AMM Vol 119/No1 – January/2012

January 2012 Contents


A Letter from the Editor

Scott Chapman

Invariant Histograms

Daniel Brinkman and Peter J. Olver

We introduce and study a Euclidean-invariant distance histogram function
for curves. For a sufficiently regular plane curve, we prove that the
cumulative distance histograms based on discretizing the curve by either
uniformly spaced or randomly chosen sample points converge to our
histogram function. We argue that the histogram function serves as a
simple, noise-resistant shape classifier for regular curves under the
Euclidean group of rigid motions. Extensions of the underlying ideas to
higher-dimensional submanifolds, as well as to area histogram functions
invariant under the group of planar area-preserving affine
transformations, are discussed.

Zariski Decomposition: A New (Old) Chapter of Linear Algebra

Thomas Bauer, Mirel Caibăr, and Gary Kennedy

In a 1962 paper, Zariski introduced the decomposition theory that now
bears his name. Although it arose in the context of algebraic geometry
and deals with the configuration of curves on an algebraic surface, we
have recently observed that the essential concept is purely within the
realm of linear algebra. In this paper, we formulate Zariski
decomposition as a theorem in linear algebra and present a linear
algebraic proof. We also sketch the geometric context in which Zariski
first introduced his decomposition.

Another Way to Sum a Series: Generating Functions, Euler, and the Dilog Function

Dan Kalman and Mark McKinzie

It is tempting to try to reprove Euler’s famous result that using power
series methods of the sort taught in calculus 2. This leads to , the
evaluation of which presents an obstacle. With two key identities the
obstacle is overcome, proving the desired result. And who discovered the
requisite identities? Euler! Whether he knew of this proof remains to
be discovered.


A Class of Periodic Continued Radicals

Costas J. Efthimiou

We compute the limits of a class of periodic continued radicals and we
establish a connection between them and the fixed points of the
Chebycheff polynomials.

A Geometric Interpretation of Pascal’s Formula for Sums of Powers of Integers

Parames Laosinchai and Bhinyo Panijpan

We present a geometric interpretation of Pascal’s formula for sums of
powers of integers and extend the interpretation to the formula for sums
of powers of arithmetic progressions. Related interpretations of a few
other formulas are also discussed.

Covering Numbers in Linear Algebra

Pete L. Clark

We compute the minimal cardinalities of coverings and irredundant
coverings of a vector space over an arbitrary field by proper linear
subspaces. Analogues for affine linear subspaces are also given.



An Introduction to the Mathematics of Money by David Lovelock, Marilou Mendel, and A. Larry Wright. Reviewed by Alan Durfee.