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Math in the Media 0899

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Tony Phillips' Take on Math in the Media A monthly survey of math news |

A new Mersenne prime! These are prime numbers of the form: the *n*-th power of 2 minus 1. For instance 3 (*n*=2), 7 (*n* = 3), 31 (*n* = 5), 127 (*n* = 7) are Mersenne primes. These were known in antiquity. More were found starting in the Renaissance, about one per century, until the 20th. Now they've been finding one a year. Last year the first 37 were known, with the largest corresponding to *n* = 3021377. On June 1 1999 a new Mersenne prime was found, corresponding to *n* = 6972593. This number has 2098960 digits in its decimal expansion. The discoverer, Nayan Hajratwala, gets a $50,000 prize for his achievement (less than 2.5 cents per digit). Hajratwala worked with the Great Internet Mersenne Prime Search which ``harnesses the power of thousands of small computers like yours to solve the seemingly intractable problem of finding HUGE prime numbers.'' You can play, too.

What's purple and commutes? An abelian grape. What's gray and does non-commutative algebra? Your unconscious brain. The May 20 *Nature* has a report ``Non-commutativity in the brain'' by an international team (Canada, Switzerland, Germany). The vestibulo-ocular reflex (VOR) is what moves your eyes in the opposite direction when your head turns. Since the algebra of rotations (under composition) is non-commutative it follows that the neural processing that computes the reflex must be non-commutative, too. Surprisingly, according to the authors this ideal behaviour is not predicted by most models of the neural circuitry underlying the VOR.

What has six legs and does path integrals? An ant. A british-french team studying the navigation of *Cataglyphis cursor* analyzes the interplay between path-integration and landmark recognition (``The use of path integration to guide route learning in ants'', *Nature*, June 24). The experiments reported are all 2-dimensional; otherwise the ants would be surely be performing non-abelian path integration.

Phase transitions in computation. In a physical system a phase transition occurs when the system switches from one pattern of behavior to another. Above and below zero centigrade water has two very different responses to changes in temperature. Freezing-melting is a phase transition. Scientists studying computational complexity have found similar phenomena in their investigation of the *K-SAT* (for ``K-satisfiability'') problem. This is reported in ``Separating the Insolvable and the Merely Difficult, '' a piece by George Johnson in the July 13 *New York Times*, picking up an article in in the July 8 *Nature* by Monasson, Zecchina, Kirkpatrick, Selman and Troyansky along with a `news and views' column by Philip Anderson. The K-SAT problem can be taken as a representative for ``a large collection of problems (several thousand in fact) from many different fields.'' What the authors find is that the variation of solution time with respect to increase in problem size shows a phase transition: a critical size (analogous to the melting point of water) above which and below which the response patterns are different. Moreover, the nature of this phase transition depends on the parameter K and on the resulting complexity of the K-SAT problem. ``Our results show that techniques from statistical physics can provide important new insights into computational phenomena, possibly leading to improved solution methods. In addition, computational problems, viewed as many-particle systems, provide models which challenge the assumptions of physics, and may shed light on the behaviour of some highly nonlinear materials now coming under study.''

Phase transitions in traffic. Ever wonder why you suddenly go from stop-and-go to smooth sailing? You are experiencing a phase transition in traffic flow. Peter Weiss in the July 3 *Science News* explains how mathematical models are indicating ``previously unrecognized complexity in highway flow.'' Some models have been linked with data from road sensors to give real-time information on traffic flow. Sample the traffic in Duisburg, Germany. Warning: not all traffic engineers dig the sophistication. ``When congestion arises for no apparent reason,'' says Carlos Daganzo of UC Berkeley, ``this just means that its cause has not been identified.''

Unemployed but busy genius. The MacArthur Foundation 1999 Fellows include Jeff Weeks, free-lance mathematician and author of *The Shape of Space*. At the Geometry Center Weeks developed the Snappea hyperbolic structures programs which among other things go from a knot you sketch with the mouse to a complete description of the canonical hyperbolic structure on the complement of that knot in 3-space. Check out his genial Web site.