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American Mathematical Monthly - October 2001

OCTOBER 2001

E.H. Moore’s Early Twentieth-Century Program for Reform in Mathematics Education
by David Lindsay Roberts
robertsdl@aol.com
E.H. Moore of the University of Chicago (1862-1932) was a productive pure mathematician and vigorous academic politician who inspired the admiration of his students and colleagues. In his retiring address as president of the American Mathematical Society, in December 1902, Moore proposed an ambitious program of educational reform for secondary schools and colleges. He championed the "laboratory method" of instruction and called for mathematicians to take a larger role in educational issues. This paper examines Moore’s reform proposals within the context of his career and in relation to the evolution of mathematics and mathematics education in his era. Although Moore’s educational activism was not without influence, and in particular helped instigate the founding of the Mathematical Association of America, mathematics education in the United States did not develop as he envisioned in the early twentieth century. The paper suggests some reasons why Moore’s program was not enacted, despite his advantages as an educational spokesman.


 

Bidiagonal Factorizations of Totally Nonnegative Matrices
by Shaun Fallat
sfallat@math.uregina.ca
Motivated by types of positivity that are closed under matrix multiplication, we introduce the class of totally nonnegative matrices. A matrix is totally nonnegative if the determinants of all of its square submatrices are nonnegative. Beginning with Gantmacher and Krein in the thirties and continuing with Karlin, totally nonnegative matrices have had an illustrious history and arise in numerous applications. There has been a recent surge of papers studying this class, most of which are motivated by a surprising matrix factorization. Loosely stated, any totally nonnegative matrix can be factored into a product of matrices in which each factor has nonnegative main diagonal entries and at most one nonzero entry off the main diagonal, which must be positive and occur on the super- or sub-diagonal. We develop this factorization result, survey the relevant history along the way, and explore several applications that demonstrate its usefulness for investigating properties of totally nonnegative matrices.

 

Sequential Searches: Proofreading, Russian Roulette, and the Incomplete q-Eulerian Polynomials Revisited
by Don Rawlings
drawling@math.calpoly.edu
There are many natural contexts in which a sequence of searches is conducted for lost objects. Ranging from the commonplace to the exhilarating to the risky, they include proofreading, treasure hunts, and the clearing of dangerously littered live munitions from a region following a war.
A single fundamental question arises in all such scenarios: What is the expected number of searches needed to find all, or some acceptable percentage, of the lost objects? Time and resources are only finite after all!
Beyond resolving the fundamental question, our approach has some amusing sidelights. First, the relevant distributions lead to a probabilistic proof of an identity for the q-Eulerian polynomials studied by Carlitz and to the discovery of a new formula for the incomplete q-Eulerian polynomials introduced by Herbranson and Rawlings. Also, the machismo factor is computed for Sandell’s fair Russian roulette with several players.

 

Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields
by Jan Holly
jeholly@colby.edu
Ultrametric spaces can be pictured as trees, which give an intuitive feel for distance. The tree picture easily displays properties such as the strong triangle inequality and the fact that every point in a given disc is a center of that disc. This paper presents the tree picture and gives an introduction to the field of p-adic numbers in the process. In addition, the tree picture serves for valued fields in general.

 

Jacobi Elliptic Functions from a Dynamical Systems Point of View
by Kenneth R. Meyer
ken.meyer@uc.edu
We present a differential equation definition of the Jacobi elliptic functions and use it to obtain many of their elementary properties. This presentation illustrates the power of the dynamical systems approach to the theory of differential equations. The basic properties of these functions are an immediate applications of the fundamental theorems on existence, uniqueness, and continuous dependence of solutions on initial conditions, and thus this presentation gives some excellent examples for a course in differential equations. These functions are used to solve the pendulum equation and the undamped Duffing equation.

 

Large Torsional Oscillations in Suspension Bridges Visited Again: Vertical Forcing Creates Torsional Response
by P.J. McKenna and Cilliam O’Tuama
mckenna@math.uconn.edu
This paper continues an investigation into the large oscillations seen in the Tacoma Narrows suspension bridge before its famous collapse in 1940. Our previous paper studied the coupling of the torsional and vertical modes of oscillation, using a piecewise linear restoring force to model the resistance of the cable to expansion from the unloaded state. This paper explores the effect of having a smoother version of the same resistance. We urge the reader to get involved in the experiment. Download a Windows version of our visualization software from http://euclid.ucc.ie/applmath/projects/bridge/ and try some experiments for yourself. Let the authors know of any interesting results.

 

Absolutely Abnormal Numbers
by Greg Martin
gerg@math.toronto.edu
Despite the fact that almost all real numbers are absolutely normal - that is, the digits in their expansion to any base occur in all possible configurations with the expected frequency - not one specific example of an absolutely normal number is known. In this paper we investigate the opposite extreme: absolutely abnormal numbers - numbers that are normal to no base whatsoever. We give a simple construction that yields such absolutely abnormal numbers in abundance.

 

Notes

Limiting Distributions for Derangments
by Christopher Stuart
christopher.stuart@enmu.edu

The Finite Field Kakeya Problem
by Keith Rogers
kmr29@cam.ac.uk

The Fundamental Theorem of Algebra Revisted
by Airton Von Sohsten de Medeiros
airton@impa.br

Other Versions of the Steiner-Lehmus Theorem
by Mowaffaq Hajja
mhajja@sharjah.ac.ae

A Simpler Proof of sin p z = p z ? (1-z2/k2)
by Wladimir de Azevedo Pribitkin
wladimir@princeton.edu

A Quick Proof for the Volume of n-Balls
by Jean B. Lasserre
lasserre@laas.fr

Reviews

Niels Henrik Abel and His Times: Called Too Soon
By Arild Stubhaug
Reviewed by O. A. Laudal
arnfinnl@math.uio.no

A First Course in Fourier Analysis
By David W. Kammler
Reviewed by O. Carruth McGehee
mcgehee@math.lsu.edu

 

Dummy View - NOT TO BE DELETED