Hi.
I was wondering about this. it appears that the Kneser tetrational satisfies the following real-analytic property on \( (-2, \infty) \):
\( \mathrm{tet}^{(2n)}(x) \) is strict-monotone increasing for \( n \ge 0 \), \( n \in \mathbb{Z} \)
\( \mathrm{tet}^{(2n+1)}(x) \) is positive for \( n \ge 0 \), \( n \in \mathbb{Z} \)
and
\( \mathrm{tet}^{(2n+1)}(x) \) is convex for \( n \ge 0 \), \( n \in \mathbb{Z} \).
(the notation denotes differentiation)
These are equivalent. Easy theorems from calculus concerning monotonicity, convexity, and derivatives and integrals shows that 1 and 2 imply each other and that 3 implies 1.
But here's the thing: could this be a uniqueness criterion for tetration? (here, I'm thinking of base \( e \)) I tried some numerical tests of Kneser's tetration solution, warping it with small \( \theta(z) \) 1-cyclic warping mappings (specifically \( \theta(z) = \frac{\sin(2\pi z)}{K} \) and \( \theta(z) = \frac{\sin(2\pi z - \pi) + 1}{K} \)) with amplitudes of down to \( 10^{-5} \) (which is \( K = 10^5 \)) and the criterion seems to fail if the derivative is high enough (for amplitudes of the given magnitude, at around the 32nd derivative). The derivative at which it fails seems to increase rapidly as \( K \) shrinks, so I'm not sure if it is singular, meaning there is a range of \( K \) for which the condition is satisfied and hence uniqueness is not obtained, or whether or not it will eventually fail no matter how small the 1-cyclic wobble is, which would mean this should provide a uniqueness condition when combined with, perhaps, analyticity or maybe even just smoothness and the usual functional equations.
What do you think?
I was wondering about this. it appears that the Kneser tetrational satisfies the following real-analytic property on \( (-2, \infty) \):
\( \mathrm{tet}^{(2n)}(x) \) is strict-monotone increasing for \( n \ge 0 \), \( n \in \mathbb{Z} \)
\( \mathrm{tet}^{(2n+1)}(x) \) is positive for \( n \ge 0 \), \( n \in \mathbb{Z} \)
and
\( \mathrm{tet}^{(2n+1)}(x) \) is convex for \( n \ge 0 \), \( n \in \mathbb{Z} \).
(the notation denotes differentiation)
These are equivalent. Easy theorems from calculus concerning monotonicity, convexity, and derivatives and integrals shows that 1 and 2 imply each other and that 3 implies 1.
But here's the thing: could this be a uniqueness criterion for tetration? (here, I'm thinking of base \( e \)) I tried some numerical tests of Kneser's tetration solution, warping it with small \( \theta(z) \) 1-cyclic warping mappings (specifically \( \theta(z) = \frac{\sin(2\pi z)}{K} \) and \( \theta(z) = \frac{\sin(2\pi z - \pi) + 1}{K} \)) with amplitudes of down to \( 10^{-5} \) (which is \( K = 10^5 \)) and the criterion seems to fail if the derivative is high enough (for amplitudes of the given magnitude, at around the 32nd derivative). The derivative at which it fails seems to increase rapidly as \( K \) shrinks, so I'm not sure if it is singular, meaning there is a range of \( K \) for which the condition is satisfied and hence uniqueness is not obtained, or whether or not it will eventually fail no matter how small the 1-cyclic wobble is, which would mean this should provide a uniqueness condition when combined with, perhaps, analyticity or maybe even just smoothness and the usual functional equations.
What do you think?