American Mathematical Monthly -February 2010

February 2010

A Dynamical System Using the Voronoï Tessellation
By: Natalie Priebe Frank and Sean M. Hart
nafrank@vassar.edu, shart@math.utexas.edu
There are many good reasons to make a tiling around some discrete set of points. For example, maybe you need to know what regions are served by some set of post offices or cell phone satellites. Once you have a tiling, there are many reasons to define a point set with it. For instance, you might want to decorate each tile with a few points to create or destroy symmetry, or to obtain combinatorial information. Since you can go from point sets to tilings and back again, you might come across some interesting ways to associate one set of points to another. Once you have a map from a class of objects back to itself, you can take a dynamical systems viewpoint to analyze the situation. This paper does exactly that, with a new dynamical system based on the vertices of Voronoï tessellations.

Variations on Kuratowski's 14-Set Theorem
By: David Sherman
dsherman@virginia.edu
Kuratowski's 14-set theorem says that in a topological space, 14 is the maximum possible number of distinct sets which can be generated from a fixed set by taking closures and complements. Here we consider the analogous question for any possible subcollection of the operations {closure, complement, interior, intersection, union}, and any number of initially given sets. Even though these problems concern point-set topology, the solutions rely instead on lattice theory and related flavors of universal algebra, with a little bit of logic for seasoning.

The Trouble with von Koch Curves Built from n-gons
By: Tamás Keleti and Elliot Paquette
tamas.keleti@gmail.com, paquette@math.washington.edu
The von Koch curve, as Helge von Koch himself observes, can be quite naturally generalized. One such family of generalizations can be built using n-gons. To build these curves, fix some positive scalar c less than one. Draw a line segment with unit length. Replace the middle c portion of the segment with the sides of a regular n-gon whose own sides are length c. This produces a total of n+1 line segments. Recursively apply this procedure to each line segment, and take the limit to produce the (n,c)-von Koch Curve. For fixed n and c sufficiently large, the curve intersects itself. Likewise, for c sufficiently small, the curve does not. The trouble with the (n,c)-von Koch curve is what happens in between. The set of c for which the curve self-intersects is not necessarily an interval, a phenomenon that we explore here.

Nemirovski's Inequalities Revisited
By: Lutz Dümbgen, Sara A. van de Geer, Mark C. Veraar, and Jon A. Wellner
duembgen@stat.unibe.ch, geer@stat.math.ethz.ch, m.c.veraar@tudelft.nl, jaw@stat.washington.edu
We study generalizations of the well-known fact that the variance of a sum of real-valued independent random variables is the sum of the variances. When the random variables take values in a Banach space or some other vector space, we study generalizations which bound the square of the norm of the sum of independent elements in terms of a constant K (which depends on the space and norm) times the sum of the expected values of the square of the norms of the independent summands. Such moment inequalities for sums of independent random vectors are important tools for statistical research. Nemirovski and coworkers (1983, 2000) and Pinelis (1994) derived one particular type of such inequalities. We present and compare three different approaches to obtain such inequalities: The results of Nemirovski and Pinelis are based on deterministic inequalities for norms. A second method involves type and cotype inequalities, a tool from probability theory on Banach spaces. Finally, we use a truncation argument plus Bernstein's inequality to obtain another version of the moment inequality above. Interestingly, all three approaches have their own merits.

Notes

An Ancient Elliptic Locus
By: John E. Wetzel
j-wetzel@illinois.edu
In the plane, two vertices P and Qof a triangular tile PQRmove respectively on two intersecting lines p and q. What is the locus of the third vertex R? Frans van Schooten the younger showed in 1646 that the locus is an ellipse. We present two proofs of this result, one an elegant deduction from the special case in which Rlies on the line PQ that was given by John Casey in 1885, the other a clever argument using motions that John Sullivan included in a 1996 paper that has never been published.

Counting Interesting Elections
By: Lara K. Pudwell and Eric S. Rowland
Lara.Pudwell@valpo.edu, erowland@math.rutgers.edu
Suppose candidates E and N are competing for a public office, and candidate E wins the election. The classical ballot problem asks for the number of ways to report the votes so that at all times during the tally N is not ahead of E, corresponding to northeast lattice paths that do not go above the line y = x. In this paper we ask more generally for the number of northeast lattice paths that are restricted to a certain region of the plane. The solution involves an elegant counting argument using elementary methods.

A Proof of Darboux's Theorem
We offer a simple, transparent proof that derivatives have the intermediate value property (Darboux's theorem) that we feel shows systematically and conceptually why the theorem is true.

On the Behavior at Infinity of an Integrable Function
By: Emmanuel Lesigne
emmanuel.lesigne@lmpt.univ-tours.fr
We prove that, in a weak sense, any integrable function on the real line tends to zero at infinity: if f is an integrable function on $\mathbb R$, then for almost all real number x, the sequence $(f(nx))_{n\in{\mathbb N}}$ tends to zero when n goes to infinity. Using Khinchin's metric theorem on Diophantine approximation, we establish that this convergence to zero can be arbitrarily slow.

Reviews

Tools of American Mathematics Teaching, 1800-2000
By: Peggy Aldrich Kidwell, Amy Ackerberg-Hastings, and David Lindsay Roberts
Reviewed by: Daniel S. Silver
silver@jaguar1.usouthal.edu