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September 2000

Updating Ramanujan. The June 17 2000 issue of Science News has a very complete and satisfying piece by Ivars Peterson about the recent discovery of new Ramanujan-type partition congruences. The n-th partition number p(n) is the number of different ways of expressing n as a sum of positive integers less than or equal to n.. So p(5) = 7, as is easy to check, but these numbers grow very rapidly with n. Ramanujan discovered for example, that p(5n + 4) is always a multiple of 5. (Thus p(4) = 5, p(9) = 30, p(14) = 135, p(19) = 490, ...). He also discovered similar congruences involving the primes 7 and 11. No one knew if those were all the possible partition congruences and if so, what was so special about 5, 7 and 11. Peterson recounts how Ken Ono, a number theorist at Penn State and Wisconsin-Madison, became interested in the problem and how he ended up proving that in fact there exist infinitely many partition congruences, work reported in the January 2000 Annals of Mathematics. Ono only gave one example: p(a n + b) is always a multiple of 13, where a = 594 x 13 and b = 111247. (This gives an idea of why such congruences had not been found before!) His work was complemented in a remarkable way by Rhiannon L. Weaver, an undergraduate at Penn State, who developed an algorithm and used it to generate over 70000 new examples. Peterson quotes Ono: "It is now apparent that Ramanujan-type congruences are plentiful. However, it is typical that such congruences are monstrous."

Making RNA think. In the first and classic example of DNA computing (The Hamiltonian Path Problem) the problem involved constructing a certain path, and so was naturally suitable for attack using strings of DNA to represent segments of the path. A new approach using RNA seems much more flexible, since the strings are abstract encodings of of configurations, and the chemical operations correspond to the logical either-or statements constituting the problem. Charles Seife reports in the 18 February 2000 Science on work by the evolutionary biologist Laura Landweber and her associates at Princeton. They unleashed their RNA on the "3x3 knight's problem:" finding all the ways of placing knights on a 3 by 3 chessboard so that no knight can attack another. The holy grail of this line of research is the nucleic-acid solution of a problem that cannot be done by a human in a reasonable amount of time.

5 + 2 = yellow. Synaesthesia is a condition in which stimuli from one sense (typically, sounds) elicit impressions related to a different sense (typically, colors). A rare form is digit-color synaesthesia, where the decimal digits 1, 2, etc. evoke colors in the mind of the synaesthete. A team at the University of Waterloo, led by Mike Dixon, set up an ingenious experiment to show that in their subject the synaesthesia was between the concept of the number (and not, for example, the shape of the digit) and the corresponding color. This subject associated seven and yellow; in the experiment she showed significantly longer reaction times in naming color patches which were not yellow when they were shown next to an equation (e.g. 5 + 2) which implied seven. This work was reported in the July 27 2000 issue of Nature.

Contentious disciples. What happens when 10 people all want to be as close as possible to a central point, but want to stay as far away from each other as possible? A team at Harvard (B. Grzybowski, H. Stone and G. Whitesides) set up an experiment using 1mm-diameter magnetizable washers floating in a dish above a rotating bar magnet. The net magnetic force attracts them to the center, but each one picks up a rotation which drags the nearby fluid into a tiny whirlpool, and these whirlpools repel each other. An article in the June 29 2000 Nature shows the resulting configurations. Some (for n = 2, 3, 4, 5 washers) are what one might expect: the vertices of a regular n-gon. But then it gets more complicated. For n = 10, 12 and 19 (as high as they could go with this set-up) there are two possible optimal configurations:

The two stable configurations ("polymorphs") of 10 mutually repelling points all attracted to the center. "... the patterns spontaneously interconvert between the polymorphs."

A Figure-8 knot in a protein. W. R. Taylor of the National Institute for Medical Research in London has discovered a Figure-8 knot deeply embedded in the string of amino acids that make up the plant protein acetohydroxy acid isomeroreductase, as reported in the 24 August 2000 Nature. "Deeply" means here that even though an open string cannot be topologically knotted, the knot is so far from the ends of the string that for practical purposes it is a topological invariant. Taylor started from the three-dimensional structure of the protein, and interpreted it as a curve in space. He then applied a smoothing algorithm just like the ones topologists use to produce nice round pictures of knots, except that he kept the free ends fixed. Taylor proposes a mechanism for how this knot could be formed during the folding of the protein.

Math in Los Angeles. David Ferrell had a very nice piece in the August 12 2000 Los Angeles Times on the giant MathFest 2000, which ended that day. He interviewed Tony Chan, head of UCLA's new Math Institute, Peter Sarnak on the Riemann Hypothesis ("Right now ... we go around trying to fix things with a screwdriver. But once we prove this we'll have a machine gun. It'll just blow away problems.") and Ed Witten. He asked Witten if he found it "distressing that so few appreciate -much less understand- the radical concepts that govern his life, that may change our whole understanding of reality?
[Witten] thought a minute. It bothers me more,' he said, that I don't understand the math.'"

-Tony Phillips
SUNY at Stony Brook

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