American Mathematical Monthly -APRIL 2001

APRIL 2001

Linear Algebra to Quantum Cohomology: The Story of Alfred Horn's Inequalities
by Rajendra Bhatia
rbh@isid.ac.in
A long-standing problem in linear algebra--Alfred Horn's conjecture on eigenvalues of sums of Hermitian matrices--has been solved recently. The solution appeared in two papers, one by Alexander Klyachko in 1998 and the other by Allen Knutson and Terence Tao in 1999. This has been followed by a flurry of activity that has brought to the mathematical centre-stage what for many years had been somewhat of a side-show. The aim of this article is to describe the problem, its origins, some of the early work on it, and some ideas that have gone into its solution.

A Proof of Thebault's Theorem
by R. Shail
r.shail@surrey.ac.uk
In the 1938 volume of the MONTHLY the prolific French geometer Victor Thébault posed a problem in Euclidean geometry concerning the collinearity of the incentre of a triangle and the centres of two circles drawn to touch a line through a vertex of the triangle, the opposite side, and the circumcircle. It was not until some 45 years later that an elaborate proof was given. We provide an elementary proof using the methods of Cartesian coordinate geometry. Computer algebra is used to assist with some of the calculations, but no advanced algebraic methods are employed.

The Convergence of an Euler Approximation of an Initial Value Problem is not Always Obvious
by Samer S. Habre, John H. Hubbard, and Beverly H. West
bhw2@cornell.edu
We consider the initial value problem dx / dt = (|x|)^1/2 with x (-2) = -1. Uniqueness of solutions fails when x = 0, and there is a continuum of solutions u(t) with u (-2) = -1, for which u (2) takes all the values between 0 and 1. But what about numerical approximations to solutions? Do they converge to a limit as ? No, contrary to the expectation even of many a professional mathematician. We show that as , the Euler approximations approach all the solutions, and how, as , the value u_h(2) oscillates between cusp values at 0 and maxima >1.

The Missing Spectral Basis in Algebra and Number Theory
by Garret Sobczyk
sobczyk@mail.udlap.mx
The fundamental concept of a spectral basis is missing in elementary mathematics! Beginning with one of the oldest and most venerated theorems in mathematics, the Euclidean Algorithm, we define the spectral basis of the modular numbers, and the spectral basis of the modular polynomials. Using a spectral basis simplifies proofs of elementary results in number theory, in factor rings of polynomials, and in linear algebra, and it gives the classical Lagrange and Hermite interpolation polynomials. While all of our results are well known in advanced mathematics, the concept of a spectral basis makes it possible to teach these ideas at an elementary level. The article is based on the author's experience in teaching this concept to undergraduate students in courses in linear algebra, modern algebra, and finite mathematics at the Universidad de las Américas-Puebla, Mexico.

The Mathematics of Musical Instruments
by Rachel W. Hall and Kresimir Josic
rhall@sju.edu
This article highlights several applications of mathematics to the design of musical instruments. In particular, we consider the physical properties of a Norwegian folk instrument called the willow flute. The willow flute relies on harmonics, rather than finger holes, to produce a scale that is related to a major scale. The pitches correspond to fundamental solutions of the one-dimensional wave equation. This "natural" scale is the jumping--off point for a discussion of several systems of scale construction-just, Pythagorean, and equal temperament--that have connections to number theory and dynamical systems and are crucial in the design of keyboard instruments. The willow flute example also provides a nice introduction to the spectral theory of partial differential equations, which explains the differences between the sounds of wind or stringed instruments and drums.

NOTES

A Simple Slide Rule for Finite Fields
by Holger Schellwat
holger.schellwat@nat.oru.se

A General Method for Establishing Geometric Inequalities in a Triangle
by Razvan Alin Satnoianu
razvansa@maths.ox.ac.uk

A Theorem of D. J. Newman on Euler's Function and Arithmetic Progressions
by J.M. Aldaz, A. Bravo, S. Guitiérrez, and A. Ubis
aldaz@dmc.unirioja.es

A Counterexample for the Two-Dimensional Density Function
by Liu Wen

The Remarkable Tetron
by N.S. Astapov and N.C. Noland
nika@hydro.nsc.ru

PROBLEMS AND SOLUTIONS

REVIEWS

The Universal History of Numbers: From Prehistory to the Invention of the Computer.
By Georges Ifrah

Reviewed by Eli Maor
emaor@suba.com

Mathematics: Frontiers and Perspectives.
Edited by V. Arnold, M. Atiyah, P. Lax, and B. Mazur

Reviewed by Harsh V. Pittie

EDITOR'S ENDNOTES