Digital Commons Network^™

A Sequel To “A Space Topologized By Functions From Omega To Omega”, Tetsuya Ishiu, Akira Iwasa

Faculty Publications

We consider a topological space ⟨𝑋, 𝜏 (ℱ)⟩, where 𝑋 = {𝑝 ∗} ∪ [𝜔 Å~ 𝜔] and ℱ ⊆ 𝜔𝜔. Each point in 𝜔 Å~ 𝜔 is isolated and a neighborhood of 𝑝∗ has the form {𝑝∗}∪{⟨𝑖, 𝑗⟩ : 𝑖 ≥ 𝑛, 𝑗 ≥ 𝑓(𝑖)} for some 𝑛 ∈ 𝜔 and 𝑓 ∈ ℱ. We show that there are subsets ℱ and 𝒢 of 𝜔𝜔 such that ℱ is not bounded, 𝒢 is bounded, yet ⟨𝑋, 𝜏 (ℱ)⟩ and ⟨𝑋, 𝜏 (𝒢)⟩ are homeomorphic. This answers a question of the second author posed in A space topologized by functions …

Information-Preserving Structures: A General Framework For Quantum Zero-Error Information, Robin Blume-Kohout, Hui Khoon Ng, David Poulin, Lorenza Viola

Dartmouth Scholarship

Quantum systems carry information. Quantum theory supports at least two distinct kinds of information (classical and quantum), and a variety of different ways to encode and preserve information in physical systems. A system’s ability to carry information is constrained and defined by the noise in its dynamics. This paper introduces an operational framework, using information-preserving structures, to classify all the kinds of information that can be perfectly (i.e., with zero error) preserved by quantum dynamics. We prove that every perfectly preserved code has the same structure as a matrix algebra, and that preserved information can always be corrected. We …

Gsu Students Teach Math To Educators, Office Of Public Affairs

Press Releases

Governors State University graduate and undergraduate students had the opportunity to teach educators recently when they presented at the 61st Annual Meeting of the Illinois Council of Teacher of Mathematics in Springfield, Illinois.

The students, all studying to be math teachers, presented math activities they designed based on the massive sculptures in the Nathan Manilow Sculpture Park. The students found that the 26 sculptures, which are located on the grounds of the university, can be used to illustrate various mathematical formulas and principles.

Developing Mathematical Content Knowledge For Teaching Elementary School Mathematics, Eva Thanheiser, Christine A. Browning, Meg Moss, Tad Watanabe, Gina Garza-Kling

Faculty and Research Publications

In this paper the authors present three design principles they use to develop preservice teachers' mathematical content knowledge for teaching in their mathematics content and/or methods courses: (1) building on currently held conceptions, (2) modeling teaching for understanding, (3) focusing on connections between content knowledge and other types of knowledge. The authors share results of individual research projects and teaching approaches focusing on helping preservice elementary teachers develop such knowledge. Specific examples from different content areas (whole number, fractions, angle, and area) are discussed.

The Impact Of Content Courses On Pre-Service Elementary Teachers’ Mathematical Content Knowledge, Michael Matthews, Janice Rech, Neal Grandgenett

Teacher Education Faculty Publications

In response to research documenting the mathematical deficiencies of pre-service elementary teachers, many teacher preparation programs are requiring mathematical content courses specifically focusing on the mathematics taught at the elementary level. This study considers what impact two such courses (one course focusing on Arithmetic, and the other course focusing on Geometry and Measurement) had on the mathematical content knowledge and attitude towards mathematics by comparing a group of pre-service elementary teachers who took these courses to a group of pre-service elementary teachers who took only a more general mathematics course (such as College Algebra). Results indicated that those teachers who …

The Impact Of Smart Board Technology On Growth In Mathematics Achievement Of Gifted Learners, Patricia Ann Riska

Doctoral Dissertations and Projects

This study examined whether SMART Board technology increased growth in mathematics performance of fourth grade gifted students. Gifted students in North Carolina were studied to determine if the use of SMART Board technology during mathematics instruction impacted their growth on standardized state tests. The sample consisted of 175 students from six elementary schools with similar populations. Three of the schools used SMART Boards during mathematics instruction, and three schools did not use SMART Board technology. All students were taught the mathematics curriculum according to the North Carolina Standard Course of Study. The instrument for evaluating growth was the state End-of-Grade …

Teacher Quality, Content Knowledge, And Self-Efficacy In One Mathematics Teach For America Cohort, Brian R. Evans

NERA Conference Proceedings 2010

The purpose of this study was to understand the relationships between mathematical content knowledge and perceptions of teaching self-efficacy in one cohort of Teach for America teachers. It was found that teachers had high levels of self-efficacy. It was also found that mathematics related majors had higher mathematical content knowledge than did business majors, but similar levels of self-efficacy. Liberal arts majors had similar content knowledge and levels of self-efficacy as did mathematics related majors.

Middle And High School Mathematics Teacher Differences In Mathematics Alternative Certification, Brian R. Evans

NERA Conference Proceedings 2010

his study examined the differences in content knowledge, attitudes toward mathematics, and concepts of teacher self-efficacy among several different types of teachers in the New York City Teaching Fellows program, and informs teacher education in mathematics alternative certification. Findings revealed that high school teachers had significantly higher content knowledge than middle school teachers. Mathematics Teaching Fellows had significantly higher content knowledge than Mathematics Immersion Teaching Fellows. Mathematics and science majors had significantly higher content knowledge than other majors. Teachers had the same high positive attitudes toward mathematics and same high concepts of self-efficacy regardless of content ability.

Project: Application Of Algebra Or Analysis, Eduardo C. Balreira

Information Literacy Resources for Curriculum Development

No abstract provided.

Mathematics Self-Efficacy Of Community College Students In Developmental Mathematics Courses, David Walker Clutts

Doctoral Dissertations and Projects

Mathematics self-efficacy was defined as an individual's beliefs about how he or she would perform a specific math task or in a specific mathematics or related course. Mathematics self-efficacy was differentiated from self-esteem. Previous literature found self-efficacy in general and mathematics self-efficacy in particular to be significantly related to enrollment, retention, and completion. This study used the Mathematics Self-Efficacy Survey to investigate whether age, gender, developmental mathematics course, or developmental mathematics grade were significantly predictive of mathematics self-efficacy among developmental mathematics students course at a Kentucky community college. Multiple linear regression found that none of these variables were statistically significant …

Cooperative Learning And The Gifted Student In Elementary Mathematics, Christine C. Hecox

Doctoral Dissertations and Projects

The research was a quantitative research project dealing with Florida Comprehensive Assessment Test (FCAT) Mathematics scores of fourth grade students, including gifted and high-achieving students, in 2008-2009 under the exposure of daily cooperative learning in mathematics. The problem statement was as follows: In Polk County, Florida, how does cooperative learning affect the FCAT Mathematics scores among fourth grade students, including gifted and high-achieving students? The purpose of the quasi-experimental study was to explore the relationship of cooperative learning versus traditional learning on their student achievement. The null hypothesis was that cooperative learning would have no effect on fourth grade gifted …

Teaching Calculus With Wolfram Alpha, Andrew Lang

College of Science and Engineering Faculty Research and Scholarship

This article describes the benefits and drawbacks of using Wolfram|Alpha as the platform for teaching calculus concepts in the lab setting. It is a result of our experiences designing and creating an entirely new set of labs using Wolfram|Alpha. We present the reasoning behind our transition from using a standard computer algebra system (CAS) to Wolfram|Alpha in our differential and integral calculus labs, together with the positive results from our experience. We also discuss the current limitations of Wolfram|Alpha, including a discussion on why we still use a CAS for our multivariate calculus labs.

Schrödinger Dispersive Dstimates For A Dcaling-Critical Class Of Potentials, Marius Beceanu, Michael Goldberg

Mathematics and Statistics Faculty Scholarship

Consider the focussing cubic nonlinear Schr\"odinger equation in R 3 :

iψ t +Δψ=−|ψ| 2 ψ.

It admits special solutions of the form e itα ϕ , whereϕ is a Schwartz function and a positive (ϕ>0 ) solution of

−Δϕ+αϕ=ϕ 3 .

The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the eight-dimensional manifold that consists of functions of the form e i(v⋅+Γ) ϕ(⋅−y,α) . We prove that any solution starting sufficiently close to a standing wave in the Σ=W 1,2 (R 3 …

Review Of: Pearls Of Discrete Mathematics By Martin Erickson, Robert A. Beezer

All Faculty Scholarship

This article reviews the book "Pearls of Discrete Mathematics," by Martin Erickson.

An Explicit Super‐Time‐Stepping Scheme For Non‐Symmetric Parabolic Problems, Stephen O'Sullivan, Katharine Gurski

Conference papers

Explicit numerical methods for the solution of a system of differential equations may suffer from a time step size that approaches zero in order to satisfy stability conditions. When the differential equations are dominated by a skew-symmetric component, the problem is that the real eigenvalues are dominated by imaginary eigenvalues. We compare results for stable time step limits for the super-time-stepping method of Alexiades, Amiez, and Gremaud (super-time-stepping methods belong to the Runge-Kutta-Chebyshev class) and a new method modeled on a predictor-corrector scheme with multiplicative operator splitting. This new explicit method increases stability of the original super-time-stepping whenever the skew-symmetric …

Is Competition Making A Comeback? Discovering Methods To Keep Female Adolescents Engaged In Stem: A Phenomenological Approach, Kathryn B. Notter

College of Education and Human Sciences: Dissertations, Theses, and Student Research

The decreasing number of women who are graduating in the Science, Technology, Engineering and Mathematics (STEM) fields continues to be a major concern. Despite national support in the form of grants provided by National Science Foundation, National Center for Information and Technology and legislation passed such as the Deficit Reduction Act of 2005 that encourages women to enter the STEM fields, the number of women actually graduating in these fields is surprisingly low. This research study focuses on a robotics competition and its ability to engage female adolescents in STEM curricula. Data have been collected to help explain why young …

Perpendicular Ion Heating By Low-Frequency Alfvén-Wave Turbulence In The Solar Wind, Benjamin D. G. Chandran, Bo Li, Barrett N. Rogers, Eliot Quataert, Kai Germaschewski

Dartmouth Scholarship

We consider ion heating by turbulent Alfvén waves (AWs) and kinetic Alfvén waves (KAWs) with wavelengths (measured perpendicular to the magnetic field) that are comparable to the ion gyroradius and frequencies ω smaller than the ion cyclotron frequency Ω. We focus on plasmas in which β < 1, where β is the ratio of plasma pressure to magnetic pressure. As in previous studies, we find that when the turbulence amplitude exceeds a certain threshold, an ion's orbit becomes chaotic. The ion then interacts stochastically with the time-varying electrostatic potential, and the ion's energy undergoes a random walk. Using phenomenological arguments, we derive an analytic expression for the rates at which different ion species are heated, which we test by simulating test particles interacting with a spectrum of randomly phased AWs and KAWs. We find that the stochastic heating rate depends sensitively on the quantity ε = δv _ρ/v _⊥, where v _⊥ (v _∥) is the component of the ion velocity perpendicular (parallel) to the background magnetic field B ₀, and δv _ρ (δB _ρ) is the rms amplitude of the velocity (magnetic-field) fluctuations at the gyroradius scale. In the case …

Interview With Chad Topaz, Professor Of Mathematics, Chad Topaz

Math, Stats, and Computer Science Department Oral Histories

No abstract provided.

Noncommutative Topology And The World’S Simplest Index Theorem, Erik Van Erp

Dartmouth Scholarship

In this article we outline an approach to index theory on the basis of methods of noncommutative topology. We start with an explicit index theorem for second-order differential operators on 3-manifolds that are Fredholm but not elliptic. This low-brow index formula is expressed in terms of winding numbers. We then proceed to show how it is derived as a special case of an index theorem for hypoelliptic operators on contact manifolds. Finally, we discuss the noncommutative topology that is employed in the proof of this theorem. The article is intended to illustrate that noncommutative topology can be a powerful tool …

A Statistical Analysis Of Defined Benefit, Defined Contribution, And Hybrid Plans, Katie Heeder

Honors Projects in Mathematics

The purpose of this study is to compare three major types of employer sponsored retirement plans, Defined Benefit (DB), Defined Contribution (DC), and hybrid, and their impact on the employee. Employee careers are simulated to understand the employee’s advantages and disadvantages of each type of plan, especially in the state of an economic depression. The study uses actuarial assumptions and the simulation varies a number of quantities to better understand the impact of employee savings. The variables which are simulated at different levels are: service start age, retirement age, current compensation, salary increase rate, rate of return on market investments, …

Σary, Minnesota State University Moorhead, Mathematics Department

Math Department Newsletters

No abstract provided.

Rate Of Change, Elizabeth Schofield

Interface Compendium of Student Work

This piece was created for an introductory course in Differential Equations. The goal of the work is to visually represent the shared context and commonalities of applications of ordinary differential equations. The piece was created with colored pencil on paper. Applications displayed include flow rate of a fluid, a mass-spring system, and a pendulum.

Ordinary Differential Equations, Taylor Mcadam

Interface Compendium of Student Work

This painting was the final project for Math 13 -- Introduction to Differential Equations. The aim of the final project was to display the knowledge we had gained throughout the course in a creative way. My painting displays plots of solutions to a variety of specific differential equations we studied in the course, as well as some more general tools we learned for solving ODEs (such as the Wronskian in the background). The piece is acrylic on canvas.

Neutrosophic Physics: More Problems, More Solutions, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

When considering the laws of theoretical physics, one of the physicists says that these laws – the actual expressions of the laws of mathematics and logics being applied to physical phenomena – should be limited according to the physical meaning we attribute to the phenomena. In other word, there is an opinion that a theoretical physicist should put some limitations onto mathematics, in order to “reduce” it to the observed reality. No doubt, we can do it. However, if following this way, we would arrive at only mathematical models of already known physical phenomena. Of course, this might be useful …

Unbreakable Codes: Rsa Public Key Cryptosystem, Razia Amzad

Honors College Theses

The purpose of this paper is to comprehend the evolution of codes and ciphers; along with understanding how to encode and decode a message using RSA coding. In this paper "Unbreakable Codes" we will highlight the historical advances of communicating secure messages by illustrating the process of RSA coding with an example.

The Life Of Evariste Galois And His Theory Of Field Extension, Felicia N. Adams

Senior Honors Theses

Evariste Galois made many important mathematical discoveries in his short lifetime, yet perhaps the most important are his studies in the realm of field extensions. Through his discoveries in field extensions, Galois determined the solvability of polynomials. Namely, given a polynomial P with coefficients is in the field F and such that the equation P(x) = 0 has no solution, one can extend F into a field L with α in L, such that P(α) = 0. Whereas Galois Theory has numerous practical applications, this thesis will conclude with the examination and proof of the fact that it is impossible …

Determining The Success Of Ncaa Basketball Teams Through Team Characteristics, Raymond Witkos

Honors Projects in Mathematics

Every year much of the nation becomes engulfed in the NCAA basketball postseason tournament more affectionately known as “March Madness.” The tournament has received the name because of the ability for any team to win a single game and advance to the next round. The purpose of this study is to determine whether concrete statistical measures can be used to predict the final outcome of the tournament. The data collected in the study include 13 independent variables ranging from the 2003-2004 season up until the current 2009-2010 season. Different tests were run in an attempt to achieve the most accurate …

Predictive Modeling Of Alumni Donor Behavior, Lauren Prue

Honors Projects in Mathematics

In recent years, college and universities have relied increasingly upon the charitable contributions of its previous graduates; as the costs of tuition rise substantially, development offices are facing the challenge of creating annual fund campaigns that are minimally expensive while providing the maximum potential for return. This study addresses the available constituent database at one University in particular in an effort to identify what criteria are the strongest predictors of donor response at a small, private university located within New England. The analysis utilized predictive modeling and data-mining largely within the software program Rapid Insight to build several models in …

Mathematics Discipline Assessment Report 2009/2010, Mathematics Discipline

Assessment of Student Learning Reports

No abstract provided.

Inter Spem Et Metum, Fiat Lux, Michael A. Mota

Honors Projects

Explores the design and development of a simple, 3D flight simulator. The resulting application allows users to pilot an abstract human avatar and to create free-hand strokes and physically-based explosions onto the environment through a ball discharge meta-game feature. Uses the C++ language, and the ancillary programming API libraries, OpenGL, GLEW, and Win32.