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 On the existence of rational operators sheldonison Long Time Fellow Posts: 638 Threads: 22 Joined: Oct 2008 12/19/2010, 08:23 PM (This post was last modified: 12/19/2010, 08:41 PM by sheldonison.) (12/14/2010, 01:41 AM)JmsNxn Wrote: And now if the critical strip of tetration is defined as: -1 <= f <= 0 b {3} f = f + 1 S(q) = q and therefore: m {q} q = m Further notes: Consider the function A(x) = m {x} n Which is a generalization of the Ackerman function, extending it to domain real. ...I'm a little slow at catching on to the jist of your post, but the Ackermann function is A(m=4,n)=2^^(n+3) - 3, or roughly base(2) tetration for m=4. So I assume you're trying to define an extension to the Ackermann function for real numbers, where A(x) = m {x} n, where "m" is the base, and x is a rational operator. So, A(x=2)= m (2) n = m*n. A(x=3)= m (3) n = m^n. A(x=4)= m (4) n = m^^n Is this the basic idea, where we are extending it to allow for for real values of "x" as well? Then Henryk's request is to see a graph of f(q) = 2 {q} 3. so f(2)=2*3=6, f(3)=2^3=8, f(4)=2^^3=16 ..... Sounds interesting! I don't think the linear approximation for the critical strip for [-1..0] for tetration is a good idea. There are many approaches to extending tetration to real numbers, that are analytic on the complex plane, and they all seem to agree with each other. - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread On the existence of rational operators - by JmsNxn - 12/14/2010, 01:41 AM RE: On the existence of rational operators - by bo198214 - 12/19/2010, 05:23 AM RE: On the existence of rational operators - by tommy1729 - 12/19/2010, 05:21 PM RE: On the existence of rational operators - by JmsNxn - 12/19/2010, 06:47 PM RE: On the existence of rational operators - by sheldonison - 12/19/2010, 08:23 PM RE: On the existence of rational operators - by JmsNxn - 12/20/2010, 02:16 AM RE: On the existence of rational operators - by sheldonison - 12/20/2010, 04:53 AM RE: On the existence of rational operators - by JmsNxn - 12/20/2010, 07:01 PM RE: On the existence of rational operators - by sheldonison - 12/20/2010, 08:28 PM RE: On the existence of rational operators - by JmsNxn - 12/20/2010, 09:38 PM RE: On the existence of rational operators - by bo198214 - 12/20/2010, 09:17 AM RE: On the existence of rational operators - by JmsNxn - 03/11/2011, 07:23 PM RE: On the existence of rational operators - by JmsNxn - 03/19/2011, 05:36 PM

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