May flowers bloom in the Monthly! Our lead article delves into the interesting aspects of the mathematics of "dark matter." Don't miss Donald Saari's explanation of these issues in his paper "Mathematics and the "Dark Matter" Puzzle." What is the probability that a subtree picked at random from a complete graph is a spanning tree? Find the surprising answer in Gary Gordon (and his 3 co-author's) paper "Pick a Tree - any Tree." Gerry Folland reviews "A History in Sum: 150 Years of Mathematics at Harvard (1825-1975)" by Steve Nadis and Shing-Tung Yau. Have some spare time on your hands now that classes are finished? Try our Problem Section. Stay tuned for June/July when Jason Siefken and Dennis Epple show us how to play "Collapse." - Scott T. Chapman, Editor
JOURNAL SUBSCRIBERS AND MAA MEMBERS:
To read the full articles, please log in to the member portal by clicking on 'Login' in the upper right corner. Once logged in, click on 'My Profile' in the upper right corner.
Volume 122, Issue 05, pp. 403 - 514
Table of Contents
Articles
Mathematics and the “Dark Matter” Puzzle
Donald G. Saari
Surprisingly, aspects of the compelling mystery of “dark matter”—the mysterious, undiscovered material that supposedly consumes most of a spiral galaxy’s mass—are mathematical issues rather than astronomical ones. When analyzed from the perspective of mathematics, doubt is cast on standard predictions about the existence of huge amounts of this material.
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.407
Pick a Tree – Any Tree
Alex Chin, Gary Gordon, Kellie MacPhee, and Charles Vincent
A subtree is picked at random from the collection of all subtrees of a complete graph. What is the probability the subtree spans? We find the surprising answer this question and a few closely related questions.
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.424
Generalizations of Wilson’s Theorem for Double-, Hyper-, Sub- and Superfactorials
Christian Aebi and Grant Cairns
We present generalizations of Wilson’s theorem for double factorials, hyperfactorials, subfactorials, and superfactorials.
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.433
A Simple Computation of ζ (2k)
Óscar Ciaurri, Luis M. Navas, Francisco J. Ruiz, and Juan L. Varona
We present a new simple proof of Euler’s formulas for ζ(2k), where k = 1, 2, 3,…. The computation is done using only the defining properties of the Bernoulli polynomials and summing a telescoping series. The same method also yields integral formulas for ζ(2k + 1).
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.444
Using Rolling Circles to Generate Caustic Envelopes Resulting from Reflected Light
Jeffrey A. Boyle
Given any smooth plane curve α (s) representing a mirror that reflects light the usual way and any radiant light source at a point in the plane, the reflected light will produce a caustic envelope. For such an envelope we show that there is an associated curve β (s) and a family of circles Cs that roll on β (s) without slipping such that there is a point on each circle that will trace the caustic envelope as the circles roll. For a given curve α (s) and for all radiants at infinity there is a single curve β (s) and family of circles Cs that roll on β (s) so that the different points on Cs will simultaneously trace out, as the circles roll, all caustic envelopes from these radiants at infinity. We explore many classical examples using this method.
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.452
A Porism for Cyclic Quadrilaterals, Butterfl y Theorems,and Hyperbolic Geometry
Ivan Izmestiev
If there exists a cyclic quadrilateral whose sides go through the given four collinear points, then there are infinitely many such quadrilaterals inscribed in the same circle. We give two proofs of this porism: one based on cross-ratios, the other on compositions of hyperbolic isometries.
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.467
Notes
Side Lengths of Equiangular Polygons (as seen by a coding theorist)
Maria Bras-Amorós and Marta Pujol
A tuple of positive real numbers is said to be equiangular if there is an equiangular polygon with consecutive side lengths . It is well known that a is equiangular if and only if the polynomial vanishes at . Here we dispense with complex numbers and borrow an idea from the theory of cyclic codes to prove that a is equiangular if and only if is divisible by .
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.476
Relationships Between the First Four Moments
Iosif Pinelis
Best possible bounds on the third moment of a random variable are given in terms of the first, second, and fourth moments.
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.479
Eigenvalues of Real Symmetric Matrices
Meinolf Geck
We present a proof of the existence of real eigenvalues of real symmetric matrices which does not rely on any limit or compactness arguments, but only uses the notions of “sup”, “inf”.
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.482
A One-Formula Proof of the Nonvanishing of L-Functions of Real Characters at 1
Bogdan Veklych
We present a simple analytic proof that L-functions of real nonprincipal Dirichlet characters are nonzero at 1.
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.484
The Volume of the Convex Hull of a Body and its Homothetic Copies
Jesús Jerónimo Castro
In this note, we prove the following result. Let K ⊂ ℝd be a convex body with the origin O in its interior. If there is a number λ ∈ (0, 1) such that the n-dimensional volume of the convex hull of the union of K with the translates of λ K, by a vector x, depends only on the Euclidean norm of x, then K is a Euclidean ball.
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.486
Explaining a Mysterious Maximal Inequality — and a Path to the Law of Large Numbers
J. Michael Steele
In 1964 A. Garsia gave a stunningly brief proof of a useful maximal inequality of E. Hopf. The proof has become a textbook standard, but the inequality and its proof are widely regarded as mysterious. Here we suggest a straightforward first step analysis that may dispel some of the mystery. The development requires little more than the notion of a random variable, and, the inequality may be introduced as early as one likes in a graduate probability course. The benefit is that one gains access to a proof of the strong law of large numbers that is pleasantly free of technicalities or tricky ideas.
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.490
The Brocard Angle and a Geometrical Gem from Dmitriev and Dynkin
Ádám Besenyei
In a celebrated paper on the eigenvalues of special matrices, N. Dmitriev and E. Dynkin formulated and proved a nice geometrical lemma. This lesser-known gem provides a simple proof of the maximal value of the Brocard angle and yields also the solution to some challenging problems from the past few decades.
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.495
Problems and Solutions
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.500
Book Review
A History in Sum: 150 Years of Mathematics at Harvard(1825-1975)By Steve Nadis and Shing-Tung Yau
Reviewed by Gerald B. Folland
To purchase the article from JSTOR: http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.508
MathBits
A New Proof of the Pythagorean Theorem
Nam Gu Heo
A One-Line Proof of the Infinitude of Primes
Sam Northshield
An Upper Bound for Some Coloring Problem
Janusz Januszewski
Using Infinitesimals to Differentiate Secant and Tangent
Michael Hardy
100 Years Ago This Month in the American Mathematical Monthly
Edited by Vadim Ponomarenko