03/30/2008, 05:55 AM

It sounds like we should be producing 2 documents. The "FAQ", and the "Reference". I believe that the "Open Problems" section should be listed in the "Reference" document, and that this should be our guide as to what problems remain unsolved. There could also be a section for "Closed Problems" to remind people what was recently solved (of course, with pointers or references to places in the document where the solution can be found). I also think that these 2 documents have been delayed in part because of a lack of structure in collaboration. For a while I was under the impression that we were to produce a single FAQ for everything, but then when I produced my version of the FAQ, there was a lot of reactions to its verbosity, indicating that we should separate those ideas into a FAQ and a Reference document. Another thing I have noticed an inconsistency in filenames. I think in terms of accessibility (to newbies to the forum) all future FAQ and Reference revisions should follow the conventions "tetration-faq-yyyymmdd" and "tetration-ref-yyyymmdd" instead of the past filenames (20070809tetrationfaq, tetration-formula, and FAQ_20080112). This would add a lot of consistency that would be beneficial for people looking for the latest revisions. I will incorporate these filenames into my future revisions.

As for the open questions mentioned in this thread, these are really good questions. I need to re-read many things before I can attack these problems. I am still having difficulty with the regular/matrix approaches. I have read Gottfried's approach many times, and each time I understand a little bit more, but I must admit that I find Aldrovandi's discussion of matrices seem to be much more understandable. What are the differences between Aldovandi's approach and Gottfried's approach? Does "diagonalization" imply the standard diagonalization? or some non-standard diagonalization? Are the diagonalization approach and Gottfried's matrix approach the same? If not, then how are these approaches different from the Carleman matrix applied to the Schroeder functional equation? When applying the Carleman matrix to the Schroeder functional equation, are the diagonals necessarily powers of the fixed point? Correspondingly, do all functions that satisfy the Schroeder functional equation require using a fixed point as a "base" of sorts (defining the Schroeder function as the base-(fixed point) exponential of the Abel function)? Is Aldrovandi's eigen-matrix approach fundamentally different than the diagonalization approach?

I still have many open questions, and these were just about matrices!

Andrew Robbins

PS. @bo198214:

I still need to do some research on the (b>1) thing, but my current ideas can be summarized by this thread which I intend on updating as soon as I have better ideas...

As for the open questions mentioned in this thread, these are really good questions. I need to re-read many things before I can attack these problems. I am still having difficulty with the regular/matrix approaches. I have read Gottfried's approach many times, and each time I understand a little bit more, but I must admit that I find Aldrovandi's discussion of matrices seem to be much more understandable. What are the differences between Aldovandi's approach and Gottfried's approach? Does "diagonalization" imply the standard diagonalization? or some non-standard diagonalization? Are the diagonalization approach and Gottfried's matrix approach the same? If not, then how are these approaches different from the Carleman matrix applied to the Schroeder functional equation? When applying the Carleman matrix to the Schroeder functional equation, are the diagonals necessarily powers of the fixed point? Correspondingly, do all functions that satisfy the Schroeder functional equation require using a fixed point as a "base" of sorts (defining the Schroeder function as the base-(fixed point) exponential of the Abel function)? Is Aldrovandi's eigen-matrix approach fundamentally different than the diagonalization approach?

I still have many open questions, and these were just about matrices!

Andrew Robbins

PS. @bo198214:

I still need to do some research on the (b>1) thing, but my current ideas can be summarized by this thread which I intend on updating as soon as I have better ideas...