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+--- Thread: Bridging fractional iteration and fractional calculus (/showthread.php?tid=1730)
Bridging fractional iteration and fractional calculus - Daniel - 03/25/2023
Shouldn't there be a bridge between fractional iteration and fractional calculus? Consider my work on fractional iteration that derives the fractional iteration of a function from a fixed point. What if abstractly the function being iterated is the differential function and the fixed point is the exponential function?
RE: Bridging fractional iteration and fractional calculus - tommy1729 - 03/25/2023
but f'(x) = f(f(x)) is a problematic equation for x near a fixpoint.
Say the fixpoint is zero.
take the truncated taylor series for an infinitesimal h that vanishes at h^n = 0.
then
f(h) = 0 + a h + b h^2 + ... + 0.
f'(h) =/= f(f(h))
not even close.
So we already have issues with integer iterations and integer derivatives.
Similarly the carleman matrix A(f) does not satisfy
D^n A(f) = A(f)^n
Or take an example without fixpoints :
t(x) = x + 1.
the derivatives are 0 eventually.
while the iterations are not.
regards
tommy1729
RE: Bridging fractional iteration and fractional calculus - tommy1729 - 03/25/2023
even if f ' (x) = f(f(x)) then f ' ' (x) = f(f(f(x))) does not hold.
For nonlinear f.
regards
tommy1729
RE: Bridging fractional iteration and fractional calculus - tommy1729 - 03/25/2023
James will probably disagree with me :p
RE: Bridging fractional iteration and fractional calculus - tommy1729 - 03/25/2023
im still thinking though .. just my first ideas.
RE: Bridging fractional iteration and fractional calculus - JmsNxn - 03/27/2023
I actually don't disagree with you, Tommy!
We have to apply the fractional derivative/integral to an AUXILIARY function; not the function itself. So if I write:
\[
\vartheta(w,\xi) = \sum_{n=0}^\infty f^{\circ n}(\xi) \frac{w^n}{n!}\\
\]
Then the fractional derivative:
\[
\frac{d^{s}}{dw^s} \Big{|}_{w=0} \vartheta(w,\xi) = f^{\circ s}(\xi)\\
\]
This converges to the standard regular iteration (schroder iteration for geometric fixed points; and abel iteration for neutral fixed points). The trouble with this method, is that it's very non-trivial to show that this integral transform converges.
ALSO! It's very important to note; that we must use the exponential differintegral (Or the Riemann-Liouville differintegral); because this is the sole differintegral that satisfies:
\[
\frac{d^s}{dw^s} e^{\lambda w} = \lambda^s e^{\lambda w}\\
\]
Which is the differintegral:
\[
\frac{d^s}{dw^s} f(w) = \frac{1}{\Gamma(-s)} \int_0^\infty f(w-y)y^{-s-1}\,dy\\
\]
Here the integral is taken along SOME PATH \(\gamma(0) = 0\) and \(\gamma(\infty) = \infty\). This definition is not path independent.
Which in our language is: \(f(\xi) = \lambda \xi\) then:
\[
\vartheta(w,\xi) = e^{\lambda w} \xi = \sum_{n=0}^\infty f^{\circ n}(\xi) \frac{w^n}{n!}\\
\]
And:
\[
\frac{d^{s}}{dw^s} \Big{|}_{w=0} \vartheta(w,\xi) = \lambda^s \xi = f^{\circ s}(\xi)\\
\]
This is pretty fucking trivial here; but if we allow for more advanced \(f\) (and not dilations); the same result happens. I've proven it in as general a manner as I could. There are still edge cases I'm not certain on; but I am certain they "converge in some manner". For example take:
\[
\vartheta(w,\xi) = \sum_{n=0}^\infty \sin^{\circ n +1}(\xi) \frac{w^n}{n!}\\
\]
Then:
\[
\frac{d^s}{dw^s}\Big{|}_{w=0} \vartheta(w,\xi) = \sin^{\circ 1+s}(\xi)\\
\]
But this is only true for \(\xi \in \mathbb{R}\). It's nowhere holomorphic at \(\xi = 0\)--where we get an asymptotic series. And outside of this area; I'm still not sure what happens--but it looks like it does converge.
If \(0<|f'(\xi_0)|<1\) then this result is "generally true"--it gets really tricky. For example; let:
\[
f(\xi) = -\frac{\xi}{2} + \xi^2\\
\]
Then we have to take:
\[
\vartheta(w,\xi) = \sum_{n=0}^\infty f^{\circ 2n}(\xi) \frac{w^n}{n!}\\
\]
Then:
\[
\frac{d^s}{dw^s}\Big{|}_{w=0} \vartheta(w,\xi) = \left(f^{\circ 2}\right)^{\circ s}(\xi)\\
\]
This is not the iteration, but it's close enough you can recover the iteration. We have to take into account the period of \(-1\). The function \(e^{\pi i s}\) has period \(2\).
\[
\frac{d^{s/2+ \pi i s}}{dw^{s/2 + \pi i s}}\Big{|}_{w=0} \vartheta(w,\xi) = f^{\circ s}(\xi)\\
\]
I may have made a typo here( hope to god not); but this creates a fractional iteration that is exactly the Schroder iteration. It's actually pretty easy to identify too. A lot of my notes are scattered on the subject, but I do have a few papers on my arxiv dealing with specific instances. I never found a "global theory" that worked; so I abandoned a lot of the work. It ended up being a case by case kind of theory; and that's ugly as fuck; so I moved away from it.
There's a lot, and I fucking mean a lot, of similarities between the exponential differintegral and local iteration (or regular iteration).
Regards, James
RE: Bridging fractional iteration and fractional calculus - JmsNxn - 03/29/2023
I thought I'd add here a relationship between "iterated matrices" and "iterated derivatives". This is a secondary thought to most of my work on fractional calculus; but it's masterful as a bridge between these operations. I am going to refer to this as Ramanujan; and "little circle method" mathematicians would look at it.
Let's let the Matrix \(A\) be a non-singular matrix; so that \(A^{-1}\) exists. To be simple; let's assume that \(A : \mathbb{C}^n \to \mathbb{C}^n\). And let's write that:
\[
e^{Ax} = \sum_{n=0}^\infty A^n \frac{x^n}{n!}\\
\]
We can safely assume that \(e^{Ax} : \mathbb{C}^n \to \mathbb{C}^n\). Where \(x\) produces a semi-group structure. From here, we can write:
\[
\frac{d^s}{dx^s} e^{Ax} = A^s e^{Ax}\\
\]
Where this is a linear operator applying \(\mathbb{C}^n \to \mathbb{C}^n\). We can set \(x=0\); and I'm just rewriting Ramanujan's master theorem as he wrote it:
\[
\Gamma(s) A^{-s} = \int_0^\infty e^{-Ax}x^{s-1}\,dx\\
\]
And we've fractionally iterated the matrix \(A: \mathbb{C}^n \to \mathbb{C}^n\). I avoided a lot of "singular moments" here. But if we can map the matrix \(A\) well enough; this discussion is entirely rigorous. It relates Daniel's work; Sheldon's work; Bo's work; Tommy's work; and all that matrix shit in a quantum physics./hilbert space shit.
Fractional calculus is just:
\[
\frac{d^s}{dA^s} : \mathbb{C}_{\Re(s) > 0} \times \mathcal{H} \to \mathcal{H}\\
\]
Where
\[
\mathcal{H} = \{ A : \mathbb{C}^n \to \mathbb{C}^n\,|\, A^{-1} :\mathbb{C}^n \to \mathbb{C}^n\}\\
\]
Then:
\[
\frac{d^s}{dA^s} e^{Ax} = x^s e^{Ax}
\]
And we can differentiate across \(A\) or \(x\); and still have the same rules.
This is the language I think in; and it's just a translation of much of the standard "tetration forum" language; and current literature.
RE: Bridging fractional iteration and fractional calculus - tommy1729 - 03/31/2023
(03/29/2023, 07:35 AM)JmsNxn Wrote: I thought I'd add here a relationship between "iterated matrices" and "iterated derivatives". This is a secondary thought to most of my work on fractional calculus; but it's masterful as a bridge between these operations. I am going to refer to this as Ramanujan; and "little circle method" mathematicians would look at it.
Let's let the Matrix \(A\) be a non-singular matrix; so that \(A^{-1}\) exists. To be simple; let's assume that \(A : \mathbb{C}^n \to \mathbb{C}^n\). And let's write that:
\[
e^{Ax} = \sum_{n=0}^\infty A^n \frac{x^n}{n!}\\
\]
We can safely assume that \(e^{Ax} : \mathbb{C}^n \to \mathbb{C}^n\). Where \(x\) produces a semi-group structure. From here, we can write:
\[
\frac{d^s}{dx^s} e^{Ax} = A^s e^{Ax}\\
\]
Where this is a linear operator applying \(\mathbb{C}^n \to \mathbb{C}^n\). We can set \(x=0\); and I'm just rewriting Ramanujan's master theorem as he wrote it:
\[
\Gamma(s) A^{-s} = \int_0^\infty e^{-Ax}x^{s-1}\,dx\\
\]
And we've fractionally iterated the matrix \(A: \mathbb{C}^n \to \mathbb{C}^n\). I avoided a lot of "singular moments" here. But if we can map the matrix \(A\) well enough; this discussion is entirely rigorous. It relates Daniel's work; Sheldon's work; Bo's work; Tommy's work; and all that matrix shit in a quantum physics./hilbert space shit.
Fractional calculus is just:
\[
\frac{d^s}{dA^s} : \mathbb{C}_{\Re(s) > 0} \times \mathcal{H} \to \mathcal{H}\\
\]
Where
\[
\mathcal{H} = \{ A : \mathbb{C}^n \to \mathbb{C}^n\,|\, A^{-1} :\mathbb{C}^n \to \mathbb{C}^n\}\\
\]
Then:
\[
\frac{d^s}{dA^s} e^{Ax} = x^s e^{Ax}
\]
And we can differentiate across \(A\) or \(x\); and still have the same rules.
This is the language I think in; and it's just a translation of much of the standard "tetration forum" language; and current literature.
In theory yeah.
But in practice ...
Differentiating a noninteger amount of times with respect to a nontrivial infinite square matrix that might not be diagonalizable ?!
That gives a non-unique infinite tensor with divergent norm ?!
Or did you mean differentiating with respect to a vector ?
And that is just the last line of your answer.
regards
tommy1729
RE: Bridging fractional iteration and fractional calculus - JmsNxn - 04/02/2023
(03/31/2023, 07:19 PM)tommy1729 Wrote:(03/29/2023, 07:35 AM)JmsNxn Wrote: I thought I'd add here a relationship between "iterated matrices" and "iterated derivatives". This is a secondary thought to most of my work on fractional calculus; but it's masterful as a bridge between these operations. I am going to refer to this as Ramanujan; and "little circle method" mathematicians would look at it.
Let's let the Matrix \(A\) be a non-singular matrix; so that \(A^{-1}\) exists. To be simple; let's assume that \(A : \mathbb{C}^n \to \mathbb{C}^n\). And let's write that:
\[
e^{Ax} = \sum_{n=0}^\infty A^n \frac{x^n}{n!}\\
\]
We can safely assume that \(e^{Ax} : \mathbb{C}^n \to \mathbb{C}^n\). Where \(x\) produces a semi-group structure. From here, we can write:
\[
\frac{d^s}{dx^s} e^{Ax} = A^s e^{Ax}\\
\]
Where this is a linear operator applying \(\mathbb{C}^n \to \mathbb{C}^n\). We can set \(x=0\); and I'm just rewriting Ramanujan's master theorem as he wrote it:
\[
\Gamma(s) A^{-s} = \int_0^\infty e^{-Ax}x^{s-1}\,dx\\
\]
And we've fractionally iterated the matrix \(A: \mathbb{C}^n \to \mathbb{C}^n\). I avoided a lot of "singular moments" here. But if we can map the matrix \(A\) well enough; this discussion is entirely rigorous. It relates Daniel's work; Sheldon's work; Bo's work; Tommy's work; and all that matrix shit in a quantum physics./hilbert space shit.
Fractional calculus is just:
\[
\frac{d^s}{dA^s} : \mathbb{C}_{\Re(s) > 0} \times \mathcal{H} \to \mathcal{H}\\
\]
Where
\[
\mathcal{H} = \{ A : \mathbb{C}^n \to \mathbb{C}^n\,|\, A^{-1} :\mathbb{C}^n \to \mathbb{C}^n\}\\
\]
Then:
\[
\frac{d^s}{dA^s} e^{Ax} = x^s e^{Ax}
\]
And we can differentiate across \(A\) or \(x\); and still have the same rules.
This is the language I think in; and it's just a translation of much of the standard "tetration forum" language; and current literature.
In theory yeah.
But in practice ...
Differentiating a noninteger amount of times with respect to a nontrivial infinite square matrix that might not be diagonalizable ?!
That gives a non-unique infinite tensor with divergent norm ?!
Or did you mean differentiating with respect to a vector ?
And that is just the last line of your answer.
regards
tommy1729
Oh yes! Tommy! I apologize; by "A" I meant non-singular; which implies diagonalizable. I can write the math for you if you'd like. But every FINITE diagonalizable matrix can be differentiated as:
\[
A^s e^{Ax} = \frac{d^s}{dx^s} e^{Ax}\\
\]
But we must choose a path of the differintegral so that it converges. For example; let:
\[
A x_j = \lambda_j x_j\\
\]
Where \(x_j\) is an eigenvector, and \(\lambda_j\) is an eigenvalue. And find a path \(\gamma\) such that \(\gamma(0) = 0\) and \(\gamma(\infty) = \infty\). Then assuming that:
\[
\int_\gamma \left|e^{-\lambda_j x}\right|\,dx < \infty\,\,\text{for all}\,\, 1 \le j \le n\\
\]
Then the differintegral always converges. Finding \(\gamma\) can be tricky. But if for the sake of the argument we assume that \(-\pi/2 < -\kappa< \arg(\lambda_j) < \kappa < \pi/2\); then choosing \(\gamma = [0,\infty]\) works fine. It gets much much trickier in a more general sense; but the idea still holds--at least in a general sense.
The differentiation by \(A\); is a formal operation. But it is entirely rigorous, though not common notation. If I take a function:
\[
f(A) = \sum_{k=0}^\infty f_k A^k\\
\]
Then:
\[
\frac{d}{dA} f(A) = \sum_{k=1}^\infty kf_k A^{k-1}\\
\]
Since \(A\) is an n'th order matrix; by the Cayley-Hamilton theorem; this is still a finite polynomial; and we're just differentiating a polynomial in \(A\).
The idea is that \(\frac{d^s}{dA^s}\) exists in the dual space; where as \(\frac{d^s}{dx^s}\) exists in the normal space. I could never be bothered to work out too many of the details; but much of it holds weight in numerical calculations.
You'll probably see this more often in functional analysis but the operation:
\[
x^{-s} e^{Ax} = \frac{1}{\Gamma(s)} \int_0^\infty e^{-Ax}A^{s-1}\,dA\\
\]
Is a perfectly valid operation.
Also, I apologize; I was mostly just spitballing a lot. So I may have screwed up some details. But the idea looks a lot like this--but still, roughly looks like this.