Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Rational operators (a {t} b); a,b > e solved
(06/06/2011, 02:45 AM)JmsNxn Wrote: Well alas, logarithmic semi operators have paid off and have given a beautiful smooth curve over domain . This solution for rational operators is given by :

Which extends the ackerman function to domain real (given the restrictions provided).
the upper superfunction of is used (i.e: the cheta function).
Hey James,

Sounds very exciting. fyi, I admit I don't yet understand your new functions, but I made some graphs, and it looks very promising. I thought I'd post the following snippet of pari-gp code, which implements the rational operator function you posted, which can be used with mylatest code, which includes support. This would also make it easier for other to try your new function. If a<e, then I used sexpeta(invsexpeta(a)+t), instead of cheta(invcheta(a)+t). I think this is legal since the two are in many ways the same function when they are being to generate fractional iterations of eta. This expansive definition eliminates your restrictions, and covers all of the reals, working seamlessly from -infinity to infinity. It also matches your second example exactly.
expeta(t,a) = {
  if (real(a)<exp(1), sexpeta(invsexpeta(a)+t), cheta(invcheta(a)+t));
fatb(a,t,b) = {
  if (t<1,  return (expeta(t,expeta(-t,a)+expeta(-t,b))));
  if (t>=1, return (expeta(t-1,b*expeta(1-t,a))));

I've gotten as far as quickly verifying that for t=0, we have addition, t=1, we have multiplication, and t=2 is exponentiation. The existing code has problems for invcheta(z) or invsexpeta(z), where z is too close to e, and invcheta(z)<=-1000, or invsexpeta(z)>1000. Other than that, it seems to work great. For example,
fatb(3,0,4)=7, which is 3+4
fatb(3,1,4)=12, which is 3x4
fatb(3,2,4)=81, which is 3^4
I wonder what it means that fatb(3,-1,4)=5.429897..?
Also, is there a smooth continuation to a function for t=3, which would be tetration?
- Sheldon

Messages In This Thread
RE: Rational operators (a {t} b); a,b > e solved - by sheldonison - 06/06/2011, 04:39 AM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Thoughts on hyper-operations of rational but non-integer orders? VSO 2 1,208 09/09/2019, 10:38 PM
Last Post: tommy1729
  Hyper operators in computability theory JmsNxn 5 5,151 02/15/2017, 10:07 PM
Last Post: MphLee
  Recursive formula generating bounded hyper-operators JmsNxn 0 1,847 01/17/2017, 05:10 AM
Last Post: JmsNxn
  holomorphic binary operators over naturals; generalized hyper operators JmsNxn 15 18,924 08/22/2016, 12:19 AM
Last Post: JmsNxn
  The bounded analytic semiHyper-operators JmsNxn 2 4,254 05/27/2016, 04:03 AM
Last Post: JmsNxn
  Bounded Analytic Hyper operators JmsNxn 25 24,165 04/01/2015, 06:09 PM
Last Post: MphLee
  Incredible reduction for Hyper operators JmsNxn 0 2,585 02/13/2014, 06:20 PM
Last Post: JmsNxn
  interpolating the hyper operators JmsNxn 3 5,833 06/07/2013, 09:03 PM
Last Post: JmsNxn
  Number theory and hyper operators JmsNxn 7 8,896 05/29/2013, 09:24 PM
Last Post: MphLee
  Number theoretic formula for hyper operators (-oo, 2] at prime numbers JmsNxn 2 4,589 07/17/2012, 02:12 AM
Last Post: JmsNxn

Users browsing this thread: 1 Guest(s)