Physical Sciences and Mathematics Commons^™

Counting On R-Fibonacci Numbers, Arthur Benjamin, Curtis Heberle

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We prove the r-Fibonacci identities of Howard and Cooper using a combinatorial tiling approach.

Existence Of Positive Solutions For A Superlinear Elliptic System With Neumann Boundary Condition, Alfonso Castro, Juan C. Cardeño

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We prove the existence of a positive solution for a class of nonlin- ear elliptic systems with Neumann boundary conditions. The proof combines extensive use of a priori estimates for elliptic problems with Neumann boundary condition and Krasnoselskii's compression-expansion theorem

Quantitative Approaches To Sustainability Seminars, Rachel Levy

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How can mathematicians contribute to education of about sustainability? Mathematicians study climate change, energy-related technologies, models of energy availability, production and consumption, and even the political and social aspects of sustainable legislation and practices. However, at this point, few courses on sustainability can be found in math department offerings. When we consider problems that our current and future students will face, energy sustainability certainly seems important. But how many of these ideas reach our classrooms?

Adventures In Teaching: A Professor Goes To High School To Learn About Teaching Math, Darryl H. Yong

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During the 2009–2010 academic year I did something unusual for a university mathematician on sabbatical: I taught high school mathematics in a large urban school district. This might not be so strange except that my school does not have a teacher preparation program and only graduates a few students per year who intend to be teachers. Why did I do this? I, like many of you, am deeply concerned about mathematics education and I wanted to see what a typical high school in my city is like. Because I regularly work with high school mathematics teachers, I wanted to experience …

Squaring, Cubing, And Cube Rooting, Arthur T. Benjamin

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We present mentally efficient algorithms for mentally squaring and cubing 2-digit and 3-digit numbers and for finding cube roots of numbers with 2-digit or 3-digit answers.

A Census Of Vertices By Generations In Regular Tessellations Of The Plane, Alice Paul '12, Nicholas Pippenger

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We consider regular tessellations of the plane as infinite graphs in which q edges and q faces meet at each vertex, and in which p edges and p vertices surround each face. For 1/p + 1/q = 1/2, these are tilings of the Euclidean plane; for 1/p + 1/q < 1/2, they are tilings of the hyperbolic plane. We choose a vertex as the origin, and classify vertices into generations according to their distance (as measured by the number of edges in a shortest path) from the origin. For all p ≥ 3 and q ≥ 3 with 1/p + 1/q ≤ 1/2, we give simple combinatorial derivations of the rational generating functions for the number of vertices in each generation.

An Amazing Mathematical Card Trick, Arthur T. Benjamin

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A magician gives a member of the audience 20 cards to shuffle. After the cards are thoroughly mixed, the magician goes through the deck two cards at a time, sometimes putting the two cards face to face, sometimes back to back, and sometimes in the same direction. Before dealing each pair of cards into a pile, he asks random members of the audience if the pair should be flipped over or not. He goes through the pile again four cards at a time and before each group of four is dealt to a pile, the audience gets to decide whether …

Mathematical Biology At An Undergraduate Liberal Arts College, Stephen C. Adolph, Lisette G. De Pillis

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Since 2002 we have offered an undergraduate major in Mathematical Biology at Harvey Mudd College. The major was developed and is administered jointly by the mathematics and biology faculty. In this paper we describe the major, courses, and faculty and student research and discuss some of the challenges and opportunities we have experienced.

Imagine Math Day: Encouraging Secondary School Students And Teachers To Engage In Authentic Mathematical Discovery, Darryl H. Yong, Michael E. Orrison Jr.

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Research mathematicians and school children experience mathematics in profoundly different ways. Ask a group of mathematicians what it means to “do mathematics” and you are likely to get a myriad of responses: mathematics involves analyzing and organizing patterns and relationships, reasoning and drawing conclusions about the world, or creating languages and tools to describe and solve important problems. Students of mathematics often report “doing mathematics” as performing calculations or following rules. It’s natural that they see mathematics as monolithic rather than an evolving, growing, socially constructed body of knowledge, because most mathematical training in primary and secondary schools consists of …

The Art Of Teaching Mathematics, Garikai Campbell, Jon T. Jacobsen, Aimee S A Johnson, Michael E. Orrison Jr.

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On June 10–12, 2007, Harvey Mudd College hosted A Conference on the Art of Teaching Mathematics. The conference brought together approximately thirty mathematicians from the Claremont Colleges, Denison, DePauw, Furman, Middlebury, Penn State, Swarthmore, and Vassar to explore the topic of teaching as an art. Assuming there is an element of artistic creativity in teaching mathematics, in what ways does it surface and what should we be doing to develop this creativity?

Teaching Time Savers: The Exam Practically Wrote Itself!, Michael E. Orrison Jr.

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When I first started teaching, creating an exam for my upper division courses was a genuinely exciting process. The material felt fresh and relatively unexplored (at least by me), and I remember often feeling pleasantly overwhelmed with what seemed like a vast supply of intriguing and engrossing exam-ready problems. Crafting the perfect exam, one that was noticeably inviting, exceedingly fair, and unavoidably illuminating, was a real joy.

Teaching Time Savers: Is Homework Grading On Your Nerves?, Lisette G. De Pillis, Michael E. Orrison Jr.

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You have probably heard it said that we learn mathematics best when we do mathematics, or that mathematics is not a spectator sport. For most of our students, this means that their mathematics courses will involve a fair amount of homework. This homework is often used to evaluate individual student progress, but it can also be used, for example, as a catalyst for discussion, to emphasize a point made in class, and to identify common misunderstandings throughout the class as a whole. There is, however, the matter of grading homework.

Teaching Time Savers: Some Advice On Giving Advice, Michael E. Orrison Jr.

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There are always a lot of questions that need to be answered at the beginning of a course. When are office hours? What are the grading policies? How many exams will there be? Will late homework be accepted? We have all seen the answers to these sorts of questions form the bulk of a standard course syllabus, and most of us feel an obligation (and rightly so) to provide such information.

Teaching Time Savers: Style Points, Michael E. Orrison Jr.

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When I began as an assistant professor, I had a pretty good sense of how much time it would take for me to prepare for each class. After a few conversations with my new colleagues, I even had a good sense of how much time I should devote to tasks like office hours and committee work. Somewhere in the middle of grading my first exam, though, it became painfully clear that I had underestimated the amount of time I would need to grade exams!

Hole Dynamics In Polymer Langmuir Films, James C. Alexander, Andrew J. Bernoff, Elizabeth K. Mann, J. Adin Mann Jr., Lu Zou

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This article develops a model for the closing of a gaseous hole in a liquid domain within a two-dimensional fluid layer coupled to a Stokesian subfluid substrate, and compares this model to experiments following hole dynamics in a polymer Langmuir monolayer. Closure of such a hole in a fluid layer is driven by the line tension at the hole boundary and the difference in surface pressure within the hole and far outside it. The observed rate of hole closing is close to that predicted by our model using estimates of the line tension obtained by other means, assuming that the …

Teaching Time Savers: A Recommendation For Recommendations, Michael E. Orrison Jr.

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I admit it — I enjoy writing recommendation letters for my students. I like
learning about their hopes and dreams, where they have been and where they want to go. A recommendation letter is an opportunity to remind myself how much my students can grow while they are in college, and how much I have grown as an instructor, advisor, and mentor.

Looking Beyond The Curriculum In Jamaica, Jon T. Jacobsen, Michael E. Orrison Jr.

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In August 2004, we had the opportunity to travel to Jamaica to lead a pilot workshop for Jamaican high school math teachers. The workshop focused on the importance of mathematical context in the teaching of mathematics. It was sponsored by the Gibraltar Institute, a Jamaica-based nongovernmental organization led by Trevor Campbell (Pomona College) and Reginald Nugent (Cal State Pomona), Jamaica’s College of Agriculture, Science and Education, and Harvey Mudd College.

Mathematical Magic, Arthur T. Benjamin

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In this paper, we present simple strategies for performing mathematical calculations that appear magical to most audiences. Specifically, we explain how to square large numbers, memorize pi to 100 places and determine the day of the week of any given date.

Math Major, Math Major, Arthur T. Benjamin

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Sung to the tune of "Matchmaker" from Fiddler on the Roof:

Math major, math major, make me some math.

Find me a prime, sketch me a path.

Math major, math major look through your books,

And make me some perfect math.

Mathematical Constance (A Poem Dedicated To Constance Reid), Arthur T. Benjamin

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Mathematical Constance (A Poem Dedicated to Constance Reid)

I think that I shall never see

A constant lovelier than e,

Whose digits are too great too state,

They're 2.71828…

And e has such amazing features

It's loved by all (but mostly teachers).

With all of e's great properties

Most integrals are done with … ease.

Theorems are proved by fools like me

But only Euler could make an e.

I suppose, though, if I had to try

To choose another constant, I

Might offer i or phi or pi.

But none of those would satisfy.

Of all the …

The Best Way To Knock 'M Down, Arthur T. Benjamin, Matthew T. Fluet '99

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"Knock 'm Down" is a game of dice that is so easy to learn that it is being played in classrooms around the world. Although this game has been effective at developing students' intuition about probability [Fendel et al. 1997; Hunt 1998], we will show that lurking underneath this deceptively simple game are many surprising and highly unintuitive results.

Bounds On A Bug, Arthur T. Benjamin, Matthew T. Fluet '99

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In the game of Cootie, players race to construct a "cootie bug" by rolling a die to collect component parts. Each cootie bug is composed of a body, a head, two eyes, one nose, two antennae, and six legs. Players must first acquire the body of the bug by rolling a 1. Next, they must roll a 2 to add the head to the body. Once the body and head are both in place, the remaining body parts can be obtained in any order by rolling two 3s for the eyes, one 4 for the nose, two 5s for the …

Sensible Rules For Remembering Duals -- The S-O-B Method, Arthur T. Benjamin

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We present a natural motivation and simple mnemonic for creating the dual LP of any linear programing problem.

Optimal Klappenspiel, Arthur T. Benjamin, Derek Stanford '93

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The game Klappenspiel ("flipping game") is a traditional German game of flipping tiles according to dice rolls. In this paper, we derive the optimal strategy for this game by using dynamic programming. We show that the probability of winning using the optimal strategy is 0.30%.

On Multiple Solutions Of Nonlinear Elliptic Equations With Odd Nonlinearities, Alfonso Castro, J. V. A. Gonçalves

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In this paper we stablish results on multiplicity of solutions for the boundary value problem where a e IR and f: R - IR is an odd continuous function.

Multiplicatively Periodic Rings, Ted Chinburg, Melvin Henriksen

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We prove a generalization of Luh's result without using Dirichlet's Theorem. We then use Theorem 1 to show that the J-subrings of a periodic ring form a lattice with respect to join and intersection (the join of two subrings is the smallest subring containing both of them). After noting that every J-ring has nonzero characteristic, we determine for which positive integers n and m there exist J-rings of period n and characteristic m. This generalizes a problem posed by G. Wene.

Calculus And The Computer: A Conservative Approach, Melvin Henriksen

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This paper describes a program for making the use of numerical methods an integral part of the freshman college course in single variable calculus.