04/11/2021, 01:01 AM
Hey, guys!
So I've fixed the error in the first iteration, which was pretty silly of me actually. I had written that the process,
\(
\tau_\lambda^{n+1}(s) = \log(\beta_\lambda(s+1) + \tau_\lambda^n(s+1)) - \beta_\lambda(s)\\
\)
produces an exponential series. This is not true at all. It just looks like an exponential series (enough for the main theorem to be unaffected); but there are a bunch of singularities. It was really silly of me. The correct statement is that,
\(
\tau_\lambda^n(s) = -\log(1+ e^{-\lambda s}) + o(e^{-\lambda s})\,\,\text{as}\,\,\Re(s)\to \infty\\
\)
This doesn't affect the main theorem at all though. I just have to be more careful when applying Banach's Fixed Point theorem. For the most part, it's much of the same proof.
I've added a total of 12 figures; which took a lot out of me. I am horrible at working out graphs. And since this construction is to do with super-exponential behaviour at infinity; I don't know how to code around the overflow errors. I'm still working on making graphs of the actual tetration (which looks like I might be able to do soonish). I just have to put on my coding hat and find an efficient manner at computing this. I've also added a bunch of commutative diagrams, in an attempt to better explain some of the morphisms I use.
I've expanded a bunch of the arguments and clarified as much of the language as I think I can. Particularly, I made sure the proof that,
\(
\beta_\lambda(s) \to \infty\,\,\text{as}\,\,\Re(s) \to \infty\\
\)
Was as solid as I could make it. I had to reference three people to make this argument--all from Milnor's book. This theorem is the crux of the method though. Where, the number one reason the construction with \( \phi \) failed was because it oscillated between \( 0 \) and \( \infty \) very rapidly. I must say, it's very satisfying to see the graphs concur with the divergence of \( \beta_\lambda \) on paper.
I tried to explain the variable changes more clearly in this iteration. I did it a tad off-hand initially. But, I imagine much of you are new to this infinite composition stuff; so the idea of changing variables in an infinite composition may seem odd. I tried to make it simpler to understand by contrasting it with commutative diagrams.
I don't think the paper is quite done yet. I am looking to revamp it one more time once I somehow manage to get some workable code to evaluate these tetrations (of which we want to limit to find the right tetration). At least, as long as I can evaluate it in a rudimentary way.
The more I work with this though, the more I'm convinced this actually constructs tetration. Plus the graphs are behaving exactly as I expected them too (even if they're not the tetration graphs yet).
Regards, James.
asymptotic_tetration__4.pdf (Size: 3.07 MB / Downloads: 811)
So I've fixed the error in the first iteration, which was pretty silly of me actually. I had written that the process,
\(
\tau_\lambda^{n+1}(s) = \log(\beta_\lambda(s+1) + \tau_\lambda^n(s+1)) - \beta_\lambda(s)\\
\)
produces an exponential series. This is not true at all. It just looks like an exponential series (enough for the main theorem to be unaffected); but there are a bunch of singularities. It was really silly of me. The correct statement is that,
\(
\tau_\lambda^n(s) = -\log(1+ e^{-\lambda s}) + o(e^{-\lambda s})\,\,\text{as}\,\,\Re(s)\to \infty\\
\)
This doesn't affect the main theorem at all though. I just have to be more careful when applying Banach's Fixed Point theorem. For the most part, it's much of the same proof.
I've added a total of 12 figures; which took a lot out of me. I am horrible at working out graphs. And since this construction is to do with super-exponential behaviour at infinity; I don't know how to code around the overflow errors. I'm still working on making graphs of the actual tetration (which looks like I might be able to do soonish). I just have to put on my coding hat and find an efficient manner at computing this. I've also added a bunch of commutative diagrams, in an attempt to better explain some of the morphisms I use.
I've expanded a bunch of the arguments and clarified as much of the language as I think I can. Particularly, I made sure the proof that,
\(
\beta_\lambda(s) \to \infty\,\,\text{as}\,\,\Re(s) \to \infty\\
\)
Was as solid as I could make it. I had to reference three people to make this argument--all from Milnor's book. This theorem is the crux of the method though. Where, the number one reason the construction with \( \phi \) failed was because it oscillated between \( 0 \) and \( \infty \) very rapidly. I must say, it's very satisfying to see the graphs concur with the divergence of \( \beta_\lambda \) on paper.
I tried to explain the variable changes more clearly in this iteration. I did it a tad off-hand initially. But, I imagine much of you are new to this infinite composition stuff; so the idea of changing variables in an infinite composition may seem odd. I tried to make it simpler to understand by contrasting it with commutative diagrams.
I don't think the paper is quite done yet. I am looking to revamp it one more time once I somehow manage to get some workable code to evaluate these tetrations (of which we want to limit to find the right tetration). At least, as long as I can evaluate it in a rudimentary way.
The more I work with this though, the more I'm convinced this actually constructs tetration. Plus the graphs are behaving exactly as I expected them too (even if they're not the tetration graphs yet).
Regards, James.
asymptotic_tetration__4.pdf (Size: 3.07 MB / Downloads: 811)
