Hey, Everyone; been a long time...

[JmsNxn]

I'd never say what you are doing is trivial! I'd say all those Art History majors I met at University of Toronto are doing trivial things! At least YOU'RE DOING MATH! Lol!

I see what you mean though, by how that would not construct tetration. But I can definitely see it giving an existence method... maybe even uniqueness. But it'd probably be one of those things where you don't really hold it in your hands; you just know it's out there, lol. But it'd definitely be great for arbitrary super-functions; giving you an existence method regardless.

Happy Hunting! I hope to read of your work soon! I'll at least make an effort to digest it!

[MphLee]

Mmh, I agree with your sentiment. Even if computation and category theory, constructive methods, are very very close actually... both are very far from me right now. Also, Analysis, don't kill me, is the art of approximating so it is no surprise that she's the first on the finish line doing computations and evaluations... the fact that to me most of it its black magic, while with algebra I have conceptual clarity over practical estimations.

Said that while I was preparing a question for you (and the Forum) about one of the protagonists of the story, i.e. the superfunction trick, I just went back to your composition integral paper. After my question I will be in position to go deeper on this but... but it struck me how many of those algebraic properties of the compositional integral are actually functorial properties...wow!

[JmsNxn]

I totally get how analysis is the art of approximating, and complex analysis is the art of approximating uniformly. Don't go thinking I'm staring at excel sheets all day though, lmao! To me it's all just the manipulation of symbols and getting a feel for them. I never numerically evaluate; which is probably a bad thing, but as long as the is there, who cares what the graph looks like or what the numbers are. Numbers can lie, can't. If all us analysts were is big calculators approximating, the riemann-hypothesis wouldn't even be a problem. (What are we at now, 200 trillion zeroes found on the critical line? surely! that's enough.) lol.

Thanks for the nod about the compositional integral. It was the most non-commutative algebra I could really get into; I probably have a few typos in that paper, it took a chunk out of me. But I'm proud of how I designed the notation, and yes it does take a lot from category theory. I had help...<_<

[MphLee]

I totally get how analysis is the art of approximating, and complex analysis is the art of approximating uniformly. Don't go thinking I'm staring at excel sheets all day though, lmao! To me it's all just the manipulation of symbols and getting a feel for them. I never numerically evaluate; which is probably a bad thing, but as long as the is there, who cares what the graph looks like or what the numbers are. Numbers can lie, can't. If all us analysts were is big calculators approximating, the riemann-hypothesis wouldn't even be a problem. (What are we at now, 200 trillion zeroes found on the critical line? surely! that's enough.) lol.

Hahaha I surely do not depict you and the other analysts of this forum like that. Writing down my questions on your superfunction trick I had to make a disclaimer on notation and I was tempted to include in it a meme about it. At the end in order to be sober I decided to avoid it, but after this provocation I guess I ll post it here.



Back to serious business...
I think I'm close to understanding the general mechanism behind this superfunction tricks that we can see from Kneser's to Tommy's 2sinh, and that you wanted to upgrade to iterated compositions.
Since I'm very ignorant of real and complex analysis here to really appreciate all the difficulties I went more algebraic on it: it is possible that I'm still blind to some fundamental gear of this mechanism.

I give you here a pdf with a LaTex version of the argument and of the "existential questions" that follow .
I hope you can skim thru it.


  2021_01_16_Generalized_superfunction_trick.pdf (Size: 307.64 KB / Downloads: 20)

To give you the summary: I'm breaking the usual method (Kneser's, principal Abel/Schroeder) in two parts: the construction of the sequence and effectiveness of the limit of the sequence to our purposes. I gave indeed a real proof of the first part (in the appendix) but I propose a formalization schema for the second part.

Anyways I'll make a separate post on it on the forum so you and the others can eventually make some constructive criticisms of it.


ADDENDUM: This is just a draft complied very quickly during the last 3 nights... hahah so forgive me for some grammar errors. I'm already seeing 4 typos...

[JmsNxn]

I like your meme, LOL!

Just so you know, branching goes brrrrrrrr! for me because I think of it as local solutions. And then, we paste local solutions together to get a **somewhat** global solution--minus the branch cuts.

[MphLee]

Just so you know, branching goes brrrrrrrr! for me because I think of it as local solutions. And then, we paste local solutions together to get a **somewhat** global solution--minus the branch cuts.

This pasting reminds me alot the definition of manifold and of sheaves: a basic example of sheaf is a procedure that instead of associating to a geometric object an algebraic information, because in some cases you can't, you consider the full lattice of the "parts" of (e.g. subspaces) and you associate to every one of them an algebraic gadget in a functorial way, i.e. if a part sits inside a part then the gadgets and must be related or there must be a transoformation between them.
In this way instead of defining a unique map we define something piecewise, over the pieces of X but with some regularity: this is just a functor. It becomes a sheaf when you also ask that if two parts of X overlaps you have some rule to glue the maps in those intersections, like we do with an atlas of charts for a top. space.