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A monthly survey of math news
Chinese characters can help you learn geometry. This is a tentative conclusion of a study published in the January Journal of Cross-Cultural Psychology and picked up in the September 28, 2001 Chronicle of Higher Education. In the study, run by John Nuttall (Boston College), Chieh Li (Northeastern) and Shuwen Zhao (Capital Normal, Beijing), Chinese-speaking Chinese-American students who wrote Chinese had their SAT math scores compared with those who did not. The male character-writers averaged 28 points better than those who could only speak (643 vs. 615), while the female differential was even larger: 703 vs. 629. According to the Chronicle, the authors ``speculate that the linearity of writing English does not allow children to develop the same spatial-reasoning skills as writing Chinese, which involves both left-right and up-down thinking.'' A related piece appears in the on-line Boston College Chronicle.
Evolution's mathematical rhythm is presented in a work by Philip Gerrish (Theoretical Biology and Biophysics, Los Alamos) entitled ``The rhythm of microbial adaptation,'' in the September 30, 2001 Nature. The methods section of the paper starts out with pure mathematics: ``Let the fitness of lineages created by contending mutations be independent, identically distributed random variables drawn from an unknown distribution. etc.'' Gerrish calculates that ``adaptive steps can have fairly strong rhythm.'' The strength of this rhythm, ``its relative temporal regularity, is equal to a constant that is the same for all microbial populations.'' In particular the scaled probability distribution of the number of accumulated adaptations in a given time frame is population-independent.
Haven't we had this conversation before? The phenomenon has been investigated by a team of Brazilian statistical physicists (Lima, Martinez, Kinouchi, ``Deterministic walks in random media;'' their Phys. Rev. Lett. article is available online.) The article was picked up in a News and Views piece ``The salesman and the tourist'' by Eugene Stanley and Sergey Budriev in the September 27 2001 Nature. The travelling tourist problem is a local optimization problem: cities are distributed at random but the tourist always visits the nearest city next. A refinement on the problem forbids the tourist from re-visiting any one of the last N cities visited. What happens? Lima, Martinez and Kinouchi show that, no matter how large N is, cyles are formed in which the tourist is trapped into eternally revisiting the same p cities in the same order. (They also show that the distribution of these p-cycles follows a power-law function.) Stanley and Budriev muse on the manifestations of traveling tourism in every-day life. ``Many humans tend, after some time, to fall into a rut -- ... even for some scientists to revisit the same set of ideas in their scholarship -- all actions seemingly occurring as if by magic when in fact there may be a simple underlying reason.'' The trick is to hold out for a large N.
Multiplication in the brain of owls, at least. A paper in the April 13 2001 Science by José Luis Peña and Masakazu Konishi entitled ``Auditory Spatial Receptive Fields Created by Multiplication'' shows how . Barn owls (Tyto alba) can locate prey in total darkness. Their ear openings are at slightly different levels on the head, and set at different angles, according to Owlpages.com. Peña and Konishi examine how barn owls process the two independent cues by which they locate the origin of a sound in space: the interaural time difference, or ITD (how much earlier it arrives at one ear than at the other) and the interaural level difference or ILD (how much louder it sounds to one ear than to the other). ``The owl's auditory system computes interaural time (ITD) and level (ILD) differences to create a two-dimensional map of auditory space. ... . A multiplication of separate postsynaptic potentials tuned to ITD and ILD, rather than an addition, can account for the subthreshold responses of these neurons to ITD-ILD pairs.'' The surprising word here is multiplication. An accompanying article by Laura Helmuth (``Location Neurons Do Advanced Math'') quotes Christof Koch, a CalTech biophysicist: "This is the cleanest evidence of multiplication in the brain." But he also says ``We don't know anything about how [the multiplication] is computed.''
Catastrophes in Nature. The October 11 2001 issue features the long review article ``Catastrophic shifts in ecosystems,'' by Marten Scheffer, Steve Carpenter, Jonathan Foley, Carl Folke and Brian Walker; the catastrophe in question is our old friend the cusp catastrophe. The authors use a large number of recent studies from many ecological contexts, grouping the references by habitat: Lakes, Woodlands, Deserts, Oceans. Under ``Lakes,'' for example, they describe ``hysteresis in the response of charophyte vegetation in the shallow Lake Veluwe [Netherlands] to increase and subsequent decrease of the phosphorus concentration,'' a classic manifestation of the cusp catastrophe. The moral of the story, although one that may be difficult to translate into practical action, is that, when deleterious alternative states are possible, one should work at keeping the system far from the catastrophe locus.
Drunk on fractals. A 40-year old conjecture on random walks (``drunkard's walks'') has recently been solved by ``an important and rigorous application of fractals to probability theory and mathematical physics.'' This from Ian Stewart's News and Views piece ``Where drunkards hang out'' in the October 18 2001 Nature. The conjecture, due to Paul Erdös and S. J. Taylor, was proved this year by Amir Dembo, Yuval Peres, Jay Rosen and Ofer Zaitouni (preprint available online) in Acta Mathematica. The conjecture involves the number of times a planar random-walking particle can be expected to revisit its most frequently visited site in the first n steps. The answer is (log n)2/pi . Fractals? According to Stewart, the particle ``makes frequent excusions away from the most frequently occupied disc, but keeps returning to it. These excursions occur on all length scales, which is where fractal geometry comes in.''