• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Rational operators (a {t} b); a,b > e solved JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 06/12/2011, 07:55 PM (06/11/2011, 02:33 PM)Gottfried Wrote: Hi James, $\hspace{48} \begin{eqnarray} a &+& b &=& a + b \\ a &+_{\tiny -1} & b &=& \log(e^a + e^b) \\ a &+_{\tiny -2} & b &=& \log(\log(e^e^a + e^e^b)) \\ \vspace8 \end{eqnarray}$ Yes this is right: $a\,\,\bigtriangleup_{-1}^e\,\,b = \ln(e^a+e^b)$ I did a lot of investigation into this operator (well, to the best that I could). Quote:and $L^{\tiny o h}(3)$ for the h-fold iterated log( 3) then Mike's limit can be expressed $\begin{eqnarray} t_1 &=& L^{\tiny o 1}(3) \\ t_2 &=& L^{\tiny o 1}(3) &+& L^{\tiny o 2}(3) \\ t_3 &=& L^{\tiny o 1}(3) &+& L^{\tiny o 2}(3) &+_{\tiny -1}& L^{\tiny o 3}(3) \\ t_4 &=& L^{\tiny o 1}(3) &+& L^{\tiny o 2}(3) &+_{\tiny -1}& L^{\tiny o 3}(3) &+_{\tiny -2}& L^{\tiny o 4}(3) \\ \vspace8 & &\\ ...& &... \\ \vspace8 & &\\ t_{n\to \infty} &\to & \text{constant} \\ \end{eqnarray}$ where the operator-precedence is lower the more negative the index at the plus is (so we evaluate it from the left). First question: is this in fact an application of your "rational operator"? Well it appears to be. I'm floored, I'm terrible at finding applications. Quote:And if it is so, then second question: does this help to evaluate this to higher depth of iteration than we can do it when we try it just by log and exp alone (we can do it to iteration 4 or 5 at max I think) ? Well, not so far since the calculations involved in lower order operators rely on iterations of exp. However, I investigated in seeing if $f(x) = a\,\,\bigtriangleup_{-1}^e\,\,x$ was analytic (which should help in calculations). But I only made it to the sixth or seventh derivative before I realized I wasn't going to recognize the pattern. The thread's here http://www.mymathforum.com/viewtopic.php?f=23&t=20993 . I assume it would be analytic, (the function looks analytic when graphed, if that's any argument). Also, I think that if $a\,\,\bigtriangleup_{-1}^e\,\,x$ is analytic, then $a\,\,\bigtriangleup_{-2}\,\,x$ is probably analytic, since it's basically the same function with just a faster convergence to y=x and a higher starting point at negative infinity. « Next Oldest | Next Newest »

 Messages In This Thread Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/06/2011, 02:45 AM RE: Rational operators (a {t} b); a,b > e solved - by sheldonison - 06/06/2011, 04:39 AM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/06/2011, 05:34 AM RE: Rational operators (a {t} b); a,b > e solved - by sheldonison - 06/06/2011, 06:02 AM RE: Rational operators (a {t} b); a,b > e solved - by sheldonison - 06/06/2011, 07:03 AM RE: Rational operators (a {t} b); a,b > e solved - by nuninho1980 - 06/06/2011, 05:16 PM RE: Rational operators (a {t} b); a,b > e solved - by bo198214 - 06/06/2011, 06:53 AM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/06/2011, 08:47 AM RE: Rational operators (a {t} b); a,b > e solved - by bo198214 - 06/06/2011, 09:23 AM RE: Rational operators (a {t} b); a,b > e solved - by tommy1729 - 06/06/2011, 11:59 AM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/06/2011, 05:44 PM RE: Rational operators (a {t} b); a,b > e solved - by bo198214 - 06/06/2011, 09:28 PM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/06/2011, 07:47 PM RE: Rational operators (a {t} b); a,b > e solved - by sheldonison - 06/06/2011, 08:43 PM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/07/2011, 02:45 AM RE: Rational operators (a {t} b); a,b > e solved - by bo198214 - 06/07/2011, 06:59 AM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/08/2011, 04:54 AM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/08/2011, 07:31 PM RE: Rational operators (a {t} b); a,b > e solved - by sheldonison - 06/08/2011, 08:32 PM RE: Rational operators (a {t} b); a,b > e solved - by bo198214 - 06/08/2011, 09:14 PM RE: Rational operators (a {t} b); a,b > e solved - by tommy1729 - 09/02/2016, 01:50 AM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/08/2011, 11:47 PM RE: Rational operators (a {t} b); a,b > e solved - by Gottfried - 06/11/2011, 02:33 PM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/12/2011, 07:55 PM RE: Rational operators (a {t} b); a,b > e solved - by Xorter - 08/21/2016, 06:56 PM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 08/22/2016, 12:36 AM RE: Rational operators (a {t} b); a,b > e solved - by Xorter - 08/24/2016, 07:24 PM RE: Rational operators (a {t} b); a,b > e solved - by Xorter - 08/29/2016, 02:06 PM RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 09/01/2016, 06:47 PM RE: Rational operators (a {t} b); a,b > e solved - by tommy1729 - 09/02/2016, 02:04 AM RE: Rational operators (a {t} b); a,b > e solved - by tommy1729 - 09/02/2016, 02:11 AM

 Possibly Related Threads... Thread Author Replies Views Last Post Thoughts on hyper-operations of rational but non-integer orders? VSO 2 650 09/09/2019, 10:38 PM Last Post: tommy1729 Hyper operators in computability theory JmsNxn 5 4,206 02/15/2017, 10:07 PM Last Post: MphLee Recursive formula generating bounded hyper-operators JmsNxn 0 1,557 01/17/2017, 05:10 AM Last Post: JmsNxn holomorphic binary operators over naturals; generalized hyper operators JmsNxn 15 16,922 08/22/2016, 12:19 AM Last Post: JmsNxn The bounded analytic semiHyper-operators JmsNxn 2 3,697 05/27/2016, 04:03 AM Last Post: JmsNxn Bounded Analytic Hyper operators JmsNxn 25 20,603 04/01/2015, 06:09 PM Last Post: MphLee Incredible reduction for Hyper operators JmsNxn 0 2,314 02/13/2014, 06:20 PM Last Post: JmsNxn interpolating the hyper operators JmsNxn 3 5,220 06/07/2013, 09:03 PM Last Post: JmsNxn Number theory and hyper operators JmsNxn 7 7,977 05/29/2013, 09:24 PM Last Post: MphLee Number theoretic formula for hyper operators (-oo, 2] at prime numbers JmsNxn 2 4,096 07/17/2012, 02:12 AM Last Post: JmsNxn

Users browsing this thread: 1 Guest(s)