The October Monthly deals a winning hand! In our lead article, Early Round Bluffing in Poker, "California" Jack Cassidy shows us whether or not early round bluffing is a good idea. October also contains our annual review of the William Lowell Putnam Mathematical Competition. Ever have trouble with the proof of the Lebesgue Decomposition Theorem? Tamás Titkos offers us a simple proof in our Notes Section. Ferebee Tunno reviews Probability Theory in Finance: A Mathematical Guide to the Black–Scholes Formula (Second edition) by Séan Dineen. Our Problem Section will keep you busy until Halloween. If you have fond memories of the famous Monthly paper "Can You Hear the Sound of a Drum?," then you will love our lead article for November, The Sound of Symmetry, by Zhiqin Lu and Julie Rowlett.- Scott T. Chapman, Editor
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Volume 122, Issue 08, pp. 713 - 812
Table of Contents
Articles
The Seventy-Fifth William Lowell Putnam Mathematical Competition
Leonard F. Klosinski, Gerald L. Alexanderson, and Mark Krusemeyer
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.715
Early Round Bluffing in Poker
California Jack Cassidy
Using a simplified form of the Von Neumann and Morgenstern poker calculations, the author explores the effect of hand volatility on bluffing strategy, and shows that one should never bluff in the first round of Texas Hold'Em.
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.726
Volume/Surface Area Relations for n-Dimensional Spheres, Pseudospheres, and Catenoids
Tom M. Apostol and Mamikon A. Mnatsakanian
Circle and sphere properties are extended to the tractrix and pseudosphere, the catenary and catenoid, and to higher-dimensional analogs.
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.745
Besicovitch, Bisection, and the Normality of 0.(1)(4)(9)(16)(25) · · ·
P. Pollack and J. Vandehey
We revisit Besicovitch's 1935 paper in which he introduced several techniques that have become essential elements of modern combinatorial methods of normality proofs. Despite his paper's influence, the results he inspired are not strong enough to reprove his original result. We provide a new proof of the normality of the constant 0.(1)(4)(9)(16)(25) … formed by concatenating the squares, updating Besicovitch's methods.
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.757
On a Recursion Formula Related to Confl uent Vandermonde
Shui-Hung Hou and Edwin Hou
In this short article, we give a novel and elegant proof of a recursion formula related to confluent Vandermonde determinants. This leads to an easy derivation of the well-known determinant identity due to Schendel.
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.766
On Devaney Chaos and Dense Periodic Points: Period 3 and Higher Implies Chaos
Syahida Che Dzul-Kifl i and Chris Good
We look at density of periodic points and Devaney Chaos. We prove that if f is Devaney Chaotic on a compact metric space with no isolated points, then the set of points with prime period at least n is dense for each n. Conversely, we show that if f is a continuous function from a closed interval to itself, for which the set of points with prime period at least n is dense for each n, then there is a decomposition of the interval into closed subintervals on which either f or f2 is Devaney Chaotic. (In fact, this result holds if the set of points with prime period at least 3 is dense.)
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.773
Notes
A Viscosity Proof of the Cauchy–Schwarz Inequality
Tadashi Tokieda
The Cauchy–Schwarz inequality for positive quadratic forms has many proofs. This note gives a new derivation that looks unusual at first, but is natural in retrospect, interpreting the quadratic form as kinetic energy and the inequality as dissipation in a viscous flow.
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.781
A Short Proof of the Fact That the Matrix Trace Is the Expectation of the Numerical Values
Tomasz Kania
Using the fact that the normalized matrix trace is the unique linear functional f on the algebra of n ×n matrices which satisfies f(I) = 1 and f(AB) = f(BA) for all n × n matrices A and B, we derive a well-known formula expressing the normalized trace of a complex matrix A as the expectation of the numerical values of A; that is the function 〈Ax, x〉, where x ranges the unit sphere of ℂn.
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.782
van der Waerden and the Primes
Levent Alpoge
In this note we prove the infinitude of the primes via an application of van der Waerden's theorem.
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.784
An Exponential Inequality for Symmetric Random Variables
Raphaël Cerf and Matthias Gorny
We prove a simple exponential inequality which gives a control on the first two empirical moments of a sequence of independent identically distributed symmetric real-valued random variables.
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.786
Pairs of Matrices in GL2(R≥0) That Freely Generate
Melvyn B. Nathanson
We present an elementary proof that certain pairs of 2 × 2 matrices with nonnegative real coordinates generate free monoids.
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.790
A Simple Proof of the Lebesgue Decomposition Theorem
Tamás Titkos
Motivated by the notion of operator, a short and simple proof of Lebesgue's decomposition theorem is presented in this note.
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.793
A New Proof of the Change of Variable Theorem for the Riemann Integral
Haryono Tandra
We give a new proof of the complete version of the change of variable theorem for the Riemann integral using several basic properties within the elementary theory.
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.795
I Prefer Pi: Addenda
Jonathan Borwein and Scott Chapman
In the rush to prepare our March 2015 article on Pi [1], several infelicities escaped our eye. Herein we repair the damage.
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Problems and Solutions
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Book Review
Probability Theory in Finance: A Mathematical Guide to the Black–Scholes Formula (Second edition) By Se´an Dineen
Reviewed by Joseph Malkevitch
To purchase the article from JSTOR: 10.4169/amer.math.monthly.122.8.809
MathBits
Funny Forms of the Mean Value Theorem
Cristinel Mortici
Pythagoras by Integral
Zsolt Lengvárszky
How to Make Equivalent Measures?
Tamás Titkos
100 Years Ago This Month in the American Mathematical Monthly
Edited by Vadim Ponomarenko