01/13/2008, 11:32 PM
Although I agree with your conclusion, I would say it different. Instead of saying:
"towers and super-roots are disjoint" I would say:
"the ranges of tetrates and super-roots are pointwise disjoint over the domain (0, 1)."
Also, a slightly more accurate statement would be \( (\mathbb{N} = \mathbb{Z}^{+}) \):
"\( \text{srt}^{(2\mathbb{N})}(x) \ <\ x^{1/x} \ <\ \text{srt}^{(2\mathbb{N}+1)}(x) \ <\ x \ <\ {}^{(2\mathbb{N}+1)}x \ <\ {}^{\infty}x \ <\ {}^{(2\mathbb{N})}x \) for all \( 0 < x < 1 \)"
Andrew Robbins
"towers and super-roots are disjoint" I would say:
"the ranges of tetrates and super-roots are pointwise disjoint over the domain (0, 1)."
Also, a slightly more accurate statement would be \( (\mathbb{N} = \mathbb{Z}^{+}) \):
"\( \text{srt}^{(2\mathbb{N})}(x) \ <\ x^{1/x} \ <\ \text{srt}^{(2\mathbb{N}+1)}(x) \ <\ x \ <\ {}^{(2\mathbb{N}+1)}x \ <\ {}^{\infty}x \ <\ {}^{(2\mathbb{N})}x \) for all \( 0 < x < 1 \)"
Andrew Robbins