User talk:R.e.b.

Dickson polynomial and Cartan subalgebra questions

Howdy, I think you've already answered my question (Talk:Dickson polynomial#Schur conjecture reference) on Dickson polynomials, but I wanted to make sure. I am not terribly familiar with these sorts of objects, and so it was not entirely clear to me exactly when they were equivalent to Chebyshev polynomials. I think you explicitly say they are equivalent over the complexes, but I think some people might believe the article is implying they are equivalent over the rationals. I think they are not equivalent over the rationals, but the Schur conjecture part seems to suggest they are. Basically, I find the article confusing, but I cannot say whether it is the text of the article or my dim mind which is the cause.

I had another question (Talk:Cartan subalgebra#Maximal abelian subalgebras) about maximal abelian subalgebras that are not cartan subalgebras. Your recent sl(2n) example indirectly proves they exist, but I think it might be nice to give one explicitly. I tried to fill in a little of the detail of the argument, but I think all I showed was that an even larger abelian subalgebra existed. I think this is a standard remark explained perhaps in Jacobson's text, but I don't have it handy. Basically, I think the example can be improved with only a minor change, but I fear I don't know how to do it. JackSchmidt (talk) 18:34, 11 February 2008 (UTC)

Dickson polynomials are more or less the same as Chebyshev ones whenever the field has enough square roots (and contains 1/2), but as you say they are not equivalent over the rationals. I have no plans to add anything more to this (rather obscure) topic; feel free to make the article clearer.

The larger abelian algebra you found is in gl_2n (and is in fact maximal in gl_2n) but is not in sl_2n, at least not when the field has characteristic 0, as sl does not contain the identity matrix. R.e.b. (talk) 19:06, 11 February 2008 (UTC)

Thanks! Your correction to Cartan subalgebra is exactly what I was looking for. Sorry, I am used to working in (finite) Lie groups, and forgot I was looking for trace zero not determinant one. I don't know how to fix Dickson polynomial. I just marked the statement with {{fact}} and a comment. I suspect the explanation is simple, and there are lots of wikipedians who know more about polynomial families than I do. JackSchmidt (talk) 19:20, 11 February 2008 (UTC)

Thanks for Bad group fix

Merging the definition into the article where it is relevant whether they exist is a very clean solution. Thanks for taking care of it; all I know about Morely rank is it clutters my google searches for finite simple groups. I'll let the bad group author know the prod was changed to a redirect. JackSchmidt (talk) 05:29, 15 February 2008 (UTC)

Colleague request!

Dear colleague , please tell me how you got the formula

$B_{n}(x)=2xU_{n-1}(x/2)-2T_{n}(x/2)$

(This is an emergent request )

please answer here!

Thank you. —Preceding unsigned comment added by 41.224.221.178 (talk) 20:18, 16 February 2008 (UTC)

I just happened to notice that the formula for B_n in the article was very similar to the usual explicit formula for Chebyshev polynomials. If you want to know how to prove it, it follows easily from the explicit formulas in Abramowitz and Stegun. R.e.b. (talk) 20:38, 16 February 2008 (UTC)

Colleague gratefulness!

Thank you R.e.b.

If only you can help us finding a 2-order differential equation for these polynomials as you did to the explicit expression above!! We can provide basical equations.. Thanks

P.S. Dear colleague can you help us writng this letter : B with the sign ~ on it?? (which is pronounced B-Tilda, in french?) Thank you for help and understanding.

You could find a 2nd order differential equation for B_n by expressing it as

$B_{n}(x)=2xT_{n}'(x/2)/n-2T_{n}(x/2)$

Differentiating twice expresses B_n and its first two derivatives as linear combinations of derivatives of the Chebyshev polynomials. The differential equation of the Chebyshev polynomial gives two relations between Chebyshev polynomials and their first 3 derivatives. This gives 5 linear equations in B_n and its first two derivatives and T_n and its first 3 derivatives. Eliminating T_n and its first 3 derivatives from these equations should give a second order differential equation for B_n.

The differential equation you get like this is probably rather complicated. You can instead write B_n as the sum 4U_n(x/2)−6T_n(x/2) for n>0, both of which satisfy well known differential equations.

<math>\tilde B_n</math> produces ${\tilde {B}}_{n}$. R.e.b. (talk) 19:39, 17 February 2008 (UTC)

Colleague gratefulness and thanks!

This is great!

Thank you Sir! —Preceding unsigned comment added by 41.224.183.78 (talk) 22:47, 17 February 2008 (UTC)

Stability theory

Hello R.e.b., I see you have been active around stability theory. I haven't got the time to finish my draft article User:Hans_Adler/Stability_spectrum right now, but since you have just created a redlink for stability spectrum I thought I should mention its existence to avoid unnecessary duplication. You are very welcome to edit the draft, and to move it to article space whenever you feel it makes sense. --Hans Adler (talk) 18:20, 19 February 2008 (UTC)

OK; your draft looks fine apart from not yet having references, so I'll probably add these and move it into article space in a few days if you don't do so first. R.e.b. (talk) 18:52, 19 February 2008 (UTC)

Emergent Scientific request

Dear Colleague: you said : ...The differential equation of the Chebyshev polynomial gives two relations between Chebyshev polynomials and their first 3 derivatives..

We found only one differential equation (for each kind):

$(1-x^{2})\,y''-x\,y'+n^{2}\,y=0\,\!$

and

$(1-x^{2})\,y''-3x\,y'+n(n+2)\,y=0\,\!$

Please can you write here the relations between Chebyshev polynomials and their first 3 derivatives ???

Thank you for patience !! —Preceding unsigned comment added by 41.224.191.69 (talk) 19:06, 19 February 2008 (UTC)

If you differentiate these differential equations you get extra linear relations between y', y'', and y''' R.e.b. (talk) 19:12, 19 February 2008 (UTC)

Emergent Scientific request, Thanks !!

Thank you for help! Can you provide links to these items??