Physical Sciences and Mathematics Commons^™

Existence Of Many Positive Nonradial Solutions For A Superlinear Dirichlet Problem On Thin Annuli, Alfonso Castro, Marcel B. Finan

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We study the existence of many positive nonradial solutions of a superlinear Dirichlet problem in an annulus in R^N. Our strategy consists of finding the minimizer of the energy functional restricted to the Nehrai manifold of a subspace of functions with symmetries. The minimizer is a global critical point and therefore is a desired solution. Then we show that the minimal energy solutions in different symmetric classes have mutually different energies. The same approach has been used to prove the existence of many sign-changing nonradial solutions (see [5]).

Optimal Token Allocations In Solitaire Knock'm Down, Arthur T. Benjamin, Mark L. Huber, Matthew T. Fluet '99

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In the game Knock 'm Down, tokens are placed in N bins. At each step of the game, a bin is chosen at random according to a fixed probability distribution. If a token remains in that bin, it is removed. When all the tokens have been removed, the player is done. In the solitaire version of this game, the goal is to minimize the expected number of moves needed to remove all the tokens. Here we present necessary conditions on the number of tokens needed for each bin in an optimal solution, leading to an asymptotic solution.

Random Approaches To Fibonacci Identities, Arthur T. Benjamin, Gregory M. Levin, Karl Mahlburg '01, Jennifer J. Quinn

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No abstract provided in this article.

Phased Tilings And Generalized Fibonacci Identities, Arthur T. Benjamin, Jennifer J. Quinn, Francis E. Su

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Fibonacci numbers arise in the solution of many combinatorial problems. They count the number of binary sequences with no consecutive zeros, the number of sequences of 1's and 2's which sum to a given number, and the number of independent sets of a path graph. Similar interpretations exist for Lucas numbers. Using these interpretations, it is possible to provide combinatorial proofs that shed light on many interesting Fibonacci and Lucas identities (see [1], [3]). In this paper we extend the combinatorial approach to understand relationships among generalized Fibonacci numbers.

Given G₀ and G₁ a generalized Fibonacci sequence G …

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

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We study the existence, multiplicity, and stability of positive solutions to -u''(x) = λf(u(x)) for x є (-1,1), u(-1) = 0 = u(1), where λ > 0 and f:[0,∞)→R is monotonically increasing and concave with f(0) < 0 (semipositone). We establish that f should be appropriately concave (by establishing conditions on f) to allow multiple positive solutions. For any λ > 0, we obtain the exact number of positive solutions as a function of f(t)/t. We follow several families of nonlinearities f for which f(∞) := lim t→∞ f(t) > 0 and study how the positive solution curves to the above problem evolve. Also, we give examples where our results apply. This work extends the work of A. Castro and R. Shivaji (1988, Proc. Roy. Soc. Edinburgh …

Counting On Continued Fractions, Arthur T. Benjamin, Francis E. Su, Jennifer J. Quinn

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No abstract provided in this article.

A Rational Solution To Cootie, Arthur T. Benjamin, Matthew T. Fluet '99

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No abstract provided in this article.

Procedural Support For Cooperative Negotiations: Theoretical Design And Practical Implementation, Matthias G. Raith, Francis Su

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We discuss the theoretical design of algorithms for solving distributional conflicts within groups. We consider an algorithm to be procedural if the implementation of the outcome requires the participation of the players, or if it can even be conducted by the players themselves without computational assistance. We compare two procedures for multilateral problems of fair division; both establish envy-freeness, given the possibility of monetary compensations between players.

Review: Cake-Cutting Algorithms: Be Fair If You Can, Francis E. Su

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No abstract provided in this article.

A Leveque-Type Lower Bound For Discrepancy, Francis E. Su

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A sharp lower bound for discrepancy on R / Z is derived that resembles the upper bound due to LeVeque. An analogous bound is proved for discrepancy on R^k / Z^k. These are discussed in the more general context of the discrepancy of probablity measures. As applications, the bounds are applied to Kronecker sequences and to a random walk on the torus.

A Look At Biseparating Maps From An Algebraic Point Of View, Melvin Henriksen, Frank A. Smith

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In [ABN], Araujo, Beckenstein, and Narici add the capstone to a series of papers by several groups of authors by showing that if ρ is a biseparating map between two algebras of all real or complex-valued functions on realcompact spaces, then it is a continuous multiple of an isomorphism between these rings. Their proof uses relatively powerful analytic and topological techniques. In what follows, the extent to which such a result can be generalized to a wider class of algebras using algebraic techniques is investigated. We are unable, however to obtain the main result of [ABN] using these techniques.

When Is |C(X X Y)| = |C(X)||C(Y)|?, O. T. Alas, W. W. Comfort, S. Garcia-Ferreira, Melvin Henriksen, R. G. Wilson, R. G. Woods

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Sufficient conditions on the Tychonoff spaces X and Y are found that imply that the equation in the title holds. Sufficient conditions on the Tychonoff space X are found that ensure that the equation holds for every Tychonoff space Y . A series of examples (some using rather sophisticated cardinal arithmetic) are given that witness that these results cannot be generalized much.