Recently I learned about a matrix called Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying.

Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations[2]

I noticed that carleman matrix may be a generalization of Maclaurin series.

  • I researched about Maclaurin series of multivariable function.

$$a_{kl}=\frac 1{(l-k)!k!} \left [ \frac {\partial^l f(x,y)}{\partial x^k\partial y^{l-k}} \right ]_{ x=0, y=0}$$

$$f(x,y)=\sum^\infty_{l=0}\sum^l_{k=0}a_{kl}x^ky^{l-k}$$

And I modified above to make it look like carleman matrix:

$$a_{jkl}=\frac 1{(l-k)!k!} \left [ \frac {\partial^l f(x,y)^j}{\partial x^k\partial y^{l-k}} \right ]_{ x=0, y=0}$$

$$f(x,y)^j=\sum^\infty_{l=0}\sum^l_{k=0}a_{jkl}x^ky^{l-k}$$.

Which gives me something like a Carleman tensor. (I COINED IT)

Is my modification correct? And if then, could you teach me how to apply this tensor?

E.g.

How can I create a carleman tensor with multivariable functions such as $f(x,y)=x+y$?

(Like the Maclaurin series of a multivariate function).

Example of desired answers:

The carleman coefficients for $f(x,y)=x+y$ are:

$$M_{xy}[x+y]_{jk}= \cdots$$

Also I want to know how to iterate functions with the multivariable carleman tensors like:

$M[f(g(x),h(y))]=M[f]\bigoplus$ Concatenate $\left(M[g], M[h] \right)$

Which might not (mostly) be correct.

Edit #2

I think I need to find out how multivariable functions can be coordinatizated in orthogonal function space.

Edit #3

I suddenly came up with an idea, namely,

Partial carlemann matrix

And

Total carlemann matrix $:=\sum \bigoplus \text{partial carlemann matrix}$.

Hope that the above idea helps as a clue..

  • I found some useful material here: sciencedirect.com/science/article/pii/S1474667017313149/… – KYHSGeekCode Aug 31 at 15:13
  • Please be free to comment on this. I'll be fully appreciated with your small comments as I am not an authority nor an expert(I'm just a high schooler). Are my expressions about multivirate maclaurin series correct?(I've just guessed from this answer ( math.stackexchange.com/a/1020025/553404) and might be incorrect) – KYHSGeekCode Sep 1 at 9:12
  • There was a critical error on this maclaurin series and I fixed it.. – KYHSGeekCode Sep 4 at 14:38
  • You have made many edits to this question during the last 7 days, which suggests that the question is not clearly formed (or that you are making minor edits needlessly) – Yemon Choi Sep 7 at 14:05
  • @YemonChoi Too many edits... I agree. I think I am somewhat too interested. – KYHSGeekCode Sep 8 at 5:10

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.