Hgintis

Terrific work on developing the dual inheritance theory article! Perhaps you would be interested in adding more detailed information on guided variation and biased transmission at the article, adaptive bias. That article currently only talks about adaptive bias within the framework of Error Management Theory, which has been developed by David Buss and Martie Haselton. (It would be interesting to see if there are any comparisons and contrasts which can be made between Error Management Theory and The Costly Information Hypothesis, as well.) Just some thoughts... EPM (talk) 20:52, 4 May 2008 (UTC)

Your addition to Elementary symmetric polynomial

I see you have added a proof for the fundamental theorem of symmetric polynomials at the mentioned article. I realize this represents quite an effort, unfortunately, since you could have spared it by reading the existing article more carefully. The proof you give is basically the same as the "alternative proof" given before it, with a slight twist since you order the monomials "backwards" (taking the exponent of the last rather than the first variable as the most significant one). Also your formulation is somewhat clumsy at certain points (for instance by suggesting that coefficients play a role in comparing monomials, which they don't).

Just for a rapid comparison, you transform a symmetric polynomial into an expression in the elementary ones by ordering monomials and repeatedly making a contribution that matches the leading term, while adding only lower ones. Thus is precisely what is described in the paragraph starting "Now one proves by induction on the leading monomial...", except that there the work remaining after a step is represented by an invocation of the induction hypothesis, which is a fairly common way to represent an iteration in a mathematical proof. In fact the idea of repeatedly matching leading terms is also quite standard, as it is done similarly in long division (of polynomials). The "alternative proof" focuses on the one aspect that needs particular attention, the fact that the product of elementary symmetric polynomials proposed has the required leading monomial (the lemma). You also show this, though more informally, by a reasoning that tracks the possible contributions to the product. It is more convincing (because less technical and dependent on details of the context) to invoke the general property that ensures the absence of higher terms: for any monomial ordering, the leading term of a product of polynomials is the product of their leading terms (this is true whenever the product of (nonzero) the leading coefficients is nonzero, as is always the case in an integral domain or (as here) if the leading coefficients are all 1). So the only essential point here is that the ordering is a monomial ordering, which is true for any lexicographic ordering.

As for the choice of the product that matches the leading monomial, you take as exponents differences of powers of the x_i, which is also done in the previous proof, albeit somewhat implicitly: to find the product giving X^λ the Young diagram with λ_i boxes in row i is formed, and each of its columns gives an elementary symmetric polynomial as factor; the number of columns of length k is λ_k − λ_k+1 (since λ_k gives the number of columns of length at least k.)

Finally note that your proof does not address uniqueness of the expression, although this can easily be obtained.

The upshot is unfortunately that your addition is superfluous. I can see if there is anything that can be merged into the previous proof (maybe mention the effectiveness of the procedure more explicitly, some example might be adapted). Anyway the alternative proof that was given should be sourced, which I shall do shortly (it occurs more or less in this form in several books; by contrast, I don't think your proof can be sourced). So please be prepared to see your work disappear soon, or at least transformed profoundly; sorry about that, but it's like this in Wikipedia. Marc van Leeuwen (talk) 10:08, 1 May 2010 (UTC)

Group Selection, Kin Selection

Hallo Hgintis, I think you've just broken some refs in Group selection. I'm also a bit unsure about the mix of ref styles in Kin selection - there seem to be both little blue numbers and (Author, Date, page no)s in parentheses, feels a bit odd. I wonder if we're allowed to rearrange it so there are only new-style refs? Chiswick Chap (talk) 20:53, 17 February 2013 (UTC)