Contents for May 2000
Three Fermat Trails to Elliptic Curves
Elliptic curves: what are they and why are they so called? This paper answers that question and gives some applications of them, including one, to congruent numbers, that may not be familiar. A congruent number is one that is the area of a right triangle whose sides have rational lengths. So, 6 (the area of the (3, 4, 5) right triangle) is a congruent number, as is 5 (the area of the (3/2, 20/3, 41/6) right triangle), but 4 is not. How to tell which integers are congruent numbers? Use elliptic curves.
Modeling Mathematics With Playing Cards
"Arrange the deck so the suits are in an order, say spades, hearts, clubs, diamonds, that repeats throughout the pack. Deal as many cards as you like to form a pile. When the pile is about the same size as the remaining portion of the deck, riffle shuffle the two portions together. If you now take cards in quadruplets from the top of the shuffled pack you will find that each set of four contains all four suits." Amazing! That is not all that Martin Gardner can do with a deck of cards.
Summing Series with Integrals
It is not generally known that can be written as a simple integral. But it (and infinitely many other series) can, making its sum, , easy to find.
On Lunda-Designs and the Construction of Associated Magic Squares of Order 4p
Lunda designs, found in Africa, can be used to construct magic squares. Not of every order, but if you feel the need for a 20 ( 20 magic square you can now satisfy it.
The Super Bowl Theory: Fourth and Long
When a team from the National Football Conference wins the Super Bowl, the stock market goes up, and when it doesn't, the stock market goes down. Usually, that is. Alas, the predictive power of this rule is on the decline and in a few years it may be down to zero. The author does not predict what the next spurious correlation will be.
455 Mathematics Majors: What Have They Done Since?
Patricia Clark Kenschaft
Why would anyone want to major in mathematics? Fewer and fewer students have been finding reasons to in recent years. This is too bad, because the study of mathematics is splendid preparation for almost anything that comes after. One survey from one school cannot prove this proposition, but it provides evidence in its favor.
The problem is to determine where an object is by taking sightings from several sites. The difficulty is that the position determined is not a point but an area. If we must report a point, which do we choose?
The Geometry of Statistics
The normal density is built into nature: as the author says, "the normal density is the native density of Euclidean space." Pick another metric and you get a different native density.
Fallacies, Flaws and Flimflam
A two-line proof that all the coefficients in a Fourier series are the same, and other items.
Eigenvalues of Matrices of Low Rank
Stewart Venit and Richard Katz
You don't need Mathematica to find the eigenvalues of a 10(10 matrix. If, that is, it has rank 2. Here's how to do it.
Binomials to Binomials
From time to time we may need to write something like in the form 38 + 17 We could use Mathematica for that, too, but it is nice to know it can be done independently of electronic devices and without the drudgery of actually expanding a ninth power.
Introducing Hyperbolicity Via Piecewise Euclidean Complexes
Jessica Benashaski, John Meier, Kevin O'Brien, Paige Reinheimer, and Margaret Skarbek
Illustrating plane or spherical geometry is easy: planes and spheres abound.
Hyperbolic geometry is tougher. Potato chips are too brittle. But Thurston paper will do the trick.
t-Probabilities as Finite Sums
Though I've never tried, I suppose that Mathematica can calculate t-probabilities along with all the other things it can do. Here is another way, more human.
Problems and Solutions
New results about 666, and thanks to referees, who are never beastly.
A Beautiful Mind
Reviewed by Peter Ross