06/06/2011, 04:39 AM
(This post was last modified: 06/06/2011, 05:54 AM by sheldonison.)
(06/06/2011, 02:45 AM)JmsNxn Wrote: Well alas, logarithmic semi operators have paid off and have given a beautiful smooth curve over domainHey James,. This solution for rational operators is given by
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Which extends the ackerman function to domain real (given the restrictions provided).
the upper superfunction ofis used (i.e: the cheta function).
.....
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 kneser.gp code, which includes
Code:
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