Mathematics Commons^™

Mathematics In The Mountains: The Park City Mathematics Institute, Andrew J. Bernoff

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It's noon. A Fields medalist, master high school teachers from the US and abroad, aspiring undergraduate and graduate students, gifted expositors of mathematics, and mathematical artists gather at tables under a tent. Lunch and so much more is served at these meetings of the minds.

Tiling Proofs Of Recent Sum Identities Involving Pell Numbers, Arthur T. Benjamin, Sean S. Plott '08, James A. Sellers

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In a recent note, Santana and Diaz-Barrero proved a number of sum identities involving the well-known Pell numbers. Their proofs relied heavily on the Binet formula for the Pell numbers. Our goal in this note is to reconsider these identities from a purely combinatorial viewpoint. We provide bijective proofs for each of the results by interpreting the Pell numbers as enumerators of certain types of tilings. In turn, our proofs provide helpful insight for straightforward generalizations of a number of the identities.

Borda Meets Pascal, Marie K. Jameson, Gregory Minton '08, Michael E. Orrison

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Every so often (especially in mathematics), unforeseen connections between different ideas arise and beg explanation. This happened to us when, in an effort to generalize the voting procedure known as the Borda count, we began to see vectors of the form (-1, 1), (1, -2, 1), (-1, 3, -3, 1), (1, -4, 6, -4, 1), and so on. As you might imagine, we were instantly intrigued by this unanticipated relationship with Pascal's triangle, and we quickly set out to find an explanation. This article describes some of our initial findings.

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 …

An Alternate Approach To Alternating Sums: A Method To Die For, Arthur T. Benjamin, Jennifer J. Quinn

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No abstract provided in this article.

The Shapley Value Of Phylogenetic Trees, Claus-Jochen Haake, Akemi Kashiwada '05, Francis E. Su

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Every weighted tree corresponds naturally to a cooperative game that we call a tree game; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley value on the edge weights of the tree, and we also compute a null space …

Traveling Waves And Shocks In A Viscoelastic Generalization Of Burgers' Equation, Victor Camacho '07, Robert D. Guy, Jon T. Jacobsen

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We consider traveling wave phenomena for a viscoelastic generalization of Burgers' equation. For asymptotically constant velocity profiles we find three classes of solutions corresponding to smooth traveling waves, piecewise smooth waves, and piecewise constant (shock) solutions. Each solution type is possible for a given pair of asymptotic limits, and we characterize the dynamics in terms of the relaxation time and viscosity.

The 99th Fibonacci Identity, Arthur T. Benjamin, Alex K. Eustis '06, Sean S. Plott '08

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We provide elementary combinatorial proofs of several Fibonacci and Lucas number identities left open in the book Proofs That Really Count [1], and generalize these to Gibonacci sequences G_n that satisfy the Fibonacci recurrence, but with arbitrary real initial conditions. We offer several new identities as well.

[1] A. T. Benjamin and J. J. Quinn, Proofs That Really Count: The Art of Combinatorial Proof, The Dolciani Mathematical Expositions, 27, Mathematical Association of America, Washington, DC, 2003

Paint It Black -- A Combinatorial Yawp, Arthur T. Benjamin, Jennifer J. Quinn, James A. Sellers, Mark A. Shattuck

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No abstract provided in this paper.

A Combinatorial Approach To Fibonomial Coefficients, Arthur T. Benjamin, Sean S. Plott '08

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A combinatorial argument is used to explain the integrality of Fibonomial coefficients and their generalizations. The numerator of the Fibonomial coeffcient counts tilings of staggered lengths, which can be decomposed into a sum of integers, such that each integer is a multiple of the denominator of the Fibonomial coeffcient. By colorizing this argument, we can extend this result from Fibonacci numbers to arbitrary Lucas sequences.

Mathematics Of Voting, Darryl H. Yong

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Voting theory is a fascinating area of research involving mathematics, political scientists, and economists. The American Mathematical Society, the American Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics chose mathematics and voting as the theme for Mathematics Awareness Month 2008. There is more information on mathematics and voting at www.mathaware.org/mam/08/. It is a mathematical topic that is rich yet accessible to students, pertinent to their lives, especially during this election year, and has the potential to draw students who may not have a strong affinity for mathematics to become interested in mathematics.

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?

Distribution Of The Number Of Encryptions In Revocation Schemes For Stateless Receivers, Christopher Eagle, Zhicheng Gao, Mohamed Omar, Daniel Panario, Bruce Richmond

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We study the number of encryptions necessary to revoke a set of users in the complete subtree scheme (CST) and the subset-difference scheme (SD). These are well-known tree based broadcast encryption schemes. Park and Blake in: Journal of Discrete Algorithms, vol. 4, 2006, pp. 215--238, give the mean number of encryptions for these schemes. We continue their analysis and show that the limiting distribution of the number of encryptions for these schemes is normal. This implies that the mean numbers of Park and Blake are good estimates for the number of necessary encryptions used by these schemes.

A Model For Rolling Swarms Of Locusts, Chad M. Topaz, Andrew J. Bernoff, Sheldon Logan '06, Wyatt Toolson '07

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We construct an individual-based kinematic model of rolling migratory locust swarms. The model incorporates social interactions, gravity, wind, and the effect of the impenetrable boundary formed by the ground. We study the model using numerical simulations and tools from statistical mechanics, namely the notion of H-stability. For a free-space swarm (no wind and gravity), as the number of locusts increases, the group approaches a crystalline lattice of fixed density if it is H-stable, and in contrast becomes ever denser if it is catastrophic. Numerical simulations suggest that whether or not a swarm rolls depends on the statistical mechanical properties of …