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American Mathematical Monthly - March 1997

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Some Probabilistic Aspects of Set Partitions Jim Pitman


Some classical combinatorial identities derived from partitions of a finite set, involving the Stirling numbers of the second kind and the Bell numbers, are related to moments of the Poisson distribution and the distribution of the number of fixed points of a random permutation.

Building an International Reputation: The Case of J. J. Sylvester (1814-1897)
Karen Hunger Parshall and Eugene Seneta

In the nineteenth century first France and then Germany dominated the mathematical scene and set the standards for high quality research and reputation. This paper analyzes the unique set of circumstances that motivated the British mathematician, James Joseph Sylvester, to seek to establish his reputation beyond the boundaries of the British Isles at a time when most communities had a national as opposed to international focus. It also examines--through a close reading of his correspondence--the various strategies he adopted in achieving this goal and his attitudes on competing in an international arena.

The Inverse-Similarity Problem for Real Orthogonal Matrices
Geoffrey R. Goodson


This paper introduces the reader to some new results in the spectral theory of unitary operators. This is done by investigating the special case of real orthogonal matrices. We give necessary and sufficient conditions for every real orthogonal similarity between a real orthogonal matrix U and its transpose to be an involution, and we investigate what happens when there exist similarities that are not involutions. We are interested in the form such similarities can have, and what they tell us about the matrix U.

Rethinking Rigor in Calculus: The Role of the Mean Value Theorem
Thomas W. Tucker


Most calculus students do not see the need to prove what one might call the Increasing Function Theorem (IFT): a function is increasing on any interval where its derivative is nonnegative. No wonder they find suspicious the standard proof that asks them to accept on faith the Mean Value Theorem, a subtle existence theorem that, in its textbook form, Cauchy himself was unable to prove. This paper proposes using the IFT as the theoretical cornerstone for introductory calculus. A short, intuitive, self-contained, constructive proof of the IFT is presented, followed by applications to results normally obtained via the Mean Value Theorem. In particular, the transparent but oft-forgotten proof of Lagrange for his error bound on Taylor polynomials is given (just antidifferentiate repeatedly an inequality bounding the (n+1)st derivative). This approach seems more sensible and intuitive than the usual unthinking dependence on the Mean Value Theorem.

Commentary on Rethinking Rigor in Calculus: The Role of the Mean Value Theorem
Howard Swann

This article attempts a counter-attack to the headlong dash backward proposed by reformers, who chastise us when we "bemoan the absence \epsilon - \delta definition of limit." With the help of some of the new technological tools, we argue that an informal discussion of the foundations of differential calculus is fascinating and accessible. As they have throughout history, why shouldn't today's students again contemplate the abyss confronting human attempts to comprehend infinity implicit in any notion of limit, particularly when Weierstrass has contrived a remarkably clever way across? The implications of Weierstrass' success so astonished Bertrand Russell that he pronounced, "The solution of the difficulties which formerly surrounded the mathematically infinite is probably the greatest achievement of which our age has to boast." We argue that learning to understand and appreciate proofs is a gradual process; it surely is imperative to introduce the notion of mathematical proof in beginning multi-semester calculus and keep it alive even though actual proofs are few. Such an introduction is essential for later mathematics courses taken by majors and non-majors alike.

Fermat's Last Theorem, The Four Color Conjecture, and Bill Clinton for April Fools' Day
Edward B. Burger and Frank Morgan


Often in mathematics we learn from mistakes. Here, in honor of April Fools' Day, we celebrate some famous 19th century mistakes in mathematics and provide "proofs" of Fermat's Last Theorem and the Four Color Conjecture. These proofs contain the errors that were originally made in the 1800's. We also illustrate the insights gained from such mistakes and the inevitable resolution of the result. Finally, we use some slight-of-logic to prove that one of us is actually Bill Clinton.