A relation between Catalan Numbers and World Series type problems

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*n*-Ellipses and the Minimum Distance Sum Problem

by Junpei Sekino

sekino@willamette.edu

Given *n *points in a plane, which we call* foci*, an * n-ellipse* is the locus of points in the plane the sum of whose distances from the foci is a constant. Like circles (1-ellipses) and ellipses (2-ellipses), * n*-ellipses are Jordan curves with convex interiors but come in a greater variety of shapes such as cross-sections of eggs and filberts; some of them even resemble contours of human faces, symmetric or asymmetric. After illustrating some of these figures (computer generated), we turn our attention to the critical points of the distance sum function whose contours are * n*-ellipses. We show that a critical point is nondegenerate if and only if the foci are noncollinear, and demonstrate that a nondegenerate critical point not only yields a unique point minimizing the distance sum function but also possesses a geometrically appealing property. The paper concludes with an optimization problem and a suggestion for a further generalization of an ellipse to an * n*-ellipse with * weights*.

**The Bieberbach Conjecture and Milin's Functionals**

by Arcadii Z. Grinshpan

azg@math.usf.edu

The Bieberbach conjecture (1916) for the Taylor coefficients of one-to-one analytic functions is one of the most famous and inspirational problems of mathematics. This conjecture remained open for nearly 70 years until in 1984 L. de Branges proved a stronger conjecture for certain logarithmic functionals proposed by I. M. Milin in 1971.

de Branges' approach was based on Loewner's parametric method (1923), a classic tool in geometric function theory, but it involved other fields of mathematics as well. Simplifications made by several authors during 1984-1997 have led to the present proof, which is now accessible to many readers.

**Decision Making: A Golden Rule**

by Dimitris A. Sardelis and Theodoros M. Valahas

dereef@hol.gr

This article is an educational exposition of a classical optimal-stopping problem, the dowry problem. Its solution is presented by building on easily understandable special cases. A pattern of optimal strategies emerges, which is ultimately expressed by general conditions that lead to the (1/e) *golden decision rule*.

**A Tale of Two Integrals**

by Vilmos Totik

totik@math.usf.edu

In the paper several approaches are presented to the following simple-looking but highly non-trivial combinatorial-analysis problem: *whose integrals ove* [0,1]* are both equal to* 1. *Show that there is some interval I * [0,1] *such that the integrals of f and g over I are both equal to 1/2*. This problem is equivalent to the following problem without integrals: *On a blackjack machine one can win or lose one dollar at a time. Suppose two players playing once per minute during a period find that eventually both of them win exactly 2N dollars. Show that there was a time interval during which both of them won exactly N dollars*.

Six solutions are presented that are based on other combinatorial or geometrical/topological results. These are:

- the Borsuk-Ulam antipodal theorem on continuous mappings of spheres;
- the mountain climbing problem on two climbers staying always at the same altitude;
- the chord theorem for parallel chords of planar curves;
- the chess king-moving theorem on king walks on cells of the same colour;
- the winding number theorem for vector fields;
- the Jordan curve theorem on components of simple curves.

**A Mathematical Excursion: From the Three-Door Problem to a Cantor-Type Set**

by Jaume Paradís, Pelegrí Viader, and Lluís Bibiloni

jaume.paradis@econ.upf.es, pelegri.viader@econ.upf.es, l.bibiloni@uab.es

Can you imagine a connection between the well-known three-door problem--also known as the Monty Hall problem--and a set with a complex structure like Cantor's set? We offer a generalization of the three-door problem, the *n*-box problem, whose solution provides us with a system for real number representation, Pierce expansions. Through these expansions we get a new enumeration of the positive rationals and also a Cantor-type set containing only transcendental numbers.

**NOTES**

**An Inequality Relating the Circumradius and Diameter of
Two-Dimensional Lattice-Point-Free Convex Bodies**

by Poh Wah Awyong

awyongpw@nievax.nie.ac.sg

**Cutting a Polyomino into Triangles of Equal Areas**

by Sherman K. Stein

stein@math.ucdavis.edu

**A Short Proof of Turaacute;nÕs Theorem**

by William Staton

mmstaton@olemiss.edu

**A Characterization of the Set of Points of Continuity of a Real Function**

by Sung Soo Kim

sskim@newton.hanyang.ac.kr

**THE EVOLUTION OF...
The Literal Calculus of Vieacute;te and Descartes**

by I. G. Bashmakova and G. S. Smirnova

**PROBLEMS AND SOLUTIONS**

**REVIEWS**

*An Accompaniment to Higher Mathematics*

By George R. Exner

*Journey Into Mathemtics: An Introduction to Proofs*

By Joseph Rotman

*Mathematical Thinking: Problem Solving and Proofs*

By John P. D'Angelo and Douglas B. West

Reviewed by Joseph H. Silverman

jhs@math.brown.edu

*The Social Life of Numbers: A Quechua Ontology of Numbers
and Philosophy of Arithmetic*

By Gary Urton

Reviewed by John Meier and Trisha Thorme

meierj@lafayette.edu, tat2@cornell.edu

**TELEGRAPHIC REVIEWS**

**EDITOR'S ENDNOTES**