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Subgroups of Finite Wreath Product Groups for p=3
Master of Science, University of Akron, Applied Mathematics
Let M be the additive abelian group of 3-by-3 matrices whose entries are from the ring of integers modulo 9. The problem of determining all the normal subgroups of the regular wreath product group P=Z9≀(Z3 × Z3) that are contained in its base subgroup is equivalent to the problem of determining the subgroups of M that are invariant under two particular endomorphisms of M. In this thesis we give a partial solution to the latter problem by implementing a systematic approach using concepts from group theory and linear algebra.

#### Committee:

Jeffrey Reidl, Dr. (Advisor); Hung Nguyen, Dr. (Other); James Cossey, Dr. (Other)

#### Subjects:

Applied Mathematics; Mathematics; Theoretical Mathematics

#### Keywords:

applied math; mathematics, group theory, abstract algebra, wreath products

An Introduction to the Generalized Riemann Integral and Its Role in Undergraduate Mathematics Education
Bachelor of Science, Ashland University, 2017, Mathematics/Computer Science
The Riemann integral is often introduced to undergraduate calculus students, as its definition and related theorems are relatively straightforward to understand. However, the Riemann integral is limited in its power to integrate a wide variety of functions. This paper introduces an alternate definition of the integral, known as the generalized Riemann integral. This version of the integral was introduced around 1960 by Ralph Henstock and Jaroslav Kurzweil, and its definition and theorems are almost as simple as the traditional Riemann integral, yet its power to integrate functions far surpasses Riemann's integral. This paper includes an overview of the most important theorems and definitions related to the generalized Riemann integral and explains how it can be used to supplement, or even replace, the Riemann integral in undergraduate calculus and analysis courses.

#### Committee:

Darren Wick, Ph.D. (Advisor); Gordon Swain, Ph.D. (Committee Member); Christopher Swanson, Ph.D. (Committee Member)

#### Subjects:

Applied Mathematics; Education; Mathematics; Mathematics Education

#### Keywords:

Riemann integral; generalized Riemann integral; integral; calculus; analysis; gauge integral; Henstock-Kurzweil integral; mathematics education; undergraduate mathematics education

Teachers’ Perceptions Regarding Financial Literacy in Kindergarten Through Grade 2
Doctor of Philosophy (PhD), Ohio University, 2016, Curriculum and Instruction Mathematics Education (Education)
Financial literacy is an important life skill, yet how are we fostering understanding in our youngest students? Unless schools begin instruction on money concepts and skills at an early age, the majority of the students will not have the needed exposure until much later in their educational career. This study used a mixed methods research approach to explore kindergarten through second grade teachers’ perspectives regarding the curriculum and instruction of financial literacy. The study had two main phases. Both phases consisted of a two-step process of data collection and analysis. Phase 1 was qualitative and comprised interviews of teachers who taught in K–Grade 2 at three schools in Ohio. The interviews were coded descriptively, and the author used codeweaving to analyze the data for common themes. From these results, an online survey was created and distributed in Phase 2. Phase 2 was quantitative and involved a survey of a broader sample of K–2 teachers in Ohio. This phase tested the veracity of the Phase 1 results. Phase 2 determined whether generalizations could be made regarding teachers’ perceptions of students’ prior knowledge and skills, and of students’ cognitive readiness to understand financial literacy content. Perceptions from the two phases were triangulated with theory and research relating to child development to explore what, when, and how teachers are teaching money concepts and skills in their classroom. The findings indicate that K–2 teachers see value in teaching financial literacy concepts and skills in their classroom, but they are unsure of the expectations for implementation. In particular, the majority of the participants were unaware of the Jump\$tart Coalition for Personal Financial Literacy’s National Standards in K–12 Personal Finance Education and demonstrated confusion on state and Common Core standard expectations. During this study, making connections and providing students with genuine experiences were frequently identified as important practices. Though teachers’ knowledge of financial literacy expectations is limited, teachers incorporate money concepts and skills into their classrooms by employing such strategies as calendar time, school stores, behavior systems, games, and centers. They use a moderate amount of technology and a variety of manipulatives to support instruction. These results indicate a need to inform teachers about the written and intended curriculum regarding financial literacy and a need to align the various sets of standards to ensure a cohesive and comprehensive K–12 financial literacy curriculum. With the proper guidance and implementation, teachers at all grade levels can experience success in preparing their students for a financially stable future.

#### Committee:

Gregory Foley (Advisor); Eugene Geist (Committee Member); Koestler Courtney (Committee Member); Machtmes Krisanna (Committee Member)

#### Subjects:

Applied Mathematics; Curricula; Early Childhood Education; Education; Educational Theory; Elementary Education; Finance; Literacy; Mathematics; Mathematics Education; Pedagogy; Teacher Education; Teaching

#### Keywords:

financial literacy; mathematics education; math; elementary; money; skills; concepts; financial skills; financial concepts; early childhood; mixed methods; child development; standards; teaching; manipulatives; Common Core; JumpStart; K-2; primary; state

Surface Nonuniformities in Waterborne Coatings due to Evaporative Mechanisms
Master of Science, University of Akron, 2016, Applied Mathematics
A model is developed for predicting the long range length scale of film thickness nonuniformities in drying latex paint films. After applying the lubrication approximation to the model equations, spatially independent base state solutions are found. These equations are solved numerically and a linear stability analysis of these base state solutions is conducted. In this case, the base state solution of the height represents a uniformly drying latex paint film with respect to time. The stability of the film is dependent on temperature, latex particle volume fraction, surface surfactant density, bulk surfactant density, and several material and environmental factors. Evaporation, slow and equal surfactant kinetics, low initial surface tension, substrate permeability and high initial latex particle volume fractions are identified as destabilizing mechanisms while fast surfactant kinetics, high initial surface tension and viscosity are identified as stabilizing mechanisms.

#### Committee:

Patrick Wilber, Dr. (Advisor); Curtis Clemons, Dr. (Committee Member); Kevin Kreider, Dr. (Committee Member)

#### Subjects:

Applied Mathematics; Chemical Engineering; Chemistry; Engineering; Fluid Dynamics; Mathematics

#### Keywords:

Asymptotic analysis, paint, linear stability analysis, latex, colloids, coating flows, drying, fluid mechanics, mathematical modeling

African American Urban Public High School Graduates’ Experiences Concerning Mathematics
Doctor of Philosophy in Urban Education, Cleveland State University, 2016, College of Education and Human Services

#### Committee:

Joanne Goodell, Ph.D. (Committee Chair); Anne Galletta, Ph.D. (Committee Member); Brian Harper, Ph.D. (Committee Member); Mittie Davis Jones, Ph.D. (Committee Member); Roland G. Pourdavood, Ph.D. (Committee Member)

#### Subjects:

African Americans; Mathematics; Mathematics Education; Secondary Education

#### Keywords:

African American; Model of Academic Choice; urban public high school; Northeast Ohio; student engagement; academic outcomes of African Americans in mathematics education; African Americans in STEM-related courses; persistence

PIEZOELECTRIC INVERSE PROBLEMS WITH RESONANCE DATA: A SEQUENTIAL MONTE CARLO ANALYSIS
Doctor of Philosophy, Case Western Reserve University, 2014, Applied Mathematics
Piezoelectricity is a property of certain materials that allows the conversion of mechanic deformation into electric voltage potential, and vice versa. The wide use of piezoelectric materials, e.g., in transducer technology and energy harvesting makes the design problem of optimizing the material parameters and geometry an important target in scientific computing. In energy harvesting in particular, the design of devices with impedance resonances in a predetermined range is of special interest: Matching the resonances with the ambient vibration frequencies may lead potentially to higher efficiency of the device. Material scientist can rely on numerical simulations in the design and production of piezoelectric devices. Numerical simulations employ numerical techniques like finite element methods to generate information about the design starting from an input of material parameters. In the context of this thesis, these material parameters include the elastic, electromagnetic and piezoelectric constants. Because the quantitative values of the material parameters are often determined from simplified experiments based on some assumptions, the reliability of the results of the simulations depends on the validity of these assumptions. Optimization based approaches to the numerical acquisition of the material parameters normally give a single set of values which, in turn, identifies one specific material as the approximation of the target. From a practical point of view this may be too restrictive, because it leaves little flexibility when trying to develop materials with a certain desired response. In this thesis we approach the inverse problem of material characterization for piezoelectric materials from a Bayesian perspective. The main question addressed in this thesis is, how to choose the elastic, electromagnetic, and piezoelectric material parameters so that a target resonance frequency is achieved, and the band-pass impedance response outside the resonance is matched to a target profile. The methodology is based on a Bayesian formulation of the problem which is then solved by means of a new Sequential Monte Carlo (SMC) sampling technique of the posterior probability density. The algorithm suggested in this work is based on a sequential approximation of the posterior density, combining the method with the auxiliary particle techniques.

#### Subjects:

Applied Mathematics; Materials Science; Mathematics

#### Keywords:

Piezoelectric;Inverse problems;Finite elements;Bayesian; Sampling;Monte Carlo

Techniques for the Study of Biological Coupled Oscillator Systems
Bachelor of Science (BS), Ohio University, 2014, Mathematics
For over forty years, biologists have observed oscillating oxygen consumption in dense cultures of yeast. The period of the oxygen oscillations divides the period of the yeast cell cycle suggesting that the cell cycle drives oxygen consumption and that the distribution of cell cycle phases is not uniform. Non-uniform phase distributions can be reproduced by ordinary differential equations models in which cells progress through the cell cycle at rates dependent on the phases of other cells. The model predicts that after a long time, cells will segregate into two or more synchronized cohorts. We develop and automate techniques for determining what phase distributions are possible and exploring how synchrony emerges from arbitrary distributions. We extend the techniques to a broad class of coupled oscillator systems.

#### Committee:

Todd Young (Advisor); Todd Eisworth (Other)

#### Subjects:

Applied Mathematics; Mathematics

The Crossover Project: A Case Study of One High School's Effort to Provide Skill-Deficient Students the Opportunity to Cross Over Into a College Preparatory Math Track
Doctor of Education, Miami University, 2011, Educational Leadership

Ability tracking is used in schools in the U.S. and other industrialized countries to stratify students into homogeneous ability groups as a method for delivery of instruction (Falkenstein, 2007). The strong tradition of tracking in America has socio-economic repercussions for the Nation and its students, and it continues to be debated because of the way it locks students into specific times at which they are exposed to defined curricular concepts and content, limiting college and career opportunities for low-tracked students.

Like most other high schools across the state and country, Fairmont High School in the Kettering City School District in southwest Ohio utilizes a tracked curriculum. Specifically in mathematics at Fairmont, the college bound track begins with Algebra I in the ninth grade, which research shows is the gatekeeper to a college bound track. Once placed in a lower track upon entering high school, students have almost no chance of moving into the higher track.

The purpose of this curriculum and administrative case study is to analyze and evaluate the effectiveness of an Algebra I mathematics program that was piloted in the 2009-2010 school year at Fairmont High School that aims to move a cohort of low-tracked ninth-graders into the college-bound math track at the high school level. It is called the Crossover Project.

This study reveals the student achievement data from the Crossover Project students compared to students not in the program and investigates the critical components for student success. The study found that under specific conditions, high school students in the lowest track can be caught up in subjects such as mathematics sufficiently to prepare them for success in the college bound track. Due to the findings of this research, the Crossover project at Fairmont High School is being expanded to include low tracked English students and a larger segment of low tracked math students in an effort to make all students college and career-ready.

The evaluation of this program may serve to provide evidence that students with skill deficits in math can “catch up” to their college bound peers in math, once at the high school level.

#### Committee:

Kate Rousmaniere, PhD (Committee Chair); Steven Thompson, PhD (Committee Member); Dennis Carlson, PhD (Committee Member); William Boone, PhD (Committee Member)

#### Subjects:

Education; Education History; Education Philosophy; Educational Leadership; Mathematics; Mathematics Education; School Administration; Secondary Education; Social Research; Teaching

#### Keywords:

tracking; math curriculum; curriculum tracking; education; curriculum; Jeannie Oakes; Tom Loveless; changing tracks; mathematics education; secondary math curriculum; tracking in high school

Gender and Cognitive Skills throughout Childhood
Doctor of Philosophy, The Ohio State University, 2009, Sociology

#### Committee:

Douglas Downey (Committee Chair); Rachel Dwyer (Committee Member); Vincent Roscigno (Committee Member)

#### Subjects:

Behaviorial Sciences; Developmental Psychology; Economics; Education; Educational Psychology; Educational Sociology; Mathematics; Mathematics Education; Psychology; Social Psychology; Social Research; Social Studies Education; Social Work; Sociology

#### Keywords:

gender differences; cognitive skills; math; reading; early childhood

A Transition from Informal to Formal Proof in K-12 Mathematics
Master of Mathematical Sciences, The Ohio State University, 2017, Mathematical Sciences
Proof plays a valuable role in increasing understanding of mathematics. Despite recent calls for proof in all grades and all math courses, constructing formal proofs is a prevalent challenge in high school mathematics, and it remains relegated mostly to geometry courses. While some informal reasoning is being integrated in elementary classrooms and beyond, students are not developing enough understanding of proof to be successful in formal reasoning. Consequently, a beneficial question is: What can be done in K-12 education to help students transition from informal to formal proof? This paper illustrates, with examples, four aspects of proof: generality, justification, precision of language, and conceptual complexity— all of which can vary across grades. Together these four aspects frame a transition from informal to formal proof in K-12 mathematics. The paper also considers implications of the transition for curriculum, for instruction in the classroom, and for teacher education, as well as directions for future research.

#### Committee:

James Fowler, Dr. (Advisor); Bradford Findell, Dr. (Committee Member); Herb Clemens, Dr. (Committee Member); Michael Battista, Dr. (Committee Member)

#### Subjects:

Mathematics; Mathematics Education

#### Keywords:

Proof; Proving; K-12 Mathematics; Prospective Teachers

A Hybrid Method For Solving A Single Nonlinear Equation
Master of Science in Mathematics, Youngstown State University, 2011, Department of Mathematics and Statistics
The purpose of this paper is to develop a root finding method for non-linear functions. The problem, f(x)=0 where x is in R, is common in many areas of mathematics and can be traced back as far as 1700 B.C. A cuneiform table in the Yale Babylonian Collection dating from that period gives a base-60 number equivalent to 1.414222 as an approximation to the square root of 2, a result accurate to within .00001 (the square root of 2 is approximately 1.414214). We wanted to develop a hybrid method that quickly produces a small interval containing the solution and then switch to a method with faster convergence. We have created a method to solve functions whose exact roots are not easy to find using common techniques learned in algebra and calculus courses. We have compiled test functions, some of our own and some from other works on the same topic. We have also compared our method with that of several other methods consisting of Secant Method, False Position, a modified version of Modified False Position, Inverse Quadratic Interpolation, Bisection and a few other hybrid methods. Our method begins with the modified version of Modified False Position, which will be discussed in more detail later, then switches to Muller's method once a certain tolerance is reached. In certain instances, our method switches back to the modified version of Modified False Position. We found our method outperformed these methods in most cases and was competitive to the other hybrid methods, and in many cases, it outperformed them as well.

#### Committee:

Jozsi Jalics, PhD (Advisor); Richard Burden, PhD (Committee Member); J. D. Faires, PhD (Committee Member)

#### Subjects:

Applied Mathematics; Mathematics

#### Keywords:

Numerical analysis; Root finding methods

On the Trajectories of Particles in Solitary Waves
Master of Science in Mathematics, Youngstown State University, 2011, Department of Mathematics and Statistics

Across the country, school students learn that ocean waves cause water particles to form looping paths, traveling in circles which become smaller as you look deeper underwater.

In this paper, we investigate the approximations which are used to make this claim. Furthermore, we investigate closer approximation techniques which show that these looping paths actually propogate forward with the wave's motion. Finally, we investigate the specific case of the soliton, which causes particles underneath to travel in a forward-moving arc, with no looping motion at all.

With this background, we examine the recent work of A. Constantin and collaborators, specifically his conclusion that our results for the soliton hold for any solitary wave.

#### Committee:

George T. Yates, PhD (Advisor); Jozsi Jalics, PhD (Committee Member); Steven L. Kent, PhD (Committee Member)

#### Subjects:

Applied Mathematics; Fluid Dynamics; Mathematics; Physics

#### Keywords:

Solitons; Fluid dynamics; Water waves; Euler equation; KdV equation; Differential equations

An Exploration of Mathematical Applications in Cryptography
Master of Mathematical Sciences, The Ohio State University, 2015, Mathematics
Modern cryptography relies heavily on concepts from mathematics. In this thesis we will be discussing several cryptographic ciphers and discovering the mathematical applications which can be found by exploring them. This paper is intended to be accessible to undergraduate or graduate students as a supplement to a course in number theory or modern algebra. The structure of the paper also lends itself to be accessible to a person interested in learning about mathematics in cryptography on their own, since we will always give a review of the background material which will be needed before delving into the cryptographic ciphers.

#### Committee:

James Cogdell (Advisor); Rodica Costin (Committee Member)

#### Subjects:

Mathematics; Mathematics Education

#### Keywords:

cryptography; cryptographic ciphers; number theory; elliptic curve cryptography

SLEEP-WAKE TRANSITION DYNAMICS AND POWER-LAW FITTING WITH AN UPPER BOUND
Doctor of Philosophy, The Ohio State University, 2014, Mathematics
Sleep and wake bout durations, for mammalian species, are known to follow lawful statistical behavior. Experiments have shown that while sleep bout durations are exponentially distributed, the distribution of wake bout durations follow a power-law. Furthermore, in infant rats, wake bout durations are exponentially distributed which turns to a power-law only when the animal develops. In this dissertation, we analyze the sleep-wake regulatory system as a competitive graph model in order to understand the relation between the degree distributions of the graph and the resulting distribution of the bout durations. For graphs with small average excitatory and inhibitory degrees, simulated sleep and wake bout durations are found to follow power-law distributions but only in finite intervals. For a careful identification of power-law distributions in finite intervals, we developed a robust method for fitting a power-law distribution with an upper bound. This new method is also applied on real sleep and wake bout durations collected from rat experiments, and the numerical analysis showed that sleep bouts are exponentially distributed only in the tail which is preceded by a power-law distributed region in a finite interval.

#### Committee:

Janet Best (Advisor); Edward Overman (Committee Member); David Sivakoff (Committee Member); Vincent Billock (Other)

#### Subjects:

Applied Mathematics; Mathematics; Neurosciences

#### Keywords:

sleep; wake; bout; power-law; exponential; KS; fitting; regulated; Brownian; bootstrap; mean; field

Laboratory and field work in plane geometry
Master of Arts, The Ohio State University, 1946, EDU Teaching and Learning
N/A

#### Subjects:

Education; Mathematics; Mathematics Education

#### Keywords:

Education; mathematics

PARAMETER CHOICES FOR THE SPLIT BREGMAN METHOD APPLIED TO SIGNAL RESTORATION
Master of Science, University of Akron, 2016, Applied Mathematics
This thesis aims to study the parameters involved in the Split Bregmann Method (SBM) when it is applied to signal restoration problems. SBM has been used increasingly on signal processing problems in recent years in particular for compressed sensing problems. SBM is an iterative method which has been used to solve l1-regularized optimization problems. Tikhonov regularization is one of the most common methods to solve discrete ill-posed problem and it appears as one of the steps inside SBM. Four standard different parameter choice methods for Tikhonov regularization were examined in order to find the best values for the regularized and threshold parameters inside SBM: L-curve, generalized cross validation (GCV), discrepancy principle (DP), and unbiased predictive risk estimation (UPRE). Several numerical experiments have been done with signals with sharp edges to compare all the different approaches.

#### Committee:

Malena Espanol (Advisor); Wilber Patrick (Committee Member); Kreider Kevin (Committee Member)

#### Subjects:

Applied Mathematics; Mathematics

#### Keywords:

Signal restoration, Deblurring problem, Split Bregman method, Image processing, regularization parameter

A cross-cultural comparison of Brazilian and American mathematics curricula
Bachelor of Science, Ashland University, 2013, Curriculum/Instruction
The study of mathematics tends to hold reputation as being universal language of studies, meaning mathematical literacy sustains value worldwide. However, mathematics cannot be determined universal by mathematical literacy alone. It is the curricular development in education that lays the groundwork in deciding what mathematical knowledge is of most value which in turn sets the tone for mathematical literacy. Specifically, the development of the scope and sequence of mathematics standards in education frame the content and order in obtaining mathematical literacy and are examined in this thesis between two different countries, Brazil and the U.S. The outcome of comparing mathematics standards of both Brazil and the U.S. is discussed with some minor curricular differences, but also with an overall cohesiveness in mathematical literacy which may indicate universality in mathematics in the two countries which could potentially translate worldwide.

#### Committee:

Jason Ellis, PhD (Advisor); David Kommer, EdD (Committee Member); Christopher Swanson, PhD (Committee Member)

#### Subjects:

Education; Mathematics; Mathematics Education

#### Keywords:

math curricula; mathematics curricula; Brazil; America; United States

Complex Dynamical Systems: Definitions of Entropy, Proliferation of Epithelia and Spread of Infections and Information
Doctor of Philosophy (PhD), Ohio University, 2018, Mathematics (Arts and Sciences)
Dynamical systems as models of complex biological systems are powerful tools that have been used to study problems in biology. This dissertation first discusses a problem in the theory of dynamical systems on the definition of topological entropy. Then dynamical systems are applied in two different fields of biology: proliferation of monolayer epithelia and the spread of infections and information. The notion of topological entropy is a measure of the complexity of a dynamical system. It can be conceptualized in terms of the number of forward trajectories that are distinguishable at resolution ε within T time units. It can then be formally defined as a limit of a limit superior that involves either covering numbers, or separation numbers, or spanning numbers. If covering numbers are used, the limit superior reduces to a limit. While it has been generally believed that the latter may not necessarily be the case when the definition is based on separation or spanning numbers, no actual counterexamples appear to have been previously known. Here we fill this gap in the literature by constructing such counterexamples. We then use dynamical systems to study the proliferation of epithelia. Epithelia are sheets of tightly adherent cells that line both internal and external surfaces in metazoans (multicellular animals). Mathematically, a cell in an epithelial tissue can be modeled as a k-sided polygon. Empirically studied distributions of the proportions of k-sided cells in epithelia show remarkable similarities in a wide range of evolutionarily distant organisms. Multiple types of mathematical models have been proposed to explain this phenomenon. Among the most parsimonious of such models are topological ones that take into account only the number of sides of a given cell and the neighborhood relation between cells. The model studied here is a refinement of a previously published such model (Patel et al., PLoS Comput. Biol., 2009). While using the same modeling framework as that paper, we introduce additional options for the choice of the endpoints of the cleavage plane in the simulated development of epithelial tissues. Some of these options appear to be more biologically realistic approximations of known mechanisms that influence the cleavage plane orientation. A comparative analysis of comprehensive simulations for relevant combinations of both previously studied and our newly designed options is reported here. We compared the outcomes with the empirical distributions of the same five organisms as in (Patel et al., 2009), Drosophila (fruit fly), Hydra (closely related to jellyfish), Xenopus (clawed frog), Cucumber, and Anagallis (a flowering plant). We found that combinations of some of our new options consistently gave better fits with each of these data sets than combinations of previously studied options for choosing the cleavage plane. We also examine ODE models of epidemic spreading with a preventive behavioral response that is triggered by awareness of the infection. Previous research on such models has mostly focused on the impact of the response on the initial growth of an outbreak and the existence and location of endemic equilibria. Here we study the question whether this type of response is sufficient to prevent future flare-ups from low endemic levels if awareness is assumed to decay over time. In the ODE context, such flare-ups would translate into sustained oscillations with significant amplitudes. Our results show that such oscillations are ruled out in Susceptible-Aware-Infectious-Susceptible models with a single compartment of aware hosts, but can occur if we consider two distinct compartments of aware hosts who differ in their willingness to alert other susceptible hosts.

#### Committee:

Winfried Just (Advisor); Todd Young (Committee Member); Martin Mohlenkamp (Committee Member); Alexander Neiman (Committee Member)

#### Subjects:

Applied Mathematics; Mathematics

#### Keywords:

dynamical systems; topological entropy; mathematical modeling; cell topology; epithelial morphogenesis; epidemic models; awareness dissemination; periodic oscillations

Generalized Non-Autonomous Kato Classes and Nonlinear Bessel Potentials
Doctor of Philosophy (PhD), Ohio University, 2005, Mathematics (Arts and Sciences)

In this dissertation, we study the scale of function spaces Mp(Rn) introduced by Zamboni. For these spaces, we get a characterization in terms of nonlinear Bessel potentials. This result is based on a known characterization of the Kato class Kn,s of order s in terms of Bessel potentials. We also introduce and study new function classes Ppn,T([0,T] x Rn) and Apn,T([0,T] x Rn) these classes generalize the classes Pn,T([0,T] x Rn) and An,T([0,T] x Rn) introduced by Gulisashvili.

#### Subjects:

Mathematics; Mathematics

#### Keywords:

Gradient estimate; Non-autonomous functions; Time-dependent functions; Bessel potentials; Riesz potentials; Kato class; Non-linear potentials

Financial Mathematical Tasks in a Middle School Mathematics Textbook Series: A Content Analysis
Doctor of Philosophy, University of Akron, 2009, Elementary Education

This content analysis examined the distribution of financial mathematical tasks (FMTs), mathematical tasks that contain financial terminology and require financially related solutions, across the National Standards in K-12 Personal Finance Education categories (JumpStart Coalition, 2007), the thinking skills as identified by A Taxonomy for Learning, Teaching, and Assessing (Anderson et al., 2001), and the National Council of Teachers of Mathematics Standards (NCTM, 2000). Two hundred seventy-eight FMTs, recording units for this study, were taken from a selected portion in each lesson within the three grade level textbooks of the middle school mathematics textbook series, Math Connects Concepts, Skills, and Problem Solving Course 1, 2, and 3 (Glencoe McGraw-Hill, 2009).

Three research questions, with corresponding coding forms, were developed for this study. After the coding forms were evaluated, the researcher trained coders, held trial codings, and conducted a pilot test to determine reliability, address validity concerns, and determine her credentials as the sole coder. As a result of the evaluations, trial codings, and pilot test, the coding forms were refined. The data analysis yielded frequency counts and percentages. None of the FMTs focused on planning a budget. The FMTs poorly addressed Create, the highest order thinking skill. The FMTs did not support the NCTM standard Representation adequately.

The findings indicate that the FMTs did not uniformly address the personal finance categories, the selected thinking skills, and the selected NCTM standards investigated in this research study. The potential is limited for middle school students to experience FMTs that contain: a balanced array of personal finance concepts and skills, challenging higher order thinking requirements, and an equal balance of the NCTM standards investigated in this research study. Among the recommendations advocated are: stabilizing the alignment of the FMTs to the personal finance categories, thinking skills, and NCTM standards, directing future research to continue investigating FMTs, focusing on worthwhile financial mathematical tasks, and investigating the potential for mathematics textbooks to be a vehicle for financial literacy education.

#### Committee:

Katharine D. Owens, PhD (Committee Member)

#### Subjects:

Education; Elementary Education; Home Economics; Literacy; Mathematics; Mathematics Education; Secondary Education

#### Keywords:

financial literacy education; financial literacy; personal finance; personal finance education; financial mathematical tasks

A Study of CS and Σ-CS Rings and Modules
Doctor of Philosophy (PhD), Ohio University, 2005, Mathematics (Arts and Sciences)

A right R-module M is called CS if every submodule of M is essential in a direct summand of M. In this dissertation, we study certain classes of CS and Σ-CS rings and modules. A ring R is called right (left) max-min CS if every maximal closed right (left) ideal with nonzero left (right) annihilator and every minimal closed right (left) ideal of R is a direct summand of R. Among other results, it is shown that if R is a nondomain prime ring, then R is right nonsingular, right max-min CS with a uniform right ideal if and only if R is a left nonsingular, left max-min CS with a uniform left ideal. This result gives, in particular, Huynh, Jain and López-Permouth Theorem for prime rings of finite uniform dimension. Also we show that a nondomain right nonsingular prime ring with a uniform right ideal is right finitely Σ-min-CS if every finitely generated right ideal of R is min CS. Jain, Kanwar and López-Permouth characterized right nonsingular semiperfect right CS rings. We obtain the structure of right nonsingular semiperfect right min CS rings with a uniform right ideal. It is shown that such rings are direct sums of indecomposable right CS rings and a ring with no uniform right ideal. As a consequence, we show that an indecomposable right nonsingular semiperfect ring is right CS if and only if it is min CS with a uniform right ideal. We generalize this result to endomorphism rings of nonsingular semiperfect progenerator min CS modules with a uniform submodule. It is known that every Σ-CS module is a direct sum of uniform modules and countably Σ-CS modules need not be Σ-CS. A sufficient condition that guarantees a countably Σ-CS module, which is a direct sum of uniform modules, to be Σ-CS has been obtained.

#### Subjects:

Mathematics; Mathematics

#### Keywords:

Ring Theory; Prime Ring; Module Theory; CS Module; Extending Module; Semiperfect Module

Academic Mathematicians' Dispositions Toward Software Use in Mathematics Instruction: What Are the Underlying Reasons?
Doctor of Philosophy (PhD), Ohio University, 2012, Curriculum and Instruction Mathematics Education (Education)

Academic mathematicians’ opinions are divided regarding software use in undergraduate mathematics instruction. This study explored these opinions through interviews and a subsequent survey of mathematicians at PhD-granting institutions in the United States regarding their dispositions and the underlying attitudes. Most prior related work had focused on mathematicians who used software in teaching, thus ignoring skeptics and critics. This investigation studied the full range of views. The research questions were

¿¿¿¿¿¿¿ What are academic mathematicians’ dispositions toward software integration in undergraduate mathematics classrooms?

¿¿¿¿¿¿¿ What are the reasons underlying academic mathematicians’ dispositions toward software integration in undergraduate mathematics classrooms?

An exploratory sequential research design built, expanded, and tested a model to explain mathematicians’ dispositions toward software use in undergraduate instruction. This model subsumed Fishbein and Ajzen’s attitude framework. The researcher reviewed anecdotal evidence, published opinions, related theories, and research results to add to this framework, thus building an initial model. Next, interview data were used to expand the model, and the data and expanded model served as bases to develop a survey instrument. Using a sample of mathematicians from 50 PhD-granting institutions, survey data tested the factors in the expanded model. The interview data and the survey data were triangulated with the reviewed literature to refine the model to include factors that emerged as the underlying reasons for the use or nonuse of software.

The triangulation process suggests that most mathematicians have a moderate and somewhat skeptical attitude toward software use in teaching. Small numbers of mathematicians either strongly oppose or strongly support software use across undergraduate instruction. Most mathematicians value the benefits of software but are concerned about its potential harm and prefer traditional instructional methods. The interviews identified 8 factors and 16 subfactors that contribute to mathematicians’ attitudes regarding software use. Among these, the triangulation process suggests that software characteristics, perceived effect on learning, and instructor’s personality were the three most influential factors. Among the remaining factors, students’ level, students’ major, instructor’s educational background, and teaching background were supported. In addition, there were inconsistent results with regard to institution and research interest as factors, which may warrant future investigation.

#### Committee:

Gregory Foley (Committee Chair); John Hitchcock (Committee Member); Sergio L¿¿¿¿pez-Permouth (Committee Member); Timothy McKeny (Committee Member)

#### Subjects:

Education; Educational Software; Educational Technology; Higher Education; Mathematics; Mathematics Education

#### Keywords:

Mathematics Teaching; Mathematical Software; Using software in mathematics instruction; Undergraduate mathematics classrooms; Mathematicians' attitude; Mathematicians' dispositions; Mathematicians' opinions about software use;

Site-Specific Point Positioning and GPS Code Multipath Parameterization and Prediction
Doctor of Philosophy, The Ohio State University, 2011, Geodetic Science and Surveying

In spite of the fact that many multipath mitigation techniques are currently implemented in GPS hardware and firmware, there remains a post-receiver-processed multipath error which plagues GPS observations stored in the RINEX files. At static Continuously Operating Receiver Station (CORS) sites, this multipath error expresses sidereal periodicity by virtue of the repeated configuration of the GPS satellite relative to the ground-borne GPS antenna and its environment.

In this study, this repeatability was capitalized upon to investigate the multipath signal (its variability inclusive) generated at a highly multipath-prone Suominet CORS site (named SG03). In this regard, the multipath was parameterized using a Fourier Analysis and a Wavelet Analysis technique. The latter was found to capture in excess of 90% of the identified signal, outperforming the Fourier parameterization.

Therefore, investigation of an optimal multipath modeling technique (in this case using a third order one-dimensional Daubechies wavelet) served as a pre-cursor to the main focus of this study, that being to construct an Integrated Point Position and Multipath Prediction (IPPMP) algorithm at a static CORS site. To this end, five different IPPMP algorithms were developed in MATLAB based on three types of prior-information models, namely, the Extended Gauss-Markov Model, the Mixed Linear Model and the Random Effects Model. While it is true that other wavelet-based mitigation schemes (such as the trademarked Wavesmooth technique) exist, the implementation of this study emphasized both the mitigation as well as the inclusion of multipath parameters in the IPPMP schemes developed.

The results of this study suggest that use of an elevation-dependent variance model for the high frequency wavelet parameters (called wavelet details) is insignificant to the position determination. It was confirmed that the use of the Random Effects Model is useful where the prior information is actually bias-prone, this model being more suitable to high precision positioning applications. Inclusion of the multipath parameters in the IPPMP models did indeed improve the position determination capability, but only where the receiver coordinates were sequentially-updated– the influence of the prior coordinate values on the solution was greater than that of the multipath prior information. The solution convergence rate was understandably-higher the more accurate the initial coordinates.

Ultimately, the results of this study suggest that there is value in parameterizing the multipath signal at stationary CORS sites with the intention of minimizing the impact of the multipath on position determination at static locations.

#### Committee:

Dorota Grejner-Brzezinska, PhD (Advisor); Alper Yilmaz, PhD (Committee Member); Andria Bilich, PhD (Committee Member)

#### Subjects:

Applied Mathematics; Civil Engineering; Engineering; Mathematics; Meteorology; Remote Sensing; Statistics

#### Keywords:

GPS; multipath; mitigation; point positioning; wavelet analysis; Fourier analysis; Extended Gauss-Markov Model; Mixed Linear Model; Random Effects Model; CORS; prior information models; least squares adjustment

A Game Theoretic Analysis and Simulation of Non-Incumbent Elections
Master of Science, University of Akron, 2014, Applied Mathematics
We develop a model that provides evidence to explain the changes in policy platform during a non-incumbent, two candidate election. We propose a modification to a well known model presented by Hummel [1], and predict the behavior of candidates in a selection of varying scenarios. We show voter support convergence for all models, and show evidence for convergence for bimodal voter ideology distributions through numerical simulations. We conclude that candidates adjust strategies in order to seek the highest local concentration of voters.

#### Committee:

Stefan Forcey, Dr. (Advisor); Francesco Renna, Dr. (Advisor); Gerald Young, Dr. (Committee Member); Curtis Clemons, Dr. (Committee Member)

#### Subjects:

Applied Mathematics; Economic Theory; Economics; Mathematics; Political Science; Public Policy

#### Keywords:

game theory; political science; elections; electoral spatial models; median voter theorem

"The effect of ability-based versus effort-based praise on task performance, task persistence, and internal factors in children identified as gifted or talented in mathematics"
Specialist in Education, Miami University, 2014, School Psychology