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

July/August 2000

* Roach Applied Math. A team at the NEC Research Institute in Princeton NJ has analyzed "the response by the wind-sensing system of the American cockroach (Periplaneta americana) to a complex hydrodynamic flow." D. Rindberg and H. Davidowitz published the results of their study in the 15 June 2000 Nature. They tapped into individual interneurons in P. americana's sensory system and determined that there is some nontrivial spectral analysis going on. "...exposing the system to narrow-band, low-frequency noise produces a strong cell response - that is, a high firing rate - whereas exposure to wide-band stimuli does not. In the limiting case of white noise, the firing rate is almost zero ..."

* Pinning down the Butterfly. In 1972, at the 139th meeting of the AAAS, Edward Lorenz gave a talk entitled, "Predictability: Does the Flap of a Butterfly's Wings in Brazil Set off a Tornado in Texas?" Lorenz's butterfly has become the standard metaphor for the central criterion of chaos: the large-scale instability of future behavior with respect to infinitesimal perturbation of initial conditions. A report in the 13 April 2000 Nature describes a (computer) experimental setup in which the tiny nucleus of future large-scale distruption can be pinned down qualitatively and quantitatively. A team led by David Egolf (Center for Nonlinear Studies, Los Alamos) investigated simulations of the Boussinesq equations, the standard hydrodynamic equations describing convection. They showed how very localized spikes in the Lyapunov vector (an indicator of exponential time divergence of similar states) correspond to topological defects: local changes in a winding number. This report was reviewed by J.P. Gollub and M.C. Cross in the same issue; their review was picked up in the June 20 edition of the web journal Science-Week.

* The Joy of Sharing. "Mathematical Devices for Getting a Fair Share" in the July-August 2000 American Scientist is a review by Ted Hill of Georgia Tech of the mathematical theory of fair division. Hill's examples range from a surprisingly subtle estate-division algorithm from a 2nd-century Babylonian Talmud, through Hugo Steinhaus' Ham-sandwich Theorem and Rob Kirby's arms-reduction system up to work of his own on cake-cutting problems when the cake has sizeable, indivisible crumbs. Recent wrinkles in the theory include "envy-free" and "super-fair" divisions, those last being divisions in which "every participant receives a share he feels is worth strictly more than a fair share."

--Tony Phillips
SUNY at Stony Brook

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