April 2010

"To a factor près": Cayley’s Partial Anticipation of the Weierstrass P-Function
By: Adrian Rice
arice4@rmc.edu
Although Arthur Cayley is perhaps best remembered for his contributions to matrix algebra and group theory, one of his prime interests was the subject of elliptic functions, among the most vibrant areas of mathematics in the nineteenth century. His publications on this topic included a little-known but interesting anticipation of an identity later made famous by Weierstrass. What is particularly pleasing about Cayley's derivation of this identity is that it relied totally on his use of invariant theory, a subject that seems, on the face of it, to have little connection to the theory of elliptic functions. In this paper, we compare Cayley's derivation to the standard Weierstrassian approach, and discuss reasons for the obscurity of the former compared to the relative fame of the latter.

Transitive Decompositions of Graphs and Their Links with Geometry and Origami
By: Geoffrey Pearce
Modular origami is a popular offshoot of the traditional Japanese art of paper folding; it involves building large and elaborate geometrical structures by fitting together a number of "modules" folded from separate squares of paper. We show how the idea of a "transitive decomposition" of a graph can be used to find highly symmetrical and decorative colorings of these structures.

Trigonometric Identities à la Hermite
By: Warren P. Johnson
wpjoh@conncoll.edu
Hermite once observed that a certain product of cotangents can be integrated by breaking it into a sum of cotangents, where the coefficients are themselves products of cotangents. Why should such an identity exist? We give two derivations, one based on the partial fractions expansion of the cotangent. Hermite seems to have used a mixture of the two. We also discuss and extend a second theorem of Hermite, which leads to generalizations of his cotangent identity. The paper veers off into determinants at the end.

Quasi-Cauchy Sequences
By: David Burton and John Coleman
dburton@franciscan.edu, jcoleman@franciscan.edu
A quasi-Cauchy sequence is one in which the distance between successive terms tends to zero. This is a far weaker property than that of being Cauchy, although students in undergraduate real analysis classes often struggle with the distinction between the two concepts. Nevertheless, quasi-Cauchy sequences have many interesting properties. In this paper we investigate such sequences in both the real number system and in general metric spaces.

p-Free lp Inequalities
By: Grahame Bennett
bennettg@indiana.edu
We show how certain simple lp inequalities may be proved by “ignoring the p.” This entails revisiting that weird and wonderful world of Hardy, Littlewood, and Pólya (wherein everything is more-or-less and nothing’s as it seems). We encounter many familiar characters along the way, among them: Cauchy’s Inequality, Theory of Means, Convexity, Rolle’s Theorem, and Descartes’ Rule of Signs. What makes our trip worthwhile is the realization that these old chestnuts still have something new to tell us: at any rate, they all appear here in novel, and sometimes surprising, ways.

Notes

A Short Proof of ζ (2) = π2/6
By: T. H. Marshall
tmarshall@aus.edu
We show that the sum $\sum 1/\log2(z)$, taken over all branches of the logarithm, is a rational function, and use this to give a short proof of \sum 1/n2=\π2/6.

The Group of Symmetries of the Tower of Hanoi Graph
By: So Eun Park
soeun.park@berkeley.edu
The Tower of Hanoi problem, one of the most famous mathematical puzzles, has many interesting aspects to study, such as the properties of its graph in the case of 3 pegs (the most widely known form of the puzzle) and the shortest paths (or geodesics) in generalized Tower of Hanoi problems.

In particular, Frame and Stewart, in response to Monthly problem 3918, suggested a way to narrow down the possible cases of shortest paths of the generalized problems with more than 3 pegs (this Monthly {48} (1941) 216--219); the problem remains unsolved.

In this note, we look at the generalized Tower of Hanoi problems, with the number of pegs greater than 3, from the perspective of graph theory. We prove that no matter how many pegs and disks we are playing with in the problem, the group of automorphisms of its graph is always isomorphic to the group of the peg permutations, i.e., the number of disks of the problem does not affect the automorphism group of the Tower of Hanoi graph.

Recurrent Proofs of the Irrationality of Certain Trigonometric Values
By: Li Zhou and Lubomir Markov
lzhou@polk.edu, lmarkov@mail.barry.edu
Consider the set of transcendental functions ${F=\{ \cos , \sin , \tan , \cosh , \sinh , \tanh , \exp \}$. If x is a nonzero real number and $f\in F$, then x2and (f(x))2 cannot both be rational. This result is known and the proofs of some of its corollaries, such as the irrationality of π, are classical. It is the purpose of our paper to offer simple new proofs of these results, accessible to a calculus student.