Poll: Do you believe this conjecture is true?You do not have permission to vote in this poll. Yes 66.67% 2 66.67% No 33.33% 1 33.33% Total 3 vote(s) 100%
 * You voted for this item.

 A conjectured uniqueness criteria for analytic tetration bo198214 Administrator Posts: 1,616 Threads: 102 Joined: Aug 2007 01/25/2017, 06:26 PM It seems like Szekeres considered this problem already but didnt come to some explicit conlusion. Look at the paper in the attachment (hopefully not getting some copyright lawsuit ...) Attached Files   Szekeres1961_150.pdf (Size: 5.15 MB / Downloads: 549) JmsNxn Ultimate Fellow Posts: 1,064 Threads: 121 Joined: Dec 2010 01/25/2017, 09:46 PM (This post was last modified: 01/25/2017, 10:07 PM by JmsNxn.) It looks like we are in good company investigating this property as a uniqueness criterion! tommy1729 Ultimate Fellow Posts: 1,746 Threads: 384 Joined: Feb 2009 02/03/2017, 01:23 PM (01/25/2017, 06:26 PM)bo198214 Wrote: It seems like Szekeres considered this problem already but didnt come to some explicit conlusion. Look at the paper in the attachment (hopefully not getting some copyright lawsuit ...) Quote from paper : [img=595x842]file:///page4image256[/img] As for the main subject of this thread that is ok. But i want to point out a few things. 1) the majority of the ideas have actually been discussed here before. 2) in particular the idea of lim a(x) / b(x) where a , b are the derivatives of the 2 abel functions. 3) the derivative of the Abel equation is called the Julia equation and its solution Julia function. When that solution has a limit towards x -> oo we call it a smooth Julia function. 4) the Smooth Julia function condition was considered b4 as no zero's for its second derivative ( Tommy 2 sinh and elsewhere both by me and others ). 5) the smooth Julia is NOT a uniqueness criterion that makes exp(x) - 1 the best method. In fact i think 2sinh also has a smooth Julia , and lim a(x)/b(x) =1 also. 6) not sure if it has been shown that koenigs Function ( for hyperb fix ) always produces a smooth Julia function. That might be intresting. 7) again Julia stuff : the Julia set of exp - 1 , exp , 2sinh has not yet been compared !! Not Trying to be hostile. Regards  Tommy1729 JmsNxn Ultimate Fellow Posts: 1,064 Threads: 121 Joined: Dec 2010 02/17/2017, 05:21 AM So I posted here the most solvable version of your question I could think of on MO http://mathoverflow.net/questions/262444...-functions If this statement turns out to be true then it unequivocably proves that not only is the schroder iteration method the only completely monotone solution to tetration; but also for pentation, sexation, and so on. I'm confident this statement follows too. This would give a really good real valued criterion for the uniqueness of $\alpha \uparrow^n x$ for $\alpha \in (1,\eta)$ and $x \in \mathbb{R}^+$. I hope someone in the more general mathematics community might be able to help us. « Next Oldest | Next Newest »

 Possibly Related Threads… Thread Author Replies Views Last Post Iteration with two analytic fixed points bo198214 62 6,368 11/27/2022, 06:53 AM Last Post: JmsNxn Qs on extension of continuous iterations from analytic functs to non-analytic Leo.W 18 2,442 09/18/2022, 09:37 PM Last Post: tommy1729 Constructing an analytic repelling Abel function JmsNxn 0 400 07/11/2022, 10:30 PM Last Post: JmsNxn Is tetration analytic? Daniel 6 786 07/08/2022, 01:31 AM Last Post: JmsNxn Uniqueness of fractionally iterated functions Daniel 7 1,076 07/05/2022, 01:21 AM Last Post: JmsNxn Universal uniqueness criterion? bo198214 57 114,681 06/28/2022, 12:00 AM Last Post: JmsNxn A question concerning uniqueness JmsNxn 4 10,282 06/10/2022, 08:45 AM Last Post: Catullus Brute force tetration A_k(s) is analytic ! tommy1729 9 4,677 03/22/2021, 11:39 PM Last Post: JmsNxn [Exercise] A deal of Uniqueness-critrion:Gamma-functionas iteration Gottfried 6 7,784 03/19/2021, 01:25 PM Last Post: tommy1729 Nixon-Banach-Lambert-Raes tetration is analytic , simple and “ closed form “ !! tommy1729 11 6,544 02/04/2021, 03:47 AM Last Post: JmsNxn

Users browsing this thread: 1 Guest(s)