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This month's topics:

The math of slithering

 Image : A snake slithers across a rough surface. Slithering is the mode of snake locomotion in which the creature seems to flow along a sine curve traced on the ground (the image on the left represents a sequence of stills from a movie available online). This motion is analyzed and explained in "The mechanics of slithering locomotion" by David L. Hu, Jasmine Nirody, Terri Scott and Michael J. Shelley of the Courant Institute, NYU (Hu is also affiliated with Georgia Tech), published in the Proceedings of the NAS (106 10081-10085, June 23, 2009). "We model snakes ... as inextensible 1-dimensional curves X(s, t) = (x(s, t), y(s, t)) of fixed length L and uniform mass per unit length ... . Here, s is the curve arc length measured from the head, and t is time." The snake's secret is the frictional anisotropy of its belly scales: the forwards, backwards and transversal friction coefficients are in the ratio μf : μb : μt = 11 : 14 : 20. The authors use a numerical solution of the force-acceleration equation to show how a traveling wave of curvature κ(s, t) = α cos k π (s + t) exploits that frictional anisotropy to produce uniform forward motion. The three coefficients of friction: μf forwards, μb backwards and μt transversal. This report was picked up in a "News and Views" piece by Andrew Clark and Adam Summers in the June 18, 2009 Nature. They point out a fundamental mechanical difference between snake locomotion and that of limbed animals. Even at high speeds, "the gravitational or frictional forces are an order of magnitude higher than the inertial forces. So ... snakes are not running but walking."(Images courtesy of David Hu.)

NYU's Abel Trifecta

"Complex Math, Simple Sum: 3 Awards in 5 Years" was the headline for Lisa Foderaro's article in the May 31 2009 New York Times. The occasion was Mikhael Gromov's Abel Prize (the Times ran a photograph of him with Queen Sonja and King Harald of Norway) which turned out to be the third won by a Courant Institute professor. The prize has only been awarded since 2003, so the Courant has won three, vs. four for the rest of the universe (their others were Peter Lax in 2005 and Srinivasa Varadhan in 2007). Foderaro sketches the mathematical biographies of all three, along with a recounting of the why-no-Nobel-in-math story: "Some speculate that Alfred Nobel, a Swedish industrialist who invented dynamite and who established the prizes, simply wasn't a math guy. Others cite a bit of lore that he harbored a grudge against math because a love interest rejected him in favor of a well-known mathematician." She ends with a quote from Varadhan: "We had a very good math teacher in high school who instilled in us the idea that math didn't have to be work. You could do it for fun."

Those evil mathematicians, again

This time it's from Time magazine (July 9, 2009, in a story by David von Drehle and Jay Newton-Small about the present and future of Sarah Palin.) "If ever there has been a time to gamble on a flimsy résumé ... this might be it." The reason: "Résumés ain't what they used to be; they count only with people who trust credentials--a dwindling breed. The mathematics Ph.D.s who dreamed up economy-killing derivatives have pretty impressive résumés." Etc.

Cortical circuits for mental arithmetic

"Throughout the history of mathematics, concepts of number and space have been tightly intertwined." This is the beginning of the abstract of a report in Science for June 19, 2009. The authors (André Knops, Bertrand Thirion, Edward M. Hubbard, Vincent Michel, Stanislas Dehaene, based at various institutions in and near Paris) "tested the hypothesis that cortical circuits for spatial attention contribute to mental arithmetic in humans." The underlying neurobiological context is that "cultural inventions such as writing and mathematics" are "too recent for natural selection to have dedicated specific brain mechanisms to them. It has therefore been suggested that they co-opt or 'recycle' evolutionarily older circuits with a related function ... ." It is known that people think of numbers as positioned on a "mental number line" with (for left-to-right readers) smaller numbers to the left and larger ones to the right. This and other pieces of evidence suggest searching in the parts of the brain related to vision for activity related to mental computation. The authors concentrated on regions associated with saccades (the quick simultaneous movements with which the eyes track moving objects; they also occur in reading), and "predicted that mental addition, which increases number size, would be associated with a rightward shift of attention and subtraction with a leftward shift." They tested this hypothesis by first training a computer to distinguish the fMRI records of subjects moving their eyes to the right, from those moving their eyes to the left; and then asking the computer to predict, from fMRI records, whether subjects were performing mental addition or mental subtraction. "Equating addition with rightward saccades and subtraction with leftward saccades, the mean accuracy for inferring whether an addition or subtraction was performed, averaged over all participants, was 55.0 ± 1.8%, which is significantly greater than chance."

Tony Phillips
Stony Brook University
tony at math.sunysb.edu