January 2012 Contents
The January 2012 issue will be the first edited by the Monthly’s new Editor Scott Chapman. In the January articles, we will learn about invariant histograms, the decompositions of Zariski, and the use of generating functions to sum a series. The Notes Section looks at periodic continued radicals, a geometric interpretation of Pascal’s formula, and covering numbers in Linear Algebra. Our book review covers An Introduction to the Mathematics of Money, by Lovelock, Mendel and Wright and as always, our Problems Section will keep you thinking.
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A Letter from the EditorScott Chapman (click here to read online)
Invariant HistogramsDaniel 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 AlgebraThomas 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 FunctionDan 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 RadicalsCostas 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 IntegersParames 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 AlgebraPete 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.