Showing papers in "American Mathematical Monthly in 1982"

Combinatorial optimization: algorithms and complexity

Abstract: This clearly written , mathematically rigorous text includes a novel algorithmic exposition of the simplex method and also discusses the Soviet ellipsoid algorithm for linear programming; efficient algorithms for network flow, matching, spanning trees, and matroids; the theory of NP-complete problems; approximation algorithms, local search heuristics for NPcomplete problems, more All chapters are supplemented by thoughtprovoking problems A useful work for graduate-level students with backgrounds in computer science, operations research, and electrical engineering Mathematicians wishing a self-contained introduction need look no further—American Mathematical Monthly 1982 ed

The Mathematical Experience.

Abstract: Winner of the 1983 National Book Award! \"...a perfectly marvelous book about the Queen of Sciences, from which one will get a real feeling for what mathematicians do and who they are. The exposition is clear and full of wit and humor...\" - The New Yorker (1983 National Book Award edition) Mathematics has been a human activity for thousands of years. Yet only a few people from the vast population of users are professional mathematicians, who create, teach, foster, and apply it in a variety of situations. The authors of this book believe that it should be possible for these professional mathematicians to explain to non-professionals what they do, what they say they are doing, and why the world should support them at it. They also believe that mathematics should be taught to non-mathematics majors in such a way as to instill an appreciation of the power and beauty of mathematics. Many people from around the world have told the authors that they have done precisely that with the first edition and they have encouraged publication of this revised edition complete with exercises for helping students to demonstrate their understanding. This edition of the book should find a new generation of general readers and students who would like to know what mathematics is all about. It will prove invaluable as a course text for a general mathematics appreciation course, one in which the student can combine an appreciation for the esthetics with some satisfying and revealing applications. The text is ideal for 1) a GE course for Liberal Arts students 2) a Capstone course for perspective teachers 3) a writing course for mathematics teachers. A wealth of customizable online course materials for the book can be obtained from Elena Anne Marchisotto (elena.marchisotto@csun.edu) upon request.

On Chebyshev-Type Inequalities for Primes

Abstract: (1982). On Chebyshev-Type Inequalities for Primes. The American Mathematical Monthly: Vol. 89, No. 2, pp. 126-129.

How to Beat Your Wythoff Games' Opponent on Three Fronts

Abstract: 1. Wythoff Games. Let a be a positive integer. Given two piles of tokens, two players move alternately. The moves are of two types: a player may remove any positive number of tokens from a single pile, or he may take from both piles, say k (> 0) from one and 1 (> 0) from the other, provided that I k 11 0, is an N-position for every a; the Next player moves to (0, 0) and wins. For a = 2, the position (1, 3) is a P-position: if Next moves to (0, 3), (0,2) or (0, 1), then Previous, using a move of the first type, moves to (0, 0) and wins. If Next moves to (1, 2) or to (1, 1), then Previous, using a move of the second type, can again move to (0, 0). The set of all P-positions is denoted by P, and the set of all N-positions by N.

On General Properties of Quadratic Systems

Abstract: (1982). On General Properties of Quadratic Systems. The American Mathematical Monthly: Vol. 89, No. 3, pp. 167-178.

Radon inversion-variations on a theme

Abstract: (1982). Radon Inversion–Variations on a Theme. The American Mathematical Monthly: Vol. 89, No. 6, pp. 377-423.

Differential and Difference Equations

Abstract: (1982). Differential and Difference Equations. The American Mathematical Monthly: Vol. 89, No. 6, pp. 402-407.

What is an Answer

Abstract: In many branches of pure mathematics it can be surprisingly hard to recognize when a question has, in fact, been answered. A clearcut proof of a theorem or the discovery of a counterexample leaves no doubt in the reader's mind that a solution has been found. But when an "explicit solution" to a problem is given, it may happen that more work is needed to evaluate that " solution," in a particular case, than exhaustively to examine all of the possibilities directly from the original formulation of the problem. In such a situation, other things being equal, we may justifiably question whether the problem has in fact been solved. Examples of this sort can turn up anywhere, but here we will concentrate on problems in combinatorial mathematics, specifically those of the type "how manyare there?" Such enumeration problems lie at the heart of the subject, and it is important to be able to recognize solutions when they appear. The point, of course, is that sometimes the "answer" is presented as a formula that is so messy and long, and so full of factorials and sign alternations and whatnot, that we may feel that the disease was preferable to the cure. An answer to such an enumeration question may be given by means of a generating function, a recurrence relation, or by an explicit formula. Each of these is, in essence, just an algorithm for the computation of the counting sequence that is to be determined. How do we judge the usefulness of such answers? Obviously we might be able to do many things with the answer, such as to make asymptotic estimates, to discover congruence relations, to delight in its elegance, and so forth. We're going to restrict attention here to the appraisal of solutions from the point of view of how easily they allow us to calculate the number of objects in the set that is being studied. The quality of such -formulas should therefore be judged by the usual combination of esthetic and quantitative benchmarks that are used on algorithms. In particular, the quantitative criterion is the computational complexity: the amount of work required to get an answer. We suggest here that the same criterion should be applied to enumeration formulas. We will see that a corollary of this attitude is that our decision as to what constitutes an answer may be time-dependent: as faster algorithms for listing the objects become available, a proposed formula for counting the objects will have to be comparably faster to evaluate. For concreteness, suppose that for each integer n > 0 there is a set Sn that we want to count. Let f(n) = I Sn I (the cardinality of Sn), for each n. Suppose further that a certain formula has been found, say

Gamma Function Derivation of n-Sphere Volumes

Abstract: (1982). Gamma Function Derivation of n-Sphere Volumes. The American Mathematical Monthly: Vol. 89, No. 5, pp. 301-302.

Mean Curvature, the Laplacian, and Soap Bubbles

Abstract: (1982). Mean Curvature, the Laplacian, and Soap Bubbles. The American Mathematical Monthly: Vol. 89, No. 3, pp. 180-198.

The Disk with the College Education.

Abstract: (1982). The Disk with the College Education. The American Mathematical Monthly: Vol. 89, No. 1, pp. 4-8.

Blowing up Singularities in Classical Mechanical Systems

Abstract: Among smooth dynamical systems, some of the most complicated arise from Newton's equations in classical mechanics. Some of these systems (including many of those studied in elementary physics courses) are integrable, and hence their phase portraits are well understood. But there are many other nonintegrable systems whose phase portraits are still far from being completely understood. Examples of these systems include the well-known n-body problem of celestial mechanics when n > 2. For these systems, the set of time-periodic solutions of the differential equation is typically dense in the phase space of the system, but there are also many other much more complicated types of solutions. These include recurrent solutions, which never close up but which return infinitely often to any prescribed neighborhood of their starting position. Sometimes these solutions wind densely about a torus; other times they fill out a dense subset of an open set in a surface of constant energy. In these cases it may be impossible to tell what type of solution will be generated by a given initial condition, no matter how accurate the analytic, qualitative or numerical techniques one uses. To make matters worse, specific Hamiltonian systems which arise in applications often suffer singularities as well. By a singularity we mean a point where the differential equation itself is undefined. A typical example of a singularity is a collision between two or more of the point masses in the Newtonian n-body problem. At collision, the differential equation breaks down: the velocities of the particles involved become undefined. A singularity or collision can create havoc among nearby solution curves. Solutions which pass near a singularity may behave in an erratic or unstable manner, and solutions which start out close to one another can end up far apart after passing by a singularity. At the other end of the spectrum in dynamical systems are the gradient-like Morse-Smale systems. These systems feature none of the complicated solution curves that appear in Hamiltoman systems. There are never any periodic or recurrent solutions. The only "interesting" solutions are the rest points or equilibrium solutions, and each of these is simple in character; either it is a sink, a source, or a saddle point. Given a particular differential equation of this type, these rest points may often be found exactly, and their character determined explicitly. Moreover, all other

Generalized Hyperbolic Functions

Abstract: (1982). Generalized Hyperbolic Functions. The American Mathematical Monthly: Vol. 89, No. 9, pp. 688-691.

On the Mean Value Theorem for Integrals

Abstract: (1982). On the Mean Value Theorem for Integrals. The American Mathematical Monthly: Vol. 89, No. 5, pp. 300-301.

Petri Nets and Marked Graphs–Mathematical Models of Concurrent Computation

Abstract: (1982). Petri Nets and Marked Graphs–Mathematical Models of Concurrent Computation. The American Mathematical Monthly: Vol. 89, No. 8, pp. 552-566.

Kantorovich-Type Inequalities

Abstract: (1982). Kantorovich-Type Inequalities. The American Mathematical Monthly: Vol. 89, No. 5, pp. 314-330.

Outer Automorphisms of S6

Abstract: (1982). Outer Automorphisms of S6. The American Mathematical Monthly: Vol. 89, No. 6, pp. 407-410.

At what Points is the Projection Mapping Differentiable

Abstract: (1982). At what Points is the Projection Mapping Differentiable? The American Mathematical Monthly: Vol. 89, No. 7, pp. 456-458.

The Width of a Chair

Abstract: Moving a large chair is not much fun. That is especially true when a door is involved. Probably the designer verifies that his chair will go through a standard door, but he never supplies instructions, and trial and error is the algorithm most frequently adopted. At present we do not know a uniformly successful alternative, but at least we can call attention to the problem and give a partial solution. In the case of a convex chair, the problem is easier but still not completely solved. We begin in two dimensions, with a compact set C and the closed interval I = {(x, y): x = 0, 0 < y < 1}. The goal is to determine necessary and sufficient conditions for C to pass through I by a continuous family R, of rigid motions-translations combined with rotations. It is like mailing a postcard of shape C into a slot I, and the motion can bring points of C back through the slot before the whole set ultimately passes through. For a convex set intuition suggests that a reversal of this kind, and even a rotation, should not be necessary: if the set can get through at all, it can do so without turning. We anticipate a single rotation, to make C as thin as possible in the vertical direction, and a single translation to put C to the left of I. Then if C can pass through I, translation in the x direction should do it. Our contribution is to confirm that this intuition is correct in the plane. In three dimensions it is not correct. Harold Stark has constructed convex chairs which can pass through a door, either square or circular, although no projection of the chair will fit in the doorway. The chair itself can go through, but it cannot be put into a box (or a cylinder) that will. I am extremely grateful to the referee who observed that my original example was unsatisfactory, and to Harold Stark and Mike Artin for correcting it. The smallest box into which it will fit is not known.