Mathematics Commons^™

Rental Harmony: Sperner's Lemma In Fair Division, Francis E. Su

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No abstract provided in this article.

Recounting Fibonacci And Lucas Identities, Arthur T. Benjamin, Jennifer J. Quinn

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No abstract provided in this article.

Stability Of Self-Similar Solutions For Van Der Waals Driven Thin Film Rupture, Thomas P. Witelski, Andrew J. Bernoff

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Recent studies of pinch-off of filaments and rupture in thin films have found infinite sets of first-type similarity solutions. Of these, the dynamically stable similarity solutions produce observable rupture behavior as localized, finite-time singularities in the models of the flow. In this letter we describe a systematic technique for calculating such solutions and determining their linear stability. For the problem of axisymmetric van der Waals driven rupture (recently studied by Zhang and Lister), we identify the unique stable similarity solution for point rupture of a thin film and an alternative mode of singularity formation corresponding to annular “ring rupture.”

Almost Periodic Factorization Of Certain Block Triangular Matrix Functions, Ilya M. Spitkovsky, Darryl H. Yong

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Let

where , and . For rational such matrices are periodic, and their Wiener-Hopf factorization with respect to the real line always exists and can be constructed explicitly. For irrational , a certain modification (called an almost periodic factorization) can be considered instead. The case of invertible and commuting , was disposed of earlier-it was discovered that an almost periodic factorization of such matrices does not always exist, and a necessary and sufficient condition for its existence was found. This paper is devoted mostly to the situation when is not invertible but the commute pairwise (). The complete description is …

Why The Player Never Wins In The Long Run At La Blackjack, Arthur T. Benjamin, Michael Lauzon '00, Christopher Moore '00

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No abstract provided in this article.

Unevening The Odds Of "Even Up", Arthur T. Benjamin, Jennifer J. Quinn

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No abstract provided in this article.

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.

Magic "Squares" Indeed, Arthur T. Benjamin, Kan Yasuda '97

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No abstract provided in this article.

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 …

On The Number Of Radially Symmetric Solutions To Dirichlet Problems With Jumping Nonlinearities Of Superlinear Order, Alfonso Castro, Hendrik J. Kuiper

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This paper is concerned with the multiplicity of radially symmetric solutions u(x) to the Dirichlet problem

Δu+f(u)=h(x)+cφ(x)

on the unit ball Ω⊂R^N with boundary condition u=0 on ∂Ω. Here φ(x) is a positive function and f(u) is a function that is superlinear (but of subcritical growth) for large positive u, while for large negative u we have that f'(u)<μ, where μ is the smallest positive eigenvalue for Δψ+μψ=0 in Ω with ψ=0 on ∂Ω. It is shown that, given any integer k≥0, the value c may be chosen so large that there are 2k+1 solutions with k or less interior nodes. Existence of positive solutions is excluded for large enough values of c.

An Inverse Function Theorem, Alfonso Castro, J. W. Neuberger

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In this note we present a local surjectivity result which is applicable to differential equations for which full boundary conditions may not be known. Our method uses continuous steepest descent and Sobolev gradients.

The Bordalo Order On A Commutative Ring, Melvin Henriksen, Frank A. Smith

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If R is a commutative ring with identity and ≤ is defined by letting a ≤ b mean ab = a or a = b, then (R,≤) is a partially ordered ring. Necessary and sufficient conditions on R are given for (R,≤) to be a lattice, and conditions are given for it to be modular or distributive. The results are applied to the rings Z_n of integers mod n for n ≥ 2. In particular, if R is reduced, then (R,≤) is a lattice iff R is a weak Baer ring, and (R,≤) is a distributive lattice iff R …