Among the pages of the June/July Monthly lies a triple threat! We start with a paper titled "Arcology" dedicated to the memory of John Nash by his colleague János Kollár. Add to this a paper coauthored by former MAA President Ron Graham, and another by Ford-Halmos Award-winning author Tadashi Tokieda, and you have an issue that will keep you turning the pages.
Finally, Leslie Hogben and Mark Hunacek review Sheldon Axler's "Linear Algebra Done Right," and don't forget our Problems and Solutions section for that long afternoon at the beach. See you in August, when we will learn about the range of a rotor walk.
- Scott T. Chapman, Editor
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Table of Contents
Jennifer M. Johnson and János Kollár
John Nash initiated the study of arcs in algebraic geometry in a quite unusual paper written in 1968, at the beginning of his long illness. This paper raised many interesting and still-open questions that can be investigated using elementary methods. The tone of the Nash article is discursive, and through a series of examples and comments it reports on “some interesting possible truths which were encountered.” We also follow this approach and present the intricacies and beauties of this subject using concrete examples. In addition we discuss some recent results and unsolved problems that are accessible to undergraduates.
The Mathematics of the Flip and Horseshoe Shuffles
Steve Butler, Persi Diaconis and Ron Graham
We consider new types of perfect shuffles wherein a deck is split in half, one half of the deck is “reversed,” and then the cards are interlaced. Flip shuffles are when the reversal comes from turning the half of the deck over so that we also need to account for face-up/face-down configurations, while horseshoe shuffles are when the order of the cards are reversed but all cards still face the same direction. We show that these shuffles are closely related to faro shuffling and determine the order of the associated shuffling groups. An application of this theory is given through a card trick based on applying different shuffles of eight cards.
Valerio De Angelis and Dominic Marcello
In a Note in this Monthly, Klazar raised the question of whether the alternating sum of the Stirling numbers of the second kind is ever zero for n ≠ 2. In this article, we present the history of this problem, and an economical account of a recent proof that there is at most one n ≠ 2 for which B± (n) = 0.
A Hyperbolic Analog of the Phasor Addition Formula
F. Adrián F. Tojo
In this article, we review the basics of the phasor formalism in a rigorous way, highlighting the physical motivation behind it and presenting a hyperbolic counterpart of the phasor addition formula.
Tiling the Plane with Different Hexagons and Triangles
Pablo A. Panzone
We prove that the plane can be tiled with equilateral triangles and regular hexagons of integer sides using exactly one of each family.
How to Win Your Betting Pool with Jensen′s Inequality and the Law of Large Numbers
Franklin H. J. Kenter
We determine the optimal strategy for a winners-take-all betting pool with sufficiently many participants using only elementary tools.
On the Occurrence of Perfect Squares Among Values of Certain Polynomial Products
We prove that the product of first n consecutive values of the polynomial P(k) = 4k4 + 1 is a perfect square infinitely often whereas the product of first n consecutive values of the polynomial Q(k) = k4 + 4 is a perfect square only for n = 2.
Expected Number of Vertices of a Hypercube Slice
Given a random k-dimensional cross section of a hypercube, what is its expected number of vertices? We show that, for a suitable distribution of random slices, the answer is 2k, independent of the dimension of the hypercube.
Quasiconvex Linear Perturbations and Convexity
Pham Duy Khanh and Marc Lassonde
It is known that a function f defined on a convex subset of a vector space is convex provided that all its perturbations f + u* by linear forms are quasiconvex. We show that, under more restrictive assumptions, the convexity of f follows from the quasiconvexity of perturbations by just a suitable one-parameter family of linear forms.
Surprises in Numerical Expressions of Physical Constants
Ariel Amir, Mikhail Lemeshko and Tadashi Tokieda
In science, as in life, “surprises” can be adequately appreciated only in the presence of a null model, what we expect a priori. In physics, theories sometimes express the values of dimensionless physical constants as combinations of mathematical constants like π or e. The inverse problem also arises, whereby the measured value of a physical constant admits a “surprisingly” simple approximation in terms of well-known mathematical constants. Can we estimate the probability for this to be a mere coincidence, rather than an inkling of some theory? We answer the question in the most naive form.
Problems and Solutions
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.6.613
Linear Algebra Done Right, Third Edition by Sheldon Axler
Reviewed by Leslie Hogben
To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.6.621
Proofs Without Using Limits of Both Calculus Power Rules
100 Years Ago This Month in the American Mathematical Monthly
Edited by Vadim Ponomarenko