Wolfram Summer School Hyperoperator Project

[Daniel]

Howdy,
Stephen Wolfram invited me to come to this year's summer school and I have accepted. I have proposed I focus on coding my work with hyperoperators so that Mathematica has built in matrix hyperoperators. 

Wolfram:
It would be really interesting to see continuous versions of such iterations.  I think it would help in understanding continuum limits of computational processes, which is very relevant for https://www.wolframphysics.org/

[MphLee]

Hi Daniel,
That was unexpected.
I've skimmed many times thru your site in the last 5 years or so but I was never able to see your work on iteration explicitly applied to the hyperoperations (with that I mean non integer ranks).

However I was puzzled by this post All Maps Have Flows & All Hyperoperators Operate on Matrices but I never had time to ask you to expand on it. Is it related? Can you expand on this?

About the project, is the syllabus, or a sketch of it, already available?

[Daniel]

Hi Daniel,
That was unexpected.
I've skimmed many times thru your site in the last 5 years or so but I was never able to see your work on iteration explicitly applied to the hyperoperations (with that I mean non integer ranks).

However I was puzzled by this post All Maps Have Flows & All Hyperoperators Operate on Matrices but I never had time to ask you to expand on it. Is it related? Can you expand on this?

About the project, is the syllabus, or a sketch of it, already available?

Hello MphLee,
After writing my first paper on tetration in 1990, I began to reflect on what I could generalize as per my conversation with Wolfram. I soon realized my technique was in no way dependent on tetration, but applied to iterated functions in general. When I refer to the hyperoperators I mean tetration, pentation and so on and not the operators in between. 

Wolfram:
Having recently been working on https://writings.stephenwolfram.com/2021/03/after-100-years-can-we-finally-crack-posts-problem-of-tag-a-story-of-computational-irreducibility-and-more/  (see particularly the later sections) I am increasingly curious about rapidly growing iterations and their relationships to axiom systems (cf https://www.wolframscience.com/nks/notes-12-9--examples-of-unprovable-statements/ )

All this is preliminary. I've kept Wolfram in the loop of what I was researching as both of us are tracking similar issues. I will let folks know how this develops.

[JmsNxn]

Just for clarification, you mean taking invertible-matrices \( A \) and producing a matrix \( e \uparrow^n A \)? Is this done using a Taylor Series approach \( f(A) \) where \( f \) is a formal power-series? Is it possible yet to produce a proof that the formal Taylor series produced by the Matrix method is convergent? I assume you are doing the same process to define a formal power-series for hyper-operations, right? I only say "formally" because I'm curious to see a proof showing the radius of convergence is one (though I have no doubt it definitely is). That sounds really fascinating though.

Forgive me for asking but what does the functional equation look like? Does it look something like this,

\(
e \uparrow^{n-1} (e \uparrow^n A) = e \uparrow^n TA\\
\)

For some matrix/operator \( T \), essentially equivalent to the transfer operator for scalars \( Tx = x+1 \)--but for matrices? Or is it something stranger? Or is the functional equation essentially nulled; and only considered as a plug in play with a flow?

Sorry for so many questions, I'm just really curious. It would be very fascinating to have a built in hyper-operations code for Wolfram, I must say. I want to know more about plugging in matrices though and their structure! Is there any other literature you have on the subject?

[Daniel]

Thanks for your interest JmsNxn. I'm suffering a bit of burnout from working to many jobs at the same time, so I'm taking a vacation now. For the time being I recommend checking out https://www.overleaf.com/project/5f802a8...0001b407be .

[JmsNxn]

Thanks for your interest JmsNxn. I'm suffering a bit of burnout from working to many jobs at the same time, so I'm taking a vacation now. For the time being I recommend checking out https://www.overleaf.com/project/5f802a8...0001b407be .

Unfortunately, I don't have permission to access that page.

[Daniel]

Thanks for your interest JmsNxn. I'm suffering a bit of burnout from working to many jobs at the same time, so I'm taking a vacation now. For the time being I recommend checking out https://www.overleaf.com/project/5f802a8...0001b407be .

Unfortunately, I don't have permission to access that page.

I'm sorry, try http://tetration.org/Extending_the_Hyperoperators.pdf .

[JmsNxn]

Hmm, you seem to have constructed a formal Taylor Series; I don't really see a proof that the Taylor series converges though. Though yes it does, and is definitely provable using current analysis. As far as I can tell this tetration would be equivalent to,

\(
F(z) = \Psi^{-1}(\lambda^z A)\\
\)

For the Schroder function \( \Psi \) about a fixed point of \( e^{z} \); where \( A \) is some constant and \( \lambda \) is the multiplier. This would definitely be analytic; I agree. It will not be real valued on the real-line; as far as I can tell it should be entire because it's analytic in a neighborhood of zero and \( \exp^{\circ n} \Psi^{-1}(z) = \Psi^{-1}(\lambda^n z) \) where since \( |\lambda| > 1 \) this will cover the plane inside of \( \Psi^{-1} \), and the iterated exponential is always valid. And then the theory is to find a fixed point of \( F \); and construct a Schroder function again about this,

\(
\Psi_1(\lambda_1^z A_1)\\
\)

In essentially the same manner as Tetration. And you add the case that if we have a neutral fixed point we'll use the Abel-form of iteration. Presuming you are avoiding super-attracting fixed points as hyper-operators should have a non-zero derivative. Whereof you continue this train of construction to create a sequence of super-functions:

\(
F_n(F_{n+1}(z)) = F_{n+1}(z+1)\\
\)

I'm curious as to how you are choosing your fixed points? And I'm curious how you are choosing your value of \( A \); or rather what your initial conditions are, \( F_n(0) = ? \). Presumably you want that it equals \( 1 \); I'm not quite too sure how you're doing this though, but I assume it's by setting \( A = \Psi(1) \)--which I can't see anything wrong with in Tetration (as the entire complex plane is the Julia set of \( e^z \), and \( \Psi^{-1} \) is an entire function). But I don't see how you are guaranteed that \( \Psi_n(1) \) is finite for higher order hyper-operations--necessarily you would need to discuss the domain of \( \Psi \) or \( \Psi^{-1} \). And largely the reason it is working for tetration is because all the fixed points of \( e^z \) are repelling \( |\lambda| > 1 \) (implying an entire inverse Schroder function). As I see it you would need the fixed point you use from tetration to construct pentation to be repelling as well (so that it's inverse Schroder function is entire).

If you were to take the alternative route, that it's geometrically attracting, you are opening yourself up to branch-cuts because \( \Psi^{-1} \) will not be entire, and instead \( \Psi \) will be holomorphic on the immediate basin of attraction sending to \( \mathbb{C} \). Which typically means that \( \Psi^{-1} \) has branch cuts; otherwise if it were entire on \( \mathbb{C} \) it would only send to the immediate basin of the fixed point, which is impossible due to Picard, unless the immediate basin is the entire complex plane minus a point (which cuts out a very large resevoir of functions and essentially means we have 2 fixed points, one attractive and one repelling).

If the fixed point is neutral you open yourselves up to similar problems as the attractive case. When you construct the Abel function of a neutral fixed point it only converges in a petal of the fixed point (not the neighborhood), and so if the inverse Abel function \( g \) were entire, it again would send to only a petal of the complex plane, which is impossible due to Picard. Again, opening yourself up to branch-cuts.

Although I agree absolutely with your paper and the construction of the formal Taylor Series; which is very impressive (something I never would've thought would be feasible), I think you're avoiding the biggest question. What's the radius of convergence of these Taylor series? For Tetration it'll be infinite, but this isn't necessarily so. Even taking something like \( \alpha^\xi \) for \( 1 < \alpha < e^{1/e} \) your theory seems to be incomplete by avoiding where in fact the iteration will converge. Though yes, I do agree, that locally everything should be fine; so long as we are not on the Julia set (boundary of an immediate basin).

I should say, it would be very impressive if you managed to derive a manner of computing the radius of convergence--though this is largely the criticism of using Faa di Bruno's method; it doesn't plug too well into Cauchy's limsup formula (or any radius of convergence method). Though, I will say that you are absolutely correct if we were to restrict your construction to the theory of Sheafs. And in fact, seems like a very smart move. As I see it, you are more aptly talking about flows on sheafs rather than flows on functions (mostly because you are treating the Taylor Series as formal objects)--though proving local convergence almost everywhere does seem feasible. Which is again, very good because it means it's a sheaf with a non-zero radius of convergence. However, when discussing as a global thing; we'd like to have our hands on exactly where it converges.

I hope I don't come off as rude or anything (rereading I'm trying not to be harsh, but I may be coming off that way, I apologize if so); this is still very impressive. I had always thought the Taylor series approach to iteration was a null method but you've definitely convinced me otherwise. But I do think we'd need a stronger sense of where these things converge, rather than simply knowing that we are guaranteed local convergence.

Regards, James

Edit: I made a minor error when discussing the neutral fixed point by not referring to repelling petals and attracting petals; if the petal is repelling the superfunction will be entire (much like repelling fixed points), so substitute my above comment with the word attracting petal. Had to double check Milnor, haven't thought about neutral fixed points in a while, lol.

[Daniel]

Hello James,
Thanks for your analysis of my work. You are not being harsh, just honest. 

Convergence - James, you are correct about there being no information on convergence in my article. Yiannis is helping out by reviewing my work with an analysis of the Faa Di Bruno algorithm. He was concerned about the weakness of my converge proofs and recommended an analytic continuation approach. Since I know the value of all the multipliers, I hope to provide an analytic continuation proof.  

Convergence proofs

• Combinatorial convergence - The Faa Di Bruno algorithm only produces a finite number of components whose values are finite, therefore must Faa Di Bruno converge.
  
• Moiré patterns - Proof by contradiction. Given two Moiré transparencies; one with infinite values and one with finite values. offsetting them creates a situation where an finite value in the first transparency becomes arbitrarily close to an infinite value in the second transparency. 
  
• Analytic continuation - this is a well accepted way to establish convergence.
  

Fixed points - I only use arbitrary fixed points.

Complex Dynamics - I've worked out much of complex dynamics before reading about it. My approach gives a non-topological way of deriving large potions of complex dynamics. I have minimal formal education, college junior, so my understanding of higher dimensional complex dynamics is limited.

[JmsNxn]

Thanks, Daniel. I really felt I was being a tad harsh, thank you for saying that I wasn't; because I honestly felt I was.

What you are doing is absolutely certainly possible, and I agree with your general conclusion. There just seem to be some gaps.

I had an idea not too dissimilar to what you are writing, that I produced mostly for fun as evidence of a sequence of entire functions that satisfy,

\(
F_n(z) : \mathbb{C} \to \mathbb{C}\,\,\text{for}\,\,n\ge 1\\
F_n(F_{n+1}(z)) = F_n(z+1)\\
\)

This was done by instead of insisting \( F_n(0) =1 \) but rather that, \( F_n(L) = L \) for a complex fixed point of \( e^z \), and an insistence that \( |F'_n(L)| > 1 \). Then starting with \( F_1(z) = e^z \), one can construct a sequence of entire functions \( F_n \), of what I called whacky hyper-operators. Where they satisfied the superfunction identity \( F_n(F_{n+1}(z)) = F_{n+1}(z+1) \) but instead of being normalized with \( F_n(0) = 1 \) were normalized to \( F_n(L) = L \) (which proves to be so much simpler!). I only did this to use as evidence that a chain of entire super functions can exist. The way I did it is largely similar to what you are doing, and based on the same principles. So I'm pretty sure your method will work.

I'm excited to see how you're going to make this work. I'm glad you're aware of the anomalies I pointed out--I kind of figured you were. I'm still curious though, is there any kind of functional equation in the matrix equation. I'm consistently visualizing this as,

\(
e \uparrow^n e\uparrow^{n+1} A = e \uparrow^{n+1} T A\\
\)

For invertible square matrices \( A,T \). I'm wondering if at all something like this happens, or if we are really just doing a plug and play with the flow. I didn't see much about the matrix element in your paper, except for the formal identification of \( e \uparrow^n A \) with a formal Taylor series in A.

Anyway, that was still a really interesting paper,

Regards, James.

P.S. Oh and don't worry about a lack of formal education. I had to leave university for health reasons; but kept on learning, and when I came back I'd gawk at most formally educated people. So, don't feel ashamed in anyway. We're both in the same boat.