| The Putnam in Time. "Crunching the Numbers" is the title of a piece by Lev Grossman, in the December 23 2002 Time magazine, about the William Lowell Putnam Mathematical Competition. "Every year," it begins, "on the first Saturday in December, 2,500 of the most brilliant college students in North America take what may be the hardest math test in the world." Grossman gives a quick survey of the history of the exam, a summary of the daunting statistics ("the median score on last year's test was 1 point. Out of a possible 120.") and a Time-like glimpse of its mystique ("think of it as a coming-out party for the next generation of beautiful minds"). He interviews Leonard Klosinski (Santa Clara; the competition director), Richard Stanley (coach of the MIT team) and Kevin Lacker, one of last year's winners, who remarks: "Doing well on the Putnam and doing good math research are two different tasks that take two different kinds of intelligence." |
The piece includes a sample problem, labeled "An Easy One." "A rightcircular cone has a base of radius 1 and a height of 3. A cube isinscribed on the cone so that one face of the cube is contained in thebase of the cone. What is the length of an edge of the cube?" CheckTime for the answer.
Too much pi? Under the title "How to Slice the Pi Very, Very Thin," theDecember 7 2002 New York Times ran an AP dispatch fromTokyoreporting on the calculation of pi to 1.24 trillion places,"six times the number of places recognized now." A ten-person teamled by Yasumasa Kanada broke the trillion-place barrier with the help of a Hitachi supercomputer at the InformationTechnology Center of Tokyo University.The report quotes David Bailey(Lawrence Berleley Lab): "It's an enormous feat of computing,not only for the sheer volume, but it's an advance in thetechnique he's using. All known techniques would exceed thecapacity of the computer he's using." Which is, we are told,two trillioncalculations a second. Note that light travels .15 mm in onetwo-trillionth of a second. This must be a very small or veryparallel computer.
The best ways to laceyour shoes has been worked out by Burkard Polster, a mathematicianat Monash University (Victoria, Australia). His report, in theDecember 5 2002 Nature, was picked up in the December 10Boston Globe (via Reuters) and in Time magazinefor December 23.
The best way to lace depends on your criteria, but in all allowablelacings each eyelet is connected to at least one eyelet on theopposite side. Thestrongest lacings with n pairs of eyelets are the "crisscross" (when the ratio h of vertical eyelet spacing to horizontal is below a certain valuehn) and the "straight" (when h isgreater than hn). The shortest lacings arethe "bowties". There is only one minimal bowtie lacing when n iseven, but there are (n+1)/2 when n is odd. Theshortest "dense" lacing (no vertical segments) is the crisscross.
Freak waves. BBC Two, on November 14, 2002, aired a program on this phenomenon and its recent mathematical analysis. Freak waves, also "rogue waves," "monster waves," are extraordinarily tall and steep waves that appear sporadically and wreck havoc with shipping. One is suspected to have washed away the German cargo München which went down with all hands in the midst of a routine voyage in 1978. More recently, the cruise ship Caledonian Star was struck by a 30m wave on March 2, 2001. The standard analysis of ocean waves predicts a Gaussian-like distribution of heights; extreme heights, although possible, should be very rare - a 30m wave is expected once in ten thousand years, according to the BBC. But these waves occur much more frequently than predicted. The program focused on new methods of analysis, and on the work of the mathematician A. R. Osborne (Fisica Generale, Torino). Osborne has applied the inverse scattering transform, which he describes as "nonlinear Fourier analysis," to the time series analysis of wave data. He conducted simulations using the nonlinear Schrödinger equation and found near agreement with the standard analysis, except that "every once in a while a large rogue wave rises up out of the random background noise." His paper, availableonline, gives an example of such a simulation:
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