Hey, everyone!
Although I haven't been too active on the forum of late; it's with very good reason. Sheldon and I have done a bunch of zoom calls and a bunch of experiments to draw out exactly how the \(\beta\) method works. From it, we've derived a whole plethora of results and limitations of the beta method. Like for instance, I've been able to show that the \(\beta\) method can at best create an asymptotic series for \(b > e^{1/e}\). This means all the fancy graphs for these bases, are at best tetration up to a \((s-s_0)^m\) term for arbitrary \(m\) and \(s_0\). So that, they cannot be holomorphic. Additionally, this shows that they are exactly \(\mathcal{C}^\infty\) on the real line.
This led me to devote the majority of the paper towards the Shell-Thron region. It is in this region that we see holomorphy much more obviously, and additionally, we can "move the period around", with only creating different fractals in the graph. Whereby, for example if \(b = \sqrt{2}\), we can be holomorphic in the right half plane only if the period of the tetration is \(-2\pi i/\log \log(2)\). We can shrink this period, but if we do we gain a bunch of branching/singularities.
I've been able to EXACTLY determine where and when a \(\beta\) function can create a tetration for arbitrarily \(\lambda, b = e^{\mu}\). Where again, the period of this tetration will be \(\ell = 2 \pi i/ \lambda\). To the extent that a natural boundary appears as \(2 \pi i/\lambda \to P\), where \(P\) is the period of the regular iteration. Whereby regular iteration I mean the \(\sqrt{2}\) tetration done through the Schroder function with trivial theta mapping. I have a lot to share, and much of it settles the questions we've had about the \(\beta\) method.
Ironically enough, this does not disprove the earlier paper I had written, it shows that there are some errors in it (largely with the assumption of holomorphy instead of asymptotic series)--but the main result is largely still possible (it just needs to be reworded). And how it needs to be reworded is pretty simple; we can approximate Kneser with \(\beta\) functions, but we cannot create a new tetration. Additionally it needs to be done more carefully than I had done. This portion will not appear in the paper; instead I focus specifically on understanding where the \(\beta\) method can be holomorphic (or produce an asymptotic series) with arbitrary period.
This report is clocked at 90 pages at the moment; and I'm just polishing up the final chapter which compares possible other beta methods and relates them to the beta method itself. Which is intended to describe how the beta method with period \(\ell\) sits with other periodic solutions of tetration with period \(\ell\). Which is basically just a controlled discussion of \(\theta\) mappings.
There are still some loose bolts in this paper. There are still things I may not have made air-tight. But I can make it all air-tight. Sheldon and I have talked and worked through a lot of code; and every result I've displayed is backed by hard empirical data. If some results appear a tad under-explained.
I have incorporated every piece of the \(\beta\) method and have fleshed out every piece we've discussed here. This is pretty much the summation of all my research precluding to infinite compositions and tetration. I have included all the coding which confirms every facet of the paper as well. I will upload the code soon; I just haven't gotten around to double and triple checking everything works--and writing a readme.txt file.
I am also waiting to finish the second part of the appendix, which is Sheldon's analysis of the case \(\lambda=1,\,\mu =1\) (\(b=e\)). This is mostly just a large empirical justification of one of the large results in this paper.
Anyway, as I'm mostly just fixing typos at this point, I figure it'd be okay to post the pdf!
Thanks a lot for the support guys! I hope this paper meets the standard of the forum!
Asymptotic_Solutions_Of_The_Tetration_Equation_In_The_Style_Of_Sterling.pdf (Size: 7.05 MB / Downloads: 718)
EDIT:
I've attached the link to the github library; it has a readme which breaks down how to use the program more. I am trying to add a helper protocol but I'm not sure what would be the right way to do it; I'm not too much of a fan of Sheldon's.
https://github.com/JmsNxn92/The-Beta-Method-Thesis
I made the code more conducive to grabbing taylor series as well; forgot to add some code in the one I uploaded. For that reason I'm reuploading and attaching to the main post; and deleting the post I made at the beginning of this thread.
I've also made a thread in computing to store this code; it can be found here.
The readme can be found there also;
Regards, James
Although I haven't been too active on the forum of late; it's with very good reason. Sheldon and I have done a bunch of zoom calls and a bunch of experiments to draw out exactly how the \(\beta\) method works. From it, we've derived a whole plethora of results and limitations of the beta method. Like for instance, I've been able to show that the \(\beta\) method can at best create an asymptotic series for \(b > e^{1/e}\). This means all the fancy graphs for these bases, are at best tetration up to a \((s-s_0)^m\) term for arbitrary \(m\) and \(s_0\). So that, they cannot be holomorphic. Additionally, this shows that they are exactly \(\mathcal{C}^\infty\) on the real line.
This led me to devote the majority of the paper towards the Shell-Thron region. It is in this region that we see holomorphy much more obviously, and additionally, we can "move the period around", with only creating different fractals in the graph. Whereby, for example if \(b = \sqrt{2}\), we can be holomorphic in the right half plane only if the period of the tetration is \(-2\pi i/\log \log(2)\). We can shrink this period, but if we do we gain a bunch of branching/singularities.
I've been able to EXACTLY determine where and when a \(\beta\) function can create a tetration for arbitrarily \(\lambda, b = e^{\mu}\). Where again, the period of this tetration will be \(\ell = 2 \pi i/ \lambda\). To the extent that a natural boundary appears as \(2 \pi i/\lambda \to P\), where \(P\) is the period of the regular iteration. Whereby regular iteration I mean the \(\sqrt{2}\) tetration done through the Schroder function with trivial theta mapping. I have a lot to share, and much of it settles the questions we've had about the \(\beta\) method.
Ironically enough, this does not disprove the earlier paper I had written, it shows that there are some errors in it (largely with the assumption of holomorphy instead of asymptotic series)--but the main result is largely still possible (it just needs to be reworded). And how it needs to be reworded is pretty simple; we can approximate Kneser with \(\beta\) functions, but we cannot create a new tetration. Additionally it needs to be done more carefully than I had done. This portion will not appear in the paper; instead I focus specifically on understanding where the \(\beta\) method can be holomorphic (or produce an asymptotic series) with arbitrary period.
This report is clocked at 90 pages at the moment; and I'm just polishing up the final chapter which compares possible other beta methods and relates them to the beta method itself. Which is intended to describe how the beta method with period \(\ell\) sits with other periodic solutions of tetration with period \(\ell\). Which is basically just a controlled discussion of \(\theta\) mappings.
There are still some loose bolts in this paper. There are still things I may not have made air-tight. But I can make it all air-tight. Sheldon and I have talked and worked through a lot of code; and every result I've displayed is backed by hard empirical data. If some results appear a tad under-explained.
I have incorporated every piece of the \(\beta\) method and have fleshed out every piece we've discussed here. This is pretty much the summation of all my research precluding to infinite compositions and tetration. I have included all the coding which confirms every facet of the paper as well. I will upload the code soon; I just haven't gotten around to double and triple checking everything works--and writing a readme.txt file.
I am also waiting to finish the second part of the appendix, which is Sheldon's analysis of the case \(\lambda=1,\,\mu =1\) (\(b=e\)). This is mostly just a large empirical justification of one of the large results in this paper.
Anyway, as I'm mostly just fixing typos at this point, I figure it'd be okay to post the pdf!
Thanks a lot for the support guys! I hope this paper meets the standard of the forum!
Asymptotic_Solutions_Of_The_Tetration_Equation_In_The_Style_Of_Sterling.pdf (Size: 7.05 MB / Downloads: 718)
EDIT:
I've attached the link to the github library; it has a readme which breaks down how to use the program more. I am trying to add a helper protocol but I'm not sure what would be the right way to do it; I'm not too much of a fan of Sheldon's.
https://github.com/JmsNxn92/The-Beta-Method-Thesis
I made the code more conducive to grabbing taylor series as well; forgot to add some code in the one I uploaded. For that reason I'm reuploading and attaching to the main post; and deleting the post I made at the beginning of this thread.
I've also made a thread in computing to store this code; it can be found here.
The readme can be found there also;
Regards, James
