Mathematics Commons^™

A Semilinear Wave Equation With Smooth Data And No Resonance Having No Continuous Solution, Jose F. Caicedo, Alfonso Castro

All HMC Faculty Publications and Research

We prove that a boundary value problem for a semilinear wave equation with smooth nonlinearity, smooth forcing, and no resonance cannot have continuous solutions. Our proof shows that this is due to the non-monotonicity of the nonlinearity.

The Journal (Wheel) Keeps On Turning, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

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 …

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 …

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 …

Variational Analysis In Semi-Infinite And Infinite Programming, Ii: Necessary Optimality Conditions, M J. Cánovas, M A. Lopez, Boris S. Mordukhovich, J Parra

Mathematics Research Reports

This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to problems of semi-infinite and infinite programming with feasible solution sets defined by parameterized systems of infinitely many linear inequalities of the type intensively studied in the preceding development [5] from our viewpoint of robust Lipschitzian stability. We present meaningful interpretations and practical examples of such models. The main results establish necessary optimality conditions for a broad class of semi-infinite and infinite programs, where objectives are generally described by nonsmooth and nonconvex functions on Banach spaces and where infinite constraint inequality systems are indexed by arbitrary sets. …

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.

The Monty Hall Problem, Brian Johnson

Department of Mathematics: Master of Arts in Teaching, Exam Expository Papers

In the game show "Let's Make a Deal", host Monty Hall would present a contestant with three doors. Behind one door is a car and behind the other two doors are goats. The contestant picks one door, after which Monty opens one of the other doors, showing a goat. Monty then offers the contestant the opportunity to keep the same door or switch to the other unopened door; the contestant will get to keep whatever is behind that door. Should the contestant switch?

Advisor: Gordon Woodward

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.

Helping Teachers Un-Structure: A Promising Approach, Eric Hsu, Judy Kysh, Katherine Ramage, Diane Resek

The Mathematics Enthusiast

The amount of overt structure in the presentation of a task affects students’ engagement, creativity, and willingness to tolerate frustration. In a professional development project, with algebra teachers from nine American schools, we tried to help teachers make judicious decisions in their use of structure by having them facilitate low-structure tasks, remove structure from overly structured tasks, and observe “at-risk” students engaged in learning through low-structure tasks. Project schools that worked on structuring generally improved their algebra passing rates, both overall and for African-American students.

Who Can Solve 2x=1? An Analysis Of Cognitive Load Related To Learning Linear Equation Solving, Timo Tossavainen

The Mathematics Enthusiast

Using 2x = 1 as an example, we discuss the cognitive load related to learning linear equation solving. In the framework of the Cognitive Load Theory we consider especially the intrinsic cognitive load needed in arithmetical, geometrical and real analytical approach to linear equation solving. This will be done e.g. from the point of view of the conceptual and procedural knowledge of mathematics and the APOS Theory. Basing on our observations, in the end of the paper we design a setting for teaching linear equation solving.

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 …

Section Abstracts: Astronomy, Mathematics, And Physics & Materials Science

Virginia Journal of Science

Abstracts of the Astronomy, Mathematics and Physics & Materials Science Section for the 87th Annual Meeting of the Virginia Academy of Science, May 27-29, 2009, Virginia Commonwealth University, Richmond VA.

Pointed Trees Of Projective Spaces, Linda Chen, A. Gibney, D. Krashen

Mathematics & Statistics Faculty Works

We introduce a smooth projective variety T(d,n) which compactifies the space of configurations of it distinct points oil affine d-space modulo translation and homothety. The points in the boundary correspond to n-pointed stable rooted trees of d-dimensional projective spaces, which for d = 1, are (n + 1)-pointed stable rational curves. In particular, T(1,n) is isomorphic to ($) over bar (0,n+1), the moduli space of such curves. The variety T(d,n) shares many properties with (M) over bar (0,n+1). For example, as we prove, the boundary is a smooth normal crossings divisor whose components are products of T(d,i) for i < n and it has an inductive construction analogous to but differing from Keel's for (0,n+1). This call be used to describe its Chow groups and Chow motive generalizing [Trans. Airier. Math. Soc. 330 (1992), 545-574]. It also allows us to compute its Poincare polynomials, giving all alternative to the description implicit in [Progr. Math., vol. 129, Birkhauser, 1995, pp. 401-417]. We give a presentation of the Chow rings of T(d,n), exhibit explicit dual bases for the dimension I and codimension 1 cycles. The variety T(d,n) is embedded in the Fulton-MacPherson spaces X[n] for any smooth variety X, and we use this connection in a number of ways. In particular we give a family of ample divisors on T(d,n) and an inductive presentation of the Chow motive of X[n]. This also gives an inductive presentation of the Chow groups of X[n] analogous to Keel's presentation for (M) over bar (0,n+1), solving a problem posed by Fulton and MacPherson.

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 …

Intuitions Of "Infinite Numbers": Infinite Magnitude Vs. Infinite Representation, Ami Mamolo

The Mathematics Enthusiast

This study examines undergraduate students’ emerging conceptions of infinity as manifested in their engagement with geometric tasks. Students’ attempts to reduce the level of abstraction of infinity and properties of infinite quantities are described. Their arguments revealed they perceive infinity as an ongoing process, rather than a completed one, and fail to notice conflicting ideas. In particular, confusion between the infinite magnitude of points on a line segment and the infinite representation of real numbers was observed. Furthermore, students struggled to draw a connection between real numbers and their representation on a number line.

Small Change - Big Difference, Ilana Lavy, Atara Shriki

The Mathematics Enthusiast

Starting in a well known theorem concerning medians of triangle and using the ‘What If Not?’ strategy, we describe an example of an activity in which some relations among segments and areas in triangle were revealed. Some of the relations were proved by means of Affine Geometry.

An Application Of Gröbner Bases, Shengxiang Xia, Gaoxiang Xia

The Mathematics Enthusiast

In this paper, we program a procedure using Maple's packages, with it we can realize mechanical proving of some theorems in elementary geometry.

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.

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 …

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?

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 …

Book Review: What's All The Commotion Over Commognition? A Review Of Anna Sfard's Thinking As Communicating, Bharath Sriraman

The Mathematics Enthusiast

If straight edge and compass constructions are the so-called “atoms” of Euclidean geometry, if sequences are the “atoms” of Analysis, then what are the “atoms” (if any) of mathematics education? Arguably mathematics education is a much wider field than Euclidean Geometry or Elementary Analysis, however there are several fundamental things that the field purports to study, chief among which is mathematical thinking or more generally “thinking”. The book under review, though it appears in a Cambridge University Press series entitled Learning in Doing: Social, Cognitive, and Computational Perspectives, is in my view situated at the intersection of Consciousness Studies, …

Tme Volume 6, Number 3

The Mathematics Enthusiast

No abstract provided.

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 …

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 …

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 …

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 …