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 On to C^\infty--and attempts at C^\infty hyper-operations JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 03/02/2021, 09:55 PM Hey Everyone. Haven't been on for a while, been a little busy.  I thought I'd post the modified form of the paper. It details how to construct $C^\infty$ hyper-operations; minus maybe a few details. But I'm confident I got everything down. In fact, I'm confident that if we do it in the manner Sheldon describes, using a sequence of infinite compositions, $ \phi_n(s) = \Omega_{j=1}^\infty \phi_{n-1}(s-j+z)\bullet z\\$ Not much really changes.  I chose to use the exponential convergents rather than the $\phi_n$ convergents, simply because it generalizes well to the construction of arbitrary super-functions. All we need really is an exponential convergents. In fact, $\phi_n$ will look pretty much similarly, because of its exponential nature. The better form of Sheldon's method is that we retain holomorphy. Since I'm only trying to show $C^\infty$, I figure holomorphy isn't really needed. I do believe we could definitely use $\phi_n(s)$ to construct $e \uparrow^n x$. I believe it's more of an aesthetic issue, and I find it a bit more natural to just use, $ \Phi_n(s) = \Omega_{j=1}^\infty e^{s-j}e \uparrow^{n-1} z\bullet z\\$ And do away with $\phi$. Also because this satisfies the more natural equation, $ \Phi_n(s+1) = e^s e \uparrow^{n-1} \Phi_n(s)\\$ And it removes a bit of the untangling if we were to use $\phi_n$. Anyway, here's what I have so far. The proof of $C^\infty$ hyper-operators is surprisingly copy/paste from the proof of $C^\infty$ tetration, so long as you pay attention to the generalization, it should be fine. Any questions, comments or the what have you are greatly appreciated. I spent a lot of time restructuring the paper, but a lot of it is similar to what I wrote before. Thanks, James Attached Files   Hyper_operators.pdf (Size: 338.85 KB / Downloads: 98) « Next Oldest | Next Newest »

 Messages In This Thread On to C^\infty--and attempts at C^\infty hyper-operations - by JmsNxn - 02/08/2021, 12:12 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by JmsNxn - 02/10/2021, 02:30 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by sheldonison - 02/10/2021, 04:09 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by JmsNxn - 02/10/2021, 08:10 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by JmsNxn - 02/16/2021, 08:40 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by sheldonison - 02/21/2021, 01:38 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by tommy1729 - 02/27/2021, 12:08 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by sheldonison - 02/27/2021, 09:57 PM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by MphLee - 03/01/2021, 11:22 PM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by sheldonison - 03/02/2021, 05:09 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by MphLee - 02/27/2021, 11:37 AM RE: On to C^\infty--and attempts at C^\infty hyper-operations - by JmsNxn - 03/02/2021, 09:55 PM

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