08/12/2007, 06:39 PM
I believe that you have a lot of good ideas, but actually I can not quite follow your explanations. Most of your formulas comes without any justifications nor proofs.
That your formula
Then I dont understand how you compute \( \log_a^{\circ n} AB=\log_a^{\circ n}(A)+\epsilon_1 \). It is always unclear what you mean by "exact" solution. Isnt everything exact in mathematics except if we say its approximate? A limit for example is exact if it exists regardless what difficulties we have with numeric computation.
A side note: if you write \log in TeX then you get the properly displayed log, instead of the multiplicaton of the letters l, o and g.
That your formula
\( {}^x a = \lim_{n \to \infty} \log_a^{\circ n}\left({}^{\left( n+x+\mu_b(a)\right)} b\right) \)
yields a proper tetration the resulting \( {}^xa \) must at least satisfy the identities \( ^{x+1}a=a^{{}^xa} \) and \( {}^1 a=a \)
if \( {}^xb \) satisfies them. Can you verify this first?Then I dont understand how you compute \( \log_a^{\circ n} AB=\log_a^{\circ n}(A)+\epsilon_1 \). It is always unclear what you mean by "exact" solution. Isnt everything exact in mathematics except if we say its approximate? A limit for example is exact if it exists regardless what difficulties we have with numeric computation.
A side note: if you write \log in TeX then you get the properly displayed log, instead of the multiplicaton of the letters l, o and g.
