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A monthly survey of math news

This month's topics:

L'affaire Bogdanov back in Le Monde

This has been going on for quite a few years, but some explicit mathematical details emerged recently. A post in Le Monde's blog En quête de sciences (In quest of science; October 16, 2010) quoted from the referees' report on Grichka Bogdanov's 2003 thesis for the high-level Docteur ès Sciences degree in mathematics. "... This construction, which has the level of a school exercise on quantum groups such as can be found in many theses for Masters or Diplôme d'Études Supérieures degrees (for which, nevertheless, a much higher quality of writing is required) is therefore the only mathematical content of the thesis." More on the brothers Bogdanov here and ici. See the full text of the referees' report.

Two (equally good) heads better than one

"Optimally Interacting Minds," due to a British/Danish team lead by Bahador Bahrami, appeared in Science for August 27, 2010. The abstract begins: "In everyday life, many people believe that two heads are better than one." This is the belief the authors set out to test, "in the context of a collective low-level perceptual decision-making task." "Collective" here means that two people were tested together. The task, briefly put, is to distinguish which of two arrays of six striped discs contains one (the "contrast oddball") that has anomalously strong contrast with the background. Clearly when the anomaly is very small the task is very difficult, but when it is large the task becomes easy, and accuracy plotted against contrast difference should follow an S-shaped curve like those plotted here.

quality of observers

For this experiment, observer sensitivity is rated by the maximum slope at which accuracy improves with contrast difference. Here the observer with slope S2 is more sensitive than the observer with slope S1. Image adapted from Bahrami et al., Science 329 1081-1085.

The authors evaluate the sensitivity of an observer by the maximum slope of the corresponding S-shaped curve. They then construct four reasonable schemes for combining the reports of two observers, examine for each how the combined sensitivity Sdyad should depend on the individual S1 and S2, and subject their predictions to experiment. Only one of these schemes turns out to be qualitively consistent with the empirical data: "weighted confidence sharing" (WCS) in which the two observers share their opinions, and also share with each other a measure of their individual confidence, "an internal estimate of the probability of being correct;" the confidence measures (which can vary with the contrast difference) enter by a simple algorithm into the determination of the collective decision. For the WCS model, the authors calculate Sdyad = (S1 + S2)/√2. So when S1 and S2 are approximately equal, Sdyad in the WCS model is a solid improvement over either one separately. But if S2 is substantially smaller than S1 (in fact, once S2/ S1 < √2 -1), then Sdyad < S1; in such cases two heads are not as good as one.

The calm before the tipping point

Martin Scheffer (Wageningen) and his co-authors showed in 2009 that "systems that are close to a tipping point become very slow in recovering from perturbations" (Nature 461 53-59).


A system in a robust equilibrium will manifest stronger "restoring force" and consequently react to small random perturbations with a quick return to the equilibrium point. A system near a tipping point will react to perturbations on the same scale with slower returns. This exemplifies the "critical slowing down" (CSD) phenomenon. Image adapted from Scheffer's "News & Views" piece (Nature 471 411-412) introducing the report presented here.

A report in the September 23 2001 Nature aims to mine this observation for statistical predictors of a tipping point to come. In "Early warning signals of extinction in deteriorating environments," John Drake (University of Georgia, Athens) and Blaine Griffen (University of South Carolina, Columbia) work with Daphnia magna, a 5mm-long water flea. Populations of Daphnia are assigned to deteriorating-environment (treatment) or constant-environment (control) groups after a 100-day initial stabilization period. (In this experiment, "detiorating environment" means less and less food). Mean time to extinction for populations in the treatment group is 297 days, and after 416 days they have all gone extinct. For Drake and Griffin, the tipping point (bifurcation) here is a population level so low that there is no chance of it rebounding; for these populations of Daphnia, it occurs between 271 and 316 days after the start of the experiment. Drake and Griffin report the discovery of four statistical indicators that "all showed evidence of the approaching bifurcation as early as 110 days (~8 generations) before the transition occurred." A caveat: they were only able to recognize these signals because they had control populations at hand. A population could be its own control if conditions are stationary before environmental degradation sets in. But otherwise "the detection of deterioration through CSD may prove to be a formidable challenge if there are not adequate baseline data or reference systems available for calibration."

Drake and Griffin remark in passing that the logistic equation (now usually taught in Calc II) P' = kP(1-P/K) for a population P, growth rate with unlimited resources kP and carrying capacity K, can already predict CSD: if we think of perturbations to the system as perturbations of P, and the deteriorating-environment treatment as letting k → 0, the equation ∂P'/∂P = k(1 - 2P/K) implies that the size of the growth response to changes in P also goes to zero.


Charles Seife's new book of that name recently scored a plug in the New York Times Magazine (October 17, 2010). The context was Ben Zimmer's weekly "On Language" column, that week devoted to "truthiness" in honor of the fifth anniversary of Stephen Colbert's introduction of the "zeitgeisty word." Those unfamiliar with this useful concept are referred to Zimmer's column. "Proofiness" is defined by Seife ("the title is very much a homage to Colbert") as "the art of using bogus mathematical arguments to prove something that you know in your heart is true --even when it's not."

Tony Phillips
Stony Brook University
tony at

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