Sat March 6th second Tetration mtg; Peter Walker's 1991 paper
Next Saturday Mar 6th, 1pm eastern, 1800 GMT, invitation to our second monthly Tetration zoom meeting. We will discuss Peter Walker's 1991 paper,
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 , 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!!!
Meeting ID: 890 3824 7428
Passcode: 322183
- Sheldon

Messages In This Thread
Sat March 6th second Tetration mtg; Peter Walker's 1991 paper - by sheldonison - 01/31/2021, 11:52 PM

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