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.

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.

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

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No abstract provided in this article.

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

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No abstract provided in this article.

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.

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 …

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 …

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.

A Minmax Principle, Index Of The Critical Point, And Existence Of Sign Changing Solutions To Elliptic Boundary Value Problems, Alfonso Castro, Jorge Cossio, John M. Neuberger

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In this article we apply the minmax principle we developed in [6] to obtain sign-changing solutions for superlinear and asymptotically linear Dirichlet problems.

We prove that, when isolated, the local degree of any solution given by this minmax principle is +1. By combining the results of [6] with the degree-theoretic results of Castro and Cossio in [5], in the case where the nonlinearity is asymptotically linear, we provide sufficient conditions for:

i) the existence of at least four solutions (one of which changes sign exactly once),

ii) the existence of at least five solutions (two of which change sign), and …

Newton's Cubic Roots, Lisette G. De Pillis

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No abstract included in this article

Average-Case Lower Bounds For Noisy Boolean Decision Trees, William Evans, Nicholas Pippenger

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We present a new method for deriving lower bounds to the expected number of queries made by noisy decision trees computing Boolean functions. The new method has the feature that expectations are taken with respect to a uniformly distributed random input, as well as with respect to the random noise, thus yielding stronger lower bounds. It also applies to many more functions than do previous results. The method yields a simple proof of the result (previously established by Reischuk and Schmeltz) that almost all Boolean functions of n arguments require $\Me(n \log n)$ queries, and strengthens this bound from the …

Positive Solutions For A Concave Semipositone Dirichlet Problem, Alfonso Castro, Ratnasingham Shivaji

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No abstract provided for this article

Convergence Of Random Walks On The Circle Generated By An Irrational Rotation, Francis E. Su

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Fix . Consider the random walk on the circle which proceeds by repeatedly rotating points forward or backward, with probability , by an angle . This paper analyzes the rate of convergence of this walk to the uniform distribution under ``discrepancy'' distance. The rate depends on the continued fraction properties of the number . We obtain bounds for rates when is any irrational, and a sharp rate when is a quadratic irrational. In that case the discrepancy falls as (up to constant factors), where is the number of steps in the walk. This is the first example of a sharp …

Tree-Based Multicasting In Wormhole-Routed Irregular Topologies, Ran Libeskind-Hadas, Dominic Mazzoni '99, Ranjith Rajagopalan '99

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A deadlock-free tree-based multicast routing algorithm is presented for all direct networks, regardless of interconnection topology. The algorithm delivers a message to any number of destinations using only a single startup phase. In contrast to existing tree-based schemes, this algorithm applies to all interconnection topologies, requires only fixed-sized input buffers that are independent of maximum message length, and uses a single asynchronous flit replication mechanism. The theoretical basis of the technique used here is sufficiently general to develop other tree-based multicasting algorithms for regular and irregular topologies. Simulation results demonstrate that this tree-based algorithm provides a very promising means of …

On Multiple Solutions Of A Nonlinear Dirichlet Problem, Alfonso Castro, Jorge Cossio, John M. Neuberger

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We prove that a semilinear elliptic boundary value problem has five solutions when the range of the derivative of the nonlinearity includes at least the first two eigenvalues. We also prove that if the region is a ball the semilinear elliptic problem has two solutions that change sign and are nonradial.

Borsuk-Ulam Implies Brouwer: A Direct Construction, Francis E. Su

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No abstract provided in this article.

A Sign-Changing Solution For A Superlinear Dirichlet Problem, Alfonso Castro, Jorge Cossio, John M. Neuberger

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We show that a superlinear boundary value problem has at least three nontrivial solutions. A pair are of one sign (positive and negative, respectively), and the third solution changes sign exactly once. The critical level of the sign-changing solution is bounded below by the sum of the two lesser levels of the one-sign solutions. If nondegenerate, the one sign solutions are of Morse index 1 and the signchanging solution has Morse index 2. Our results extend and complement those of Z.Q. Wang [12].

Quasi-Steady Monopole And Tripole Attractors In Relaxing Vortices, Louis F. Rossi, Joseph F. Lingevitch, Andrew J. Bernoff

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Using fully nonlinear simulations of the two-dimensional Navier–Stokes equations at large Reynolds number (Re), we bracket a threshold amplitude above which a perturbed Gaussian monopole will relax to a quasi-steady, rotating tripole, and below which will relax to an axisymmetric monopole. The resulting quasi-steady structures are robust to small perturbations. We propose a means of measuring the decay rate of disturbances to asymptotic vortical structures wherein streamlines and lines of constant vorticity correspond in some rotating or translating frame. These experiments support the hypothesis that small or moderate deviations from asymptotic structures decay through inviscid and viscous mixing.

Positive Solution Curves Of Semipositone Problems With Concave Nonlinearities, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji

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We consider the positive solutions to the semilinear equation:

-Δu(x) = λf(u(x)) for x ∈ Ω

u(x) = 0 for x ∈ ∂Ω

where Ω denotes a smooth bounded region in R^N (N > 1) and λ > 0. Here f :[0, ∞)→R is assumed to be monotonically increasing, concave and such that f(0) < 0 (semipositone). Assuming that f'(∞) ≡ lim t→∞ f'(t) > 0, we establish the stability and uniqueness of large positive solutions in terms of (f(t)/t)'. When Ω is a ball, we determine the exact number of positive solutions for each λ > 0. We also obtain the geometry of the branches of positive solutions completely and establish how …

The Interaction Of A Point Vortex With A Wall-Bounded Vortex Layer, Oliver V. Atassi, Andrew J. Bernoff, Seth Lichter

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The interaction of a point vortex with a layer of constant vorticity, bounded below by a wall and above by an irrotational flow, is investigated as a model of vortex–boundary layer interaction. This model calculates both the evolution of the interface which separates the vortex layer from the irrotational flow and the trajectory of the vortex. In order to determine the conditions which lead to sustained unsteady interaction, three cases are investigated where the mutual interaction between the vortex and interface is initially assumed to be weak. (i) When a weak point vortex lies outside the layer, the vortex moves …

Positive Solutions For A Semilinear Elliptic Problem With Critical Exponent, Ismail Ali, Alfonso Castro

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No abstract provided in article.

Determinants Of The Tournaments, Clifford A. Mccarthy '94, Arthur T. Benjamin

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No abstract provided in this article.

Radial Solutions To A Dirichlet Problem Involving Critical Exponents When N=6, Alfonso Castro, Alexandra Kurepa

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In this paper we show that, for each λ>0, the set of radially symmetric solutions to the boundary value problem

-Δu(x) = λu(x) + u(x)|u(x)|, x ε B := {x ε R⁶:|x|<1},

u(x) = 0, x ε ∂B

is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.

Combinatorics And Campus Security, Arthur T. Benjamin

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One day I received electronic mail from our director of campus security [Gilbraith 1993]:

"I have a puzzle for you that has practical applications for me. I need to know how many different combinations there are for our combination locks. A lock has 5 buttons. In setting the combination you can use only 1button or as many as 5. Buttons may be pressed simultaneously and / or successively, but the same button cannot be used more than once in the same combination.

I had a student (obviously not a math major) email me that there are only 120 possibilities, but …

Meyer Jerison, 1922-1995, Leonard Gillman, Melvin Henriksen

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Memories and biography of mathematician Meyer Jerison (b. November 28, 1922 - d. March 19, 1995).

Branches Of Radial Solutions For Semipositone Problems, Alfonso Castro, Sudhasree Gadam, Ratnasingham Shivaji

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We consider the radially symmetric solutions to the equation −Δu(x) = λƒ(u(x)) for x ∈ Ω, u(x) = 0 for x ∈ ∂Ω, where Ω denotes the unit ball in R^N (N > 1), centered at the origin and λ > 0. Here ƒ: R→R is assumed to be semipositone (ƒ(0) < 0), monotonically increasing, superlinear with subcritical growth on [0, ∞). We establish the structure of radial solution branches for the above problem. We also prove that if ƒ is convex and ƒ(t)/(tƒ'(t)−ƒ(t)) is a nondecreasing function then for each λ > 0 there exists at most one positive solution u such that (λ, u) belongs to the unbounded branch of positive solutions. Further when ƒ(t) = tp − k, k > 0 and 1 < p < (N + 2)/(N − 2), we prove that the set of positive solutions is connected. Our results are motivated by and extend the developments in [4].