January 2012 Contents

ARTICLES

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.

NOTES

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.

PROBLEMS AND SOLUTIONS

REVIEWS

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

Link:

http://www.mediafire.com/?zxmf5fzff38qc1x