Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Rational operators (a {t} b); a,b > e solved
#19
(06/08/2011, 07:31 PM)JmsNxn Wrote: I can't believe I overlooked this! This is huge!
....
therefore:


And therefore is potentially analytic, and isn't piecewise, for all a, and b = e,


This is crazy.
So we have a definition for t=0..2 (addition, multiplication, exponentiation), for all values of a. Nice!


So, now, I'm going to define the function f(b), which returns the base which has the fixed point of "b".
, since e is the fixed point of b=
, since 2 is the lower fixed point of b=sqrt(2)
, since 3 is the upper fixed point of this base
, since 4 is the upper fixed point of b=sqrt(2)
, since 5 is the upper fixed point of this base
I don't know how to calculate the base from the fixed point, but that's the function we need, and we would like the function to be analytic! This also explains why the approximation of using base eta pretty well, since the base we're going to use isn't going to be much smaller than eta, as b gets bigger or smaller than e.

Now, we use this new function in place of eta, in James's equation. Here, f=f(b).

- Sheldon
Reply


Messages In This Thread
RE: Rational operators (a {t} b); a,b > e solved - by sheldonison - 06/08/2011, 08:32 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Thoughts on hyper-operations of rational but non-integer orders? VSO 2 774 09/09/2019, 10:38 PM
Last Post: tommy1729
  Hyper operators in computability theory JmsNxn 5 4,434 02/15/2017, 10:07 PM
Last Post: MphLee
  Recursive formula generating bounded hyper-operators JmsNxn 0 1,622 01/17/2017, 05:10 AM
Last Post: JmsNxn
  holomorphic binary operators over naturals; generalized hyper operators JmsNxn 15 17,365 08/22/2016, 12:19 AM
Last Post: JmsNxn
  The bounded analytic semiHyper-operators JmsNxn 2 3,825 05/27/2016, 04:03 AM
Last Post: JmsNxn
  Bounded Analytic Hyper operators JmsNxn 25 21,419 04/01/2015, 06:09 PM
Last Post: MphLee
  Incredible reduction for Hyper operators JmsNxn 0 2,375 02/13/2014, 06:20 PM
Last Post: JmsNxn
  interpolating the hyper operators JmsNxn 3 5,356 06/07/2013, 09:03 PM
Last Post: JmsNxn
  Number theory and hyper operators JmsNxn 7 8,164 05/29/2013, 09:24 PM
Last Post: MphLee
  Number theoretic formula for hyper operators (-oo, 2] at prime numbers JmsNxn 2 4,199 07/17/2012, 02:12 AM
Last Post: JmsNxn



Users browsing this thread: 1 Guest(s)