Showing papers in "American Mathematical Monthly in 1996"

Sums of Powers of Integers

Abstract: (1996). Sums of Powers of Integers. The American Mathematical Monthly: Vol. 103, No. 3, pp. 201-213.

The argument principle for harmonic functions

Abstract: (1996). The Argument Principle for Harmonic Functions. The American Mathematical Monthly: Vol. 103, No. 5, pp. 411-415.

On the Cyclotomic Polynomial Φpq (X)

Abstract: (1996). On the Cyclotomic Polynomial Φpq (X) The American Mathematical Monthly: Vol. 103, No. 7, pp. 562-564.

Catalan's conjecture

Abstract: It may be observed that 8 and 9 are consecutive numbers in this sequence. The first problem is: Are there any other consecutive integers in the above sequence? How many pairs of consecutive integers? Finitely many? Infinitely many? I may also consider the sequence of all proper powers, which includes also 5th powers, 7th powers, llth powers, etc... (note that powers with even exponents are squares, powers with exponent multiples of 3 are cubes . . . ) The same question may be asked: Are there consecutive powers other than 8 and 9? But for the sequence of all powers, a new problem makes sense: Are there three consecutive integers that are proper powers? Since powers grow very fast, lists of powers are necessarily very limited and, besides 8 and 9, no consecutive powers have ever been observed. This is an indication to keep in mind, but one should be careful before jumping to any conclusion. Just think, for example, that up to 100, 10% of the numbers are squares, up to 10,000, 1% are squares, up to 1,000,000, 1 in 1,000 are squares, and so on. Yet, Lagrange proved that despite the increasing scarcity of squares, every natural number is the sum of at most 4 squares. The squares seem to occupy strategic positions. Of course, ours is a different problem. Similar problems may be asked with the following sequence. Let a, b be integers, 1 < a < b and consider the sequence of all powers of a or of b. For example, if a = 2, b = 3, it is the sequence

Return to the Riemann Integral

Abstract: (1996). Return to the Riemann Integral. The American Mathematical Monthly: Vol. 103, No. 8, pp. 625-632.

Giuga's Conjecture on Primality

Abstract: G. Giuga conjectured that if an integer n satisfies \sum\limits_{k=1}^{n-1} k^{n-1} \equiv -1 mod n, then n must be a prime. We survey what is known about this interesting and now fairly old conjecture. Giuga proved that n is a counterexample to his conjecture if and only if each prime divisor~$p$ of~$n$ satisfies (p-1) \mid (n/p-1) and p \mid (n/p-1). Using this characterization, he proved computationally that any counterexample has at least 1000 digits; equipped with more computing power, E. Bedocchi later raised this bound to 1700 digits. By improving on their method, we determine that any counterexample has at least 12000 digits. We also give some new results on the second of the above conditions. This leads, in our opinion, to some interesting questions about what we call Giuga numbers and Giuga sequences.

Transitive cellular automata are sensitive

Abstract: The notion of chaos is a very appealing one, and it has intrigued several scientists (see [1, 2, 5, 7] for some work on the properties that characterize a chaotic process). There are simple deterministic dynamical systems that exhibit unpredictable behavior. Though counter-intuitive, this fact has a very clear eEplanation. The lack of infinite precision causes a loss of information which is dramatic for some processes which quickly lose their deterministic nature to assume a non deterministic (unpredictable) one. This observation leads to the intuition that investigations about chaos are intrinsically of an interdisciplinary nature. Indeed, the study of chaos draws its deeper methods of analysis from mathematics, it owes to physics a treasure of important problems, and it brings challenges to the science of computing. The reason for this last fact relies on the above observation that a chaotic behavior lies in between two different modes of computation, determinism and nondeterminism, whose quantitative comparison is central to the main open questions in the theory of computing [6]. A chaotic phenomenon can indeed be viewed as a deterministic one, in the presence of infinite precision, and as a nondeterministic one, in the presence of finite precision constraints (see Figure 1). Thus one should look at chaotic processes as at processes merged into time, space, and precision bounds, which are the key resources in the science of computing.

A simple proof of the gale-ryser theorem

Abstract: (1996). A Simple Proof of the Gale-Ryser Theorem. The American Mathematical Monthly: Vol. 103, No. 4, pp. 335-337.

A Geometric Interpretation of the Solution of the General Quartic Polynomial

Abstract: (1996). A Geometric Interpretation of the Solution of the General Quartic Polynomial. The American Mathematical Monthly: Vol. 103, No. 1, pp. 51-57.

On a Converse of Sharkovsky's Theorem

Abstract: (1996). On a Converse of Sharkovsky's Theorem. The American Mathematical Monthly: Vol. 103, No. 5, pp. 386-392.

The Binomial Coefficient Function

Abstract: (1996). The Binomial Coefficient Function. The American Mathematical Monthly: Vol. 103, No. 1, pp. 1-17.

All the Way with Gauss-Bonnet and the Sociology of Mathematics

Abstract: I was stimulated to write this story by the discussion in The American Mathematical Monthly between Peter Hilton and Jean Pederson on the one hand and Branko Grunbaum and G. C. Shephard on the other hand [HP] [GS]. The discussion as well as my story involves the Euler-Poincare Number, alias the Euler Characteristic. The discussion centers on whether the Euler-Poincare Number should be discussed in a historical way without mentioning the vast and dramatic generalization and depth of understanding that this most interesting invariant has acquired in this century. My position in this discussion is that Topology should not be viewed as an advanced subject whose theorems and concepts should be avoided until graduate school. Rather it is the study of continuity, and thus underlies the most basic geometric results. In this paper I show how the basic concept of angle leads naturally to the basic topological ideas of degree of mapping and of the EulerPoincare Number. My stozy spans the history of mathematics. It concerns what may be the most widely known non-obvious theorem of mathematics and it contains the same stunning generalization that characterizes the recent history of the Euler-Poincare number. In fact, it concerns one of the most important and earliest of the applications of the Euler-Poincare number. It shows the fickleness of mathematical fame, it shows the unreasonable power of unreasonable points of view, and it shows how easy it is for mathematicians to miss and forget beautiful and important theorems as well as simple and revealing points of view. This is a history of the Gauss-Bonnet theorem as I see it. I am not a mathematical historian. I quote only secondary sources or first hand papers that I quickly scanned, and I did not conduct any thorough interviews. Nonetheless, I am writing this history because I have contributed the last sentence to it (for the moment). I especially want to acknowledge the help of Hans Samelson. His scholarship greatly altered the thrust of earlier versions of this paper. He discovered Satz VI. He informed me of many points in this history; about Gauss' work, Descartes work, and Hopf's work. And he was a student of Hopf who generalized the GaussBonnet theorem himself.

A Note on “Impossible” Paper Folding

Abstract: (1996). A Note on “Impossible” Paper Folding. The American Mathematical Monthly: Vol. 103, No. 3, pp. 240-241.

On accumulation points of ratio sets of positive integers

Abstract: [BHP] E. Bombieri, D. C. Hunt, and A. J. van der Poorten, Determinants in the study of Thue's method and curves with prescribed singularities, J. Experimental Mathematics, to appear. [D] C. L. Dodgson, Condensation of Determinants, Proceedings of the Royal Society of London 15 (1866), 150-155. [M] P. A. MacMahon, CombinatoryAnalysis, vol. 2, Cambridge University Press, 1918. [reprinted by Chelsea, 1984].

Polytope Projection and Projection Polytopes

Abstract: Imagine yourself as the commander of a space ship. Liftoff was a piece of cake, and since then you have been gliding merrily along. But then comes the bad news: A Klingon ship is approaching, and you must prepare for the attack. More bad news: Your batteries are running low! The good news is that your solar cells are working and you are close to a bright star. Thus you can recharge your batteries, but you must certainly do that as quickly as possible. You analyze the situation. Since the solar cells are distributed evenly over the surface of the ship, you decide that you should rotate the ship so that its "face area" is maximized with respect to the light source (assuming that you are still so far from the star that the incoming rays are practically parallel). A similar but opposite problem arises when you approach a star that emits harmful radiation. You then want to minimize the exposure to the radiation and therefore to minimize the face area in the direction of the star. In these problems, you are in control of a body in 3, and you want to turn the body so as to maximize or minimize its "shadow area"with respect to a particular direction of projection (the direction of the incoming rays). In a mathematically equivalent formulation, you may regard the body as being fixed and then look for a direction that maximizes or minimizes the area of the body's projection on a plane orthogonal to the direction. Projections belong to the basic tools in many areas of mathematics. While the projection on a given subspace can be expressed as a simple matrux operation applied to the original body, it is not so clear how to find projections that are "optimal"with respect to an application that one may have in mind. Problems of this kind occur in a great variety of situations with a similarly great variety of (more or less explicit) criteria for what is a good projection. Examples include the analysis of statistical, astronomical or linguistic data, and also the design and analysis of algorithms for manifold applications. We do not want to elaborate on these applications here; the goal of this paper really is to present some of the (as we hope the reader will agree) beautiful mathematics underlying the special projection problems of maximizing or minimizing the "shadow area" and their higherdimensional analogues involving orthogonal projections of a body in 114Z1 onto an (n1)-dimensional subspace. We assume that the body in question is an ndimensional convex polytope. When n = 3, this seems to be a reasonable assumption in the case of the space ship (see Figure 1). It is not hard to see that when n = 2 (so that we are projecting a convex polygon P onto various lines), the maximum projection-length is equal to P's diameter and the minimum projection-length is equal to P's width (the minimum distance between two parallel supporting lines of P) (see Figure 2). Thus the n-dimensional task considered here is one of several ways of extending to 114Z1 the classical Euclidean task of computing the diameter and the width of a polygon.

Reverend Charles to the aid of Major Percy and Fields-Medalist Enrico

Abstract: Voltaire said that Archimedes had more imagination than Homer. Unfortunately, most of mathematicians' creativity can be appreciated only by mathematicians themselves. Sometimes, however, mathematicians employ their imagination to do non-mathematical activity. Notable examples are Multi-Millionaire Richard Garfield, the Reverend Charles Dodgson, and Major Percy MacMahon, who respectively developed: 'Magic: The Gathering' (the game of our decade), Alice, and an early version of Instant Insanity. This is not to say that their imagination did not also help mathematics proper. In this quickie, I observe how Dodgson's rule for evaluating determinants [D] (for any n x n matrix A, let Ar(k, I) be the r x r submatrix whose upper leftmost corner is the entry ak l),

When Does A*A = B*B and Why Does One Want to Know?

Abstract: (1996). When Does A*A = B*B and Why Does One Want to Know? The American Mathematical Monthly: Vol. 103, No. 6, pp. 470-482.

A Hundred Years of Prime Numbers

Abstract: (1996). A Hundred Years of Prime Numbers. The American Mathematical Monthly: Vol. 103, No. 9, pp. 729-741.

The Relationship between AB and BA

Abstract: (1996). The Relationship between AB and BA. The American Mathematical Monthly: Vol. 103, No. 7, pp. 578-582.

Random Triangles in n Dimensions

Abstract: (1996). Random Triangles in n Dimensions. The American Mathematical Monthly: Vol. 103, No. 4, pp. 308-318.

Tantrices of Spherical Curves

Abstract: While evety loop cJ on s2 with non-vanishing speed defines a "tantricial" loop in this way, not every loop is tantricial. Here, we will completely expose the non-obvious but lovely obstruction to this converse. To begin, note that if the speed of cr never vanishes, neither will that of x. In fact, the speed of , computed relative to arclength along a, gives the curvature of cr as a curve in R3. But cr, lying on S2, can nowhere approximate a straight line to second order: its cuIvature the speed of X will never vanish. Referring non-experts to the sidebar discussions of "immersion" and "ardength" for further details, we conclude more precisely that if v immerses the circle in S2, so will .

The Euler-Gergonne-Soddy Triangle of a Triangle

Abstract: (1996). The Euler-Gergonne-Soddy Triangle of a Triangle. The American Mathematical Monthly: Vol. 103, No. 4, pp. 319-329.

What is a surface

Abstract: A search for a good definition of surface leads to the rectifiable currents of geometric measure theory, with interesting advantages and disadvantages. For details and references see Morgan's Geometric Measure Theory: a Beginner's Guide, Academic Press, 2nd edition, 1995. Figures by James F. Bredt. 1. WHAT IS AN INCLUSIVE DEFINITION OF A GENERAL SURFACE IN R3? We want to include smooth embedded manifolds with boundary, as in Figure 1, and we want to be able to allow singularities, as in the cube and cone of Figure 2.

Iteration of quaternion functions

Abstract: (1996). Iteration of Quaternion Functions. The American Mathematical Monthly: Vol. 103, No. 8, pp. 654-664.

A Geometric Approach to Determinants

Abstract: We are all happy to use pictures when we first introduce students to calculus. Why not take the same approach in linear algebra? While good progress has been made in this direction in recent years, determinants seem to have escaped this trend. Most textbooks still introduce them via cofactor expansions (see [FB] and [N] for example), the permutation definition ([AK], [Se]), or via their alternating multilinear form properties ([DL], [St]). In this article I propose a geometric introduction to determinants. The details are not new, though they are well scattered through the literature. For example, a geometric view of the 2 x 2 case is used as motivation for an algebraic approach in [DL] and [O]. What perhaps is new (or at any rate, has not been fashionable for at least a couple of generations) is that I am suggesting that the geometric view be given a defining role similar to that given the "area under the graph" definition of the integral, which we routinely use in beginning calculus courses. Before you push the panic button, I'm not suggesting that rigorous algebraic approaches be abandoned. What I am saying is that, particularly in the case of determinants, this approach is not veIy suitable for students who are meeting linear algebra for the first time. In fact, many textbooks implicitly recognize this problem by relegating some key proofs to later sections or appendices, so that students may avoid them or, at any rate, take them on trust (see [FB], [N]). In my own institution, I see a geometric approach as being appropriate for our first year linear algebra students, while an algebraic approach is more appropriate for our advanced courses. Just as in the calculus context, geometry helps students to form mental images or constructs that they can use to help them understand what determinants are all about. I have anecdotal evidence that it encourages students to engage in what Blum and Kirsch call "preformal" proving [BK]. In other words, students can see or conjecture properties of the determinant, along with (geometric) explanations appropriate to their level of mathematical development. Another reason for this approach is that I want a treatment that focuses on the important properties of the determinant, without getting involved in issues that I see as being peripheral for first year students, most of whom will not major in mathematics. Despite this, I like to think that my shopping list will keep most mathematicians happy, too. Here is my list of "what every first year student should know about determinants"