George T. Gilbert

Wagering in Final Jeopardy

In the popular television game show Jeopardy! three contestants accumulate points by giving the correct questions corresponding to the answers provided by the emcee. While the contestants show a great deal of self-confidence in wagering in Final Jeopardy!, the last stage of the game, they appear to give little thought to betting strategy. In this article, we look for a reasonable strategy for wagering in Final Jeopardy! We consider betting strategies from two perspectives. First, we describe a practical strategy to use against two contestants who behave in a way predicted by actual game data. After this, we determine how the game might be played if all of the contestants were knowledgeable in game theory.

Zeros of symmetric, quasi-definite, orthogonal polynomials

Abstract It is well-known that all zeros of real polynomials, orthogonal with respect to a positive-definite moment functional, are real. In the first section of this paper, we show that the zeros of certain symmetric, quasi-definite, orthogonal polynomials are real or purely imaginary. In the second section, we apply our finding to Mellin transforms of Laguerre polynomials, generalizing a result of Bump and Ng [1].

Sliding piece puzzles with oriented tiles

In this paper, we consider n identical tiles which are placed on the n + 1 vertices of a graph and which move along the edges of the graph. The tiles come with an “orientation”, an element of an arbitrary finite group H. Moving a tile along a given edge into the empty vertex changes the orientation of the tile in a prescribed way. We study the group of oriented positions of the tiles achievable from an initial position which fix the empty vertex. It may be thought of as a subgroup of the semidirect product Hn?Sn or the wreath product H wr Sn.

Cuspidal coverings for pairs of congruence subgroups

A formula is developed for the number of cusps in certain congruence subgroups equivalent to a given cusp in certain larger subgroups. In particular, necessary and sufficient conditions for this number to be independent of the particular cusp are given. Applications to statistical mechanics are surveyed.

Mellin transforms of a generalization of Legendre polynomials

Abstract Oberle, M.K., S.L. Scott, G.T. Gilbert, R.L. Hatcher and D.F. Addis, Mellin transforms of a generalization of Legendre polynomials, Journal of Computational and Applied Mathematics 45 (1993) 367–369. We show that the zeros and poles of the Mellin transforms of polynomials on [−1, 1] orthogonal with respect to the weight> brvbar;x¦2r, with r gt; − 1 2 , are real and simple.