·

the xyplane, and ·

the plane tangent to the graph of F at (x_k,y_k,F(x_k,y_k)).
Halley: The next iterate is the zero of the Mobius transformation that osculates the level set of f at z_k.
The Rook on the HalfChessboard, or How Not to Diagonalize a Matrix
by Kiran S. Kedlaya and Lenhard L. Ng
[email protected], [email protected]
We study a simplelooking family of matrices whose eigenvalues appear equally innocuous. The diagonalization of these matrices is not quite so innocuous; we present two approaches, one of which requires a fourvariable combinatorial identity with a somewhat roundabout proof. We then apply this result to explore two related random walks, summarized as "the rook on the halfchessboard." Given an nbyn chessboard from which all squares above (but not including) the northwestsoutheast diagonal have been removed, a rook moves with probability 1/2 to some square in its row, and with probability 1/2 to some square in its column. We determine the speeds at which this walk, and a related, more efficient walk, become random.
The Last Round of Betting in Poker
by Jack Cassidy
[email protected]
Mathematical analyses of poker usually assume an unrealistic set of circumstances that never occur in real play. Chief among these is that all players have equal hands going into the final draw.
This paper assumes generalized, unequal hand distributions for two players in the final round of betting. We develop six simple equations for optimal strategies when raising is not allowed, then look at how things change when raising is allowed.
We apply the equations to a sample hand of Seven Card Stud, and give pointers on how to apply the optimal strategies to reallife poker games.
Applications of the Universal Subjectivity of the Cantor Set
by Yoav Benyamini
[email protected]
Every compact metric space is a continuous image of the Cantor set, i.e., for each compact metric space K there is a continuous map from the Cantor set onto K.
This classical theorem of Alexandroff and Hausdorff can heuristically be rephrased as saying that "compact sets of data can be continuously encoded by the Cantor set". This principle is a very powerful tool that can be applied to solve a variety of unrelated problems in topology, geometry, and analysis. In this article we present several such applications. The following two examples are typical.
(i) (R. Grzaslewicz) For each d 1 there is a compact convex set B in R^{^}(d+2) with the property that each compact convex subset of the ddimensional unit cube is congruent to a face of B.
(ii) There is a realvalued, bounded, and continuous function f on the real line R with the property that for each doubly infinite sequence (y_n)_(n \ in Z) of real numbers satisfying y_n £ 1 for all n, there is a point t \in Rsuch that y_n = f(t + n) for all n \in Z.
NOTES
The Group Z_2 x Z_n and Regular Polygonal Paths
by Apostolos Thoma
[email protected]
A Quick CayleyHamilton
by Busiso P. Chisala
[email protected]
The Probability of a Tie in an nGame Match
by J. Marshall Ash
[email protected]
The Converse of the Mean Value Theorem May Fail Generically
by J. M. Borwein and Xianfu Wang
[email protected]
A Trio of Triangular Number Theorems
by John A. Ewell
[email protected]
THE EVOLUTION OF...
Exceptional Objects
by John Stillwell
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PROBLEMS AND SOLUTIONS
REVIEWS
Goodbye, Descartes
By Keith Devlin.
Reviewed by Dan Schnabel
[email protected]
Fourier Analysis and Boundary Value Problems
By Enrique A. GonzálezVelasco.
Reviewed by Jeffrey Nunemacher
[email protected]
Fair Division: From CakeCutting to Dispute Resolution
By Steven J. Brams and Alan D. Taylor.
Reviewed by William F. Lucas
[email protected]
TELEGRAPHIC REVIEWS