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American Mathematical Monthly: November, 1997

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November, 1997

Mathematics, Statistics, and Teaching
by George W. Cobb and David S. Moore


In the ideal Platonic world of mathematics, we can start with a probabilistic chicken and use deductive logic to lay a statistical egg, but in the messier world of empirical science, we must start with the egg as observed data and construct a prior probabilistic chicken as an inference.

How does statistical thinking differ from mathematical thinking? What is the role of mathematics in statistics? If you purge statistics of its mathematical content, what intellectual substance remains? This paper offers some answers to these questions and illustrates them using a sequence of examples that provide an overview of current statistical practice. Along the way, and especially toward the end, we point to some implications for the teaching of statistics.

When is a Linear Operator Diagonalizable?
by Marco Abate


Have you ever wondered whether that mystifying 10 x 10 matrix were diagonalizable? Computing the characteristic polynomial is useless; there are no self-evident eigenvalues in view. And you don't know how to write a program to make the computer do the work. And you are losing your sleep about it (well, almost). Grieve no more! We are proud to present an explicit pen-and-paper procedure to let you decide whether any given square matrix is diagonalizable, both over the complex and over the real numbers! Read and try yourself; your sleep won't be troubled anymore.

Energy Arguments in the Theory of Algorithms
by Eric Bach


The theme of this article is that conservation laws provide an interesting way of thinking about discrete algorithms. In particular, several classic results in this area are easily verified once one has an appropriate physical model. The article also introduces the reader to the potential-function method of algorithm analysis.

Quine's NF--60 Years On
by Thomas Forster


In this brief retrospective the author motivates and outlines the progress of research in a branch of set theory called "NF" after its first appearance 60 years ago in this Monthly article entitled "New Foundations in Mathematical Logic."

Remarks on Sharkovsky's Theorem
by Michal Misiurewicz


If we iterate a continuous map of an interval into itself and take the periods of all its periodic points, what set can we get? Over 30 years ago, Sharkovsky's Theorem provided a full answer to this question. Although some parts of this theorem are widely known, it is good to recall what was proved then, and where to find simple proofs now.

Correction to: Zaphoid Beeblebrox's Brain and the Fifty-Ninth row of Pascal's Triangle
by Andrew Granville


Pascal's triangle modulo primes is well-known to exhibit a self-similar structure that is easily described. In our 1992 paper we showed how to describe the more complicated self-similar structure modulo prime powers. Going into specifics modulo powers of two, we generalized Glaisher's result that the number of odd entries in every row is a power of two, by showing that the number of entries in every row that are $a$ mod $q$, is either zero or a power of 2 for $a$ mod $q$ = 1 mod 4, or 3 mod 4, or 1 mod 8, or 3 mod 8, or 5 mod 8, or 7 mod 8. Unfortunately the induction hypothesis proving this result modulo 8 went awry in our 1992 paper; so in this corrigendum we give a correct induction hypothesis (this time, with a complete proof) based on those same ideas.


A Simple Formula for pi
by Victor Adamchik and Stan Wagon

Borsuk-Ulam Implies Brouwer: A Direct Construction
by Francis Edward Su

The Evolution of...

Does Mathematics Distinguish Certain Dimensions of Spaces
by Zdzislaw Pogoda and Leszek M. Sokolowski

Problems and Solutions


Knots and Surfaces: A Guide to Discovering Mathematics.
By David W. Farmer and Theodore B. Stanford

Knots and Surfaces.
By N. D. Gilbert and T. Porter

Reviewed by William D. Dunbar

The Book of Numbers.
By John Horton Conway and Richard K. Guy

Reviewed by Andrew Bremner

Telegraphic Reviews

The Authors