# Tetration Forum

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Next Saturday Mar 6th, 1pm eastern, 1800 GMT, invitation to our second monthly Tetration zoom meeting. We will discuss Peter Walker's 1991 paper, http://eretrandre.org/rb/files/Walker1991_111.pdf
https://us02web.zoom.us/j/89038247428?pwd=UFo1dmVGT21YTHpSbTNqUjMyazUzQT09
Meeting ID: 890 3824 7428
Passcode: 322183

I would like to discuss Peter Walker's 1991 paper.  Peter Walker's paper discusses generating the Abel function for iterating $\exp(x)\mapsto\exp(x)-1$, and then using it to generate tetration/slog for base e.  The paper also includes what was reinvented by Andrew as the matrix equation slog, which is the most accessible version of tetration/slog I know of.  And Peter Walker asks whether these two functions agree with Kneser's conformal mapping tetration function.  I would like to revisit Peter Walker's paper discussing other developments on this forum since that time, especially discussing the last paragraph with some updated results on these two methods from the past 10 years mostly from this forum.

Quote:These differences cannot be explained without at least a proof of convergence
of the matrix method. And we cannot identify our function defined by the
iteration method ... with Kneser's function defined by conformal mappings,
without an extension of the domain of the function h to include nonreal values.
Until both these difficulties have been overcome, the possibility remains that
either two or three distinct generalized logarithms have been constructed.
• The conjecture that Walker's method is an infinitely differentiable but nowhere analytic function defined only at the real axis.
There are many other similar tetration functions, conjectured to be infinitely differentiable but nowhere analytic, such as the "base change" function.
• Does Walker's matrix method converge?  How do the individual solutions behave?
• How does Jay's accelerated matrix slog technique work, and does the convergence problem remain?
Continuing on, we could venture into a discussion of Kneser's conformal mapping, 1-cyclic mappings, Henryk's uniqueness criterion, a discussion of large numbers from Robert Munafo's website, Kouznetsov's contour integral technique, and recent papers by William Paulsen.  That is enough material to keep us busy for many sessions, which is why I view these Tetration meetings as monthly.  I think its a good idea to record the zoom meeting; I'll do that next time.

First meeting invitation (saved)

Next Saturday Feb 6th, 1pm eastern, 1800 GMT,James Nixon will be presenting the paper he has been working on.  My plan it to make this a monthly meeting, first Saturday of each month, so we'll also topics for the 2nd meeting in March.  We could discuss Peter Walker's 1991 paper for our 2nd meeting.  I look forward to meeting as many of you who are able to attend online!!!

https://us02web.zoom.us/j/89038247428?pwd=UFo1dmVGT21YTHpSbTNqUjMyazUzQT09
Meeting ID: 890 3824 7428
Passcode: 322183
Should be fun, pressure's on. At this point I think I have all the pieces to the jigsaw, but I'm not sure if my solution to the jigsaw is the same on the box. I'll try to come more prepared with an action plan this time, Sheldon.
(02/02/2021, 05:49 AM)JmsNxn Wrote: [ -> ]Should be fun, pressure's on. At this point I think I have all the pieces to the jigsaw, but I'm not sure if my solution to the jigsaw is the same on the box. I'll try to come more prepared with an action plan this time, Sheldon.

Hi everyone; reminder you are all invited; Sat Feb 6th, 1800 GMT, James Nixon will be his most recent paper.   Work has been extremely busy this week, so I haven't had time to James's paper updates, but I will do so before the Saturday meeting.  I'm looking forward to meeting as many of our Tetration friends who are able to attend, so the first 10-15minutes will just be a chance to say hello.  Thanks in advance for attending!!!
https://us02web.zoom.us/j/89038247428?pwd=UFo1dmVGT21YTHpSbTNqUjMyazUzQT09
Meeting ID: 890 3824 7428
Passcode: 322183
Sadly I won't be there because I'm busy... but it is a great idea. I'd be able to understand much faster the key concept listening to a presentation from the author. Too bad.
I hope you will update the forum if some new insight  or progress will be achieved during the zoom call.
(02/05/2021, 03:02 PM)MphLee Wrote: [ -> ]Sadly I won't be there because I'm busy... but it is a great idea. I'd be able to understand much faster the key concept listening to a presentation from the author. Too bad.
I hope you will update the forum if some new insight  or progress will be achieved during the zoom call.

Thanks James for a superbly written paper, especially the first half of your paper where you carefully prove the convergence of $\phi(s)$, and that it is entire. We had a good meeting with Sheldon, James, Henryk Trapmann, Gottfried Helms, and Cheetahs.  James wasn't feeling well so we didn't go through his paper in much detail.  It was good to actually see and talk to Henryk and Gottfried, and of course James.  I'll setup a second zoom meeting Sat Mar 6th, also 1800 GMT; 1pm east coast time.

https://us02web.zoom.us/j/89038247428?pwd=UFo1dmVGT21YTHpSbTNqUjMyazUzQT09
Meeting ID: 890 3824 7428
Passcode: 322183

I would like to discuss Peter Walker's 1991 paper.  Peter Walker's paper discusses generating the Abel function for iterating $\exp(x)\mapsto\exp(x)-1$, and then using it to generate tetration/slog for base e.  The paper also includes what was reinvented by Andrew as the matrix equation slog, which is the most accessible version of tetration/slog I know of.  And Peter Walker asks whether these two functions agree with Kneser's conformal mapping tetration function.  I would like to revisit Peter Walker's paper discussing other developments on this forum since that time, especially discussing the last paragraph with some updated results on these two methods from the past 10 years mostly from this forum.

Quote:These differences cannot be explained without at least a proof of convergence
of the matrix method. And we cannot identify our function defined by the
iteration method ... with Kneser's function defined by conformal mappings,
without an extension of the domain of the function h to include nonreal values.
Until both these difficulties have been overcome, the possibility remains that
either two or three distinct generalized logarithms have been constructed.
Unfortunately I was not there during the zoom.

I wish everyone good health and succes with the zoom meetings.

I assume it is not recorded ?

I had the idea of a tetation meeting in 2020 in Belgium but yeah we know that timing was not very good.
Maybe later ?

Anyway.
What was the outcome of the zoom meeting ?
I would like some info about it.

Also as for the matrix equation slog, I want to point out that I posted a forced convergeance method for infinite linear systems.
So we can find at least one solution. However I was not able to prove a nonzero radius.
I think that deserves more attention.

see for instance here :

https://math.eretrandre.org/tetrationfor...p?tid=1081

Or a similar method where there is no minimization and we solve square matrices of growing sizes.
( the sizes grow exponential like about 7*2^n )
The disadvantage of no minization is a possible smaller radius , but it is easier.

Another promising thing that has not gotten attention is the following taylor method :

https://math.eretrandre.org/tetrationfor...hp?tid=791

where the helping function could be related to phi or 2sinh for instance.

regards

tommy1729

Tom Marcel Raes
(02/07/2021, 04:18 PM)tommy1729 Wrote: [ -> ]Unfortunately I was not there during the zoom.

I wish everyone good health and succes with the zoom meetings.

I assume it is not recorded ?

I had the idea of a tetation meeting in 2020 in Belgium but yeah we know that timing was not very good.
Maybe later ?

Anyway.
What was the outcome of the zoom meeting ?
I would like some info about it.
.... Tom Marcel Raes

Hi Tommy,

It was a lot of fun to say hello to others interested in tetration.  James was on the call, but he was not feeling well so we mostly just thanked James for writing his paper.  Because James wasn't feeling well, we didn't actually review his paper, but I did share a couple of graphs of James's phi function which I also posted online.  I look forward to meeting you Tommy, maybe at our next zoom meeting.  I'm thinking the next zoom meeting will be largely a historical overview, starting with Walker's 1991 paper, and continuing from there.  We could venture into a discussion of Kneser's conformal mapping, 1-cyclic mappings, Henryk's uniqueness criterion, Jay's accelerated technique, nowhere analytic functions and the base change function, a discussion of large numbers from Robert Munafo's website, Kouznetsov's contour integral technique, and recent papers by William Paulsen.  That is enough material to keep us busy for many sessions, which is why I view these Tetration meetings as monthly.  I think its a good idea to record the zoom meeting; I'll do that next time.

I would like to interleave historical overviews of Tetration with member's presentations of their ideas.  The last meeting covered James Nixon's paper.  The next monthly meeting is a historical overview starting with Walker's paper.  Maybe the one after that will be be a presentation by MphLee or Tommy.

I was inspired enough by the zoom meeting and talking with James and Henryk to finally get MikTex installed and working on my laptop.  So now I'm working on my long overdue paper that will include also include a rigorous proof that one of my pari-gp matrix based slog programs converges to Kneser's conformal mapping of the Schroeder equation!  I might be ready to present and/or publish in about two to three months.  I've been thinking about this paper for at least four years, ever since I added a Matrix slog solution to fatou.gp with Kneser's conformal mapping of the Schroeder equation embedded inside of it.  I had to overcome dozens of hurdles before I could imagine publishing a rigorous proof of convergence and in the end the proof applies to a different simpler pari-gp matrix superlog program

I look forward to seeing everyone again next month; our second zoom meeting will be Sat Mar 6th, also 1800 GMT; 1pm east coast time.

https://us02web.zoom.us/j/89038247428?pwd=UFo1dmVGT21YTHpSbTNqUjMyazUzQT09
Meeting ID: 890 3824 7428
Passcode: 322183
(02/09/2021, 04:58 PM)sheldonison Wrote: [ -> ]I would like to interleave historical overviews of Tetration with member's presentations of their ideas.  The last meeting covered James Nixon's paper.  The next monthly meeting is a historical overview starting with Walker's paper.  Maybe the one after that will be be a presentation by MphLee or Tommy.
Thanks for the report. Would be an honor but I guess I don't deserve it nor I have any smart enough thing to say. Not to mention that my English is garbage haha.

Anyways I went back at Walker paper. Is very interesting. I hope I'll be able to attend and learn something from your review of it.
(02/27/2021, 11:42 AM)MphLee Wrote: [ -> ]Thanks for the report. Would be an honor but I guess I don't deserve it nor I have any smart enough thing to say. Not to mention that my English is garbage haha.

Anyways I went back at Walker paper. Is very interesting. I hope I'll be able to attend and learn something from your review of it.

MphLee,

I look forward to seeing you and anyone else who would like to discuss Walker's paper this Saturday Mar 6th, also 1800 GMT; 1pm east coast time!

https://us02web.zoom.us/j/89038247428?pwd=UFo1dmVGT21YTHpSbTNqUjMyazUzQT09
Meeting ID: 890 3824 7428
Passcode: 322183
I suggest you to create a new cross-referenced post with the new date in the title.
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