Tommy's Gaussian method.

[JmsNxn]

Again I want to give some more details about my thoughts.

considering 2 things.

First a note on functional inverse.

I estimated the bounds of ln ln ... a^b^...(A)

but let B = ln ln ... a^b^...(A)

then A = ln_ln(...)(...ln_ln(b)(ln_ln(a)(exp(exp...exp(B))))))

notice how this gives similar bounds by the same method !!

so the function is bounded and its inverse is also bounded !!

That is a powerful thing !!

Also This is a nice alternative way of looking at it , that might convince ppl who were perhaps confused or arguing " chaos as objection " for one them.

***

The second thing is this thought experiment :

Let abs(s) < sqrt(2)

Then how does ln^[n](exp^[n](s) + C) behave ?

It turns out for real s this converges nice and any real C > 0 this converges fast.

However when s or C are complex things are slightly different !

This is hard to test numerically because of overflow or even with calculus.

But it is clear that when C goes to zero ( as function of n ) FAST ENOUGH , then it converges to s.

so how fast is fast ??

Well for abs(s) < sqrt(2) and sufficiently large n ( usually 2,3  or 4 will already do ) :

ln^[n](exp^[n](s) + exp(- n^2) ) = s + o(exp(- n^2)) + o(exp(- n^2)) i

(if we take the correct branches).

Now this looks familiar not ??



This strenghtens my previous ideas and bound arguments ofcourse.



regards

tommy1729

Yes Tommy;

and for large \( s \) we'll get that

\(
u(s) = \text{Tom}_A(s) - \beta(s) = \mathcal{O}(e^{-\sqrt{s}});
\)

And so we should get that,

\(
\log^{\circ n} \text{Tom}_A(s+n) = \log^{\circ n} \beta(s+n) + u(s+n) = \log^{\circ n} (\beta(s+n)) + e^{-\sqrt{n}} \mathcal{O}(e^{-\sqrt{s}})\\
\)

Or something that looks like this. The errors are something like this; these are just rough estimates. Still haven't found a good way to compare.

I'm glad you're on the same page as me. I do feel your approximation is definitely better; but I'm of the view that this is all just the beta method. The trouble with your method though is that it doesn't create the periodic tetrations; which I think are very important--and can be used to classify tetrations.

I love the idea of using a gaussian as opposed to a logistic approach.

Regards, James