Mathematics Commons^™

A Math Major, Polya, Invention, And Discovery, Susan D'Agostino

Journal of Humanistic Mathematics

An assistant professor of mathematics presents a nonmathematical application of George Polya’s problem-solving strategies. In doing so, she suggests that Polya’s ideas concerning invention and discovery apply to the world beyond the math classroom.

Pedagogy On The Ethnomathematics--Epistemology Nexus: A Manifesto, Ilhan M. Izmirli

Journal of Humanistic Mathematics

In this paper, we will elaborate on a pronouncement that should be at the onset of any study in epistemology and ethnomathematics, namely, we will argue that learners do think mathematically and it is our responsibility as educators to recognize and appreciate their modes of mathematical reasoning.

We will conduct our study in five parts. Following a brief introduction, in the second part, we will briefly discuss some of the critical tenets of epistemology especially as it applies to mathematics. The third part will be devoted to elucidating the basic nomenclature and hypotheses associated with ethnomathematics. In the fourth part …

The Elliptic Curve Discrete Logarithm And Functional Graphs, Christopher J. Evans

Mathematical Sciences Technical Reports (MSTR)

The discrete logarithm problem, and its adaptation to elliptic curves, called the elliptic curve discrete logarithm problem (ECDLP) is an open problem in the field of number theory, and its applications to modern cryptographic algorithms are numerous. This paper focuses on a statistical analysis of a modification to the ECDLP, called the x-ECDLP, where one is only given the xcoordinate of a point, instead of the entire point. Focusing only on elliptic curves whose field of definition is smaller than the number of points, this paper attempts to find a statistical indication of underlying structure (or lack thereof) in the …

Structure And Randomness Of The Discrete Lambert Map, Jingjing Chen, Mark Lotts

Mathematical Sciences Technical Reports (MSTR)

We investigate the structure and cryptographic applications of the Discrete Lambert Map (DLM). The mapping is closely related to the Discrete Log Problem, but has received far less attention since it is considered to be a more complicated map that is likely even harder to invert. However, this mapping is quite important because it underlies the security of the ElGamal Digital Signature Scheme. Using functional graphs induced by this mapping, we were able to find non-random properties that could potentially be used to exploit the ElGamal DSS.

The Square Discrete Exponentiation Map, A Wood

Mathematical Sciences Technical Reports (MSTR)

We will examine the square discrete exponentiation map and its properties. The square discrete exponentiation map is a variation on a commonly seen problem in cryptographic algorithms. This paper focuses on understanding the underlying structure of the functional graphs generated by this map. Specifically, this paper focuses on explaining the in-degree of graphs of safe primes, which are primes of the form p = 2q + 1, where q is also prime.

Operation Comics: The Story Continues, Bruce Kessler, Tressa Tullis

Mathematics Faculty Publications

This talk was given, with Tressa Tullis as the main presenter and Bruce Kessler as a minor co-presenter, at the 2011 Bridges Conference in Coimbre, Portugal, on the current developments on our Operation Comics project with Cumberland Trace Elementary.

Distributed Algorithms For Initialization And Topology Control In Wireless Ad Hoc Networks., Subhasis Bhattacharjee Dr.

Doctoral Theses

Wireless ad hoc networking is an upcoming communication technology that makes exchange of information possible without any pre-existing infrastructure. Over the last decade it has grabbed tremendous interest in the research community due to its easy deployability and high flexibility, with numerous applications to social, industrial and personal uses. In this thesis, we designed effcient light weight distributed algorithms based on minimal local information to resolve the problems related to the initialization and topology configuration of wireless ad hoc networks with special emphasis on optimal utilization of limited resources. Once the ad hoc nodes with in-built radio transceivers are deployed …

A Parallel Robin-Robin Domain Decomposition Method For The Stokes-Darcy System, Wenbin Chen, Max Gunzburger, Fei Hua, Xiaoming Wang

Mathematics and Statistics Faculty Research & Creative Works

We propose a new parallel Robin-Robin domain decomposition method for the coupled Stokes-Darcy system with Beavers-Joseph-Saffman-Jones interface boundary condition. in particular, we prove that, with an appropriate choice of parameters, the scheme converges geometrically independent of the mesh size. © 2011 Society for Industrial and Applied Mathematics.

Flow-Induced Channel Formation In The Cytoplasm Of Motile Cells, Robert D. Guy, Toshiyuki Nakagaki, Grady Wright

Mathematics Faculty Publications and Presentations

A model is presented to explain the development of flow channels within the cytoplasm of the plasmodium of the giant amoeba Physarum polycephalum. The formation of channels is related to the development of a self-organizing tubular network in large cells. Experiments indicate that the flow of cytoplasm is involved in the development and organization of these networks, and the mathematical model proposed here is motivated by recent experiments involving the observation of development of flow channel in small cells. A model of pressure-driven flow through a polymer network is presented in which the rate of flow increases the rate …

Covariant Representations Of C*-Dynamical Systems Involving Compact Groups, Firuz Kamalov

Department of Mathematics: Dissertations, Theses, and Student Research

Given a C*-dynamical system (A, G, σ) the crossed product C*-algebra A x _σG encodes the action of G on A. By the universal property of A x _σG there exists a one to one correspondence between the set all covariant representations of the system (A, G, σ) and the set of all *-representations of A x _σG. Therefore, the study of representations of A x _σG is equivalent to that of covariant representations of (A, G, σ).

We study induced covariant representations of systems involving compact groups. We prove that every irreducible (resp. factor) covariant …

On Khovanov-Seidel Quiver Algebras And Bordered Floer Homology, Denis Auroux, J. Elisenda Grigsby, Stephan M. Wehrli

Mathematics - All Scholarship

We discuss a relationship between Khovanov- and Heegaard Floer-type homology theories for braids. Explicitly, we define a filtration on the bordered Heegaard-Floer homology bimodule associated to the double-branched cover of a braid and show that its associated graded bimodule is equivalent to a similar bimodule defined by Khovanov and Seidel.

Crouzeix's Conjecture And The Gmres Algorithm, Sarah Mcbride Luo

Theses and Dissertations

This thesis explores the connection between Crouzeix's conjecture and the convergence of the GMRES algorithm. GMRES is a popular iterative method for solving linear systems and is one of the many Krylov methods. Despite its popularity, the convergence of GMRES is not completely understood. While the spectrum can in some cases be a good indicator of convergence, it has been shown that in general, the spectrum does not provide sufficient information to fully explain the behavior of GMRES iterations. Other sets associated with a matrix that can also help predict convergence are the pseudospectrum and the numerical range. This …

3d Image Reconstruction And Level Set Methods, Spencer R. Patty

Theses and Dissertations

We give a concise explication of the theory of level set methods for modeling motion of an interface as well as the numerical implementation of these methods. We then introduce the geometry of a camera and the mathematical models for 3D reconstruction with a few examples both simulated and from a real camera. We finally describe the model for 3D surface reconstruction from n-camera views using level set methods.

Stability For Traveling Waves, Joshua W. Lytle

Theses and Dissertations

In this work we present some of the general theory of shock waves and their stability properties. We examine the concepts of nonlinear stability and spectral stability, noting that for certain classes of equations the study of nonlinear stability is reduced to the analysis of the spectra of the linearized eigenvalue problem. A useful tool in the study of spectral stability is the Evans function, an analytic function whose zeros correspond to the eigenvalues of the linearized eigenvalue problem. We discuss techniques for numerical Evans function computation that ensure analyticity, allowing standard winding number arguments and rootfinding methods to be …

Quasisymmetric Graphs And Zygmund Functions, Leonid V. Kovalev, Jani Onninen

Mathematics - All Scholarship

A quasisymmetric graph is a curve whose projection onto a line is a quasisymmetric map. We show that this class of curves is related to solutions of the reduced Beltrami equation and to a generalization of the Zygmund class lambda. This relation makes it possible to use the tools of harmonic analysis to construct nontrivial examples of quasisymmetric graphs and of quasiconformal maps.

The Constrained Isoperimetric Problem, Minh Nhat Vo Do

Theses and Dissertations

Let X be a space and let S ⊂ X with a measure of set size |S| and boundary size |∂S|. Fix a set C ⊂ X called the constraining set. The constrained isoperimetric problem asks when we can find a subset S of C that maximizes the Følner ratio FR(S) = |S|/|∂S|. We consider different measures for subsets of R^2,R^3,Z^2,Z^3 and describe the properties that must be satisfied for sets S that maximize the Folner ratio. We give explicit examples.

Stone-Weierstrass Approximation Theorem, A Constructive Approach, Orhan Kaplan

Student Theses and Dissertations

The Weierstrass approximation theorem is well known in analysis. This theorem states that on a closed interval we can find a sequence of polynomials that comes closer and closer to any continuous real function. The importance of this theorem is that it is valid for both differentiable and non-differentible functions. In this paper we review and prove this theorem in one and higher dimensions by a constructive method and then give examples. An idea - different from the one used in Walter Rudin's book "Principles of Mathematical Analysis" - enables us to prove this theorem in one and higher dimensions …

An Algebra Isomorphism For The Landau-Ginzburg Mirror Symmetry Conjecture, Jared Drew Johnson

Theses and Dissertations

Landau-Ginzburg mirror symmetry takes place in the context of affine singularities in CN. Given such a singularity defined by a quasihomogeneous polynomial W and an appropriate group of symmetries G, one can construct the FJRW theory (see [3]). This construction fills the role of the A-model in a mirror symmetry proposal of Berglund and H ubsch [1]. The conjecture is that the A-model of W and G should match the B-model of a dual singularity and dual group (which we denote by WT and GT). The B-model construction is based on the Milnor ring, or local algebra, of the singularity. …

Maximal Unramified Extensions Of Cyclic Cubic Fields, Ka Lun Wong

Theses and Dissertations

Maximal unramified extensions of quadratic number fields have been well studied. This thesis focuses on maximal unramified extensions of cyclic cubic fields. We use the unconditional discriminant bounds of Moreno to determine cyclic cubic fields having no non-solvable unramified extensions. We also use a theorem of Roquette, developed from the method of Golod-Shafarevich, and some results by Cohen to construct cyclic cubic fields in which the unramified extension is of infinite degree.

Parts Of The Whole: An Algebra Lesson, Dorothy Wallace

Numeracy

This column draws on research of Eon Harper to demonstrate how an understanding of his proposed stages of algebra acquisition would inform a systemic overhaul of algebra education. Harper's stages also explain why students may pass a series of algebra courses yet still be unable to make sense of calculus, as well as offering insight on what aspects of algebra support quantitative literacy.

Reducing Math Anxiety: Findings From Incorporating Service Learning Into A Quantitative Reasoning Course At Seattle University, Allison Henrich, Kristi Lee

Numeracy

How might one teach mathematics to math-anxious students and at the same time reduce their math anxiety? This paper describes what we found when we incorporated a service learning component into a quantitative reasoning course at Seattle University in Fall 2010 (20 students) and Spring 2011 (28 students). The course is taken primarily by humanities majors, many of whom would not take a course in math if they didn’t need to satisfy the university’s core requirement. For the service learning component, each student met with and tutored children at local schools for 1-2 hours per week (total about 15 service …

Quantitative Reasoning In The Contemporary World, 3: Assessing Student Learning, Stuart Boersma, Caren Diefenderfer, Shannon W. Dingman, Bernard L. Madison

Numeracy

In this third paper in a series describing the Quantitative Reasoning in the Contemporary World course, the authors provide an adaptation of the Association of American Colleges and Universities quantitative literacy VALUE rubric. Describing achievement levels in six core competencies (interpretation, representation, calculation, analysis/synthesis, and communication), the resulting Quantitative Literacy Assessment Rubric (QLAR) is applicable to grading student work and has exhibited a high degree of reliability in two separate scoring tests (97% and 88% respectively). The distribution of the six core competencies across the 24 case studies in the authors’ quantitative reasoning casebook shows that interpretation, calculation, and analysis/synthesis …

Constructivist And Behaviorist Approaches: Development And Initial Evaluation Of A Teaching Practice Scale For Introductory Statistics At The College Level, Rossi A. Hassad

Numeracy

This study examined the teaching practices of 227 college instructors of introductory statistics from the health and behavioral sciences. Using primarily multidimensional scaling (MDS) techniques, a two-dimensional, 10-item teaching-practice scale, TISS (Teaching of Introductory Statistics Scale), was developed. The two dimensions (subscales) are characterized as constructivist and behaviorist; they are orthogonal. Criterion validity of the TISS was established in relation to instructors’ attitude toward teaching, and acceptable levels of reliability were obtained. A significantly higher level of behaviorist practice (less reform-oriented) was reported by instructors from the U.S., as well as instructors with academic degrees in mathematics and engineering, whereas …

Quantitative Literacy At Michigan State University, 2: Connection To Financial Literacy, Dennis Gilliland, Vince Melfi, Alla Sikorskii, Edward Corcoran, Eleanor Melfi

Numeracy

The lack of capability of making financial decisions has been recently described for the adult United States population. A concerted effort to increase awareness of this crisis, to improve education in quantitative and financial literacy, and to simplify financial decision-making processes is critical to the solution. This paper describes a study that was undertaken to explore the relationship between quantitative literacy and financial literacy for entering college freshmen. In summer 2010, incoming freshmen to Michigan State University were assessed. Well-tested financial literacy items and validated quantitative literacy assessment instruments were administered to 531 subjects. Logistic regression models were used to …

Quantitative Literacy At Michigan State University, 1: Development And Initial Evaluation Of The Assessment, Alla Sikorskii, Vince Melfi, Dennis Gilliland, Jennifer Kaplan, Suzie Ahn

Numeracy

Development, psychometric testing, and the results of the administration of a quantitative literacy (QL) assessment to undergraduate students are described. Three forms were developed covering a wide range of skills, contexts, and quantitative information presentation formats. Following item generation and revision based on preliminary testing and cognitive interviewing, a total of 3,701 consented undergraduate students at Michigan State University completed one of the three forms. Two of the forms contained 14 multiple-choice items, and one form contained 17 multiple-choice items. All forms were completed by students in less than 30 minutes. Evidence of validity and reliability were obtained for the …

Development Of An Assessment Of Quantitative Literacy For Miami University, Rose Marie Ward, Monica C. Schneider, James D. Kiper

Numeracy

Quantitative Literacy is a competence as important as general literacy; yet, while writing requirements are seemingly ubiquitous across the college curriculum, quantitative literacy requirements are not. The current project provides preliminary evidence of the reliability and validity of a quantitative literacy measure suitable for delivery online. A sample of 188 undergraduate students from Miami University, a midsize university in the midwestern U.S., participated in the current study. Scores on the measure, were inversely related to statistical/mathematical anxiety measures, directly related to subjective assessment of numeracy, and did not differ across gender or year in school. The resulting measure provides a …

Communicating Quantitative Literacy: An Examination Of Open-Ended Assessment Items In Timss, Nals, Ials, And Pisa, Karl W. Kosko, Jesse L. M. Wilkins

Numeracy

Quantitative Literacy (QL) has been described as the skill set an individual uses when interacting with the world in a quantitative manner. A necessary component of this interaction is communication. To this end, assessments of QL have included open-ended items as a means of including communicative aspects of QL. The present study sought to examine whether such open-ended items typically measured aspects of quantitative communication, as compared to mathematical communication, or mathematical skills. We focused on public-released items and rubrics from four of the most widely referenced assessments: the Third International Mathematics and Science Study (TIMSS-95): the National Adult Literacy …

A Quantitative Literacy View Of Natural Disasters And Nuclear Facilities, C. B. Connor

Numeracy

The March 11, 2011 earthquake, tsunami, and nuclear disaster in Tohoku, Japan, highlights the need to improve quantitative literacy (QL) in natural hazard assessment. A critical understanding of natural hazard assessments requires a sophisticated perspective on the mathematical and statistical tools used to estimate the odds of disaster, and the roles of data quality, model development, and subjective probability in estimation of uncertainty. Thus, improved QL is a basic requirement for improved decision-making about the safety of critical infrastructure, such as nuclear facilities.

A Leap Forward For Quantitative Literacy, H. L. Vacher

Numeracy

The Association of American College and Universities’ Learning Education and America’s Promise (LEAP) initiative has identified quantitative literacy (QL) as one of its Essential Learning Outcomes and classified it amongst five other Intellectual and Practical Skills such as inquiry and analysis, critical and creative thinking, and written and oral communication. This brings to mind a spreadsheet in which these transdisciplinary intellectual and practical skills are rows and academic disciplines are columns. With the view that the learning outcome QL is a row crossing mathematics and other disciplinary columns, this editorial considers how the papers in this and previous issues of …

2011 (Summer), University Of Dayton. Department Of Mathematics

Colloquia

Abstracts of the talks given at the 2011 Summer Colloquium.