Monthly Archives: August 2012

Plectics, Statistical mechanics

Currently Reading

A. Franceschini et al. (2011), “Transverse Alignment of Fibers in a Periodically Sheared Suspension: An Absorbing Phase Transition with a Slowly Varying Control Parameter” Physical Review Letters 107, 25: 250603. DOI: 10.1103/PhysRevLett.107.250603.

Abstract: Shearing solutions of fibers or polymers tends to align fiber or polymers in the flow direction. Here, non-Brownian rods subjected to oscillatory shear align perpendicular to the flow while the system undergoes a nonequilibrium absorbing phase transition. The slow alignment of the fibers can drive the system through the critical point and thus promote the transition to an absorbing state. This picture is confirmed by a universal scaling relation that collapses the data with critical exponents that are consistent with conserved directed percolation.

Evolution, Statistical mechanics

Evolutionary Ecology Readings

Last October, a paper I co-authored hit the arXivotubes (1110.3845, to be specific). This was, on reflection, one of the better things which happened to me last October. (It was, as the song sez, a lonesome month in a rather immemorial year.) Since then, more relevant work from other people has appeared. I’m collecting pointers here, most of them to freely available articles.

I read this one a while ago in non-arXiv preprint form, but now it’s on the arXiv. M. Raghib et al. (2011), “A Multiscale maximum entropy moment closure for locally regulated space-time point process models of population dynamics”, Journal of Mathematical Biology 62, 5: 605–53. arXiv:1202.6092 [q-bio].

Abstract: The pervasive presence spatial and size structure in biological populations challenges fundamental assumptions at the heart of continuum models of population dynamics based on mean densities (local or global) only. Individual-based models (IBM’s) were introduced over the last decade in an attempt to overcome this limitation by following explicitly each individual in the population. Although the IBM approach has been quite insightful, the capability to follow each individual usually comes at the expense of analytical tractability, which limits the generality of the statements that can be made. For the specific case of spatial structure in populations of sessile (and identical) organisms, space-time point processes with local regulation seem to cover the middle ground between analytical tractability and a higher degree of biological realism. Continuum approximations of these stochastic processes distill their fundamental properties, but they often result in infinite hierarchies of moment equations. We use the principle of constrained maximum entropy to derive a closure relationship for one such hierarchy truncated at second order using normalization and the product densities of first and second orders as constraints. The resulting `maxent’ closure is similar to the Kirkwood superposition approximation, but it is complemented with previously unknown correction terms that depend on on the area for which third order correlations are irreducible. This region also serves as a validation check, since it can only be found if the assumptions of the closure are met. Comparisons between simulations of the point process, alternative heuristic closures, and the maxent closure show significant improvements in the ability of the maxent closure to predict equilibrium values for mildly aggregated spatial patterns.

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Bad Math, Bibliophilia

Precalculus -> Statistics

Now that 2.2 metric Ages of Internet Time have passed since Andrew Hacker’s ill-advised “math is hard!!” ramble, I figure it’s a good day to propose my own way of improving high-school mathematics education. Be advised: this is a suggestion about the curriculum, not about how to train teachers, buy books and all that un-TED-friendly stuff which reformers happily gloss over. And I’ll be talking about changes late in the game, which won’t address problems at the “why can’t Johnny add?” level.

When I was in high school—at a pretty well-supported public school, out in the ‘burbs at the comparatively unimpoverished end of town—I took a “precalculus” class my eleventh-grade year. Most of the advanced-track students I knew did the same thing. (If you’d gotten yourself on the even-more-advanced track back in eigth grade, you took precalculus in tenth.) This was supposed to prepare us for taking the AP Calculus class our senior year, which would allow us to get college credit. Instead, it was a thoroughgoing waste of time. The content was a repeat of Algebra II/Trigonometry, which we’d taken the year before, with two exceptions thrown in. The first, probability, was a topic our teacher didn’t know how to teach. In fact, she admitted as much: “I don’t know how to teach probability, so you’re all going to read the book today.” The second, limits, served no purpose. I’ll explain why in a moment.

I suggest the following: scrap “precalculus” and replace it with a year-long statistics course. This plan has several advantages:
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