• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 The bounded analytic semiHyper-operators JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 05/27/2016, 04:03 AM Hey Mphlee. I corrected some of the mistakes I made on my old arxiv page. They were rather small (the same result holds), the new version is also being edited by my professor, and just some double checking being done by him. And to your question on the super function operator, I have added a little bit about this. If $H$ is a set in which all its elements $\phi(z)$ are holomorphic for $\Re(z) > 0$, take the real positive line to itself, have a fix point $x_0 \in \mathbb{R}^+$ such that $0<\phi'(x_0) < 1$ and $\phi^{\circ n}(x) \to x_0$ for all $x\in[0,x_0]$ then.. There is a semi group of operators $S$ such that $S$ acts on $H$. Where more beneficially, the semi group is isomorphic to $\{\mathbb{R}^+,+\}$ This semi group being $\uparrow^t$ for $t \ge 0$ where $\uparrow^t \uparrow^{s} = \uparrow^{t+s}$ and if $F_t(x) = \uparrow^t \phi(x)$ then $F_t(F_{t+1}(x)) = F_{t+1}(x+1)$ (05/23/2016, 07:25 PM)MphLee Wrote: I would like to ask you: have you some ideas/intuition on the behavior of this map $\uparrow:H\to H$ and its dynamics in general? I have no idea to be honest. This is where I'm looking. The analysis is extensively simplified using ramanujan's master theorem. and some key points arise. I have a few well thought out points but nothing too expansive. Quote: Is it injective? $\alpha \uparrow^t z$ is injective in $t \ge 0$ and $0 < \Im(z) < \ell_t$ (where $\ell_t$ is the imaginary period in $z$. The operator $\uparrow^t$ is injective as well on the function space $H$ as defined above. Quote:Has it fixed points? The only fixed point of $\uparrow$ I can think of are constant functions. Since $\uparrow \phi(z) = \phi^{\circ z}(1)$ and if $\phi(z) = \alpha$ then surely $\phi^{\circ z}(1)= \alpha$ It always follows that $\uparrow^n \phi(x) \to \phi(1)$ though. This is exemplified by $\alpha \uparrow^n x \to \alpha$ « Next Oldest | Next Newest »

 Messages In This Thread The bounded analytic semiHyper-operators - by JmsNxn - 05/06/2016, 06:30 PM RE: The bounded analytic semiHyper-operators - by MphLee - 05/23/2016, 07:25 PM RE: The bounded analytic semiHyper-operators - by JmsNxn - 05/27/2016, 04:03 AM

 Possibly Related Threads... Thread Author Replies Views Last Post [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 950 03/20/2018, 12:16 AM Last Post: tommy1729 Analytic matrices and the base units Xorter 2 2,087 07/19/2017, 10:34 AM Last Post: Xorter Non-analytic Xorter 0 1,246 04/04/2017, 10:38 PM Last Post: Xorter A conjectured uniqueness criteria for analytic tetration Vladimir Reshetnikov 13 9,908 02/17/2017, 05:21 AM Last Post: JmsNxn Hyper operators in computability theory JmsNxn 5 3,442 02/15/2017, 10:07 PM Last Post: MphLee Recursive formula generating bounded hyper-operators JmsNxn 0 1,330 01/17/2017, 05:10 AM Last Post: JmsNxn Is bounded tetration is analytic in the base argument? JmsNxn 0 1,164 01/02/2017, 06:38 AM Last Post: JmsNxn Are tetrations fixed points analytic? JmsNxn 2 2,548 12/14/2016, 08:50 PM Last Post: JmsNxn Rational operators (a {t} b); a,b > e solved JmsNxn 30 35,345 09/02/2016, 02:11 AM Last Post: tommy1729 holomorphic binary operators over naturals; generalized hyper operators JmsNxn 15 15,286 08/22/2016, 12:19 AM Last Post: JmsNxn

Users browsing this thread: 1 Guest(s)