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This month's topics:
Penfield NY, a front in "the nationwide math wars"
Samuel G. Freedman's On Education column in the November 9 2005 New York Times reports from Penfield NY, a community which "has become one of the most obvious fronts in the nationwide math wars." These are the wars "that pit progressives against traditionalists, with nothing less than America's educational and economic competitiveness at stake." Freedman talked to parents, like
What has happened to math instruction in the Penfield Central School District? Freedman also spoke with district officials, who explained that
In fact "Penfield has begun supplementing the constructivist classes with lessons in computation," according to the assistant superintendent, but Samuel G. Freedman does not believe in compromise: "... in the math wars, tweaking around the edges does not settle the issue. The dispute is fundamental." [This issue deserves much more careful reporting and analysis. -TP]Glacial climate cycles and the least common multiple
In part of the late glacial period severe climate oscillations occurred with a period of almost exactly 1470 years; these are documented by ice-core samples from Greenland, and are called Dansgaard-Oescher (DO) events. The period of these oscillations has been mysterious, because there are no excitations of that frequency either in the solar record or in the variation of the Earth's inclination and orbit. Holger Braun and his colleagues in Heidelberg, Potsdam and Bremerhaven report in the November 10 2005 Nature on a possible explanation. There are two "pronounced and stable centennial-scale solar cycles," the DeVries-Suess (period 210 years) and the Gleissberg (86.5 years); the German group designed a model to test the hypothesis that the sum of these two excitations could be driving the DO oscillations. In general two different periodic astronomical phenomena will have irrationally related frequencies unless there is "phase locking," but there are approximate common periods: waiting long enough one can get the two back as close as one wants to their initial relative position. It turns out that for the DeVries-Suess and Gleissberg cycles, 1470 years is a very good approximation to a common period (it equals 7 x 210 and almost exactly 17 x 86.5). The team used CLIMBER-2, a global climate and biosphere simulation model that has been around since 1998, forcing it with
F(t) = -A1 cos(ω1t + φ1) - A2 cos(ω2t + φ2) + K
where ω1 and ω2 are the DeVries-Suess and Gleissberg frequencies, and K represents changes in the background climate compared with a baseline.
The response of the model for different combinations of periodic excitation amplitude A = A1 = A2 (vertical) and K -- a baseline measurement of the general warmth of the climate (horizontal). The pale green squares represent the parameter ranges for which the model manifests a 1470-year periodicity. Image from Nature 438 208, used with permission. Click for full-scale image and legend.
The results of the simulation show a region where the 1470-year period would be stable under perturbation. As the authors remark, the simulation also shows that similar oscillations could not happen today.Math on the Millennium Bridge
The Millennium Bridge, a 325-meter footbridge spanning the Thames in London, opened on June 10, 2000. The November 3 2005 Nature ran a Brief Communication entitled "Crowd synchrony on the Millennium Bridge," describing what happened and giving a mathematical analysis. "Soon after the crowd streamed on ... , the bridge started to sway from side to side; many pedestrians fell spontaneously into step with the bridge's vibrations, inadvertently amplifying them." This is not the classical example of marchers across a bridge exciting a resonance of the structure. Rather there was a positive feedback loop in which the bridge invited the initially unorganized pedestrians into synchrony. The authors of the Nature communication - a five-man team led by Steven Strogatz of Cornell - modeled the phenomenon by "adapting ideas originally developed to describe the collective synchronization of biological oscillators such as neurons and fireflies." Their model starts with the differential equation for a forced, damped harmonic oscillator:
M d2X/dt2 + B dX/dt + K X = G(sinΘ1 + ... + sinΘN)
where X(t) is the lateral displacement, and each pedestrian "imparts an alternating sideways force G sinΘi to the bridge; ... Θi(t) increases by 2π during a full left/right walking cycle." What you wouldn't have seen in Introductory Differential Equations is feedback. Since feedback works through the phase difference between the natural oscillation of the bridge and the gait of the pedestrian, the authors make the pair (X,dX/dt) into an angular variable by setting X=AsinΨ, dX/dt=√(K/M)AcosΨ. Then the feedback is expressed in the set of equations
dΘi/dt = Ωi + CAsin(Ψ-Θi+α);
here Ωi is the natural walking rhythm of the i-th pedestrian and α is a phase lag. The model, once tuned by the adjustment of the parameter C, gives a close simulation of the actual event: as the number of pedestrians increases, nothing untoward happens until a critical number is reached, "when the bridge starts to sway and the crowd starts to synchronize, with each process pumping the other in a positive feedback loop."
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