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This month's topics:Lévy flight for fishPaul Lévy (18861971) is considered to be one of the founders of modern probability theory. Among his innovations is what is now called "Lévy flight," a random walk in which the probability of step length L is given by a powerlaw distribution: in simplest terms, the probability that L is greater than a is proportional to a^{k}, for some k > 0. (For this simple form to work L must have a positive minimum L_{min}, and then the proportionality constant is L_{min}^{k}). A more familiar random walk is Brownian motion, but there the probability of long steps decreases exponentially. The path of a Lévy flight. Note the characteristic sporadic large steps. Image from Wikipedia Commons. As explained in the notes for the Yale course on Fractal Geometry, a familiar example comes from the game of hideandseek:
The "Lévy flight foraging hypothesis" predicts that in nature "predators should adopt search strategies known as Lévy flights where prey is sparse and distributed unpredictably, but that Brownian movement is sufficiently efficient for locating abundant prey." A report in the June 24 2010 Nature ("Environmental context explains Lévy and Brownian movement patterns of marine predators" by Nicolas E. Humphries, David W. Sims and 14 coauthors) found "strong support" for this hypothesis "across 14 species of openocean predatory fish (sharks, tuna, billfish and ocean sunfish)..." They attached electronic tags to 55 individuals from the 14 species to obtain timestamped depth records. From these 1dimensional data were selected 129 "behaviourally consistent" sections, which were then statistically analyzed (maximumlikelihood fitting methods, goodnessoffit tests) for manifestation of a power law. They found in particular "Lévy behaviour to be associated with less productive waters (sparser prey) and Brownian movements to be associated with productive shelf or convergencefront habitats (abundant prey)." For example, in productive waters of the equatorial convergence front of the central North Pacific, the entire track of a silky shark (Carcharhinus falciformis) was best fitted by an exponential model, whereas for another silky shark tracked farther north in oligotrophic [= lacking in plant nutrients] waters, the best fit was a truncated power law with an exponent of 2.02 ..." (truncation because of constraints from ocean surface and floor; this exponent is our k plus 1). They conclude: ".. not only do our results lend strong support to the contention that Lévy flights occur in freeranging animals, but our observations of pattern switching between Lévy and Browniantype motion suggest that searching animals adaptively adjust their optimal patterns of movement to different environmental resource distributions." This work was picked up by DiscoveryNews on June 10 with the title "Sharks have math skills." Hardy and Ramanujan at Lincoln CenterThe Complicite production of "A Disappearing Number" (covered on this site when it opened in London in 2007) played five days in July at the Lincoln Center Festival, New York. It won a glowing review from Charles Isherwood in the July 17 2010 New York Times. "Sure, there is some daunting talk of string theory and convergent series and the cosine of half pi Z, not to mention the varieties of infinity. But 'A Disappearing Number' ... is lucid, dynamic and continuously engaging. It's not fundamentally about numbers, either, but about the search for meaning and the consoling satisfaction of finding the patterns that define and describe both the physical universe and individual human lives." He sketches out the story of Hardy ("a renowned mathematics lecturer at Cambridge") and Ramanujan ("the story of his career is indeed remarkable, and among the chief pleasures of the play is its sympathetic exploration of his personal odyssey") and how it interweaves with the other main plot line (a "fairly conventional love story"): Ruth, who teaches math at a British university, becomes involved with Al, an American hedge fund manager with connections to India. "All beautiful theorems require a very high degree of economy, unexpectedness and inevitability," says one of the characters. Isherwood: "That's not a bad recipe for beautiful theater either." "A Disappearing Number" will be shown live from Theatre Royal Plymouth on October 14 as part of the Fall 2010 season of National Theatre Live ("will bring the best of the National's theatre to cinema screens around the world"). Electoral math in The New YorkerAnthony Gottlieb reviews in The New Yorker (July 26, 2010) a book by George Szpiro: "Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present," just published by Princeton. Gottlieb does not spend much time on the book itself, except to remark that Szpiro "is more interested in math than in politics." But he gives the New Yorker readers a lively survey of the topic (after a detailed look at the bewilderingly complex series of ballots and lotteries by which the Venetians elected their doges). "The history of voting math comes mainly in two chunks: the period of the French Revolution, when some members of France's Academy of Sciences tried to deduce a rational way of conducting elections, and the nineteenfifties onward, when economists and game theorists set out to show that this was impossible." From the first period we meet JeanCharles de Borda (method "still used in some sporting competitions, in the Eurovision Song Contest, and to elect the Parliament of Nauru, a Pacific atoll"), Condorcet and Laplace. And from the second there is Kenneth Arrow who "examined a set of requirements that you'd think any reasonable voting system could satisfy, and proved that nothing can meet them all when there are more than two candidates," which won him a Nobel Prize in 1972. Much of the review dwells on the "winner takes all" system used in Britain and in "some of its former colonies including the United States" but in few other developed countries. This system has serious flaws. "Say that fiftythree percent of American voters are Democrats and fortysix percent are Republicans (mirroring the vote in last year's Presidential election). Then imagine that the supporters of the two parties were spread evenly throughout the country's districts ... . In that case, the Democrats would win every seat and there would be no Republicans in Congress." The system may be replaced in Britain (a referendum on the topic is scheduled for 2011), possibly giving hope for voting reform elsewhere. Gottlieb leaves us with a cautionary note. No matter how mathematically satisfactory a voting scheme may appear, "... political parties will change their campaign strategies, and voters the way they vote, to adapt to the new rules, and such variables put us in the realm of behavior and culture." The math of elections was the subject of feature columns by Joe Malkevitch on this site: Voting Games Part I and Part II.
Tony Phillips 
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