Education Commons^™

Rademacher-Type Formulas For Restricted Partition And Overpartition Functions, Andrew Sills

Department of Mathematical Sciences Faculty Publications

A collection of Hardy-Ramanujan-Rademacher type formulas for restricted partition and overpartition functions is presented, framed by several biographical anecdotes.

Voices, Echoes, And Narratives: Multidimensional Experiences Of Three Teachers Immersed In Ethnomathematical Encounters In Morocco, Mekyah Q. Mcqueen, Stanley F. H. Shaheed, Curtis V. Goings, Iman C. Chahine

Middle-Secondary Education and Instructional Technology Faculty Publications

No abstract provided.

How Is It That One Particular Statement Appeared Rather Than Another?: Opening A Different Space For Different Statements About Urban Mathematics Education, David W. Stinson

Middle-Secondary Education and Instructional Technology Faculty Publications

In this editorial, the author applies Michel Foucault's concept of "discursive formations" to examine fictions, fantasies, and power relationships in mathematics education research.

The Cohomology Of Modules Over A Complete Intersection Ring, Jesse Burke

Department of Mathematics: Dissertations, Theses, and Student Research

We investigate the cohomology of modules over commutative complete intersection rings. The first main result is that if M is an arbitrary module over a complete intersection ring R, and if one even self-extension module of M vanishes then M has finite projective dimension. The second main result gives a new proof of the fact that the support variety of a Cohen-Macaulay module whose completion is indecomposable is projectively connected.

Teaching Research: Encouraging Discoveries, Francis E. Su

All HMC Faculty Publications and Research

What does it take to turn a learner into a discoverer? Or to turn a teacher into a co-adventurer? A handful of experiences—from teaching a middle-school math class to doing research with undergraduates—have changed the way that I would answer these questions. Some of the lessons I’ve learned have surprised me.

Note On Gradient Estimates Of Heat Kernel For Schrödinger Operators, Shijun Zheng

Department of Mathematical Sciences Faculty Publications

Let H = -Δ+V be a Schrödinger operator on Rⁿ. We show that gradient estimates for the heat kernel of H with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for L^p and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists V in the Schwartz class such that the long time gradient heat …

Technology Integration In Secondary Mathematics Classrooms: Effect On Students’ Understanding, Megan Sheehan, Leah Nillas

Scholarly Publications

Technology use in secondary mathematics courses has the potential to bring about broad changes in learning environment and teaching pedagogy, allowing students to communicate and collaborate in new ways and to conjecture, justify, and generalize findings. However, this potential is only realized when teachers use technology in ways encouraging these outcomes (Galbraith, 2006). The purpose of this study is to examine the integration of technology in secondary mathematics classrooms and to evaluate the effectiveness of its use in relation to students’ learning outcomes. This self study research was conducted in honors geometry and AP calculus classes. Data sources included transcripts …

An Attempt To Get And Keep Women Involved In Physics, Jim Crumley, Kristen Nairn, Lynn Ziegler

MapCores Faculty Publications

In this talk I will briefly review some of the obstacles to the full participation of women in the STEM disciplines. In order to increase the number of women in physics, computer science, and mathematics we have started a cohort-based program with curricular and scholarship components for women in these majors. I will present the results of our program so far and offer some advice based on our experiences.

Mathematical Manipulative Models: In Defense Of "Beanbag Biology", John R. Jungck, Holly Gaff, Anton E. Weisstein

Biological Sciences Faculty Publications

Mathematical manipulative models have had a long history of influence in biological research and in secondary school education, but they are frequently neglected in undergraduate biology education. By linking mathematical manipulative models in a four-step process-1) use of physical manipulatives, 2) interactive exploration of computer simulations, 3) derivation of mathematical relationships from core principles, and 4) analysis of real data sets-we demonstrate a process that we have shared in biological faculty development workshops led by staff from the BioQUEST Curriculum Consortium over the past 24 yr. We built this approach based upon a broad survey of literature in mathematical educational …

"Beyond Bio2010: Celebration And Opportunities" At The Intersection Of Mathematics And Biology, John R. Jungck, Holly D. Gaff, Adam P. Fagen, Jay B. Labov

Biological Sciences Faculty Publications

With this special edition of CBE-LSE, which focuses on connections between and integration of the biological and mathematical sciences, it is especially fitting that we report on an important symposium, Beyond BIO2010: Celebration and Opportunities,1 which was held at the National Acad- emy of Sciences (NAS) in Washington, D.C. on May 21–22, 2010. This symposium was organized to assess what progress has been made in addressing the challenges and recommendations in the National Research Council’s (NRC) report: BIO2010: Transforming Undergraduate Education for Future Research Biologists (NRC, 2003a). Most of the presen- tations and posters at this event emphasized the increasing …

Impact Of Interdisciplinary Undergraduate Research In Mathematics And Biology On The Development Of A New Course Integrating Five Stem Disciplines, Lester Caudill, April L. Hill, Kathy Hoke, Ovidiu Z. Lipan

Biology Faculty Publications

Funded by innovative programs at the National Science Foundation and the Howard Hughes Medical Institute, University of Richmond faculty in biology, chemistry, mathematics, physics, and computer science teamed up to offer first- and second-year students the opportunity to contribute to vibrant, interdisciplinary research projects. The result was not only good science but also good science that motivated and informed course development. Here, we describe four recent undergraduate research projects involving students and faculty in biology, physics, mathematics, and computer science and how each contributed in significant ways to the conception and implementation of our new Integrated Quantitative Science course, a …

Impact Of Interdisciplinary Undergraduate Research In Mathematics And Biology On The Development Of A New Course Integrating Five Stem Disciplines, Lester Caudill, April L. Hill, Kathy Hoke, Ovidiu Z. Lipan

Department of Math & Statistics Faculty Publications

Funded by innovative programs at the National Science Foundation and the Howard Hughes Medical Institute, University of Richmond faculty in biology, chemistry, mathematics, physics, and computer science teamed up to offer first- and second-year students the opportunity to contribute to vibrant, interdisciplinary research projects. The result was not only good science but also good science that motivated and informed course development. Here, we describe four recent undergraduate research projects involving students and faculty in biology, physics, mathematics, and computer science and how each contributed in significant ways to the conception and implementation of our new Integrated Quantitative Science course, a …

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.

Applications Of Linear Programming To Coding Theory, Nathan Axvig

Department of Mathematics: Dissertations, Theses, and Student Research

Maximum-likelihood decoding is often the optimal decoding rule one can use, but it is very costly to implement in a general setting. Much effort has therefore been dedicated to find efficient decoding algorithms that either achieve or approximate the error-correcting performance of the maximum-likelihood decoder. This dissertation examines two approaches to this problem.

In 2003 Feldman and his collaborators defined the linear programming decoder, which operates by solving a linear programming relaxation of the maximum-likelihood decoding problem. As with many modern decoding algorithms, is possible for the linear programming decoder to output vectors that do not correspond to codewords; such …

Vanishing Of Ext And Tor Over Complete Intersections, Olgur Celikbas

Department of Mathematics: Dissertations, Theses, and Student Research

Let (R,m) be a local complete intersection, that is, a local ring whose m-adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of Tor(M, N) and Ext(M, N). In this context, M satisfies Serre's condition (S_{n}) if and only if M is an nth syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r-1 for all sufficiently large n. We use this notion of …

The Sixth International Mathematics Education And Society Conference: Finding Freedom In A Mathematics Education Ghetto, David W. Stinson

Middle-Secondary Education and Instructional Technology Faculty Publications

In this editorial, the author relates his experiences at the Sixth International Mathematics Education and Society Conference, held March 2010 in Berlin, Germany, and explores whether urban mathematics educators can navigate historically marginalized racial, ethnic, religious, cultural, gendered, sexual, intellectual, and other communities to find freedom in a what he sees as a "mathematics education ghetto."

The Nuts And Bolts: A Review Of Culturally Specific Pedagogy In The Mathematics Classroom: Strategies For Teachers And Students, Shonda Lemons-Smith

Middle-Secondary Education and Instructional Technology Faculty Publications

The author reviews Jacqueline Leonard's Culturally Specific Pedagogy in the Mathematics Classroom: Strategies for Teachers and Students.

Collaborative Evaluative Inquiry: A Model For Improving Mathematics Instruction In Urban Elementary Schools, Iman C. Chahine, Lesa M. Covington Clarkson

Middle-Secondary Education and Instructional Technology Faculty Publications

In this article, the authors describe the cyclical process of a collaborative evaluative inquiry project and the data collected throughout the project—data that not only informed "next steps" during the project but also show promise in documenting the benefits of such projects. Over a period of 18 months, seven elementary teachers from a K–6 urban elementary school collaborated with university personnel using Parsons’s (2002) Evaluative Inquiry Model, a 5-stage, cyclical model that includes defining, planning, and investigating challenges; collecting, analyzing, and synthesizing data; and communicating findings that transpire through collaborative inquiry. Overall, the project focused on improving the elementary teachers’ …

On The Characteristic Polynomial Of Regular Linear Matrix Pencil, Yan Wu, Phillip Lorren

Department of Mathematical Sciences Faculty Publications

Linear matrix pencil, denoted by (A,B), plays an important role in control systems and numerical linear algebra. The problem of finding the eigenvalues of (A,B) is often solved numerically by using the well-known QZ method. Another approach for exploring the eigenvalues of (A,B) is by way of its characteristic polynomial, P(λ)=A − λB. There are other applications of working directly with the characteristic polynomial, for instance, using Routh-Hurwitz analysis to count the stable roots of P(λ) and transfer function representation of control systems governed by differential-algebraic equations. In this paper, we …

A Unified Theory Of Function Spaces And Hyperspaces: Local Properties, Szymon Dolecki, Frédéric D. Mynard

Department of Mathematical Sciences Faculty Publications

Many classically used function space structures (including the topology of pointwise convergence, the compact-open topology, the Isbell topology and the continuous convergence) are induced by a hyperspace structure counterpart. This scheme is used to study local properties of function space structures on C(X,R), such as character, tighntess, fan-tightness, strong fan-tightness, the Fr{\'e}chet property and some of its variants. Under mild conditions, local properties of C(X,R) at the zero function correspond to the same property of the associated hyperspace structure at X. The latter is often easy to characterize in terms of covering properties …

Generalized Complex Hamiltonian Torus Actions: Examples And Constraints, Thomas Baird, Yi Lin

Department of Mathematical Sciences Faculty Publications

Consider an effective Hamiltonian torus action T×M→M on a topologically twisted,generalized complex manifold M of dimension 2n. We prove that the rank(T)≤n−2 and that the topological twisting survives Hamiltonian reduction. We then construct a large new class of such actions satisfying rank(T)=n−2, using a surgery procedure on toric manifolds.

Fejér Polynomials And Control Of Nonlinear Discrete Systems, Dmitriy Dmitrishin, Paul Hagelstein, Anna Khamitova, Anatolii Korenovskyi, Alexander M. Stokolos

Department of Mathematical Sciences Faculty Publications

We consider optimization problems associated to a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing T-cycles of a differentiable function f : R → R of the form x(k + 1) = f(x(k)) + u(k) where u(k) = (a₁−1)f(x(k))+a₂f(x(k−T))+· · ·+a_N f(x(k−(N −1)T)) , with a₁ + · · · + a_N = 1. Following an approach of Morgul, we associate to each periodic orbit of f, N ∈ N, and a₁, . . . …

Matching Functions And Graphs At Multiple Levels Of Bloom’S Revised Taxonomy, Kris H. Green

Mathematical and Computing Sciences Faculty/Staff Publications

This paper illustrates the power of Bloom's revised taxonomy for teaching, learning and assessing [3] in aligning our curriculum expectations and our assessment tools in multivariable calculus. The particular assessment tool considered involves a common matching problem to evaluate students' abilities to think about functions from graphical and formulaic representations. Through this analysis we gain additional understanding of why students may have difficulty in performing well on certain activities.

If Mowat And Davis Are Correct, Then Teaching Is Hard: A Response To Elizabeth Mowat & Brent Davis, Kris H. Green, Bernard P. Ricca

Mathematical and Computing Sciences Faculty/Staff Publications

In lieu of an abstract, below is the article's first paragraph.

Mowat & Davis (this issue) present a model of learning mathematics that relies heavily on ideas from network (or graph) theory. The important questions (to us, at least) concern the dynamics of the nodes and links. Answers – even tentative ones such as we present here – to these questions lead to a second set of questions concerning the implications of these answers to teachers and researchers.

Towards An Automation Of The Circle Method, Andrew Sills

Department of Mathematical Sciences Faculty Publications

The derivation of the Hardy-Ramanujan-Rademacher formula for the number of partitions of n is reviewed. Next, the steps for finding analogous formulas for certain restricted classes of partitions or overpartitions is examined, bearing in mind how these calculations can be automated in a CAS. Finally, a number of new formulas of this type which were conjectured with the aid of Mathematica are presented along with results of a test for their numerical accuracy.

Some Implications Of Chu's 10Ψ10 Generalization Of Bailey's 6Ψ6 Summation Formula, James Mclaughlin, Andrew Sills, Peter Zimmer

Department of Mathematical Sciences Faculty Publications

Lucy Slater used Bailey's 6ψ6 summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type.

In the present paper we apply the same techniques to Chu's 10ψ10 generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs.

In re-examining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find new …

Meaningful Distributed Instruction— Conceptual Previews For Symbolic Procedures, Edward C. Rathmell

Faculty Publications

Understanding a symbolic procedure means far more than “getting the right answer.” A mathematical symbolic procedure or written skill involves step-bystep thinking that leads from a computational problem to a solution. Memorizing this step-by-step procedure may enable a student to answer the problem, even answer it correctly. Yes, that is important, but understanding means much more.

Properties Of The Generalized Laplace Transform And Transport Partial Dynamic Equation On Time Scales, Chris R. Ahrendt

Department of Mathematics: Dissertations, Theses, and Student Research

In this dissertation, we first focus on the generalized Laplace transform on time scales. We prove several properties of the generalized exponential function which will allow us to explore some of the fundamental properties of the Laplace transform. We then give a description of the region in the complex plane for which the improper integral in the definition of the Laplace transform converges, and how this region is affected by the time scale in question. Conditions under which the Laplace transform of a power series can be computed term-by-term are given. We develop a formula for the Laplace transform for …

Rademacher-Type Formulas For Partitions And Overpartitions, Andrew Sills

Department of Mathematical Sciences Faculty Publications

A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of n and the Zuckerman formula for the Fourier coefficients of ϑ4_0 | τ_−1 is presented.