User talk:Hybrid Hippocampus

Difference of 2ω and ω2
I saw that you were wondering why people wrote \(2 \omega\) (instead of \(\omega 2\)). Well, \(2 \omega\) coincides with \(\omega\), but does not with \(\omega 2\). The reason why people express \(\omega\) in non-normal forms like \(1 + \omega\) or \(2 \omega\) is because they want to specify the choice of fundamental sequences. Namely, \(1 + \omega\) is \(\omega\) equipped with fundamental sequence \(\omega[n] := 1+n\), and \(2 \omega\) is \(\omega\) equipped with fundamental sequence \(\omega[n] := 2n\). They sometimes need such fundamental sequences in order to analyse large functions in FGH in precise ways. For example, FGH corresponding to \(2 \omega\) is much faster than the one corresponding to \(1 + \omega\), which is much faster than the one corresponding to \(\omega\) equipped with the usual fundamental sequences, i.e. Wainer hierarchy. Of course, they are bounded by FGH corresponding to \(\omega + 1\) in Wainer hierarchy.

p-adic 23:15, March 19, 2019 (UTC)