Pseudoalgebra tommy1729 Ultimate Fellow     Posts: 1,372 Threads: 336 Joined: Feb 2009 10/05/2016, 12:21 PM (This post was last modified: 10/08/2016, 11:54 AM by tommy1729.) Consider iterations of exponentials of a fixed height. Also called growth. For instance semi-exponentials. In algebra the main thing is Sum and product. Kinda. When we consider asymptotics i call it pseudoalgebra. So for semiexp we get the natural questions such as The best fit ( given by the symbol = ) Where y is the value we seek and x > 1. This is - for clarity - an asymptotic equation for bases ( 2,3,5,y). It reminds me of base change and others. How about these ? How to find such identities ? Regards Tommy1729 tommy1729 Ultimate Fellow     Posts: 1,372 Threads: 336 Joined: Feb 2009 10/08/2016, 12:22 PM (This post was last modified: 10/08/2016, 12:27 PM by tommy1729.) @ means approximation. Lemma From there we get Where d is ( notice this can be rewritten with 1 semi-exp and 2 semi-logs too ) But this is not the full story ofcourse. We need proofs. Perhaps consider other ways to handle the issue. And a qualitative understanding of the formula for d such as d ~ (qs)^2 or the alike. I wonder if you would have done it differently ? Also a table would be nice. Still alot of work to do. Regards Tommy1729 The master tommy1729 Ultimate Fellow     Posts: 1,372 Threads: 336 Joined: Feb 2009 10/13/2016, 02:32 AM Im afraid the strategy fails. For exp_b^[a] <*> and a < 1 we get <*> @ = exp^[a] ( T x ) Where T Goes to 1 as x grows and for a >= 1 , T Goes to ln(b). Proof sketch S commutes with exp. S(T x) = ln S ( T b^x) / ln(B). = S ln ( T b^x) / ln(B) = S ( ln T + ln B x ) / ln B Maybe. Still thinking ... Regards Tommy1729 tommy1729 Ultimate Fellow     Posts: 1,372 Threads: 336 Joined: Feb 2009 10/19/2016, 08:47 AM So originally i tried to work from " the inside " like but from " the outside " like we got already the following result. ( i Will omit x sometimes , since it Goes to oo ) For Now z > 1 must be true. Simplify Since z > 1 and we get and . -- Notice for integer n > 0 we get by the above and induction ~~ I assume it holds for n = 0 , that would imply that powers dominate bases for subexponential tetration. In other words Conjecture for p > 1 : -- However we need much better understanding and approximations. We are not close to answering semiexp_q * semiexp_s ~ semiexp_d ^ R For a given pair (q,s) and a desired best fit (d,R). I considered the base change but without succes. The approximation slog - slog_b ~~ constant is insufficient. See also http://math.stackexchange.com/questions/...ase-a-e1-e Although that might be hard to read. Regards Tommy1729 sheldonison Long Time Fellow    Posts: 641 Threads: 22 Joined: Oct 2008 10/23/2016, 09:17 PM (This post was last modified: 10/23/2016, 09:25 PM by sheldonison.) (10/05/2016, 12:21 PM)tommy1729 Wrote: .... So for semiexp we get the natural questions such as The best fit ( given by the symbol = ) Mick's question on mathstack exchange is related to this post. In my answer, I considered and . See math.stackexchange.com If one uses the analytic solution for the half iterates of base_2 and the half iterate of base_e (ignoring the conjectured nowhere analytic basechange type solutions), the fractional exponentials are not at all well ordered. If a 