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American Mathematical Monthly: August- September, 1997

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August - September, 1997

The Many Avatars of a Simple Algebra
by S. C. Coutinho


When quantum mechanics was born in the 1920s, it seemed to suffer from a terrible flaw: the multiplication of 'quantum amplitudes' did not commute. It did not take long before physicists realised that this was no quirk of the formalism, but a fundamental fact. This led Dirac to introduce his quantum algebra. It is generated by the quantum equivalents of position and momentum, which, however, do not commute; indeed their commutator is 1, not zero. This algebra is now called the Weyl algebra, and it plays a key role in many areas of mathematics, notably the theory of D-modules and the representation theory of Lie algebras. This paper is a historical tour of the various formalisms of quantum mechanics with special regard to the way in which the Weyl algebra was incarnated in each one of them. Beginning with Heisenberg's inspired bout of hay fever of 1925, we follow the development of matrix mechanics, Dirac's formalism, and wave mechanics, and end the tour with the deformation theoretic approach that has become so popular nowadays.

Multiple Integrals of Symmetric Functions
by Tiberiu Trif


A unitary treatment of a class of multiple Riemann integrals of symmetric functions (i.e., functions that do not change their values under any permutation of the variables) is illustrated. The examples given in the paper could be used as classroom applications.

Some Inequalities for Principal Submatrices
by John Chollet

Let A be a Hermitian or positive semidefinite Hermitian matrix, A[\omega] a principal submatrix of A, and f(A) a primary matrix function. We discuss inequalities of the form f(A)[\omega] > f (A[\omega]) and survey some pertinent results on monotone and convex matrix functions.

Two Applications of Calculus to Triangular Billiards
by Eugene Gutkin


This paper is an informal and brief introduction to periodic billiard orbits in polygons. We point out that the basic questions in the subject remain open. We investigate the extrema and the mean value for a particular type of billiard orbits, which are present in every acute triangle: the Fagnano periodic orbits. Our approach is in the spirit of J. F. F. Fagnano (late 18-th century), who along with H. A. Schwarz and L. Fejer (late 19-th -- early 20-th centuries) did, albeit unknowingly, primeval research on polygonal billiards.

Early Transcendentals
by Steven H. Weintraub


Several current calculus texts have "early transcendental" versions, in which the exponential and logarithm functions are introduced early in the text. These functions are usually justified by various "hand-waving" arguments. The point of this article is to show how they may be introduced rigorously.

The Logical Structure of Computer-Aided Mathematical Reasoning
by Keith Devlin


Over the past decade or so, the professional mathematician has changed from being a person who sits at a desk working with paper and pencil to a person who spends a lot of time sitting in front of a computer terminal. This rapid transformation of mode of working has changed the nature of doing mathematics in a fundamental way. The computer does not simply "assist" the mathematician in doing business as usual; it changes the nature of what is done. In particular, the logical structure of mathematical reasoning carried out with the aid of an interactive computer system is different from the structure of the more traditional form of mathematical reasoning. Classical mathematical logic provides a model of (an idealized form of) traditional mathematical reasoning. Using a relatively new branch of mathematics called situation theory, the paper analyzes the logical structure of mathematical reasoning carried out with the aid of a computer.


The Wallet Paradox
by Kent G. Merryfield, Ngo Viet, Saleem Watson

The Weierstrass Approximation Theorem and Large Deviations
by Henryk Gzyl and JosÂŽ Luis Palacios


On the Historical Development of Infinitesimal Mathematics. Part II. The Conceptual Thinking of Cauchy.
by Detlef Laugwitz



The Sheer Joy of Celestial Mechanics. By Nathaniel Grossman
Reviewed by J. N. Anthony Danby

Strength in Numbers: Discovering the Joy and Power of Mathematics in Everyday Life. By Sherman K. Stein
Reviewed by Jennifer R. Galovich