*[Corrected, thanks – T.]*

*[Corrected, thanks – T.]*

the others and the ‘s are fixed.

You’ll get a determinant with rows 1 to reading , the other rows as usual,

divided by the Vandermonde of the ‘s times the Vandermonde of the non coalescing ‘s. ]]>

we noticed that the HCIZ formula holds also for degenerate eigenvalues of either matrices A and B. However in this case, both the numerator and the denominator are zero and an implicit cancellation occurs. Do you know whether a close formula exists once the cancellation has been explicitly performed?

Best Regards

Pietro Rotondo

I asked you solve the definite integral of methods oscillatory

Integrate [(-8796541 * Pi ^ 12) / 322 * Sin [x ^ 4], {x, 74.37, 97.28}] or Integrate [(-8,796,541 * Pi ^ 12) / 322 * Sin [x * x * x * x], {x, 74.37, 97.28}]

and that your making the approximate with just pencil and paper and a pocket calculator,,,,, I hope do your answer,,,,clarified that the exercise proposed by my above,,, I do not need approximation of the number,,, I need are the procedures and series that you use to solve it, and that these series can be replaced in a pocket calculator,,,thanks

att

jefferson alexander vitola

att

jefferson alexander vitola

That’s a nice argument! I hadn’t realised that the moment map equation can be used even before one has fully verified that one has a symplectic structure. I’ve updated the text accordingly in order to incorporate this argument.

]]>First, you need to establish that is -invariant.

Second, observe that the fundamental vector fields (i.e. the vector fields that arise from the infinitesimal action of an element of the lie algebra) span the tangent space of the coadjoint orbit at each point.

Third, you need to prove the defining equation of the momentum map, i.e. show that

holds for . Here, is the fundamental vector field associated to .

Then we have

.

The last equal sign follows from (by -invariance) and E. Cartan’s homotopy formula.

As the fundamental vector fields span the tangent space, we have , as desired.

mathematician, and Brezin, the physicist. Here we are talking about

Edouard Brezin…

*[Corrected, thanks – T.]*

Thanks for the references and early history of this result! I have adjusted the text accordingly.

Regarding the Brezin-Hikami-Johansson formula, in Johansson’s paper, two proofs of this formula are given. The first, as you say, uses the HCIZ formula and would thus of course be unusable for the purposes of this blog post. But the second proof (given in page 689 of that paper) proceeds by an analysis of Dyson Brownian Motion instead of HCIZ, and Johansson comments in that paper that one can use that proof to give a non-circular proof of HCIZ, basically along the lines given in this post. Of course, this is more or less what you do in your paper with Itzykson (the heat equation weighted by the Vandermonde determinant being essentially the Fokker-Planck equation for the Dyson Brownian motion), and is also the approach I took in this previous blog post about three years ago. So I guess the summary of the situation is that the HCIZ and Brezis-Hikami-Johansson formulae are logically equivalent to each other, and either formula can be proven by solving the relevant heat equation or Brownian motion.

]]>