On my old fractional calculus approach to hyper-operations

[JmsNxn]

Omg... At first I was starting panicking... you were not talking about kernel of operators. Everywhere in algebra, linear algebra, functional analysis, matrix theory, groups, monoids and everywhere, even Hilb the cat. of hilbert spaces, kernel has an universal meaning: it is the set of vectors on which the operator vanishes. It is the set of zeroes of the operator.

In fact you're talking about the integral kernel or Hilbert–Schmidt kernel of an operator!!!

Damn, you are pushing my knowledge always to my limits. I'm seriously shocked by this actually. I was COMPLETELY unaware of this integral presentation and I wasn't expecting that hit... I mean I was wondering since day one what a (infinity x infinity) matrix  would look like but damn...  this is far worse than your Cauchy black-magic.
Heisenberg vs Schrodinger has to do with some very fundamental duality in mathematics, I red it somewhere multiple times... So if now you are telling me that on one side of the coin there is the matrix formalism for iteration... and on the other side there are integrals.... I'm done. I can feel that this links up heavily with the composition integral too...

My head is exploding... I feel like I can see the holy grail... that's heavy stuff for our iteration business.
Those damn quantum-mechanics guys were fkn paranormal 0.0

ps:
About priorities... idk too. I know less than you about how this works... But I can't unsee what you hinted 0.0
But I'm sure I won't be dangerous for the careers of your colleagues at U of T xD hahah I'll never publish something worthy.

I thought I'd add that of course you know what I just wrote now. I just don't want to over elaborate to spoil someone else's research. So I don't want to talk about it more. But my paper is an example of a "proof of concept" that we can iterate iterates. As much of this is my research so idc; but it's good to remember the following.

If \( F = \alpha \uparrow^2 z \in E_\theta \) then,

\(
\alpha \uparrow^2 z= F(z) = \frac{d^z}{dw^z}|_{w=0} \sum_{n=0}^\infty F(n)\frac{w^n}{n!}\\
\alpha \uparrow^2 \alpha \uparrow^2 z= F^{\circ 2}(z)= \frac{d^z}{dw^z}|_{w=0}\sum_{n=0}^\infty F^{\circ 2}(n)\frac{w^n}{n!}\\
\alpha \uparrow^2 \alpha \uparrow^2 \alpha \uparrow^2 z= F^{\circ 3}(z)= \frac{d^z}{dw^z}|_{w=0}\sum_{n=0}^\infty F^{\circ 3}(n)\frac{w^n}{n!}\\
\vdots\\
F^{\circ k}(z) = \frac{d^z}{dw^z}|_{w=0}\sum_{n=0}^\infty F^{\circ k}(n)\frac{w^n}{n!}\\
\)

And if we now sum across \( k \),

\(
\text{tet}_\alpha^{s}(z) = \frac{d^s}{du^s}|_{u=0}\frac{d^z}{dw^z}|_{w=0} \sum_{k=0}^\infty \sum_{n=0}^\infty F^{\circ k}(n)\frac{u^kw^n}{k!n!}\\
\)

And now we can repeat this process to get pentation, then hexation, so on and so forth because each one belongs in \( E_\theta \). This creates what is more what they mean by hyper-operation chain; we add variables, not just complexity. And then... we attempt to make an \( \omega \) kinda ordinal move and start doing,

\(
\alpha \uparrow^s z = \frac{d^{s-2}}{du^{s-2}}|_{u=0} \frac{d^z}{dw^z}|_{w=0} \sum_{k=0}^\infty \sum_{n=0}^\infty \alpha \uparrow^{k+2} n \frac{u^kw^n}{k!n!}\\
\)

Now this is a similar construction of a "hyper-operator" chain. And there's a sneaky lemma to make this work. But remember, this is technically a product,

\(
\prod_n \frac{d^z}{dw_n^z}\\
\)

Or something like that... Honestly Mphlee, it's on the verge of being solved. I'm pissed I'm not the one to do it; but, it's there. This is again, though; more applicable to Hilbert spaces than I care to admit--because I missed most of that entirely... Worst part about smart people looking at your research is when they elaborate about something right in front of your nose, lol

But there's a diagonalization process being encoded analytically here; if you can't see it, I'll write out the sums upon sums upon sums which elaborate. This is an infinite infinite sum; remember that. There are an infinite number of nested infinite sums. lol. I tend to brush that off but it's really cool to think about.

Regards, James

---

Btw, I believe the notation they had was,

\(
\alpha \uparrow^n (z_1,z_2,...,z_n) = \alpha \uparrow^n \overline{\bf{z}}\\
\)

Where,

\(
\alpha \uparrow^{n-1}(z_1,z_2,...,z_{n-1}) = \alpha \uparrow^n (z_1,z_2,...,z_{n-1},1)\\
\)

And,

\(
\alpha \uparrow^{n-1} (z_1,...,\alpha \uparrow^{n}(z_1,...,z_{n-1},z_n)) = \alpha \uparrow^{n} (z_1,...,z_{n-1},z_n+1)\\
\)

It was something like that...

So when we make \( \alpha\uparrow^s z \) it's kind of a implicit function from infinite variables. It's super cool, Mphlee, just wait a while. It'll rock ur sox.