# American Mathematical Monthly -May 2010

## May 2010

The Evolution of Markov Chain Monte Carlo Methods
By: Matthew Richey
richeym@stolaf.edu
This paper describes the story of the evolution of Markov chain Monte Carlo (MCMC) methods from the perspective of the four seminal papers. In the 50 years since their discovery by physicists at Los Alamos right after WWII, MCMC methods have revolutionized modern computational statistics and applied mathematics. Their development is a fascinating interplay between physics, applied mathematics, computer science, and statistics.

On Some Coverings of the Euclidean Plane with Pairwise Congruent Circles
By: A. B. Kharazishvili and T. Sh. Tetunashvili
kharaz2@yahoo.com, stetun@rmi.acnet.ge
A well-known theorem of S. Mazurkiewicz states the existence of a subset of the Euclidean plane that meets every line of this plane in precisely two points. We consider some assertions of elementary geometry which are dual to the above-mentioned Mazurkiewicz result. We also compare these assertions to each other from the set-theoretical point of view.

Transversals in Rectangular Arrays
By: Sherman Stein
stein@math.ucdavis.edu
For positive integers m and n an m-by-n array consists of mn cells arranged in mrows and n columns. Each cell contains one symbol. A transversal in such an array consists of the minimum of m and n cells, no two from the same row, or from the same column, or containing the same symbol. A near-transversal has one less cell than a transversal, but has the same three properties. This paper makes the following conjecture and provides evidence for it, including work of E. T. Parker and A. Drisko.

Conjecture: If no two cells in any column in an m-by-n array contain the same symbol, then (1) For n , the array has a transversal. (2) For m \leq n \leq {2m-2}, the array has a near-transversal. (3) For n \geq {2m-1}, the array has a transversal.

It shows that if the special case m = n+1 were settled, the entire conjecture would be settled. Moreover, this case would imply that a latin square has a near-transversal.

Product Approximations via Asymptotic Integration
By: Cristinel Mortici
cmortici@valahia.ro
We introduce a new approach to the asymptotic evaluation of products to deduce the series associated with some approximation formulas. As basic examples, we construct Stirling's series, Burnside's series, and the series associated with the Glaisher-Kinkelin constant and Wallis formula solving an infinite triangular linear system.

Notes

On Riemann’s Rearrangement Theorem for the Alternating Harmonic Series
By: Francisco J. Freniche
freniche@us.es
It is a well-known result by B. Riemann that the terms of a conditionally convergent series of real numbers can be rearranged in a permutation such that the resulting series converges to any prescribed sum s: add p1consecutive positive terms until their sum is greater than s; then subtract q1 consecutive negative terms until the sum drops below s, and so on. For the alternating harmonic series, with the aid of a computer program, it can be noticed that there are some fascinating patterns in the sequences pn and qn. For example, if s=\log 2+ (1/2) \log (38/5) the sequence pn is 5,7,8,7,8,7,8,8,7,8,7,8,\dots in which we notice the repetition of the pattern 8,7,8,7,8, while if s=\log 2+ (1/2) \log (37/5) the sequence pn is 5,7,7,7,8,7,8,7,7,8,7,8,\dots in which the pattern is 7,7,8,7,8.

Where do these patterns come from? Let us observe that 38/5= 7+3/5$and$37/5 = 7+2/5. The length of the repeating pattern is the denominator 5, the values of pn, at least from some n on, are 7 and 8, and the number 8 appears 3 times in the pattern of the first example, and 2 times in that of the second one. These are not coincidences: we explain them in this paper.

Modular Divisor Functions and Quadratic Reciprocity
By: Richard Steiner
r.steiner@maths.gla.ac.uk
It is well known that a positive integer is a perfect square if and only if it has an odd number of positive integer divisors. In this note, it is shown that quadratic residues modulo an odd prime can be characterized in a similar way. This characterization is used to give a particularly direct proof of the law of quadratic reciprocity.

Another Application of Siebeck's Theorem
By: Peter R. Mercer
mercerpr@math.buffalostate.edu
A beautiful result due to J. Siebeck (1864) is as follows: If p is a polynomial of degree 3 with complex coefficients having 3 noncollinear zeros, then the zeros of p' are precisely the foci of the Steiner ellipse for the triangle determined by these zeros. (The Steiner ellipse is the unique ellipse inscribed in the triangle, which is tangent to its three sides at their midpoints.) Here we obtain an analogue of Siebeck's result for a 3x3 normal matrix having 3 noncollinear eigenvalues.

The Fundamental Theorem of Algebra via the Fourier Inversion Formula
By: Alan C. Lazer and Mark Leckband
leckband@fiu.edu
The Fundamental Theorem of Algebra (FTA) is proved from the Fourier transform and its inversion formula (FIF). We provide four proofs using the formula of the Fourier transform and its uniqueness property. The FTA is commonly derived as a consequence of Liouville’s theorem. Thus, we also derive Liouville’s theorem via the FIF.

Reviews

Elementary Functional Analysis
By: Barbara D. MacCluer

Reviewed by: Carol S. Schumacher
schumacherc@kenyon.edu