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

This month's topics:

Math on the Wild Side

Olivia Judson's weekly New York Times blog "The Wild Side" is being taken over for three weeks by the Steven Strogatz of Cornell, who refers to himself as "(gasp!) a mathematician." Strogatz has posted two columns so far, on May 19 and May 26, 2009.

  • May 19: "Math and the City" (available online). Strogatz starts with the observation "One of the pleasures of looking at the world through mathematical eyes is that you can see certain patterns that would otherwise be hidden." This first column focuses on various studies of scaling, which reveal "Manhattan and a mouse to be variations on a single structural theme." He starts with Zipf's law. George Zipf, a linguist, noticed that if you order the words in a language by the frequency with which they occur (in speech, say), with the most frequent first, then the nth word on the list has 1/n the frequency of the first word, to a good approximation. This distribution, called "Zipf's Law" turns up in many contexts. For example, Zipf "noticed that if you tabulate the biggest cities in a given country and rank them according to their populations, the largest city is always about twice as big as the second largest, and three times as big as the third largest, and so on. In other words, the population of a city is, to a good approximation, inversely proportional to its rank. Why this should be true, no one knows." Now for the scaling: "For instance, if one city is 10 times as populous as another one, does it need 10 times as many gas stations? No. ... The number of gas stations grows only in proportion to the 0.77 power of population." And here's where the mice--and elephants--come in: "on a pound for pound basis, the cells of an elephant consume far less energy than those of a mouse. The relevant law of metabolism, called Kleiber's law, states that the metabolic needs of a mammal grow in proportion to its body weight raised to the 0.74 power." 0.77, 0.74, "Coincidence? Maybe, but probably not. There are theoretical grounds to expect a power close to 3/4." The column ends with a useful set of references.
  • May 26: "Loves Me, Loves Me Not (Do the Math)" (available online). Here Strogatz makes his point ("the laws of nature are written as differential equations") by modeling the ups and downs of a tempestuous romantic relationship. He posits a theoretical Romeo and Juliet, with interlocking behavior patterns: "The more Romeo loves her, the more she wants to run away and hide. But when he takes the hint and backs off, she begins to find him strangely attractive. He, on the other hand, tends to echo her: he warms up when she loves him and cools down when she hates him." (The DEs are given in an appendix: dR/dt = aJ, dJ/dt = -bR with a,b > 0, along with the qualitative description of the general solution: "Romeo and Juliet behave like simple harmonic oscillators"). There's a nice riff on the three-body problem at the end involving both Isaac Newton and Strogatz's college girlfriend's old boyfriend. Again, a good set of references.

 

"Intimidatingly awesome" knot theory project

Last April 7 Cory Doctorow posted on the "boingboing" blog ("a directory of wonderful things") an item with a link to Sana Raoof's presentation of her Intel-award winning project, "Computation of the Alexander-Conway Polynomial on the Chord Diagrams of Singular Knots." Ms. Raoof, then a Senior at Jericho High School on Long Island, now at Harvard and presumably a Sophomore, won an Intel Foundation Young Scientist Award at the Intel International Science and Engineering Fair in Atlanta, May 2008. Her amazingly articulate 3-minute presentation sketches her result (the Delta-polynomial, the formal logarithm of the Alexander-Conway polynomial, can be used as a complete invariant of singular knots whose chord diagram is a complete bipartite graph -- see Michael Luby's explanation on the NSF-supported National Science Digital Library website) and emphasizes possible applications to the biology of tangled molecules. Sana Raoof is in fact awesome, and worth watching.

Beyond Turing machines

Nature for May 20, 2009 carries a "News and Views" piece by Philippe Binder (University of Hawaii, Hilo and Kavli Institute) entitled "Computation: The edge of reductionism." The question is: to what extent can the behavior of a physical system be computed, e.g. modeled accurately on a digital computer? It is motivated by the recent publication of "More really is different" by Gu, Weedbrook, Perales and Nielsen (Physica D 238 835-839) which proves "that many macroscopic observable properties of a simple class of physical systems (the infinite periodic Ising lattice) cannot in general be derived from a microscopic description." Binder goes over the distinction between irreducible and undecidable systems, taking up work of Stephen Wolfram. A system is reducible if there is a mathematical shortcut, for example a closed formula, predicting its behavior (Binder mentions the cosine function for a simple harmonic oscillator). As an example of an undecidable system Binder gives the 1-dimensional cellular automaton "elementary rule 110" ("two states are allowed ('0' or '1'), and any cell will evolve to 0 if either its state and that of its right-neighbour cell are 0, or if its state and those of both its immediate neighbours are 1 -- otherwise it will evolve to 1.") with a very simple program but with the property (established by Wolfram and Matthew Cook) that it is universal (equivalent to a Turing machine), so as a system it is indecidable. What Gu et al. have achieved is to map an undecidable cellular automaton onto a physical model (an infinite, periodic Ising system) and to deduce that the long-term behavior of that system is also undecidable.

One does not often encounter "Alas" in Nature but here it is: "Alas, their results apply only to infinite lattices, and hence seem of limited use." Nevertheless Binder is optimistic "that finite objects may, after all, have undecidable properties." One of the hints he sees is related to work of Lenore Blum, Stephen Smale, and Mike Shub (Bull. Amer. Math. Soc, 21 1 (1989)), which describes computations over the real numbers.

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