Mathematics Commons^™

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

All HMC Faculty Publications and Research

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.

Siegel’S Lemma Outside Of A Union Of Varieties, Lenny Fukshansky

CMC Faculty Publications and Research

Lecture given at the AMS Special Session on Number Theory, October 2008.

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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.

All HMC Faculty Publications and Research

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 …

On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky

CMC Faculty Publications and Research

A lattice is called well-rounded if its minimal vectors span the corresponding Euclidean space. In this paper we completely describe well-rounded full-rank sublattices of Z², as well as their determinant and minima sets. We show that the determinant set has positive density, deriving an explicit lower bound for it, while the minima set has density 0. We also produce formulas for the number of such lattices with a fixed determinant and with a fixed minimum. These formulas are related to the number of divisors of an integer in short intervals and to the number of its representations as a sum …

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 …

On Distribution Of Well-Rounded Sublattices Of Z², Lenny Fukshansky

CMC Faculty Publications and Research

Lecture given at Institut de Mathématiques in Bordeaux, France, June 2008.

Uniform Uncertainty Principle And Signal Recovery Via Regularized Orthogonal Matching Pursuit, Deanna Needell, Roman Vershynin

CMC Faculty Publications and Research

This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements—L1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of L1-minimization. Our algorithm, ROMP, reconstructs a sparse signal in a number of iterations linear in the sparsity, and the reconstruction is exact provided the linear measurements satisfy the uniform uncertainty principle.

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

All HMC Faculty Publications and Research

No abstract provided in this article.

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

No abstract provided in this paper.

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

All HMC Faculty Publications and Research

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.

Frobenius Number, Covering Radius, And Well-Rounded Lattices, Lenny Fukshansky, Sinai Robins

CMC Faculty Publications and Research

Lecture given at the Joint Mathematics Meeting in San Diego, January 2008.

Review: Lie Structure In Semiprime Superalgebras With Superinvolution, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Mathematics Of Voting, Darryl H. Yong

All HMC Faculty Publications and Research

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.

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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

All HMC Faculty Publications and Research

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 …

Small Zeros Of Quadratic Forms Over The Algebraic Closure Of Q, Lenny Fukshansky

CMC Faculty Publications and Research

Let N >= 2 be an integer, F a quadratic form in N variables over (Q) over bar, and Z subset of (Q) over bar (N) an L-dimensional subspace, 1 <= L <= N. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z, F). This provides an analogue over (Q) over bar of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over (Q) over bar. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over (Q) over bar. This extends previous results of the author over number fields. All bounds on height are explicit.

Review: Hypercyclic Pairs Of Coanalytic Toeplitz Operators, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Review: Shift-Type Invariant Subspaces Of Contractions, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.

Interpolation And Complex Symmetry, Stephan Ramon Garcia, Mihai Putinar

Pomona Faculty Publications and Research

In a separable complex Hilbert space endowed with an isometric conjugate-linear involution, we study sequences orthonormal with respect to an associated bilinear form. Properties of such sequences are measured by a positive, possibly unbounded angle operator which is formally orthogonal as a matrix. Although developed in an abstract setting, this framework is relevant to a variety of eigenvector interpolation problems arising in function theory and in the study of differential operators.

Review: Some Remarks On Quantized Lie Superalgebras Of Classical Type, Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Review: Degenerate Series Representations Of The Q-Deformed Algebra Soq′(R,S), Gizem Karaali

Pomona Faculty Publications and Research

No abstract provided.

Intrinsic Linking And Knotting Are Arbitrarily Complex, Erica Flapan, Blake Mellor, Ramin Naimi

Pomona Faculty Publications and Research

We show that, given any n and alpha, any embedding of any sufficiently large complete graph in R³ contains an oriented link with components Q₁,...,Q_n such that for every i not equal to j, Ilk(Q_i,Q_j)I greater than or equal to alpha and la₂(Q_i)l greater than or equal to alpha, where a₂(Q_i) denotes the second coefficient of the Conway polynomial of Q_i.

Mathematicians Playing A Role In Math Education: What We Learned At The Ime/Mime Workshop, Anna Bargagliotti, Rama Chidambaram, Gizem Karaali

Pomona Faculty Publications and Research

In Hollywood, some actors are regularly cast as mean, others as sweet and endearing, and some typically play innocent big-eyed youths who inevitably succeed after awakening to the particular facts of life that their producer wants them to awaken to. It is unusual and difficult for actors to cross the bridge between different types on a regular basis. However, there are always exceptions to the rule.

In the seemingly unrelated world of academics, mathematics faculty may find themselves playing different roles. People with different skills and interests strive to balance their careers in ways that will be uniquely fulfilling to …

Review: On Rank-One Perturbations Of Normal Operators, Stephan Ramon Garcia

Pomona Faculty Publications and Research

No abstract provided.