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American Mathematical Monthly -February 2006

MONTHLY, February 2006

Wild and Wooley Numbers
By Jeffrey C. Lagarias
lagarias@umich.edu
This paper studies the multiplicative semigroup of positive rational numbers generated by the numbers (3n+2)/(2n+1) for all positive n, which we call the Wooley semigroup, and the larger multiplicative semigroup obtained by including the additional generator 1/2, which we call the wild semigroup. The question addressed is: Which integers belong to these semigroups? This question is related to the notorious 3x+1 problem. Irreducible elements in the wild integer subsemigroup and Wooley integer subsemigroup are termed wild numbers and Wooley numbers, respectively. The paper conjecturally characterizes the wild numbers, but Wooley numbers seem more elusive.

 

Is Trivial Dynamics That Trivial?
by Alejo Barrio Blaya and Víctor Jiménez López
alejobar@um.es, vjimenez@um.es
As a rule, dynamics of continuous interval mappings whose only periodic points are fixed points (so-called type-1 mappings) are dismissed as trivial. This is just natural, for the orbits of all points of the interval under such a function converge to its fixed points. However, as we show in this paper, things are not that clear: we construct a type-1 mapping with the property that almost all points of the interval (in the sense of Lebesgue measure) are sensitive to initial conditions, that is, for almost every point x there are a positive number and points y is close to x as required with the property that x and y are separated by a distance greater than under some iterate of the map. Yet this is a rather pathological situation. We prove that the set of sensitive points for any type-1 mapping is of the first Baire category and, under some mild additional assumptions (satisfied, for instance, by all piecewise monotone mappings) that it is countable and cannot be dense.

 

Waiting times for patterns and a method of gambling teams
by Vladimir Pozdnyakov and Martin Kulldorff
vladimir.pozdnyakov@uconn.edu, martin_kulldorff@hms.harvard.edu
A monkey types letters of the alphabet in a random fashion. The monkey stops when it finally types one of the following three words: END, STOP, or DONE. On average, how long does it take to get to that point? A simple but elegant argument based on martingale theory and a method of gambling teams is presented.

 

Simpson’s Rule is Exact for Quintics
by Louis A. Talman
talmanl@mscd.edu
In this article, we use tools accessible to freshman calculus students to develop exact–though usually uncomputable–expressions for the error that results in replacing a definite integral with its midpoint rule, trapezoidal rule, or Simpson’s rule approximation. Among the tools we use is an extended version of the first mean value theorem for integrals. We obtain not only the classical estimates that appear in calculus books, but estimates for functions less smooth than the classical results require. We show, in particular, how to compute the exact error for a Simpson’s rule approximation to an integral of a quintic polynomial.

 

Notes

A Sequence of Polynomials for Approximating Arctangent
by Herbert A. Medina
hmedina@lmu.edu

A Cantor-Bernstein Theorem for Paths in Graphs
by Reinhard Diestel and Carsten Thomassen
C.Thomassen@mat.dtu.dk, diestel@math.uni-hamburg.de

A Mean Paradox
by John Konvalina, Jack Heidel, and Jim Rogers
johnkon@unomaha.edu

Another Elementary Proof of the Nullstellensatz
by Enrique Arrondo
Enrique_Arrondo@mat.ucm.es

The Sphere is Not Flat
by P. L. Robinson
robinson@math.ufl.edu

A Note on -Permutations
by Daniel J. Velleman
djvelleman@amherst.edu

Problems and Solution

Reviews

R. L. Moore: Mathematician and Teacher
by John Parker.
Reviewed by David E. Zitarelli
david.zitarelli@temple.edu

 

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