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American Mathematical Monthly - April 2003

APRIL 2003

An Application of Markov Chain Monte Carlo to Community Ecology
by George W. Cobb
gcobb@mtholyoke.edu

What must surely be history's most acrimonious academic exchange about the meaning of (0,1)-matrices provides a context for introducing some ideas of Markov chain Monte Carlo. As the trigger for three decades of vituperation among community ecologists, the matrices record presence (1) and absence (0) on islands (rows) of various animal species (columns). As a source of mathematics, these same matrices serve as vertices of a very large graph, one whose order exceeds 1017. Walking at random on the graph generates Markov chains whose limiting behavior can be used for a variety of statistical purposes, such as testing hypotheses. This article also includes some new variations on these ideas, developed as part of a Research Experiences for Undergraduates program at Mount Holyoke College.

 

On the Pointwise Limit of Complex Analytic Functions
by Alan F. Beardon and David Minda
afb@dpmms.cam.ac.uk, David.Minda@math.uc.edu

This article is a survey of results concerning the limit of a pointwise convergent sequence of analytic functions. Two examples are given in which the limit function is not analytic, and these are simple enough to be included in a first course in complex analysis.

 

The Quest for Quotient Rings (of Noncommutative Noetherian Rings)
by S. C. Coutinho and J. C. McConnell
collier@impa.br, j.c.mcconnell@leeds.ac.uk

The theory of noncommutative Noetherian rings is now an established branch of algebra with applications to many other areas of mathematics such as Lie algebras and quantum groups. The seeds out of which this theory developed were the theorems on quotients rings of rings with an ascending chain condition proved by Alfred Goldie in the late 1950s. In this paper we follow Goldie from his time as a student at Cambridge University, through the war years, to his work on quotient rings. We watch as he solves, one by one, the problems that would lead him to his key theorems. The paper also contains detailed, but short, proofs of the theorems.

 

Fractals With Positive Length and Zero Buffon Needle Probability
by Yuval Peres, Károly Simon, and Boris Solomyak
peres@stat.berkeley.edu, simonk@math.bme.hu, solomyak@math.Washington.edu
 

A celebrated theorem of Besicovitch (1938) guarantees the existence of compact sets Λ in the plane that have positive "length" (i.e., positive one-dimensional Hausdorff measure) yet are disjoint from the line y = ax + b for Lebesgue almost every (a,b) (equivalently, the orthogonal projection of Λ on almost every line through the origin has zero length). One such set Λ is the "four-corner" set, which is the Cartesian square of the Cantor middle-half set. We give an elementary proof for this and related examples.

 

Problems and Solutions

Notes
 

Nested Square Roots of 2
by L. D. Servi
servi@ll.mit.edu

Strategies for the Weakest Link
by Nigel Boston
boston@math.wisc.edu

Proofs of Korovkin’s Theorems via Inequalities
by Mitsuru Uchiyama
uchiyama@fukuoka-edu.ac.jp

The Zeros of the Partial Sums of the Exponential Series
by Peter Walker
peterw@aus.ac.ae

Reviews

Symmetry in Mechanics: A Gentle, Modern Introduction.
by Stephanie Frank Singer
Reviewed by Richard Montgomery
rmont@math.ucsc.edu

Wave Motion.
by J. Billingham and A. C. King
Reviewed by Jeffrey Rauch
rauch@math.lsa.umich.edu

Editor’s Endnotes

 

Dummy View - NOT TO BE DELETED