Dear Sir,

Phi_i(xk) stands for the (i,k) element f the matrix PHI. Dont know if i is the row and k is the column or vice-versa. To me (but this is rather discutable), it makes most sense if i is the column, and k is the row. Then, Phi_i becomes a comumn vector, and

PHI = (phi_1, phi_2 ... phi_n). phi_1(xk) is the k-the element of the column vector.

I have the impression there is a mistake in (41).  The author writes x, where in fact it needs to be xk. I think its normal you got confused. To know for sure, of course, you should contact the author.

Best Regards and Cheers,

Henri.

In Theorem 2, \phi_i(x_j) defines the ijth element of an NxN matrix \Phi(x). More precisely, the ith row of \Phi(x) is associated with a function \phi_i and the jth column with variable x_j (i.e., the jth element in vector x). This is easiest to understand if we consider (9) as a special case where \phi_i(x_j) = x_j^{i-1}. This takes us directly to (43).

But if you consider (43), x_k is just a dummy variable for integration. So we can simply use x as long as we keep in mind what we are doing (often used approach in tensor notation), that is, taking products of integrals with appropriate indexing. Finally, the brackets in (41) define a rank 3 tensor as in Definition 1 so that we may use notation (39) to express (43).

What is the result good for? Well, it means that you can evaluate a difficult multidimensional integral of some determinants (and a function \xi_k) by summations and multiplications of single-dimensional integrals of functions that define the matrices involved (and the same function \xi_k as before). If this additional function \xi_k happens to be nice and does not depend on the index k, it is in fact enough to calculate some simple integrals and a determinant of a matrix formed by them (Corollary 1 and Corollary 2). The latter is indeed what is needed in Section IV (Section V needs a bit more sophisticated result given in Theorem 3).

best,

mikko

Dear Mikko, all,

- All you say is correct, however, the formulae (41) in the document is still wrong.

- You can indeed, for the reasons you mention, replace xk by x, and dxk by dx, but then also the border condition needs to change.

- If you use xk (as I suggest), then the border condition is  k = 1 .. N

- If you use instead x (as done, and as you suggest), the border condition becomes

k = 1 .. N and x = xk.

- If you do this all day, the latter (x=xk) may be obvious, but the latter for sure is nowhere precised in the document, resulting this omission makes the document difficult to understand for somebody more junior. Mathematically, its also incorrect, as I just equally well, being totally ignorant, may assume x=x1, or x=x(k-1), or anything.

Cheers,

Henri.

Dear Henri,

I'm afraid I cannot follow the argument on border conditions. How the paper is written, if we accept (43) then we have to accept (41) as well. There is no information carried in the index k in the variable x_k in (43) since the product is outside. Therefore, the integrals can be taken independently of each other and all information is in the functions \phi_i, \psi_j and \xi_k, that depend on their respective indexes. These functions can take any dummy variable as an argument when considered in the context of (41) and (43). One just has to remember what \mathcal{T}(...) means in here.

I do not claim (41) is correct since I haven't checked it (although I believe it is), my comment was only about the notation asked by Di Zhang. But if (43) is assumed to be correct then (41) is correct as well (by assumption). In fact, one advantage of the results provided in Appendix is that they get rid of the vector x and "replace" it by a scalar. If you look at Section IV, for example, you'll notice that the result involves only powers of a scalar variable x, and those powers come from the indexing of "row functions" \phi_i and \psi_j. That's why calculating the end result is possible.

best,

mikko

Hi, Mikko, Henri and all,

Thank you so much for your kindly, patient reply and question following here!

And may sorry for my late reply here because those days, I also trying to figure it out by myself.

I can understand the proof now while taking the  \phi_i, \psi_j and \xi_k as some dummy variables. Actually, I also infer them as some dummy variables but not so sure. Thank you so much for your reply here. Indeed, it will be a good tools to calculate some intractable multidimensional integral of this kind. 

Actually, in my ongoing work, I am also facing such a multiplied multidimensional determinants with a function x_k. Thus I refer to this paper for the conclusion.

For the border condition, while taking three of them within the integral functions as some dummy variables of x in (41), as  the definition of \Phi(x), \Psi(x) are two $N \times N$ matrices, thus the border of i,j,k should be 1,...,N as the paper going there, in my opinion. Is that correct?

Best Wishes to your work!

Di

Hi all,

yes the indexing is given both in Definition 1 and (41). The paper uses shorthand notation i,j,k=1,...,N to mean i=1,...,N and j=1,...,N and k=1,...,N. In fact, the problem setup in Theorem 2 already tells us that this is the only valid choice.

BTW. I did not mean that \phi_i, \psi_j and \xi_k are dummy variables - just that their arguments are. Indeed, \phi_i, \psi_j define the matrices involved as we saw in my first reply. In any case, hope this helped :-)

best,

mikko

Hi Mikko,

You tell 'there is no information in the index k in x_k'. I am afraid I dont understand that. I think you need to explain me a little more in detail.  I mean, my understanding was

that \phi_i(x) is associated with row i, and to denote the j-th element of this row (the j-th column), we write \phi_i(x_j).

This is how the notation is used in formulae (43). We see there \phi_sigmak(xk), selecting row sigma_k and column k.

Now, if we go back to (41), we see the notation \phi_i(x), meaning row i, and the column is not precised. Now, how should I know which column to take from the matrix \phi ?. From context its quite clear its column k (to me, Di Zang had some more troubles with that), but if I take this formulae (41) in isolation, I mean I drop all the rest of the paper, just take (41) and its boundary conditions, how should I know its the k-the column ?.  I mean, why cant I take the i-the column, or the j-the column, or the column denoted with sigma(k), permutating row and column in equation (43) ?.

Best Regards,

Henri.

Hi Henri,

there are no columns in the final formula, just two row functions of the matrices \Phi and \Psi (+ the extra function \xi). This is the whole point of the formulation and it works only because the matrices involved are fully defined by their respective row functions \phi and \psi (and the dimension N). The columns "vanish" because after integration there is nothing to separate x_1 from x_2.

I suggest you try to check the results in Appendix in relation with the results in Section IV. This might help you to understand the point. I'm afraid I am not able to explain this in any other way.

EDIT: I should have said "fully defined by their respective functions after integration over the argument" since in general the matrix depends on the argument vector x. Sorry about the mistake. 

best,

mikko