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Mathematics Magazine - February 2004


Emil Post and His Anticipation of Gödel and Turing
John Stillwell
Emil Post was one of the great mathematical logicians of the 20th century, the discoverer of several results generally credited to Gödel and Turing. This article outlines Post's life and career, explains his distinctive approach to the famous incompleteness and unsolvability theorems, and compares it with the approaches of Gödel and Turing.

Enter, Stage Center: The Early Drama of the Hyperbolic Functions
Janet Heine Barnett
The treatment of the hyperbolic functions found in many current calculus textbooks typically includes two common features: a comment on the applicability of these functions to certain physical problems, and a remark on the analogies that exist between properties of the hyperbolic functions and properties of the trigonometric functions. Implicit in this treatment is a suggestion that the eighteenth century mathematicians who first developed these functions were interested in them in order to solve physical problems such as the catenary. Left hanging is the question of whether hyperbolic functions were developed in a deliberate effort to find functions with trig-like properties that were required by the physical problems, or whether these trig-like properties were unintended and unforeseen by-products of the solutions to these physical problems. This article traces the drama of the early years of the hyperbolic functions, and finds it to be far richer than either of these plot lines would suggest.

Cauchy-Schwarz Inequality
Claudio Alsina
Proof Without Words: We present a sequence of pictures showing Cauchy-Schwarz inequality in the plane by means of some appropriate area-preserving transformations.

Simpson Symmetrized and Surpassed
Daniel J. Velleman
Simpson's rule is a well-known numerical method for approximating definite integrals. Like many other numerical integration methods, such as the midpoint rule and the trapezoid rule, it involves evaluating the function to be integrated at a sequence of sample points. One unusual feature of Simpson's rule is that it treats the even- and odd-numbered sample points differently, giving twice as much weight to the odd-numbered points. This seems counterintuitive; the sample points are evenly spaced, so once one gets away from the endpoints of the interval, every sample point looks very much like every other one. Why should adjacent sample points be treated so differently? In this paper we show that symmetrizing the treatment of even-and odd-numbered sample points can lead to improvements in Simpson's rule.


Tiling Deficient Rectangles with Trominoes
J. Marshall Ash and Solomon W. Golomb
A (right) tromino is a 2 x 2 square with a 1 x 1 corner removed. Let m and n be integers ≥ 2. A dog-eared rectangle is an m x n rectangle with a 1 x 1 corner removed. A dog-eared rectangle can be tiled with trominoes exactly when 3 divides its area mn-1. This is not always true of m x n rectangles that have one square removed from some place other than the corner; exactly when it is not true is determined. When two squares are removed from a rectangle, the situation is less clear. For every rectangle there is a pair of 1 x 1 squares whose removal produces a shape that cannot be tiled by trominoes, but what happens in general is not known.

Proof Without Words: Z x Z Is a Countable Set
Des MacHale
A winding path starting at the origin shows that the Cartesian product of two copies of the integers is a countable set.

Extremal Curves of a Rotating Ellipse
Carl V. Lutzer and James E. Marengo
Draw an ellipse and then carefully cut it out. Put a pin through the center, attach it to a backing, and then mark the “lowest” point. Turn it a little and then mark the new location of the lowest point. After rotating several times and marking the location of the lowest point at each step, you notice that the shape you're making appears to be an ellipse that is parallel and tangent to the original. We demonstrate that the so-called “extremal curve” parameterized by the lowest point of a rotating ellipse is not, itself, an ellipse, derive a Cartesian equation describing it, and investigate some of its interesting properties.

A Question of Limits
Paul H. Schuette
Limits are a staple of calculus courses, but the treatment of limits in multivariable calculus tends to be somewhat minimal, with most texts restricting their attention to rational functions. We shall first consider the limit of a nonrational function whose numerator is sin x + sin y and whose denominator is + y, as (x,y) → (0,0) and x + y ≠ 0 .

This limit is evaluated and generalized in two dimensions. We shall also characterize a class of functions for which a specific type of limit exists, and make a connection with the strong derivative of a function. We also extend and generalize our results from 2 to n dimensions, noting that there are significant differences in behavior of limits between the two dimensional case and that of three or more dimensions.