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This paper contains an exposition of two theorems on Banach spaces. Let X and Y be real Banach spaces and let f be a map from X to Y such that f (0) = 0. The Mazur-Ulam Theorem says that if such a map is isometric (distance-preserving) and surjective, then it is linear. In general, it is necessary to assume that f is surjective. However, for a very large class of spaces this assumption is not necessary for the conclusion of the theorem. Let epsilon be a given positive number. An epsilon-isometry is a map from X to Y that preserves distances to within epsilon. It is known that if f is such a map between real Banach spaces, if f(0) = 0, and if f is surjective, then there exists a linear isometry g such that f and g are uniformly close. This was proved in 1945 by Hyers and Ulam for the special case of Hilbert spaces, and then extended to Banach spaces over the years by several authors. Here the assumption that f be surjective is necessary even when X and Y are Euclidean spaces.
These two theorems are explained in this paper. It is also shown that if X is a Euclidean space, and Y=X, then the conclusion of the Hyers-Ulam Theorem is unaffected if surjectivity is dropped from the hypotheses. The method of proof uses the Hyers-Ulam ideas, but includes modifications that lead to sharp bounds.
How To Do Monthly Problems With Your Computer
by Herbert S. Wilf, IstvÂ‡n Nemes, Marko Petkovsek, and Doron Zeilberger firstname.lastname@example.org, Istvan.Nemes@risc.uni-linz.ac.at, Marko.Petkovsek@mat.uni-lj.si, email@example.com
The problem of finding simple evaluations of major classes of sums that involve factorials, binomial coefficients, and their q-analogues, has been completely solved. Sums that have the rather general form specified in Section 3 can all be done algorithmically, that is to say, you can do them on your own PC. Your computer evaluates the sum as a simple formula, if that's possible, and gives you a proof that you can check, or gives you a proof that your sum cannot be "done" in simple closed form, if that is the case.
We first briefly describe the algorithms and the theory that have achieved this goal. Second, to illustrate both the scope of the method and the fact that in some interesting cases human intervention still helps, we show how these computer methods would have fared in attacking 27 problems that have appeared over the years in the Problems section of this MONTHLY.
We study under what conditions on the real number c, a c b, and on the function w on [a,b], can w be simultaneously even (or odd) on the three intervals [a,c], [c,b], and [a,b]. We prove that this "simultaneous symmetry" on all three intervals greatly restricts the function. For example, if c is not rationally related to b - a, then the function must be constant. If we consider only two of the three intervals, it is not difficult to see that there exist many functions that are simultaneously even (odd) on the two intervals. We prove that simultaneous symmetry on two of these three intervals gives a "near invariance" of the function as well as other restrictions. These restrictions have surprising connections to questions that arise naturally in ergodic theory--in particular, the study of additive cocycles and coboundaries.
by Richard A. Mollin
From the recreational mathematician to the research mathematician, prime producing quadratic polynomials have held a longstanding fascination. These polynomials have been ubiquitous in the literature for centuries, but quite often they appear merely as curiosities, or with explanations that are incomplete. This article is intended to explain the reasons behind this prime production to anyone from the uninitiated reader to the expert.
We take the reader from the basic idea of a quadratic field, through the arithmetic of ideals therein. This provides the tools to explain, for example, the (well-known) reasons why Euler's polynomial x^2 + x + 41 is prime for x = 0, 1,..., 39 (the record holder for centuries as a consecutive, distinct quadratic prime-producer for an initial range of input values), and the (not so well-known) reasons why 36x^2 -810x + 2753 is prime for x = 0, 1,..., 44 (the newly discovered record holder). The reasons are given in terms of class group structures of quadratic fields, which the reader is brought to understand via the aforementioned development from the basics. Furthermore, complete lists are given of quadratic polynomials, having negative discriminant, that generate consecutive, distinct primes for an initial range, and these lists are shown to be complete under the assumption of a suitable Riemann hypothesis. Attendant topics are also discussed in detail, such as the "density" of primes produced by such polynomials and the current record holder in that regard, as well as material buried at various depths throughout the literature. We conclude with a discussion of a related MONTHLY article by Boston and Greenwood [102 (1995) 595-599].
The PoincarÂŽ-Miranda Theorem
by Wladyslaw Kulpa
A direct proof is given of the PoincarÂŽ generalization to n dimensions of Bolzano's intermediate value theorem. Relationships with the Brouwer fixed point theorem, invariance of domain, and dimension theory are given.
Lecturing at the 'Bored'
by Melanie A. King
After spending much time in thoughtful reflection about ways to improve my lecturing, I posed a different challenge to myself. Could I justify the time I spend in front of the classroom, "lecturing at the bored?" In other words, what is it that keeps me "onstage", telling my students about mathematics, even after witnessing their academic successes with alternative pedagogical modalities? In this article I discuss my reasons for lecturing, and determine if they are valid enough to keep me in the limelight.
A Generalization of Wolstenholmes Theorem
by M. Bayat
A Note on the Mean Value Theorem for Integrals
by Zhang Bao-lin
When Is There a Latin Power Set?
by J. DÂŽnes
PROBLEMS AND SOLUTIONS
The Life of Stefan Banach. By Roman Kaluza
by Sheldon Axler
101 Careers in Mathematics. Edited by Andrew Sterrett
by J. Kevin Colligan