Physical Sciences and Mathematics Commons^™

Cosamp: Iterative Signal Recovery From Incomplete And Inaccurate Samples, Deanna Needell, J. A. Tropp

CMC Faculty Publications and Research

Compressive sampling offers a new paradigm for acquiring signals that are compressible with respect to an orthonormal basis. The major algorithmic challenge in compressive sampling is to approximate a compressible signal from noisy samples. This paper describes a new iterative recovery algorithm calledCoSaMP that delivers the same guarantees as the best optimization-based approaches. Moreover, this algorithm offers rigorous bounds on computational cost and storage. It is likely to be extremely efficient for practical problems because it requires only matrix–vector multiplies with the sampling matrix. For compressible signals, the running time is just O(Nlog²N), where N is the length …

Lyapunov Exponents And Invariant Manifold For Random Dynamical Systems In A Banach Space, Zeng Lian

Theses and Dissertations

We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. Then, we use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.

The Tropical Jacobian Of A Tropical Elliptic Curve Is S^1(Q), Darryl Gene Wade

Theses and Dissertations

We establish consistent definitions for divisors, principal divisors, and Jacobians of a tropical elliptic curve and show that for a tropical elliptic cubic C , the associated Jacobian (or zero divisor class group) is the group S^1(Q).

Complete Tropical Bezout's Theorem And Intersection Theory In The Tropical Projective Plane, Gretchen Rimmasch

Theses and Dissertations

In this dissertation we prove a version of the tropical Bezout's theorem which is applicable to all tropical projective plane curves. There is a version of tropical Bezout's theorem presented in other works which applies in special cases, but we provide a proof of the theorem for all tropical projective plane curves. We provide several different definitions of intersection multiplicity and show that they all agree. Finally, we will use a tropical resultant to determine the intersection multiplicity of points of intersection at infinite distance. Using these new definitions of intersection multiplicity we prove the complete tropical Bezout's theorem.

Rational Schur Rings Over Abelian Groups, Brent L. Kerby

Theses and Dissertations

In 1993, Muzychuk showed that the rational S-rings over a cyclic group Z_n are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Z_n. This idea is easily extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational S-rings over G in a natural way. Our main result is that any finite group may be represented as the automorphism group of such a rational S-ring over an abelian p-group. In order to show this, we first …

A Universal Theory Of Decoding And Pseudocodewords, Nathan Axvig, Deanna Dreher, Katherine Morrison, Eric T. Psota, Lance C. Pérez, Judy L. Walker

Department of Mathematics: Faculty Publications

The discovery of turbo codes [5] and the subsequent rediscovery of low-density parity-check (LDPC) codes [9, 18] represent a major milestone in the field of coding theory. These two classes of codes can achieve realistic bit error rates, between 10−5 and 10−12, with signalto- noise ratios that are only slightly above the minimum possible for a given channel and code rate established by Shannon’s original capacity theorems. In this sense, these codes are said to be near-capacity-achieving codes and are sometimes considered to have solved (in the engineering sense, at least) the coding problem for the additive white Gaussian noise …

Homomorphisms Into The Fundamental Group Of One-Dimensional And Planar Peano Continua, Curtis Andrew Kent

Theses and Dissertations

Let X be a planar or one-dimensional Peano continuum. Let E be a Hawaiian Earring with fundamental group H. We show that every homomorphism from H to the fundamental group of X is conjugate to a homomorphism which is induced by a continuous function.

Parts Of The Whole: Teachers And The Work They Do, Dorothy Wallace

Numeracy

This introductory column argues the need to study our system of education with the same care we would give any complex system, pointing out some ways in which the educational system functions differently from other managed systems, and concluding that a careful study of the forces shaping education will lead to insight into what makes educational change successful.

Review Of Quantitative Reasoning: Tools For Today's Informed Citizen By A. Sevilla And K. Somers, Aaron G. Montgomery

Numeracy

Alicia Sevilla and Kay Somers, Quantitative Reasoning: Tools for Today’s Informed Citizen. (Emeryville CA: Key College Publishing, 2007). 626 pp. Softcover with Student CD. $79.95 (USA) ISBN 1-931914-90-1. http://www.keycollege.com/catalog/titles/quantitative_reasoning.html

From charts and graphs (Topic 1) to decision making (Topic 21), Quantitative Reasoning offers a good selection of topics that students in a general education mathematics course and other individuals in our society should understand. Organizing the 21 chapters (topics) under the headings of numerical reasoning, logical reasoning, and statistical reasoning, Sevilla and Somers focus their chapters on examples and devote the last 224 pages to 21 Excel activities. Parallel graphic-calculator …

Word Problems: Reflections On Embedding Quantitative Literacy In A Calculus Course, Gizem Karaali

Numeracy

Even though Quantitative Literacy (QL) programs are currently being developed and implemented in several colleges and universities, most schools still depend on traditional mathematics courses to reach their quantitative literacy goals. This note is a case study of how a traditional mathematics course sequence intended for students majoring in social and life sciences may be modified and adapted to at least partially fulfill the need in the absence of a stand-alone QL program. In particular, we focus on a freshman-level mathematics course sequence that serves various client departments. This sequence covers the traditional content of a first-year calculus sequence along …

False Positives And Referral Bias: Content For A Quantitative Literacy Course, Stuart Boersma, Teri Willard

Numeracy

An extended study of accuracy in medical screening is presented as a useful application to increase students’ quantitative reasoning skills. Two detailed examples are presented. The first explores the frequency of obtaining false positive results from a medical screening tool while the second examines the issue of referral bias and its effect on the apparent sensitivity and specificity of the screening tool. Results from student assessments indicate that the activity increases one’s ability to define terms such as “false positive” and “false negative” and increases one’s ability to read and compute with information obtained from a two-way table. Teacher assessment …

Using Local Data To Advance Quantitative Literacy, Stephen Sweet, Susanne Morgan, Danette Ifert Johnson

Numeracy

In this article we consider the application of local data as a means of advancing quantitative literacy. We illustrate the use of three different sources of local data: institutional data, Census data, and the National College Health Assessment survey. Our learning modules are applied in courses in sociology and communication, but the strategy of using local data can be integrated beyond these disciplinary boundaries. We demonstrate how these data can be used to stimulate student interests in class discussion, advance analytic skills, as well as develop capacities in written and verbal communication. We conclude by considering concerns that may influence …

Quantitative Literacy Across The Curriculum: A Case Study, Benjamin Steele, Semra Kiliç-Bahi

Numeracy

We describe a quantitative literacy (QL) program at Colby-Sawyer College, a small, residential, liberal arts college in New Hampshire. This program has grown rapidly from a traditional math curriculum to a college-wide understanding of quantitative literacy and voluntary participation by many faculty members in all departments. More than 80% of the faculty agreed that it would be useful for students to be able to use quantitative skills in their courses, but only 24 % thought students were capable of doing very well in mathematics. Twenty-three faculty members attended a summer workshop, funded by NSF, DUE # 0633133, in which they …

Quantitative Literacy On The Web Of Science, 1: The Bibliography And Its Role In The History Of This Journal, H L. Vacher, Todd Chavez

Numeracy

Prior to deciding to propose in 2006 that the National Numeracy Network (NNN) publish a new journal for quantitative literacy with their support, the University of South Florida Libraries investigated the publication environment of the field on the Web of Science®. Reproducing part of that study in this paper, we present findings from topic searches (March 2008) for “numeracy,” “quantitative literacy,” and “statistical literacy.” These updated results include a combined bibliography of 338 peer-reviewed articles amongst 210 different journals, by 748 authors from 321 institutions in 25 countries, in a total of 87 subjects (34% of the subject classes in …

Abstracts For Numeracy, H L. Vacher

Numeracy

No abstract provided.

Incoherence Of A Concept Image And Erroneous Conclusions In The Case Of Differentiability, Antti Viholainen

The Mathematics Enthusiast

The level of the coherence of a concept image conveys how well the cognitive structure concerning the concept is organized. This study considers the relationship between deficiencies in the coherence of the concept image and erroneous conclusions in the case of differentiability. The study is based on an interview where the student made conclusions contradictory to the formal theory of mathematics. He used an erroneous method to study the differentiability of piecewise defined functions. This method became the key factor which maintained the internal coherence of the concept image. It made it possible to build a cognitive structure whose basis …

Main Points Of Objections To The "No Child Left Behind", Mike O'Lear, Bettina Dahl

The Mathematics Enthusiast

Adequate Yearly Progress (AYP):

--The guidelines fail to distinguish between successful schools and unsuccessful schools

--Between schools under-performing in just one area vs. schools likely to need a complete overhaul

--Does not give credit for student growth toward a high standard

--Need to establish AYP levels which make a distinction between struggling schools and those needing only limited assistance

The Political Context Of The National Mathematics Advisory Panel, Eric Rico Gutstein

The Mathematics Enthusiast

The National Mathematics Advisory Panel needs to be situated in its broader political context to more fully understand it. Who created it, for what purpose, and who will (and will not) benefit from it are key questions I address in this article. My argument is that the NMAP, as part of a larger initiative undertaken by the Bush Administration and US financial/corporate elites, serves capital’s efforts to shore up the US’s weakening economic global position and does not benefit the majority of the US people—particularly marginalized and excluded students of color and low-income students.

Associative Operations On A Three-Element Set, Friðrik Diego, Kristín Halla Jónsdóttir

The Mathematics Enthusiast

A set with a binary operation is a fundamental concept in algebra and one of the most fundamental properties of a binary operation is associativity. In this paper the authors discuss binary operations on a three-element set and show, by an inclusion-exclusion argument, that exactly 113 operations out of the 19,683 existing operations on the set are associative. Moreover these 113 associative operations are accounted for by means of their operation tables.

Nonbinary Quantum Error-Correcting Codes From Algebraic Curves, Jon-Lark Kim, Judy L. Walker

Department of Mathematics: Faculty Publications

We give a generalized CSS construction for nonbinary quantum error-correcting codes. Using this we construct nonbinary quantum stabilizer codes from algebraic curves. We also give asymptotically good nonbinary quantum codes from a Garcia- Stichtenoth tower of function fields which are constructible in polynomial time.

Binary quantum error-correcting codes have been constructed in several ways. One interesting construction uses algebraic-geometry codes [2], [6], [7], [12], with the main idea being to apply the binary CSS construction [4], [5], [16] to the asymptotically good algebraic-geometry codes arising from the Garcia-Stichtenoth [11] tower of function fields over F_q2 (where q is a …

Matlab Report, Nancy Hale

Cornerstone 3 Reports : Interdisciplinary Informatics

The objective of this effort is to create a multidisciplinary approach to problem solving using technology. Through Thinkfinity, the Seidenberg School of Computer Science and Information Systems is collaborating with the science and quantitative faculty to create lessons that can be used by high school and first -year college students to work interactively with data to solve problems. MatLab is the tool of choice. It gives the user the ability to create visual models of large data sets and is used in the science and finance areas.

The key features of MatLab:

• Interactive tools for iterative exploration, design, and …

Confluent Mappings And Arc Kelley Continua, W. J. Charatonik, Janusz R. Prajs, J. J. Charatonik

Mathematics and Statistics Faculty Research & Creative Works

A Kelley continuum X, also called a continuum with the property of Kelley, such that, for each p X, each subcontinuum K containing p is approximated by arc-wise connected continua containing p, is called an arc Kelley continuum. A continuum homeomorphic to the inverse limit of locally connected continua with confluent bonding maps is said to be confluently LC-representable. The main subject of the paper is a study of deep connections between the arc Kelley continua and confluent mappings. It is shown that if a continuum X admits, for each ε > 0, a confluent ε-mapping onto a(n) (arc) Kelley continuum, …

Boundary Data Smoothness For Solutions Of Nonlocal Boundary Value Problems For Nth Order Differential Equations, Johnny Henderson, Britney Hopkins, Eugenie Kim, Jeffrey W. Lyons

Mathematics Faculty Articles

Under certain conditions, solutions of the boundary value problem y(n)=f(x,y,y′,…,y(n−1)), y(n)=f(x,y,y′,…,y(n−1)), y(i−1)(x1)=yiy(i−1)(x1)=yi for 1≤i≤n−11≤i≤n−1, and y(x2)−∑mi=1riy(ηi)=yny(x2)−∑i=1mriy(ηi)=yn, are differentiated with respect to boundary conditions, where a

Necessary Conditions For Nonsmooth Optimization Problems With Operator Constraints In Metric Spaces, Boris S. Mordukhovich, Libin Mou

Mathematics Research Reports

This paper concerns nonsmooth optimization problems involving operator constraints given by mappings on complete metric spaces with values in nonconvcx subsets of Banach spaces. We derive general first-order necessary optimality conditions for such problems expressed via certain constructions of generalized derivatives for mappings on metric spaces and axiomatically defined subdifferentials for the distance function to nonconvex sets in Banach spaces. Our proofs arc based on variational principles and perturbation/approximation techniques of modern variational analysis. The general necessary conditions obtained are specified in the case of optimization problems with operator constraints dDScribcd by mappings taking values in approximately convex subsets of …

Mathematics And The World: What Do Teachers Recognize As Mathematics In Real World Practice?, Barbara Garii, Lillian Okumu

The Mathematics Enthusiast

Elementary school teachers are encouraged to better integrate appropriate mathematics pedagogy with deeper, more relevant mathematics content. However, many teach a mathematics they do not fully understand to students who see, recognize, and use less mathematics than ever before. Both teachers and students struggle to articulate the role mathematics plays in society as mathematics becomes more embedded into our technology. In this study, we asked teachers to record the mathematics they used on a daily basis during a 1-week period. Their responses indicate that they do not recognize that mathematics plays any important role in technological and professional practices. This …

E(Race)Ing Race From A National Conversation On Mathematics Teaching And Learning: The National Mathematics Advisory Panel As White Institutional Space, Danny Bernard Martin

The Mathematics Enthusiast

In this paper, I wish to argue that several factors support a characterization of the National Mathematics Advisory Panel as an instantiation of the white institutional space (Moore, 2008) that characterizes mathematics education research and policy contexts more generally. In particular, my analysis highlights that mathematics education research and policy contexts such as the National Mathematics Advisory Panel are not immune to the structural and institutional racism that characterize many other areas of U.S. society.

The Brachistochrone Problem: Mathematics For A Broad Audience Via A Large Context Problem, Jeff Babb, James Currie

The Mathematics Enthusiast

Large context problems (LCP) are useful in teaching the history of science. In this article we consider the brachistochrone problem in a context stretching from Euclid through the Bernoullis. We highlight a variety of results understandable by students without a background in analytic geometry. By a judicious choice of methods and themes, large parts of the history of calculus can be made accessible to students in Humanities or Education.

Comparison Of Geometric Figures, Spyros Glenis

The Mathematics Enthusiast

Although the geometric equality of figures has already been studied thoroughly, little work has been done about the comparison of unequal figures. We are used to compare only similar figures but would it be meaningful to compare non similar ones? In this paper we attempt to build a context where it is possible to compare even non similar figures. Adopting Klein’s view for the Euclidean Geometry, we defined a relation “≤ ” as: S1 ≤ S2 whenever there is a Euclidean isometry f :R² →R² , so that . This relation is not an order because there are …

How Can Science History Contribute To The Development Of New Proposals In The Teaching Of The Notion Of Derivatives?, Arnaud Mayrargue

The Mathematics Enthusiast

The 18th century was a milestone for the incorporation of mathematics into physics. By this time already seen in Newton’s work, we know that a great deal of progress had found the light of day concerning the relationship between physics and mathematics, particularly as the latter began to deal with universal gravitation and optics. Based on his theory of monochromatic light, which used mathematics to describe optical phenomena, Newton became interested in the behavior of the various colors which, according to him, compose white light. Furthermore, toward the end of the 17th century, scientists had a remarkable mathematical tool at …

History Of Mathematics In Mathematics Education: A Saussurean Perspective, Michael Fried

The Mathematics Enthusiast

It is not only because of a certain eclecticism in mathematics education research that semiotic ideas have begun to take root there: it is also because of the dawning recognition that key areas of interest in mathematics education genuinely have a semiotic nature. For this, one need only point to research focused on meaning, communication, language, and culture (e.g., Presmeg, 1997; Ernest,1997; Radford, 2001; Brown, 2001). The ways in which semiotics informs cultural aspects of mathematics education, particularly those connected with the history of mathematics, were highlighted in a discussion session at the 2003 meeting of the International Group for …