Contents for March 1998
An Interview with Lars V. Ahlfors
Donald J. Albers
In the middle half of this century the subject of complex function theory was virtually defined by the work of Lars V. Ahlfors (1907-1996). In this 1994 interview he recalls his early years, and surveys his career from a personal perspective. (More biographical information about Ahlfors and his mathematical work appears in the February, 1998 issue of the Notices of the American Mathematical Society.)
Two Historical Applications of Calculus
Alexander J. Hahn
Two examples are given to illustrate how calculus has been energized by problems from basic science and engineering, and in turn the mathematics has enlightened and clarified these fields. The first is a statics problem from the first calculus textbook, by the Marquis De L'Hospital. It quickly translates to finding the maximum value of an algebraic function, a problem quite suitable for today's students. The second example analyzes a page in Galileo's notebooks where he records an experiment with balls rolling down an inclined plane. By constructing a mathematical model for the motion, which involves finding and applying the moment of inertia of a ball, a convincing case is made that Galileo's data are the record of a genuine experiment, and not the result of a thought experiment as some historians had once maintained.
Numerically Parametrizing Curves
Many computer software systems will make accurate plots of curves, in the plane or in space, that are defined by parametric equations. Some graphics systems will plot an implicitly defined plane curve f(x,y) = 0 , but such plots rarely show correctly the behavior near points of self-intersection. Few if any systems will plot the curve of intersection of two implicitly defined surfaces in space. This article derives a system of differential equations whose solution through a given point gives parametric equations for a given implicitly defined curve in the plane or in space. Usually these systems cannot be solved exactly, but numerical methods provide approximate solutions that can be used by the parametric plotting routines to produce accurate plots. Ideas from multivariable calculus, differential equations and linear algebra are used to derive the systems of differential equations. Many examples are worked out to explain how to implement this method for specific curves, with or without singular points. Missing Graphics
Singles in a Sequence of Coin Tosses
David M. Bloom
In a sequence of n independent tosses of a fair coin, the number of singles, or runs of length one, is a random variable . The exact probability density function for is determined, and its mean and variance are found by a clever recursion argument. In his 1990 Pólya award winning paper, "The Longest Run of Heads," M. Schilling presented two sequences of H's and T's, one the result of a random process and the other created by a person trying to simulate a random sequence. The sequence that Schilling showed had a suspiciously short longest run of H's is shown here to be quite typical with respect to its number of singles, whereas the sequence whose longest run of heads was near the expected value had so few singles that the occurrence of such a sequence by a random process would be a very rare event. Thus in judging which of these sequences is random the conclusion depends very much on which statistical test one uses.
Looking at Order of Integration and a Minimal Surface
Thomas Hern, Cliff Long, and Andy Long
When attempting to make a computer plot of a standard counterexample to Fubini's theorem on interchanging the order of integration in iterated integrals (a simple rational function of two variables), one of the authors noticed the similarity of the surface to a recently discovered minimal surface (the genus-1 Costa/Hoffman/Meeks surface). The Fubini surface can be deformed in a visually appealing way to become the minimal surface, and this deformation gives insight into the beautiful shape of this minimal surface. With striking images the two surfaces are shown and the deformation, which requires animation for optimal effect, is indicated by selected still views.
Graphics Supplement for this article http://184.108.40.206/minimal/
Leonard Gillman, Revisiting Arc Length
The article "On Arc Length," by P.D. Barry in the November 1997 issue of the CMJ, used an axiom of Archimedes about curves with the same concavity, to find upper bounds for arc length. Here an alternative approach is given, based on the geometrically appealing axiom that of two continuously differentiable functions, one of whose derivative is larger in absolute value than that of the other throughout an interval, the graph of the one with the greater derivative has the greater arc length. This axiom, together with the additivity of arc length, leads immediately to the conclusion that the arc length of a continuously differentiable function f(x) on [a,b] lies between every lower sum and every upper sum of
Conversely, the integral formula, together with additivity of arc length, implies this axiom.
James E. Mann, Jr., The Buckled Rail: Three Formulations
Imagine a steel rail one mile long with fixed ends that is heated so that its length increases by one inch, causing it to buckle upwards. Three cases are considered: the buckled shape is an isosceles triangle, an arc of a circle, or one arch of a sinusoidal curve. In each case, what is the height of the center point? Finding the answer requires different mathematics in the three cases, with the most difficult (and physically interesting) third case involving integration of a Taylor series expansion to approximate an arc-length integral.
Viet Ngo and Saleem Watson, Who Cares if has a solution?
Four answers are given, which teachers might use in reply to a student who asks the question in the title. The first explains how Bombelli used complex numbers to find real roots of certain cubics by Cardano's formula. Two replies explain the simplification that results when one uses complex exponential functions rather than products of real exponentials and trigonometric functions in solving certain elementary differential equations. The final reply explains how considering complex values of x in the power series expansion of why the radius of convergence of this series is just 1, although the function is analytic on the real line.
Cheng-Shyong Lee, Polishing the Star
A recent Menelaus-type theorem of Hoehn about pentagrams is shown to be an immediate consequence of the law of sines.
David Callan, When is "Rank" Additive?
It is well known that the matrix rank is subadditive; that is, (A + B) (A) + rank (B) Here it is shown that equality holds if and only if the column spaces of the matrices A and B are disjoint and the row spaces are disjoint.
Charles R. Johnson and Brenda K. Kroschel, Clock Hands Pictures for Real 2 x 2 matrices
The clock hands picture for a real 2 x 2 matrix A is an animated plot of a unit vector along with the vector For invertible A, as the vector sweeps out the unit circle, the image sweeps out an ellipse. This movie makes several characteristics of the matrix visible. For example, unit eigenvectors corresponding to real eigenvalues are those that are collinear with their image , the length and direction of showing the eigenvalue. The lengths of the semiaxes of the ellipse are the singular values of A, and the corresponding (right) singular vectors lie on the semiaxes. These and other features of the clock hands pictures are discussed.
Sergey Markelov, Geometric Characterization of the Shortest Path in a Tetrahedron
The problem of finding the closed path of minimum length that touches all four faces of a regular tetrahedron ABCD was solved using analytic geometry and calculus in a Computer Capsule in the November 1997 issue of the CMJ, but the solution did not provide a geometric characterization of the minimal path. This problem appeared on the 1993 Moscow Mathematical Olympiad, and the author presents a geometrical solution, showing that the minimal path is obtained by minimizing the distance between successive medians of the faces: from the median CL of face ABC to the closest point on median BK of BCD, then to the closest point on median DL of face ABD, then to the closest point on median AK of ACD and from there back to the closest point on CL, where the path began.