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This month's topics:

The Euclidean metric and jail time

Clyde Haberman's "NYC" column in the New York Times for November 29, 2005, tells the story of one James Robbins, who learned the hard way about the difference between the Euclidean metric and the Taxicab metric. Mathematically speaking, the Euclidean distance between two points in Manhattan is the length of the straight line segment between them, whereas the Taxicab distance is the length of the shortest possible path between them along the streets. The two metrics are comparable, in that E.d.(x,y) ≤ T.d.(x,y) ≤ E.d.(x,y)√2 (at least in the rectangular part of town) but the two distances are not the same; and when the law says "within 1000 feet of a school" it means within a Euclidean radius of 1000 feet. Mr. Robbins' lawyer argued that the the spot where his client had been found guilty of selling drugs was actually 1254 feet away from Holy Cross School: 764 feet north along Eighth Ave. and 490 feet west along 43rd St., and that therefore his client should not get the extra penalties allotted for selling near a school. Do the math: he lost.

Mathematical patterns in asthma attacks

A mathematically powered breakthrough in the study of the incidence of asthma attacks, with potentially important therapeutic implications, was reported in the December 1 2005 Nature. Urs Frey (University Hospital of Berne) works in pediatric respiratory medicine; Béla Suki (Boston University) is a physicist who "analyses complex nonlinear systems, such as the factors that contribute to avalanches" (quote from an "Authors" sketch at the beginning of the issue). With their collaborators, they analyzed the records of a "previously published, randomised, placebo-controlled, double-blind crossover study" following 80 asthmatic subjects for 3 six-month treatment periods. In that study, the PEF (peak expiratory flow) of each subject was measured twice daily; the subject was also assigned a daily asthma symptom score. The team's strategy was to "examine whether the statistical and correlation properties of the time series of PEF recordings can be used to predict the risk of subsequent exaggeration of airway instability." They can. To disentangle the effects the authors created a "nonlinear stochastic model of the PEF fluctuations" ("a cascade connection of a linear dynamic system followed by a second order nonlinear system with no memory. ...") They were able to tune this model to match the statistical characteristics of the experimental data, and then use it to measure the impact of the characteristics separately. One startling conclusion from their analysis is that short-acting bronchodilators, such as the popular drug albuterol, can actually aggravate medium-term risk of an asthmatic attack.

Atiyah pulls the strings

Sir Michael Atiyah appears in Nature for December 22, 2005 as the author of a review ("Pulling the Strings") of Lawrence Krauss's "Hiding in the Mirror: The Mysterious Lure of Extra Dimensions, from Plato to String Theory and Beyond." Sir Michael uses the occasion to share some of his understanding of math, physics, imagination, reality, and string theory.

• "At every major step physics has required, and frequently stimulated, the introduction of new mathematical tools and concepts. Our present understanding of the laws of physics, with their extreme precision and universality, is only possible in mathematical terms."

• "The mathematical take-over of physics has its dangers, as it could tempt us into realms of thought which embody mathematical perfection but might be far removed, or even alien to, physical reality. Even at these dizzying heights we must ponder the same deep questions that troubled both Plato and Immanuel Kant. What is reality? Does it lie in our mind, expressed by mathematical formulae, or is it 'out there'."

• " ... string theorists can explain plausible models of a unified universe, but unfortunately they cannot explain why we inhabit a particular one."

• "The mathematics involved in string theory is quite remarkable by any standards. In subtlety and sophistication it vastly exceeds previous uses of mathematics in physical theories. Almost every part of contemporary mathematics is involved somewhere in the story. Even more remarkable is that string theory has led to a whole host of amazing results in mathematics in areas that seem far removed from physics. To many this indicates that string theory must be on the right track. ... Time will tell."

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