August-September 2011 Contents
In this month's issue, you'll find out why Benford's law applies to quantities that grow exponentially, how solutions to Diophantine equations seem to "repel" each other, how the Fourier transform can be used to derive the spherical means solution to the wave equation, and much more.
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Benford’s Law, A Growth IndustryKenneth A. Ross
Often data in the real world have the property that the first digit 1 appears about 30% of the time, the first digit 2 appears about 17% of the time, and so on with the first digit 9 appearing about 5% of the time. This phenomenon is known as Benford’s law. This paper provides a simple explanation, suitable for nonmathematicians, of why Benford’s law holds for data that have been growing (or shrinking) exponentially over time. Two theorems verify that Benford’s law holds if the initial values and rates of growth of the data appear at random.
A Repulsion Motif in Diophantine EquationsGraham Everest and Thomas Ward
Problems related to the existence of integral and rational points on cubic curves date back at least to Diophantus. A significant step in the modern theory of these equations was made by Siegel, who proved that a nonsingular plane cubic equation has only finitely many integral solutions. Examples show that simple equations can have inordinately large integral solutions in comparison to the size of their coefficients. Nonetheless, a conjecture of Hall suggests a bound on the size of integral solutions in terms of the coefficients of the defining equation. It turns out that a similar phenomenon seems, conjecturally, to be at work for solutions which are close to being integral in another sense. We describe this conjecture as an illustration of an underlying motif—repulsion—in the theory of Diophantine equations.
The Fourier Transform and the Wave EquationAlberto Torchinsky
We solve the Cauchy problem for the n-dimensional wave equation using elementary properties of the Fourier transform.
Counterexamples to the Hasse PrincipleW. Aitken and F. Lemmermeyer
This article explains the Hasse principle and gives a self-contained development of certain counterexamples to this principle. The counterexamples considered are similar to the earliest counterexample discovered by Lind and Reichardt. This type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in Tate–Shafarevich groups.
On Eventually Expanding Maps of the Interval
In this paper we conjecture that the piecewise linear map f
(x) = px for for which has an expanding, onto branch and a contracting branch, is eventually piecewise expanding. We give a partial proof of hte conjecture, in particular for values of p and s such that .
A Simpler Proof of the Von Neumann Minimax TheoremHichem Ben-El-Mechaiekh and Robert W. Dimand
This note provides an elementary and simpler proof of the Nikaidô-Sion version of the von Neumann minimax theorem accessible to undergraduate students. The key ingredient is an alternative for quasiconvex/concave functions based on the separation of closed convex sets in finite dimension, a result discussed in a first course in optimization or game theory.
Probabilistically Proving that ζ (2) = π2/6Luigi Pace
We give a short proof of the identity ζ (2) = π2/6 using tools from elementary probability. Related identities, due to Euler, are also briefly discussed.
On a Problem of RudermanM. Ram Murty and V. Kumar Murty
In 1974, Harry Ruderman proposed the following problem in the problems section of this MONTHLY: if m > n ≥ 0 are integers such that 2m – 2n divides 3m – 3n , then show that 2m – 2n divides xm – xn for all natural numbers x. This problem is still open. We prove that there are only finitely many pairs of natural numbers m, n such that 2m – 2n divides 3m – 3n . Since the proof involves the Schmidt subspace theorem, our bounds on m and n are ineffective. We discuss how an effective version of the abc conjecture can be used to derive effective bounds on m and n.
Cyclic Absolute Differences of IntegersAlan F. Beardon
We give an elementary proof of the convergence, or nonconvergence, to zero in the “four numbers game.”
PROBLEMS AND SOLUTIONS
REVIEWSFamous Puzzles of Great Mathematicians
By: Miodrag S. Petkovič
Reviewed by: Peter Winkler