (2006b) found that SBMI in a detracked setting improves students’ beliefs about mathematics. The purpose of this descriptive case study was to illustrate the phenomenon of incongruent written curricula commitments across grade bands in a school district by examining small-group discourse and students’ mathematical related beliefs. This study compared students’ …

Recruiting Minoritized Mathematics Preservice Teachers With Retention In Mind, Nicholas Scott Kim

Doctoral Dissertations

The primary purpose of this research is to provide insight into the narratives and experiences of minoritized preservice teachers (PST) interested in teaching mathematics and how those experiences impact recruitment and retention of those minoritized teachers. “Minoritized,” as used here, is defined as one who identifies as a person from a historically underserved community based on race or ethnicity. As a result, this study is positioned to provide researchers, educators, and administrators with much needed direction into how to diversify the teaching force beyond the dominant storyline of the White majority, and why it is important to do so.

Narrative …

Examining The Specialized Math Content Knowledge Of Elementary Teachers In The Age Of The Common Core, Stephanie Purington

Doctoral Dissertations

Mathematical standards for students have increased with the development of the Common Core State Standards for Mathematics and its accompanying high stakes testing. Teachers need strong conceptual knowledge of the mathematics they teach in order to give students the opportunity to learn that math deeply. An earlier study (Ma, 1999) found that US elementary teachers lack the deep knowledge to teach math conceptually. Given the mathematics standards movements of the last two decades, it is plausible that the knowledge base of teachers has changed. Using the framework of Specialized Content Knowledge (SCK), which is the knowledge required to teach math …

A Semiotic Analysis Of Linguistic And Conceptual Development In Mathematics For English Language Learners, Hyunsook Shin

Doctoral Dissertations

This study explores how an elementary mathematics teacher supported English language leaners’ (ELLs’) academic language and concept development in the context of current high- stakes school reform. The conceptual frameworks informing this study include Halliday’s theory of systemic functional linguistics (e.g., Halliday & Matthiessen, 2014) and Vygotsky’s sociocultural theory of concept development (Vygotsky, 1986). Specifically, this study analyzes the interplay between academic and everyday language and how this interplay can facilitate the development of what Vygotsky referred to as “real” or complete concepts as students shift from “spontaneous” to more “scientific” understanding of phenomenon (Vygotsky, 1986, p.173). This year-long qualitative …

Effective Techniques In Reverse Mathematics, David Nichols

Doctoral Dissertations

We develop the theory of strong reductions in the reverse mathematics zoo and show by means of novel tree labeling constructions that strong computable reduction is sufficient to separate four stable relatives of Ramsey's theorem for 2-colorings of pairs.

A Complete Characterization Of Near Outer-Planar Graphs, Tanya Allen Lueder Genannt Luehr

Doctoral Dissertations

A graph is outer-planar (OP) if it has a plane embedding in which all of the vertices lie on the boundary of the outer face. A graph is near outer-planar (NOP) if it is edgeless or has an edge whose deletion results in an outer-planar graph. An edge of a non outer-planar graph whose removal results in an outer-planar graph is a vulnerable edge. This dissertation focuses on near outer-planar (NOP) graphs. We describe the class of all such graphs in terms of a finite list of excluded graphs, in a manner similar to the well-known Kuratowski Theorem for planar …

Investigating The Parent Functions: African-American Families And Schools Collaborating To Improve Secondary Mathematics Achievement, Frank E. Staples Jr.

Doctoral Dissertations

The purpose of this study was to identify and evaluate specific strategies that encouraged school personnel and African-American family members to collaborate in a spirit of mutual respect to promote and support student learning and development in high school mathematics. Three research questions were used to (a) determine if strategies identified by the participants were effective or non-effective, (b) determine if there were any similarities or differences with respect to how family members and school personnel think about or assess specific strategies, and (c) assess the results of implemented strategies. This goal was accomplished by (1) convening a group of …

Brauer-Picard Groups And Pointed Braided Tensor Categories, Costel Gabriel Bontea

Doctoral Dissertations

Tensor categories are ubiquitous in areas of mathematics involving algebraic structures. They appear, also, in other fields, such as mathematical physics (conformal field theory) and theoretical computer science (quantum computation). The study of tensor categories is, thus, a useful undertaking.

Two classes of tensor categories arise naturally in this study. One consists of group-graded extensions and another of pointed tensor categories. Understanding the former involves knowledge of the Brauer-Picard group of a tensor category, while results about pointed Hopf algebras provide insights into the structure of the latter.

This work consists of two main parts. In the first one we …

A Dynamical-Systems Approach To Understanding Turbulence In Plane Couette Flow, Mimi Szeto

Doctoral Dissertations

Dynamical systems theory is used to understand the dynamics of low-dimensional spatio-temporal chaos. Our research aimed to apply the theory to understanding turbulent fluid flows, which could be thought of as spatio-temporal chaos in a very-high dimensional space. The theory explains a system's dynamics in terms of the local dynamics of its periodic solutions; these are the periodic orbits in state space. We considered the development of a model for the dynamics of plane Couette flow based on the theory. The proposed model is essentially a set of low-dimensional models for the local dynamics of the periodic orbits of the …

A Beurling Theorem For Noncommutative Hardy Spaces Associated With A Semifinite Von Neumann Algebra With Various Norms, Lauren Beth Meitzler Sager

Doctoral Dissertations

We prove Beurling-type theorems for H-invariant spaces in relation to a semifinite von Neu-mann algebra M with a semifinite, faithful, normal tracial weight τ, using an extension of Arveson’s non-commutative Hardy space H-. First we prove a Beurling-Blecher-Labuschagne theorem for H-invariant subspaces of L p (M,τ) when 0 < p ≤ -. We also prove a Beurling-Chen-Hadwin-Shen theorem for H -invariant subspaces of L a (M,τ) where a is a unitarily invariant, locally k 1 -dominating, mutually continuous norm with respect to &\tau;. For a crossed product of a von Neumann algebra M by an action β, M o β Z, we are able to completely characterize all H-invariant subspaces of L a (Mo β Z,t) using our results. As an example, we completely characterize all H-invariant subspaces of the Schatten p-class, S p (H) (0 < p ≤ -), where H - is the lower tri-angular subalgebra of B(H). We also characterize the non-commutative Hardy space H -invariant subspaces in a Banach function space I(τ) on a semifinite von Neumann algebra M.

Assessing The Impact Of Picture Books In Primary Grades Mathematics Instruction, Jessica Stone

Doctoral Dissertations

This study documents an educational field experiment evaluating the effects of picture books on primary students’ mathematical achievement and their dispositions towards mathematics. The study involved 136 primary grade students from one elementary school in the southeastern region of the United States. The student population had an overrepresentation of students from minority backgrounds (91%), low socioeconomic status (93%) and English Language Learners (47%). During the 18-week treatment period, teacher participants from the treatment group received bi-weekly collaborative professional development regarding the use of picture books in mathematics instruction. The teachers in the control group followed their district’s mathematics curriculum.

To …

On Tensor Autoequivalences Of Graded Fusion Categories, Ian Swenson Marshall

Doctoral Dissertations

Fusion categories generalize the representation theory of finite groups. The simplest examples of fusion categories come from finite groups, their representations, and their cohomology. In general, it is useful to examine group theoretical features of fusion categories such as groups of (isomorphism classes of) tensor invertible objects, and gradings by finite groups. Indeed, every fusion category has a maximal pointed subcategory (generated by tensor invertible objects) and a universal grading by a finite group. We use such features to study tensor autoequivalences.

Pointed fusion categories: categories for which all simple objects are tensor invertible, provide our prototype for graded fusion …

Porous Medium Convection At Large Rayleigh Number: Studies Of Coherent Structure, Transport, And Reduced Dynamics, Baole Wen

Doctoral Dissertations

Buoyancy-driven convection in fluid-saturated porous media is a key environmental and technological process, with applications ranging from carbon dioxide storage in terrestrial aquifers to the design of compact heat exchangers. Porous medium convection is also a paradigm for forced-dissipative infinite-dimensional dynamical systems, exhibiting spatiotemporally chaotic dynamics if not ``true" turbulence. The objective of this dissertation research is to quantitatively characterize the dynamics and heat transport in two-dimensional horizontal and inclined porous medium convection between isothermal plane parallel boundaries at asymptotically large values of the Rayleigh number $Ra$ by investigating the emergent, quasi-coherent flow. This investigation employs a complement of direct …

Fully Coupled Fluid And Electrodynamic Modeling Of Plasmas: A Two-Fluid Isomorphism And A Strong Conservative Flux-Coupled Finite Volume Framework, Richard Joel Thompson

Doctoral Dissertations

Ideal and resistive magnetohydrodynamics (MHD) have long served as the incumbent framework for modeling plasmas of engineering interest. However, new applications, such as hypersonic flight and propulsion, plasma propulsion, plasma instability in engineering devices, charge separation effects and electromagnetic wave interaction effects may demand a higher-fidelity physical model. For these cases, the two-fluid plasma model or its limiting case of a single bulk fluid, which results in a single-fluid coupled system of the Navier-Stokes and Maxwell equations, is necessary and permits a deeper physical study than the MHD framework. At present, major challenges are imposed on solving these physical models …

The Impact Of Analyzing Correct Versus Incorrect Student Work Samples On Students’ Learning Mathematics, Lauren Jeneva Moseley

Doctoral Dissertations

The purposes of this study are to determine if learning differs when calculus learners analyze correct or incorrect work samples and to investigate students’ perceptions of the effect of analyzing work samples on their learning of mathematics. Calculus students were randomly assigned to two groups: one group analyzing correct work samples and one group analyzing incorrect work samples. Data from enrollees in 10 sections of Basic Calculus at a large university was analyzed using ANCOVA, independent-samples t-test, and inductive analysis (Hatch, 2002). Results suggest that when students analyze incorrect work samples of moderate difficulty, they are less likely to …

What Do Students Do In Self-Formed Mathematics Study Groups?, Gillian E. Galle

Doctoral Dissertations

An implicit assumption of many university classes is that students will spend a large amount of time outside the classroom refining their understanding of the material to develop mastery of the concepts. This is especially true in first year mathematics courses at the undergraduate level. However, little is known about what students do to fulfill this didactical contract with their instructors. The currently available research relies primarily on self-reported data from the students collected through questionnaires or interviews. This study sought to start describing what students do while studying mathematics in a self-created group outside of the classroom setting through …

The Multifaceted Nature Of Mathematics Knowledge For Teaching: Understanding The Use Of Teachers' Specialized Content Knowledge And The Role Of Teachers' Beliefs From A Practice-Based Perspective, Lauren E. Provost

Doctoral Dissertations

This work investigates middle school teachers' mathematics knowledge for teaching (MKT) as defined by Hill (2007). Within this two-part dissertation, the level of MKT was considered as well as the role of teacher beliefs in actual specialized content knowledge (SCK) use, a specific type of mathematics knowledge for teaching vital in quality mathematics instruction. Additionally, the model of MKT knowledge was explored through confirmatory factor analysis on a large, national dataset of middle school mathematics teacher survey responses involving mathematics knowledge for teaching. SCK was found to be vital in quality mathematics instruction yet not sufficient. Teacher beliefs about the …

The Engineering Design Process As A Model For Stem Curriculum Design, Krystal Sno Corbett

Doctoral Dissertations

Engaging pedagogics have been proven to be effective in the promotion of deep learning for science, technology, engineering, and mathematics (STEM) students. In many cases, academic institutions have shown a desire to improve education by implementing more engaging techniques in the classroom. The research framework established in this dissertation has been governed by the axiom that students should obtain a deep understanding of fundamental topics while being motivated to learn through engaging techniques. This research lays a foundation for future analysis and modeling of the curriculum design process where specific educational research questions can be considered using standard techniques. Further, …

The Impact Of Singapore Math On Student Knowledge And Enjoyment In Mathematics, Jenny Taliaferro Blalock

Doctoral Dissertations

The purpose of this study was to investigate the effect of the Singapore Math curriculum and approach to teaching on enjoyment and knowledge in mathematics in one rural school district in north Louisiana. The quantitative data used were collected from a Math Enjoyment Inventory and a mixed skills pre and posttest. Additional supplemental data were collected from a Teacher Response Form. All data were gathered in the 2010–2011 school year, and an intact population was utilized. Participants were categorized into two groups, the Singapore Math group and the traditional math group. The participating district implemented Singapore Math in the fall …

A Validation Of The Monitoring Academic Progress Mathematics: An Experimental Multidimensional Group Administered Curriculum-Based Measure Of Mathematics Fluency And Problem Solving, Michael Brandon Hopkins

Doctoral Dissertations

The study investigated the psychometric properties of a newly developed math curriculum-based measure, the Monitoring Academic Progress: Mathematics (MAP:M), through examination of its internal consistency, alternate-form, slope, and test-retest reliability and validity. Participants included 1688 first through third-grade students from a school district in Northeast Tennessee. Application of Generalizability Theory produced reliability coefficients, score variances, and standard-error-of-measures (SEM) for both absolute and relative decisions based on a particular number of probes. MAP:M reliability coefficients for relative decisions ranged from .67 to .97 across eleven probes. The highest percentage of score variance at all three grades was attributed to the Person …

On Decompositions And Connes's Embedding Problem Of Finite Von Neumann Algebras, Jinsong Wu

Doctoral Dissertations

A longstanding open question of Connes asks whether every finite von Neumann algebra embeds into an ultraproduct of finite-dimensional matrix algebras. As of yet, algebras verified to satisfy Connes's embedding property belong to just a few special classes (e.g. amenable algebras and free group factors). In this dissertation we establish Connes's embedding property for von Neumann algebras satisfying Popa's co-amenability condition. Some decomposition properties of finite von Neumann algebras are also investigated.

Chapter 1 reviews von Neumann algebras, completely bounded mappings, conditional expectations, tensor products, crossed products, direct integrals, and Jones basic construction.

Chapter 2 introduces new decompositions of finite …

Proof And Reasoning In An Inquiry-Oriented Class: The Impact Of Classroom Discourse, Susan D. Generazzo

Doctoral Dissertations

Over the past decade, mathematics educators and researchers have become increasingly aware of the impact of social interactions on students' learning (NCTM, 2000; Bowers & Nickerson, 2001; Forman, 2003). Current research indicates that the classroom environment, including the activities and discussions that take place, can have a significant effect on the ways students make sense of mathematical concepts (Yackel, 2001). Understanding mathematics involves knowing how to make sense of key concepts through the processes of reasoning and justification. Educators and researchers agree on the importance of providing students with opportunities in class to explore, conjecture, and prove in order to …

Homotopy Mapping Spaces, Jeremy Brazas

Doctoral Dissertations

In algebraic topology, one studies the group structure of sets of homotopy classes of maps (such as the homotopy groups pin( X)) to obtain information about the spaces in question. It is also possible to place natural topologies on these groups that remember local properties ignored by the algebraic structure. Upon choosing a topology, one is left to wonder how well the added topological structure interacts with the group structure and which results in homotopy theory admit topological analogues. A natural place to begin is to view the n-th homotopy group pi n(X) as the quotient space of the iterated …

Models And Methods For Computationally Efficient Analysis Of Large Spatial And Spatio-Temporal Data, Chengwei Yuan

Doctoral Dissertations

With the development of technology, massive amounts of data are often observed at a large number of spatial locations (n). However, statistical analysis is usually not feasible or not computationally efficient for such large dataset. This is the so-called "big n problem".

The goal of this dissertation is to contribute solutions to the "big n problem". The dissertation is devoted to computationally efficient methods and models for large spatial and spatio-temporal data. Several approximation methods to "the big n problem" are reviewed, and an extended autoregressive model, called the EAR model, is proposed as a parsimonious model that accounts for …

The Effects Of Math Instruction On Fifth Grade Elementary Students' Math Performance Across Four Inclusive General Education Classrooms Infused With The Afrocultural Dimensions Of Communalism, Orality, And Movement, Rosalind J. Simpson

Doctoral Dissertations

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The Impact Of Secondary Mathematics Methods Courses On Preservice Secondary Teachers’ Beliefs About The Learning And Teaching Of Mathematics, Ronald Gene Smith Ii

Doctoral Dissertations

The Comprehensive Framework for Teacher Knowledge provides a model that describes an approach to the secondary mathematics methods course, as described by Robert Ronau and P. Mark Taylor. The model includes the orientation of preservice teachers toward mathematics and the teaching of mathematics, which includes the beliefs of the preservice teachers. The first questions deal with identifying the methods used in the methods course to address beliefs. The second set of questions deal with the effects of the methods course on the beliefs that preservice teachers hold on the learning and teaching of mathematics.

The study included 16 different universities …

Affective Socialization Processes In Mathematics Doctoral Study: Gaining Insight From Successful Students, Lauren L Wagener

Doctoral Dissertations

Mathematics has the highest attrition rate among all liberal arts disciplines (and among all disciplines, except for health professions) and the second highest attrition rate of all doctoral programs in the United State. In order to prevent the loss of so many students, mathematics departments must consider the root causes for attrition and determine what individual skills and knowledge and departmental systems and support will help more mathematics doctoral students to succeed. The purpose of this qualitative interview study was to explore the interactions mathematics doctoral candidates at one institution have had during graduate school and the value that the …

Creating And Validating An Instrument To Measure Middle School Mathematics Teachers’ Technological Pedagogical Content Knowledge (Tpack), Geri A. Landry

Doctoral Dissertations

Due to the pervasiveness of technology, the role and preparation of teachers as they strategically use technology for teaching mathematics needs to be examined. Technological pedagogical content knowledge (TPACK) is a framework for knowledge as teachers develop meaningful learning experiences for their students while integrating strategic use of technology (Mishra & Koehler, 2006). The purpose of this study was to develop a survey for measuring mathematics teachers’ Mathematical Technological Pedagogical Content Knowledge (M-TPACK). The survey measures the domains of mathematics content, pedagogy and technology. This mixed methods study first examined middle school mathematics teachers’ TPACK through the use of an …

Bimodule Categories And Monoidal 2-Structure, Justin Greenough

Doctoral Dissertations

We define a notion of tensor product of bimodule categories and prove that with this product the 2-category of C -bimodule categories for fixed tensor C is a monoidal 2-category in the sense of Kapranov and Voevodsky ([KV91]). We then provide a monoidal-structure preserving 2-equivalence between the 2-category of C -bimodule categories and Z( C )-module categories (module categories over the center of C ). The (braided) tensor structure of C1⊠D C2 for (braided) fusion categories over braided fusion D is introduced. For a finite group G we show that de-equivariantization is equivalent to the tensor product over Rep( G). …