Physical Sciences and Mathematics Commons^™

Fan Cohomology And Its Application To Equivariant K-Theory Of Toric Varieties, Suanne Au

Department of Mathematics: Dissertations, Theses, and Student Research

Mu-Wan Huang, Mark Walker and I established an explicit formula for the equivariant K-groups of affine toric varieties. We also recovered a result due to Vezzosi and Vistoli, which expresses the equivariant K-groups of a smooth toric variety in terms of the K-groups of its maximal open affine toric subvarieties. This dissertation investigates the situation when the toric variety X is neither affine nor smooth. In many cases, we compute the Čech cohomology groups of the presheaf K_q^T on X endowed with a topology. Using these calculations and Walker's Localization Theorem for equivariant K-theory, we give explicit formulas …

Mathematical Beauty And Its Characteristics- A Study On The Student's Point Of View, Astrid Brinkmann

The Mathematics Enthusiast

Based on the statement, that the experience of mathematical beauty has a positive influence on students’ motivations and attitudes towards mathematics and its study, the focus of this paper is the aesthetic component of mathematics. First, the role of aesthetics for perception and education is addressed. The appreciation of the beauty of mathematics is one of the wellsprings of this subject, not only in research but also in school education. This should have implications for the teaching of mathematics. However the beauty making elements have not been very well analysed. In particular, it is not clear to what extent the …

The Contributions Of Comprehension Tests To Cognitive And Affective Development Of Prospective Teachers: A Case Study, Yüksel Dede

The Mathematics Enthusiast

The aim of this study was to investigate the role of comprehension tests while teaching algebra and its effects on students’ success. This study was carried out with 108 third year undergraduate students enrolled in math education in faculty of education. Several data collection instruments were used for gathering data from the participants such as; comprehension test, written documents, semi-structured interviews schedule, and participant and nonparticipant observations sheets. Collected data were subjected to content analysis and triangulation among the data was ensured. Results indicated that three different major categories emerged from the content analysis of the data: (1) measure of …

Cubism And The Fourth Dimension, Elijah Bodish

The Mathematics Enthusiast

When one looks into the subject of geometries that attempt to explain fourthdimensional space, it is inevitable that one encounters references to Cubism. The purpose of this paper is to find what the similarities between this mathematical concept and cubism are. There are many historical arguments as to how the cubists encountered literature about the fourth-dimension, and whether they were exposed to it at all, which I will for the most part omit and instead let the art speak for itself. It is important to see how two fields are interrelated in order to gain a better understanding of both …

If Mathematics Is A Language, How Do You Swear In It?, Dave Wagner

The Mathematics Enthusiast

Swears are words that are considered rude or offensive. Like most other words, they are arbitrary symbols that index meaning: there is nothing inherently wrong with the letters that spell a swear word, but strung together they conjure strong meaning. This reminds us that language has power. This is true in mathematics classrooms too, where language practices structure the way participants understand mathematics and where teachers and students can use language powerfully to shape their own mathematical experience and the experiences of others.

From Trapezoids To The Fundamental Theorem Of Calculus, William Gratzer, Srilal Krishnan

The Mathematics Enthusiast

The philosophy of Mathematics Education undergoes changes from the school to college level and students generally have a tough time coping with the transition. It is our endeavor to impress the importance of introducing college level topics at an early stage, so that students are not lost in the transition. Keeping this in mind, we suggest an early exposure to an important topic from Calculus; approximating the area of a planar region. Traditionally this topic is introduced using Riemann Sums but in this paper we try to follow a student’s natural inclination in approximating areas and explain how this approach …

Sum Of "N" Consecutive Integers, Steve Humble

The Mathematics Enthusiast

For all n , it is always possible to find at least one sum of n consecutive numbers with an equivalent sum of n - 1 consecutive numbers?

On The Use Of Realistic Fermi Problems For Introducing Mathematical Modelling In School, Jonas Bergman Ärlebäck

The Mathematics Enthusiast

In this paper uses an analytical tool refereed to as the MAD (Modelling Activity Diagram) framework adapted from Schoenfeld’s parsing protocol coding scheme to address the issues of how to introduce mathematical modelling to upper secondary students. The work of three groups of students engaged in solving so called realistic Fermi problems were analysed using this framework, and it was observed that the processes involved in a typical mathematical modelling cycle were richly represented in the groups’ solving processes. The importance of the social interactions within the groups was noted, as well as the extensive use of extramathematical knowledge used …

Harmonic Functions On R-Covered Foliations And Group Actions On The Circle, Sergio Fenley, Renato Feres, Kamlesh Parwani

Kamlesh Parwani

Let (M,F) be a compact codimension-one foliated manifold whose leaves are equipped with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of F. If every such function is constant on leaves we say that (M,F) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. Related results for R-covered foliations, as well as for discrete group actions and discrete harmonic functions, are also established.

Two Applications Of Art To Geometry, Viktor Blåsjö

The Mathematics Enthusiast

Geometry and art exploit the same source of human pleasure: the exercise of our spatial intuition. It is not surprising, then, that interconnections between them abound. Applications of geometry to art, of which we shall indicate a few, go back at least to Alberti’s De Pictura (1435). But although geometry started out, as it so often does, as a most courteous suitor in its relationship with art, it was soon to be affectionately rewarded. We shall study two of these rewards.

Mathematical Curiosities About Division Of Integers, Jérôme Proulx, Mary Beisiegel

The Mathematics Enthusiast

As mathematics educators, our focus of attention is mainly placed on the learning and teaching of mathematics. But, as we study phenomena of mathematical learning and teaching, we often come across intriguing mathematical phenomena that capture our interest. We find ourselves often bouncing mathematical ideas back and forth, not just looking for (new/better) ways of teaching or presenting a mathematical concept, but also of uncovering and discovering potential understandings of the concept. These mathematical issues we encounter represent for us a significant aspect of our work, and are also very stimulating. One of these issues arose for us as we …

Graph Isomorphisms And Matrix Similarity: Switching Between Representations, Thierry Dana-Picard

The Mathematics Enthusiast

A proof whether two graphs (possibly oriented graphs or multigraphs, etc.) are isomorphic or not can be derived by various methods. Some of them are reasonable for small numbers of vertices and/or edges, but not for larger numbers. Switching from iconic representation to a matrix representation transforms the problem of Graph Theory into a problem in Linear Algebra. The support provided by a Computer Algebra System is analyzed, in particular with regard to the building of new mathematical knowledge through a transition from graphical to algebraic representation. Moreover two important issues are discussed: a. the need for more than one …

Period-Doubling Bifurcation In An Array Of Coupled Stochastically-Excitable Elements Subjected To Global Periodic Forcing, Xiaohua Cui, Robert J. Rovetti, Ling Yang, Alan Garfinkel, James N. Weiss, Zhilin Qu

Mathematics, Statistics and Data Science Faculty Works

The collective behaviors of coupled, stochastically-excitable elements subjected to global periodic forcing are investigated numerically and analytically. We show that the whole system undergoes a period-doubling bifurcation as the driving period decreases, while the individual elements still exhibit random excitations. Using a mean-field representation, we show that this macroscopic bifurcation behavior is caused by interactions between the random excitation, the refractory period, and recruitment (spatial cooperativity) of the excitable elements.

Variational Theory Of Balance Systems, Serge Preston

Mathematics and Statistics Faculty Publications and Presentations

In this work we apply the Poincare-Cartan formalism of the Classical Field Theory to study the systems of balance equations (balance systems). We introduce the partial k-jet bundles of the configurational bundle and study their basic properties: partial Cartan structure, prolongation of vector fields, etc. A constitutive relation C of a balance system is realized as a mapping between a (partial) k-jet bundle and the extended dual bundle similar to the Legendre mapping of the Lagrangian Field Theory. Invariant (variational) form of the balance system corresponding to a constitutive relation C is studied. Special cases of balance systems -Lagrangian systems …

An Adaptive Method For Calculating Blow-Up Solutions, Charles F. Touron

Mathematics & Statistics Theses & Dissertations

Reactive-diffusive systems modeling physical phenomena in certain situations develop a singularity at a finite value of the independent variable referred to as "blow-up." The attempt to find the blow-up time analytically is most often impossible, thus requiring a numerical determination of the value. The numerical methods often use a priori knowledge of the blow-up solution such as monotonicity or self-similarity. For equations where such a priori knowledge is unavailable, ad hoc methods were constructed. The object of this research is to develop a simple and consistent approach to find numerically the blow-up solution without having a priori knowledge or resorting …

A Study Of Decision Analysis Methods In Aerospace Technology Assessments, Sharon Monica Jones

Engineering Management & Systems Engineering Theses & Dissertations

Managers of aerospace technology programs and projects are faced with the challenge of making technology portfolio decisions under conditions of limited data, rapidly changing macro level factors and organizational uncertainties. To help make these technology investment decisions, some aerospace managers and analysts have used techniques from the field of decision analysis. In addition, there have been a limited number of research studies of real decision problems.

This dissertation presents the results of a non-experimental examination of the use of decision analysis methods for the assessment of aerospace technology portfolios. A web-based survey instrument was developed based on the results of …

Parts Of The Whole: Approaching Education As A System, Dorothy Wallace

Numeracy

An educational system is a highly coupled complex system of inputs, outputs, sensors and actuators. Using an engineering perspective, this column begins the process of naming and categorizing parts of the system. It then focuses on teachers as one part of a large system, and analyzes the forces that influence how teachers work, and that draw or repel individuals to a teaching career. The growing shortage of qualified teachers can be explained by properties of the system as a whole that determine the context in which teachers do their job.

Review Of Super Crunchers By Ian Ayers, Eric Gaze

Numeracy

Ayers, I. Super Crunchers: Why Thinking-by-Numbers Is the New Way to be Smart. (Bantam Dell Publishing Group, 2007). 272 pp. Hard cover $25 ISBN 978-0-553-80540-6.

Super Crunchers tells the story of how analyzing data is changing the ways in which decisions are made. We in the National Numeracy Network make a case for the importance of quantitative literacy by referring to how much quantitative information is now available to each of us: “a world awash in numbers.” Ian Ayres zeroes in on the people who are making a living crunching all of these data. From the seemingly innocuous (how …

Review Of Stat-Spotting: A Field Guide To Identifying Dubious Data By Joel Best, Joe Swingle

Numeracy

Best, Joel. Stat-Spotting: A Field Guide to Identifying Dubious Data. (Berkeley: University of California Press, 2008) 144 pp. $19.95. ISBN 1-978-0-520-25746-7.

Stat-Spotting is a practical, do-it-yourself manual for detecting questionable claims reported in the media. Using examples drawn mostly from mass media sources, Stat-Spotting provides readers with a number of useful tips for identifying potentially problematic statistics. The author’s skillful analyses and explanations presented in clear and concise prose make Stat-Spotting an ideal guide for anyone who reads a newspaper, watches television, or surfs the Web. In short, everyone.

Review Of Calculation Vs. Context: Quantitative Literacy And Its Implications For Teacher Education By Bernard L. Madison And Lynn Arthur Steen (Editors), Maura B. Mast

Numeracy

Madison, Bernard L. and Steen, Lynn Arthur (Eds.). Calculation vs. Context: Quantitative Literacy and Its Implications for Teacher Education. (Washington, DC: Mathematical Association of America, 2009). 197 pp. Softcover. ISBN 978-0-88385-908-7. Available free on the MAA website at http://www.maa.org/ql/calcvscontext.html

The papers in Calculation vs. Context discuss the role of quantitative literacy in the K-12 curriculum and in teacher education. The papers present a varied set of perspectives and address three themes: the changing environment of education in American society; the challenges, and the necessity, of preparing teachers to teach quantitative literacy and of including quantitative literacy in the K-12 education; …

Engaging Math-Avoidant College Students, M. Paul Latiolais, Wendi Laurence

Numeracy

This paper is an informal, personal account of how we, as two college teachers, became interested in math anxiety, decided to explore it amongst students at our institution in order to inform our teaching, and became convinced that the massive problem is math avoidance. We tried discussion groups, but few students attended, although those that did made useful suggestions. Thus informed, we designed an innovative course, Confronting College Mathematics as a Humanities course with the possibility of credit toward the math requirement, but it was undersubscribed in its first offering and had to be canceled. How can we get college …

Measuring Resource Inequality: The Gini Coefficient, Michael T. Catalano, Tanya L. Leise, Thomas J. Pfaff

Numeracy

This paper stems from work done by the authors at the Mathematics for Social Justice Workshop held in June of 2007 at Middlebury College. We provide a description of the Gini coefficient and some discussion of how it can be used to promote quantitative literacy skills in mathematics courses. The Gini Coefficient was introduced in 1921 by Italian statistician Corrado Gini as a measure of inequality. It is defined as twice the area between two curves. One, the Lorenz curve for a given population with respect to a given resource, represents the cumulative percentage of the resource as a function …

Quantitative Literacy Assessments: An Introduction To Testing Tests, Dorothy Wallace, Kim Rheinlander, Steven Woloshin, Lisa Schwartz

Numeracy

This paper describes how professional evaluators construct assessment instruments that work properly to measure the right thing. Constructing an assessment tool begins with getting feedback from relevant experts on the content of questions. The tool is developed and refined through comparison with existing instruments, focus groups and cognitive interviews. The final instrument is formally tested for content validity, usability, reliability and construct validity through a variety of statistical measures. This process of construction is illustrated by two examples relevant to quantitative literacy: the Medical Data Interpretation Test and the Math Attitudes Survey.

Integration With Writing Programs: A Strategy For Quantitative Reasoning Program Development, Nathan D. Grawe, Carol A. Rutz

Numeracy

As an inherently interdisciplinary endeavor, quantitative reasoning (QR) risks falling through the cracks between the traditional “silos” of higher education. This article describes one strategy for developing a truly cross-campus QR initiative: leverage the existing structures of campus writing programs by placing QR in the context of argument. We first describe the integration of Carleton College’s Quantitative Inquiry, Reasoning, and Knowledge initiative with the Writing Program. Based on our experience, we argue that such an approach leads to four benefits: it reflects important aspects of QR often overlooked by other approaches; it defuses the commonly raised objection that QR is …

Assessing Quantitative Reasoning, Corrine H. Taylor

Numeracy

No abstract provided.

On The Existence And Uniqueness Of The Maximum Likelihood Estimators Of Normal And Lognormal Population Parameters With Grouped Data, Jin Xia, Jie Mi, Yan Yan Zhou

Department of Mathematics and Statistics

Lognormal distribution has abundant applications in various fields. In literature, most inferences on the two parameters of the lognormal distribution are based on Type-I censored sample data. However, exact measurements are not always attainable especially when the observation is below or above the detection limits, and only the numbers of measurements falling into predetermined intervals can be recorded instead. This is the so-called grouped data. In this paper, we will show the existence and uniqueness of the maximum likelihood estimators of the two parameters of the underlying lognormal distribution with Type-I censored data and grouped data. The proof was first …

On The Existence And Uniqueness Of The Maximum Likelihood Estimators Of Normal And Lognormal Population Parameters With Grouped Data, Jin Xia, Jie Mi, Yanyan Zhou

Department of Mathematics and Statistics

Lognormal distribution has abundant applications in various fields. In literature, most inferences on the two parameters of the lognormal distribution are based on Type-I censored sample data. However, exact measurements are not always attainable especially when the observation is below or above the detection limits, and only the numbers of measurements falling into predetermined intervals can be recorded instead. This is the so-called grouped data. In this paper, we will show the existence and uniqueness of the maximum likelihood estimators of the two parameters of the underlying lognormal distribution with Type-I censored data and grouped data. The proof was first …

Fusion Of The Parastrophic Matrix And Weak Cayley Table, Nathan C. Perry

Theses and Dissertations

The parastrophic matrix and Weak Cayley Tables are matrices that have close ties to the character table. Work by Ken Johnson has shown that fusion of groups induces a relationship between the character tables of the groups. In this paper we will demonstrate a similar induced relationship between the parastrophic matrices and Weak Cayley Tables of the fused groups.

Discrete Variable Methods For Delay-Differential Equations With Threshold-Type Delays, Z. Jackiewicz, Barbara Zubik-Kowal

Mathematics Faculty Publications and Presentations

We study numerical solution of systems of delay-differential equations in which the delay function, which depends on the unknown solution, is defined implicitly by the threshold condition. We study discrete variable numerical methods for these problems and present error analysis. The global error is composed of the error of solving the differential systems, the error from the threshold conditions and the errors in delay arguments. Our theoretical analysis is confirmed by numerical experiments on threshold problems from the theory of epidemics and from population dynamics.

Stability And Asymptoticity Of Volterra Difference Equations: A Progress Report, Saber Elaydi

Mathematics Faculty Research

We survey some of the fundamental results on the stability and asymptoticity of linear Volterra difference equations. The method of Ζ-transform is heavily utilized in equations of convolution type. An example is given to show that uniform asymptotic stability does not necessarily imply exponential stabilty. It is shown that the two notions are equivalent if the kernel decays exponentially. For equations of nonconvolution type, Liapunov functions are used to find explicit criteria for stability. Moreover, the resolvent matrix is defined to produce a variation of constants formula. The study of asymptotic equivalence for difference equations with infinite delay is carried …