October 23

Difficult part: It took me a fair amount of time to understand why N2 in figure 6.3 is an equilibrium point, though I think I realize why now (it should be because qEN is being subtracted from the other plot, and thus the result will be 0 at N2, indicating an equilibrium point). Other than that, I generally understood the reading, though "unit effort" struck me as an odd concept.

Reflective part: It's interesting that despite the depensation model seeming to account for damaged fish populations, Myers et al. found this generally not to be the case (assuming the validity of the study). Also, as I implied earlier, I'd be interested to see how exactly E translates into the real world.

October 16

Difficult part: for the most part, the reading seemed fairly straightforward, especially since I've already had differential equations. That said, I'm not sure where the constants in equations (1), (2), and (3) in Cayrel et al. come from.

Reflective part:
This seems somewhat different from what we've done thus far, in that the underlying math is not terribly complicated, but the scientific principles from which the math is derived are fairly involved and, in the case of Cayrel et al., comprise the bulk of the paper (contrast to the phyllotaxis paper, which focused almost exclusively on the math, while hand-waving the science). Cayrel et al. seems fairly obtuse, but much of the difficulty comes from astrophysics, rather than the math.

October 4th, 2007

Difficult part: the most difficult part was easily the derivation of the generalized Zipf's law. In particular, I'm not entirely sure what the normalization condition mentioned in 3 is (though I have some idea).

Reflective part: the two papers seem to demonstrate, quite effectively, the perils of neglecting to thoroughly examine one's model to make sure it really supports what one wants to say. Zipf assumed that the correlation seen between word rank and frequency was due to some "law of economy", when in fact it derived inherently from the nature of his model.

September 27

Difficult part:

I'm a little confused by the factoring done in the Vblood equation at the bottom of page 33, but other than that, I think I generally understood the math.

Reflective part:

On the one hand, the way much of life seems to obey simple scaling laws for metabolism and similar factors is impressive; on the other hand, the controversy around much of this, as well as some of the initial, incorrect suppositions of a 2/3 scaling factor shows that math cannot make predictions in a vacuum, even if the predictions are convenient.

September 25

Difficult part: overall, the section seems pretty easy to understand, but I was a little unsure of what the metabolism plot would look like on a linear scale; it should be some kind of geometric curve, right?

Reflective part: the regularity with which metabolism scales and the prevalence is quite interesting. This seems to be another example where mathematics can provide an explanation for how something occurs, even when biology is unable to explain why.

September 18

Difficult Part:

I found the most difficult part to be pages 658-660 of Atela et. al. I'm unsure how the quadrilateral Qm,n is defined and what points m,n refer to (arbitrary ones or specific ones). In addition, I was fairly confused by their definition of φ, in relation to S.

Reflective Part:

I find it interesting that, given the relatively simple Hofmeister rule, the paper's authors were able to extrapolate the math involved into something quite complex and byzantine (this, in turn, describes a fairly simple to describe phenomenon).

Second Post

Most difficult part: On the whole, I found the trickiest part to be understanding May's explanation of the existence of two cycles once a>3. I'm not completely sure why the slope of F^2(x) is the square of the slope of F(x), and I think Figures 2 and 3 might have been switched (it looks like Figure 3 has one point of intersection with the 45 degree line, whereas Figure 2 has two), unless I've misread this section.

Also, I found the derivation on Strogatz pg. 350 of the deviation near a fixed point to be a bit confusing; does the O(n) (not the actual symbol, but close enough) term refer to standard big O notation, or does it mean something else? If the former, where does that term come from?

Reflective part: This section seems quite similar to the information on bifurcation covered in my Differential Equations class last semester. However, this seems a bit different since it deals with plotting a variable against itself, with a time variable being implicit (I think; if not, I really got thrown by the reading and/or my differential equations abilities are rustier than they should be), and this was enough to throw me off a bit.