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Stephen Wolfram was interested in extending tetration because of what it might tell us about tell about fractional iteration. There are two solid mathematical systems for doing physics, PDE's and iterated functions. The problem is that time in physics appears to be continuous while in iterated functions it appears discrete. Thus is the importance of the work we do here in the context of physics.
Quoting Arnold and Avez, 1968 in E. Lee Jackson's Perspectives of nonlinear dynamics, Vol 1., pg. 51
Quote:Let M be a smooth manifold, a measure on M defined by a continuous positive density,
\(f^t:M \to M\)
a oneparameter group of measurepreserving diffeomorphisms. The collection
\((M,\mu, f^t)\) is called a classical dynamical system.
Daniel
Posts: 1,179
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Joined: Dec 2010
Could you explain how this is represented?
Would it be something like:
$$
f^t(x) = \int_0^x e^{tu} \, d\mu(u)\\
$$
I fail to understand what these dynamics mean?
I understand the Manifold comment. So let's assume that the manifold is \([0,\infty)\), and that we are creating some kind of iteration on here. Then we are writing:
$$
f^t(x)\\
$$
How are we using an integral interpretation?
I'm sorry, but I'm curious and I don't have that book or access to any article relating to it. Context would do me well?
Is this an extension of the Borel measure for regular iteration? Where there's a measure \(\mu\) such that:
$$
\sqrt{2} \uparrow \uparrow t = \int_0^\infty e^{tx}\,d\mu(x)\\
$$
I'm sorry Daniel, but could you please elaborate.
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07/26/2022, 07:14 AM
(This post was last modified: 07/26/2022, 07:19 AM by Daniel.)
(07/25/2022, 11:49 PM)JmsNxn Wrote: Could you explain how this is represented?
Would it be something like:
$$
f^t(x) = \int_0^x e^{tu} \, d\mu(u)\\
$$
I fail to understand what these dynamics mean?
I understand the Manifold comment. So let's assume that the manifold is \([0,\infty)\), and that we are creating some kind of iteration on here. Then we are writing:
$$
f^t(x)\\
$$
How are we using an integral interpretation?
I'm sorry, but I'm curious and I don't have that book or access to any article relating to it. Context would do me well?
Is this an extension of the Borel measure for regular iteration? Where there's a measure \(\mu\) such that:
$$
\sqrt{2} \uparrow \uparrow t = \int_0^\infty e^{tx}\,d\mu(x)\\
$$
I'm sorry Daniel, but could you please elaborate.
First let me say I'm pushing my minimal understanding of physics to the limit here. Kousnetsov could probably explain it better.
Consider the iterated function \(f^t(x)\) in quantum mechanics. Let \(f(x) = Hx\) where x is an infinite column vector and H is an infinite matrix. The vector x is the initial state of the system and H is the laws of physics. Then the state of the system from instant to instant is \(x, H x, H^2 x, \cdots, H^tx.\)
The constraint that f is measure preserving translates to a system having a unit multiplier, that it is on the higher dimensional analog of the ShellThron boundary.
Second example is the Feynman Path Integral, the heart of quantum field theory. Once again the system is computed from one instant to the next. But here the function f integrates and then exponentiates the results. Without the integral the expression would reduce to tetration.
Daniel
Posts: 1,179
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Joined: Dec 2010
07/27/2022, 11:35 PM
(This post was last modified: 07/27/2022, 11:40 PM by JmsNxn.)
Oh yes, this is Von Neumann's stuff. Ya, I know this. Okay, so yes, it will be something:
$$
\begin{align}
H\psi = \int_{\infty}^\infty e^{ixy}\psi(y)\,d\mu\\
H^t\psi = \int_{\infty}^\infty k(t,y)\psi(y)\,d\mu\\
\end{align}
$$
For some kernel \(k(t,y)\). Usually they use the Fourier transform, especially when we're talking about neutral/unit circle operations.
This is actually closely related to work I did when I was at U of T, I looked a lot at iterating Linear operators on Hilbert spaces (infinite square matrices acting on an infinite vector). The neutral case is by far the most interesting, but also the most difficult.
IT tends to extend pretty naturally from the \(n x n\) case. If you have some square matrix \(A\), and if you take the exponential:
$$
e^{At}x = \sum_{n=0}^\infty A^nx \frac{t^n}{n!}\\
$$
You can take the Mellin transform in specific cases (you need a specific bounded lemma), where then:
$$
\Gamma(1z) A^{z1} x = \int_0^\infty e^{At}x t^{z}\,dt\\
$$
Then in many cases you can re represent this using a Fourier transform or a Laplace transform, just becomes a kind of elaborate change of variables.
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Joined: Aug 2007
(07/27/2022, 11:35 PM)JmsNxn Wrote: Oh yes, this is Von Neumann's stuff. Ya, I know this. Okay, so yes, it will be something:
$$
\begin{align}
H\psi = \int_{\infty}^\infty e^{ixy}\psi(y)\,d\mu\\
H^t\psi = \int_{\infty}^\infty k(t,y)\psi(y)\,d\mu\\
\end{align}
$$
For some kernel \(k(t,y)\). Usually they use the Fourier transform, especially when we're talking about neutral/unit circle operations.
This is actually closely related to work I did when I was at U of T, I looked a lot at iterating Linear operators on Hilbert spaces (infinite square matrices acting on an infinite vector). The neutral case is by far the most interesting, but also the most difficult.
IT tends to extend pretty naturally from the \(n x n\) case. If you have some square matrix \(A\), and if you take the exponential:
$$
e^{At}x = \sum_{n=0}^\infty A^nx \frac{t^n}{n!}\\
$$
You can take the Mellin transform in specific cases (you need a specific bounded lemma), where then:
$$
\Gamma(1z) A^{z1} x = \int_0^\infty e^{At}x t^{z}\,dt\\
$$
Then in many cases you can re represent this using a Fourier transform or a Laplace transform, just becomes a kind of elaborate change of variables.
Yes, you get where I'm going with this. Ironically, I have always received more support from physicist than mathematicians. Part of my interest in a Taylor's series approach to iterated functions and tetration is that complex numbers and even matrices can be plugged in. Maybe even infinite matrices, but proving their convergence is beyond me. One of my favorite papers is what I believe to be the first paper devoted to general fractional(continuous) iteration:
R. Aldrovandi and L. P. Freitas,
Continuous iteration of dynamical maps
J. Math. Phys. 39, 5324 (199
It makes use of infinite Bell matrices which are a variation of Carleman matrices.
Daniel
Posts: 1,179
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Joined: Dec 2010
07/28/2022, 08:51 AM
(This post was last modified: 07/28/2022, 09:08 AM by JmsNxn.)
(07/28/2022, 08:12 AM)Daniel Wrote: Yes, you get where I'm going with this. Ironically, I have always received more support from physicist than mathematicians. Part of my interest in a Taylor's series approach to iterated functions and tetration is that complex numbers and even matrices can be plugged in. Maybe even infinite matrices, but proving their convergence is beyond me. One of my favorite papers is what I believe to be the first paper devoted to general fractional(continuous) iteration:
R. Aldrovandi and L. P. Freitas,
Continuous iteration of dynamical maps
J. Math. Phys. 39, 5324 (199
It makes use of infinite Bell matrices which are a variation of Carleman matrices.
Ironically, in the same vein as you're talking, most of the mathematicians who I've talked to and given me the time of day are analytic number theorists. This somehow related to how I performed mellin transforms and blah blah blah. Iterating a super function relationship is far more interesting. No one cares about that though. So I made a couple of cool observations at u of t involving analytic number theorybut all I cared about was tetration/iteration/recursion.
I got some love from physicist's at u of t too, but that was mostly transferred from the number theory guys. The way I manipulated number theory functions, was applicable to schrodinger solutions.
Nothing to do with tetration. So I don't give a fuck. And I didn't give a fuck.
But if we're talking background. You say physics, and what I'm presuming is more classical (not quantum mechanics). I have a good background in analytic number theory; and oddly enoughit transfers to tetration.
I bet if we work together we'll all make leaps in tetration.
EDIT:
It's also bugging me that I screwed up the above equation. It should be:
$$
\Gamma(1z)A^{z1}x = \int_0^\infty e^{At}xt^{z}\,dt\\
$$
I forgot a dumb negative sign.
Either way, this stuff appears all the time in advance number theory stuff....
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07/29/2022, 06:04 AM
(This post was last modified: 07/29/2022, 07:09 AM by JmsNxn.)
AHHH!!!
So I tracked down the exact theorem.
Let's assume that \(A = \mathbb{C}^{\infty \times \infty}\) is an infinite square matrix. Which acts on an infinite vector \(\bf{x} \in \mathbb{C}^\infty\). If all the eigenvalues \(\lambda\) of \(A\) satisfy \(\Re(\lambda) > 0\)then:
$$
\begin{align}
\left(\Gamma(1z) A^{z1}\right) {\bf x} &= \left(\int_0^\infty e^{At}t^{z}\,dt\right){\bf x}\\
\Gamma(1z) A^{z1}&=\int_0^\infty e^{At}t^{z}\,dt\\
\end{align}
$$
You can perform this on sectors as well, and arbitrary sectors in any complex direction. So you can make the exact same formula, upto a change of variables, so long as \(\theta < \arg(\lambda) < \theta'\).
This is equivalent to Von Neumann's iteration of operators on hilbert spaces, but is just written differently. Von Neumann focuses on the Fourier transform, and I spent most of my time using Mellinbut I did prove novel results about Von Neumann's iteration; but it was mostly just Ramanujan + Von Neumann added together to write straightforward obvious results, lol. I mostly just justified some results Ramanujan just fucking knew somehow, but I did it for linear operators acting on hilbert spaces.
Lmao, and if I'm bragging, I hope you know it's okay to brag a bit, Daniel
Also! You can use an analytic continuation formula by Euler to put it alllllllll together:
$$
\Gamma(1z) A^{z1} = \sum_{n=0}^\infty A^n \frac{(1)^n}{(z1n)n!} + \int_1^\infty e^{At}t^{z}\,dt\\
$$
This constructs a holomorphic iteration \(A^z\) for \(\Re(z) > 1\). This is valid if we assume the spectrum \(\sigma(A) \subset \Re(z) > 0\).
Essentially if the spectrum \(\sigma(A)\) is in a sector of \(\mathbb{C}\), meaning \(\lambda \in \sigma(A)\) and \(\theta < arg(\lambda) < \theta'\) while additionally \(\theta'  \theta \le \pi\). Then we can fractionally iterate the linear operator \(A\) on the hilbert space it acts on. This is just an overly complicaed version of Ramanujan's Master Theorem, lol.
