American Mathematical Monthly -November 2008

November 2008

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Complex Analysis as Catalyst
By: Steven G. Krantz
skrantz@aimath.org
This article explores contexts in which complex analysis can be used as a device to set up interesting problems which can then be solved using techniques from other parts of mathematics. Illustrative examples are given from hard analysis, differential equations, topology, algebra, group theory, and geometry. The article introduces the reader to some new points of view, and also to a new way of thinking about complex function theory.

New Descriptions of Conics via Twisted Cylinders, Focal Disks, and Directors
By: Tom M. Apostol and Mamikon A. Mnatsakanian
apostol@caltech.edu, mamikon@caltech.edu
From any point of an ellipse the sum of distances to its two foci is constant. For a hyperbola the absolute difference of these lengths is constant. We generalize these properties and obtain a unified characterization of all conics, including the parabola! We inscribe Dandelin-type spheres in a twisted cylinder (hyperboloid) rather than a cone, and pierce the cutting plane to produce focal disks. Focal distances are replaced by lengths of tangent segments to these disks. For each conic and any pair of focal disks, the sum of tangent distances is a constant on some portions of the conic, while on the remaining portions their difference is the same constant! Each conic now has infinitely many focal disks, resulting in a rich variety of configurations, some of which cannot occur on a cone. Special cases reveal many surprises. For example, the absolute difference of tangent distances can be constant everywhere on an ellipse! Our approach also leads naturally to a generalization of the classical focus-directrix-eccentricity description of conics.

Projectile Motion: Resistance is Fertile
By: William W. Hackborn
hackborn@ualberta.ca
The nature of projectile motion lies near the heart of the scientific revolution. Galileo, Newton, Johann Bernoulli, Euler, and even Littlewood, as a second lieutenant during World War I, applied themselves to this problem. Projectile motion in the real world is complicated by air resistance, but modeling air resistance as proportional to the projectile's speed (as we often do in calculus and differential equations courses) is almost always totally inadequate. In reality, as Newton demonstrated in the Principia, the drag experienced by bodies moving through air at subsonic speeds is well modeled as proportional to the square of the body's speed. This paper considers the motion of a projectile subject to resistance of this kind. An elementary approximation for the trajectory at launch angles up to moderate size is rediscovered and shown to lie between the exact trajectory and Galileo's parabolic trajectory; other elementary approximations for the trajectory are briefly examined. Remarks about the fascinating history of this problem and some contributions to it by mathematical giants are also made.

k-Dependence and Domination in Kings Graphs
By: Eugen J. Ionascu, Dan Pritikin, and Stephen E. Wright
ionascu_eugen@colstate.edu, pritikd@muohio.edu, wrightse@muohio.edu
What is the maximum density of kings that can be placed on a rectangular chessboard so that no king "attacks" more than a specified number, k, of other kings? This article describes the struggle to provide good upper and lower bounds on the maximum and limiting values (as the board grows) of these densities. Rival techniques employed for different values of k make elementary use of graph theory, number theory, group theory, real analysis, and integer linear programming. Exact limiting densities are determined for the cases where k is 1, 2, 3, 6, and 7, while conjectures are made for the cases where k is 4 and 5. The treatment also considers multidimensional boards and toroidal boards allowing "wrap-around."