Ellipses and Finite Blaschke Products
by Ulrich Daepp, Pamela Gorkin and Raymond Mortini
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The geometric properties of Moebius transformations are often the focal point of a first course in complex analysis, while the same properties of products of such transformations are not. We take a closer look at products of Moebius transformations that fix the point zero, or Blaschke products of finite degree. For degree two, we consider products with two zeros: one zero at zero, and one nonzero zero. Given a point w on the circle, the Blaschke product will map exactly two points on the circle to w. Any line joining those two points will pass through the nonzero zero of the Blaschke product. Drawing several of these line segments simultaneously produces an attractive asterisk with all lines passing through the nonzero zero of B. Does anything interesting happen for higher degree? The answer is affirmative. The goal of this paper is to present a surprising geometric property for Blaschke products of degree three, and an unusual algebraic formula for Blaschke products of general finite degree.
A Cohomological Viewpoint on Elementary School Arithmetic
by Daniel C. Isaksen
From finite group theory to algebraic geometry to complex analysis, cohomological methods play a major role in modern mathematics. Mathematicians usually are surprised to learn that the traditional addition algorithm from primary school is also linked to cohomology. The goal is to present some introductory notions of cohomology from familiar principles, not from sophisticated abstract principles. The reader needs only a familiarity with the basic notions of finite group theory, such as homomorphisms and quotient groups.
The Fundamental Theorem of Calculus in Two Dimensions
by Bennett Eisenberg and Rosemary Sullivan
The fundamental theorem of calculus is divided into two parts, a static part and a dynamic part. Many calculus books state that Green’s theorem is a two-dimensional generalization of the static part, i.e., the part that says that an integral of a function over a fixed interval can be found by evaluating the anti-derivative of the function at the end points of the interval. In this paper we present a two-dimensional generalization of the dynamic part, the part that says that the derivative of the integral of a function over increasing intervals is the function. This generalization follows from what we call “the little coarea theorem,” a simple version of the coarea formula from geometric measure theory. We use our results in turn to derive a generalization of the shell method for finding volumes of revolution. We also derive some interesting connections between Green’s theorem and derivatives of integrals of functions over increasing regions in the plane.
Cutting Polyominos into Equal-Area Triangles
by Iwan Praton
Can you cut a square into an odd number of equal-area triangles? (Try it!) The answer is no, but a proofÂ—finally provided by Monsky in 1970Â—is surprisingly difficult to find. In a 1999 MONTHLY article, Stein asked this question for polyominos, i.e., he wondered whether a polyomino can be cut into an odd number of equal-area triangles. He conjectured that the answer is no, and he proved the conjecture for polyominos with odd area. In this article I present a proof for all polyominos.
Problems and Solutions
Transitivity Implies Period Six: A Simple Proof
by Chun-Hung Hsu and Ming-Chia Li
A New Proof of Cavalieri’s Quadrature Formula
by N. J. Wildberger
On Euler’s ConstantÂ—Calculating Sums by Integrals
by Yingying Li
Life on the Edge
by Alf J. van der Poorten
by Oliver Aberth
Reviewed by Fred Richman