(05/26/2021, 02:34 AM)JmsNxn Wrote: As to your first point. I shared a lot of work at U of T; and they started calling these things hyper-operation chains (at least the people I talked to). There isn't anything published as of yet; but they've done quite a few things similarly to me. They never published, I presume because I have priority over these fractional calculus things; at least from their perspective. I kind of left the scene for a while; and they were upset I never published half the things they sort of knew about through me. It's why I've started publishing all over again; sort of like a code dump of everything I've done. Largely because some professors told me to.
They did do some stuff with,
\(
\alpha \uparrow^s z\\
\)
But I rarely see them (especially with covid right now); and I presume it's slow going. But they seemed confident my original formula for it 5 or 6 years ago is the correct one (but my original proof is incorrect):
\(
\alpha \uparrow^s z = \frac{d^{s-1}}{dw^{s-1}} \frac{d^{z-1}}{du^{z-1}} ||_{w=0}_{u=0} \sum_{n=0}^\infty \sum_{k=0}^\infty \alpha \uparrow^{n+1}(k+1) \frac{w^nu^k}{n!k!}\\
\)
They were also the ones who encouraged me to publish all this Infinite composition stuff. They were pretty shocked when I explained the residual theorem to them (just like you were).
Thank you, now I understand all the business with the theta angles.
About you peers at U of T, it is remarkable if they found that interesting and surprising that "they've done quite a few things similarly to me". I never had many chances to talk with mathematicians... and all the hints tell me that this topic is completely unknown and irrelevant. Obviously, as you can imagine from my last draft and recent threads, I can already, at least partially, trace back this topic to the backbone of mathematics I already begin to see that it touches many mainstream topics.
What do you think about the centrality of hyper-operations chain-like objects and their possible continuous extension to non-discrete chains (paths?)?
How do you think the reception of these ideas was and what was the atmosphere around those chain objects? Did they treat them as exotic and niche items?
Addendum
I red the last sections of the paper. I really can't find obscure points. All the machinery lies in the first two theorems. I'll give myself time to digest it.
Thank you!
Best regards!
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)
