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Articles 1 - 12 of 12
Full-Text Articles in Physical Sciences and Mathematics
Sums Of Evenly Spaced Binomial Coefficients, Arthur T. Benjamin, Bob Chen '10, Kimberly Kindred
Sums Of Evenly Spaced Binomial Coefficients, Arthur T. Benjamin, Bob Chen '10, Kimberly Kindred
All HMC Faculty Publications and Research
We provide a combinatorial proof of a formula for the sum of evenly spaced binomial coefficients. This identity, along with a generalization, are proved by counting weighted walks on a graph.
Teaching Research: Encouraging Discoveries, Francis E. Su
Teaching Research: Encouraging Discoveries, Francis E. Su
All HMC Faculty Publications and Research
What does it take to turn a learner into a discoverer? Or to turn a teacher into a co-adventurer? A handful of experiences—from teaching a middle-school math class to doing research with undergraduates—have changed the way that I would answer these questions. Some of the lessons I’ve learned have surprised me.
"Toward Integration: From Quantitative Biology To Mathbio-Biomath?", Pat Marsteller, Lisette G. De Pillis, Ann Findley, Karl Joplin, John Pelesko, Karen Nelson, Katerina Thompson, David Usher, Joseph Watkins
"Toward Integration: From Quantitative Biology To Mathbio-Biomath?", Pat Marsteller, Lisette G. De Pillis, Ann Findley, Karl Joplin, John Pelesko, Karen Nelson, Katerina Thompson, David Usher, Joseph Watkins
All HMC Faculty Publications and Research
In response to the call of BIO2010 for integrating quantitative skills into undergraduate biology education, 30 Howard Hughes Medical Institute (HHMI) Program Directors at the 2006 HHMI Program Directors Meeting established a consortium to investigate, implement, develop, and disseminate best practices resulting from the integration of math and biology. With the assistance of an HHMI-funded mini-grant, led by Karl Joplin of East Tennessee State University, and support in institutional HHMI grants at Emory and University of Delaware, these institutions held a series of summer institutes and workshops to document progress toward and address the challenges of implementing a more quantitative …
Existence Of Solutions For A Semilinear Wave Equation With Non-Monotone Nonlinearity, Alfonso Castro, Benjamin Preskill '09
Existence Of Solutions For A Semilinear Wave Equation With Non-Monotone Nonlinearity, Alfonso Castro, Benjamin Preskill '09
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For double-periodic and Dirichlet-periodic boundary conditions, we prove the existence of solutions to a forced semilinear wave equation with asymptotically linear nonlinearity, no resonance, and non-monotone nonlinearity when the forcing term is not flat on characteristics. The solutions are in L∞ when the forcing term is in L∞ and continous when the forcing term is continuous. This is in contrast with the results in [4], where the non-enxistence of continuous solutions is established even when forcing term is of class C∞ but is flat on a characteristic.
Combinatorial Trigonometry With Chebyshev Polynomials, Arthur T. Benjamin, Larry Ericksen, Pallavi Jayawant, Mark Shattuck
Combinatorial Trigonometry With Chebyshev Polynomials, Arthur T. Benjamin, Larry Ericksen, Pallavi Jayawant, Mark Shattuck
All HMC Faculty Publications and Research
We provide a combinatorial proof of the trigonometric identity cos(nθ) = Tncos(θ),
where Tn is the Chebyshev polynomial of the first kind. We also provide combinatorial proofs of other trigonometric identities, including those involving Chebyshev polynomials of the second kind.
Combinatorially Composing Chebyshev Polynomials, Arthur T. Benjamin, Daniel Walton '07
Combinatorially Composing Chebyshev Polynomials, Arthur T. Benjamin, Daniel Walton '07
All HMC Faculty Publications and Research
We present a combinatorial proof of two fundamental composition identities associated with Chebyshev polynomials. Namely, for all m, n ≥ 0, Tm(Tn(x)) = Tmn(x) and Um-1 (Tn(x))Un-1(x) = Umn-1(x).
Voting In Agreeable Societies, Deborah E. Berg '06, Serguei Norine, Francis E. Su, Robin Thomas, Paul Wollan
Voting In Agreeable Societies, Deborah E. Berg '06, Serguei Norine, Francis E. Su, Robin Thomas, Paul Wollan
All HMC Faculty Publications and Research
No abstract provided in this article.
Two-Player Envy-Free Multi-Cake Division, John Cloutier '03, Kathryn L. Nyman, Francis E. Su
Two-Player Envy-Free Multi-Cake Division, John Cloutier '03, Kathryn L. Nyman, Francis E. Su
All HMC Faculty Publications and Research
We introduce a generalized cake-cutting problem in which we seek to divide multiple cakes so that two players may get their most-preferred piece selections: a choice of one piece from each cake, allowing for the possibility of linked preferences over the cakes. For two players, we show that disjoint envy-free piece selections may not exist for two cakes cut into two pieces each, and they may not exist for three cakes cut into three pieces each. However, there do exist such divisions for two cakes cut into three pieces each, and for three cakes cut into four pieces each. The …
Stability And Dynamics Of Self-Similarity In Evolution Equations, Andrew J. Bernoff, Thomas P. Witelski
Stability And Dynamics Of Self-Similarity In Evolution Equations, Andrew J. Bernoff, Thomas P. Witelski
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A methodology for studying the linear stability of self-similar solutions is discussed. These fundamental ideas are illustrated on three prototype problems: a simple ODE with finite-time blow-up, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, the use of dimensional analysis to derive scaling invariant similarity variables is discussed, as well as the role of symmetries in the context of stability of self-similar dynamics. The …
Local Versus Global Search In Channel Graphs, A.H. Hunter, Nicholas Pippenger
Local Versus Global Search In Channel Graphs, A.H. Hunter, Nicholas Pippenger
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Previous studies of search in channel graphs has assumed that the search is global; that is, that the status of any link can be probed by the search algorithm at any time. We consider for the first time local search, for which only links to which an idle path from the source has already been established may be probed. We show that some well known channel graphs may require exponentially more probes, on the average, when search must be local than when it may be global.
Recognizing Graph Theoretic Properties With Polynomial Ideals, Jesus A. De Loera, Christopher J. Hillar, Peter N. Malkin, Mohamed Omar
Recognizing Graph Theoretic Properties With Polynomial Ideals, Jesus A. De Loera, Christopher J. Hillar, Peter N. Malkin, Mohamed Omar
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Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.
Mathematical Biology At An Undergraduate Liberal Arts College, Stephen C. Adolph, Lisette G. De Pillis
Mathematical Biology At An Undergraduate Liberal Arts College, Stephen C. Adolph, Lisette G. De Pillis
All HMC Faculty Publications and Research
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.