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 Bummer! Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 04/18/2009, 11:24 AM (This post was last modified: 04/18/2009, 12:24 PM by Kouznetsov.) bo198214 Wrote:... the conjecture about the equality of the 3 methods of tetration is shattered. I received an e-mail of Dan Asimov where he mentions that the continuous iterations of $b^x$ at the lower and upper real fixed points, $b=\sqrt{2}$, differ! He, Dean Hickerson and Richard Schroeppel found this arround 1991, however there is no paper about it. ...Shame! 18 years without advances. There should be a paper about it. Let us submit one right now! bo198214 Wrote:... The numerical computations veiled this fact because the differences are in the order of $10^{-24}$. ...It is beacuse you stay at the real axis. Get out from the real axis, and you have no need to deal with numbers of order of $10^{-24}$. bo198214 Wrote:... So the first lesson is: dont trust naive numerical verifcations. We have to reconsider the equality of our 3 methods and I guess there will show up differences too.What about the range of holomorphism of each of the 3 functions you mention? How about their periodicity? Do they have periods? Below, for base $b=sqrt{2}$, I upload the plots of two functions: $F_{b,4}$ which is $(\mathbb{C}, 0 \mapsto 3)$ superfunciton of $\exp_b$ such that $F_{b,4}(z^*)=F_{b,4}(z)^* ~\forall z \in \mathbb{C}$ and $F_{b,4}(z+T_4)=F_{b,4}(z)^*~\forall~ z \in \mathbb{C}$ where $T_4=2\pi i/ \ln(2\ln(2))$ . $F_{b,2}$ which is $(D, 0 \mapsto 3)$ superfunciton of $\exp_b$ such that $F_{b,2}(z^*)=F_{b,2}(z)^* ~\forall z \in D$ and $F_{b,2}(z+T_2)=F_{b,2}(z)^*~\forall~ z \in D$, where $T_2=2\pi i/ \ln(\ln(2))$ ; at least for $D=\{ z\in \mathbb{C}:~\Re(z)>2 \}$ .         [attachment=480]     In the first plot, the lines $p=\Re(F_{b,4}(x+ i y)=$const $q=\Im(F_{b,4}(x+ i y)=$const are shown. Thick curves correspond to integer valuse of p and q. In the second plot, the lines $p=\Re(F_{b,2}(x+ i y)=$const $q=\Im(F_{b,2}(x+ i y)=$const are shown. Thick curves correspond to integer valuse of p and q. The dashed lines show the cuts. On the third plot, the difference $F_{b,4}(x)-F_{b,2}(x))$ is shown in the same notations. The plot of this difference along the real axis is below:     Dashed: $y=F_{b,4}(x)$ Thin: $y=F_{b,2}(x)$ Thick: My approximation for $y=10^{25}(F_{b,4}(x)-F_{b,2}(x))$ I suspect, each of functions $F_{b,4}$ and $F_{b,2}$ is unique. P.S. Henryk, could you please help me to handle the sizes of the figures? I think, the same size would be better. « Next Oldest | Next Newest »

 Messages In This Thread Bummer! - by bo198214 - 10/05/2007, 10:18 AM RE: Bummer! - by Gottfried - 10/05/2007, 10:56 AM RE: Bummer! - by bo198214 - 10/05/2007, 11:17 AM RE: Bummer! - by bo198214 - 10/06/2007, 07:05 AM RE: Bummer! - by bo198214 - 10/06/2007, 09:18 AM RE: Bummer! - by nuninho1980 - 05/29/2011, 09:37 PM RE: Bummer! - by bo198214 - 05/29/2011, 10:11 PM RE: Bummer! - by nuninho1980 - 05/30/2011, 01:13 AM RE: Bummer! - by sheldonison - 05/30/2011, 03:25 PM RE: Bummer! - by bo198214 - 05/31/2011, 09:05 AM RE: Bummer! - by JmsNxn - 05/31/2011, 09:06 PM RE: Bummer! - by sheldonison - 06/01/2011, 03:03 AM RE: Bummer! - by bo198214 - 06/01/2011, 01:16 PM RE: Bummer! - by jaydfox - 10/07/2007, 04:26 PM RE: Bummer! - by bo198214 - 10/07/2007, 06:16 PM RE: Bummer! - by jaydfox - 10/07/2007, 07:48 PM RE: Bummer! - by jaydfox - 10/15/2007, 08:36 PM RE: Bummer! - by bo198214 - 11/02/2007, 08:30 PM RE: Bummer! - by jaydfox - 11/02/2007, 10:31 PM RE: Bummer! - by bo198214 - 11/02/2007, 11:06 PM RE: Bummer! - by jaydfox - 11/07/2007, 02:22 PM RE: Bummer! - by bo198214 - 11/07/2007, 02:27 PM RE: Bummer! - by jaydfox - 11/26/2007, 04:37 PM RE: Bummer! - by jaydfox - 11/04/2007, 02:24 AM RE: Bummer! - by bo198214 - 11/06/2007, 11:33 AM RE: Bummer! - by jaydfox - 11/04/2007, 02:32 AM RE: Bummer! - by jaydfox - 11/06/2007, 01:34 PM RE: Bummer! - by bo198214 - 11/06/2007, 02:06 PM RE: Bummer! - by Gottfried - 11/07/2007, 08:32 AM RE: Bummer! - by jaydfox - 11/08/2007, 02:16 AM RE: Bummer! - by bo198214 - 11/08/2007, 01:14 PM RE: Bummer! - by jaydfox - 11/09/2007, 04:59 AM RE: Bummer! - by jaydfox - 11/09/2007, 05:12 AM RE: Bummer! - by bo198214 - 11/12/2007, 08:45 PM RE: Bummer! - by jaydfox - 11/13/2007, 08:36 AM RE: Bummer! - by jaydfox - 11/09/2007, 07:04 AM RE: Bummer! - by jaydfox - 11/13/2007, 01:47 AM RE: Bummer! - by bo198214 - 11/13/2007, 10:36 AM RE: Bummer! - by jaydfox - 11/13/2007, 02:23 PM RE: Bummer! - by bo198214 - 11/13/2007, 02:41 PM Bummer conclusio - by bo198214 - 03/12/2008, 09:20 PM RE: Bummer! - by Kouznetsov - 04/18/2009, 12:46 PM RE: Bummer! - by andydude - 04/21/2009, 08:28 PM RE: Bummer! - by bo198214 - 04/21/2009, 09:02 PM RE: Bummer! - by andydude - 04/22/2009, 11:33 PM RE: Bummer! - by bo198214 - 04/23/2009, 08:39 AM RE: Bummer! - by bo198214 - 04/23/2009, 09:01 AM RE: Bummer! - by Kouznetsov - 04/18/2009, 11:24 AM regular iteration at two different fixed points - by bo198214 - 06/19/2009, 08:51 AM RE: regular iteration at two different fixed points - by Kouznetsov - 06/19/2009, 11:41 AM RE: regular iteration at two different fixed points - by bo198214 - 06/19/2009, 12:30 PM

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