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American Mathematical Monthly -December 2009

December 2009

Rethinking the Lebesgue Integral
By: Peter D. Lax
This article describes a new approach to the description of the space of Lebesgue integrable functions L1(K), where K is a ball in |n. The traditional approach enlarges the concept of integration and the class of integrable functions. As an afterthought (the Riesz-Fischer theorem), it is shown that the space of integrable functions is complete in the L1 norm. Today we know that the primary goal of the theory is to create a complete space, because most of the theorems of functional analysis require completeness. Accordingly we define L1 as the abstract completion in the L1 norm of the space C(K) of continuous functions. The elements of this completion are equivalence classes of Cauchy sequences of continuous functions; the remaining task is to assign to each element f of this abstract L1 a function f(x) defined almost everywhere, that is, except on a set that can be enclosed in an open set of arbitrary small volume. We say that f(x) represents f if there is a Cauchy sequence in f that converges to f(x) ae. We prove that every f is represented by some f(x), and that f(x) and g(x) are equal ae if and only if they represent the same element of L1. We then show how to derive the usual results of Lebesgue theory.

In this approach measure is a derived concept; a set S is measurable if its characteristic function represents some element of L1. The usual properties of measurable sets follow.

I hope to see this approach adopted in the teaching of the Lebesgue theory.


Computer Algebra in Systems Biology
By: Reinhard Laubenbacher and Bernd Sturmfels,
Systems biology focuses on the study of entire biological systems rather than on their individual components. With the emergence of high-throughput data generation technologies for molecular biology and the development of advanced mathematical modeling techniques, this field promises to provide important new insights. At the same time, with the availability of increasingly powerful computers, computer algebra has developed into a useful tool for many applications. This article illustrates the use of computer algebra in systems biology by way of a well-known gene regulatory network, the Lac Operon in the bacterium E. coli.


A Disorienting Look at Euler's Theorem on the Axis of a Rotation
By: Bob Palais, Richard Palais, and Stephen Rodi,,
In 1775 Euler showed that no matter how you rotate a sphere about its center, two points must end up where they began, so the result is equivalent to a rotation about the axis they determine. This was likely the first fixed point theorem, with many repercussions and generalizations. We give a determinant-free analysis of all 3x3 orthogonal matrices and obtain a similar result, then survey many other known proofs. We start by providing the first English translation of Euler's elegant geometric proof from the Latin, in which the essential hypothesis of orientation preservation is only mentioned implicitly. Expanding on this observation, we show how our analysis of general orthogonal transformations would look using Euler's geometric methods. Other proofs we examine are based on topology, determinants, Rodrigues' formula for quaternion multiplication (before Hamilton described quaternions!), Riemannian geometry, and Lie groups.


How to Recognize a Parabola
By: Bettina Richmond and Tom Richmond,
Parabolas have many interesting properties which were perhaps more well known in centuries past. Many of these properties hold only for parabolas, providing a characterization which can be used to recognize (theoretically, at least) a parabola. Here, we present a dozen characterizations of parabolas, involving tangent lines, areas, and the well-known reflective property. While some of these properties are widely known to hold for parabolas, the fact that they hold only for parabolas may be less well known.



On Orders of Subgroups in Abelian Groups: An Elementary Solution of an Exercise of Herstein
By: Robert Beals
I. N. Herstein's Topics in Algebra contains an exercise for which Herstein acknowledges he doesn't know of a solution using only material developed to that point in the text. The problem is to show that, if an abelian group G contains subgroups of orders m and n, then it contains a subgroup whose order is the least common multiple of m and n.

This appears as Exercise 26 in Section 2.5 of the second edition of Topics in Algebra (it is Exercise 11 in the first edition). We present a solution using material from Section 2.5 and earlier.

A Nonmeasurable Set from Coin Flips
By: Alexander E. Holroyd and Terry Soo,
In this note we give an example of a nonmeasurable set in the probability space for an infinite sequence of coin flips. The example arises naturally from the notion of an equivariant function, and serves as a pedagogical illustration of the need for measure theory.

A Note on Euler's Factoring Problem
By: John Brillhart
This note consists of a brief introduction to Euler's factoring problem and his results, as well as a complete and elegant solution to the problem given by Lucas and Matthews about a century later.

π p, the Value of π in
By: Joseph B. Keller and Ravi Vakil
The circumference of the "circle" in the plane under the metric, divided by twice the radius, may reasonably be called π p. A remarkable symmetry may be observed experimentally, π p = π q when 1/p+1/q=1, and Adler and Tanton asked why this was true. We explain why in an elementary manner. Behind our argument is geometric motivation, and in particular the notion of "polarity."

The Associativity of the Pythagorean Law
By: Lucio R. Berrone
The composite-iterative functional equation

arises from an iteration of the Pythagorean law expressing the length of the hypotenuse of a right-angled triangle as a function of the legs. An operation defined on and a two-variables real function are the unknowns in (1). Under the hypotheses of continuity, homogeneity, and associativity on the operation and of reflexivity on the function (among other ones of more technical characteristics), it is shown that the large class of pairs giving the general solution to equation (1) collapses to the (Pythagorean) pair .


Music: A Mathematical Offering
By: David J. Benson
Musimathics: The Mathematical Foundations of Music
By: Gareth Loy
Reviewed by: Michael Henle