Wops, Tommy, you unintentionally did a double-post with the same content.
This way to frame it is really interesting Tommy, it's the same as Trapmann's 2008 proposal (see link) to define ranks as the iteration height of the "superfunction operator" on analytic functions (later reworked by James and Tommy aswell).
This seems very fruitful, and I'm investigating this approach since then. The problem is... it is not technically an operator in the strict sense imho, we lack the right linear structure. So it nonlinear. We need exotic and advanced methods to treat non-linear operators, and I'm ignorant af. James did some research on this under his differ-integral program. Basically he tried to conjugate the superfunction to the differential operator in order to turn fractional differentiation to fractional iteration of the superfunction operator.
So here James is trying to define an good interpolation for \(s\) between \(0\) and \(2\) and then do a piecewise extension, in order to avoid the iteration of non-linear operators.
Regards.
Quote:[...] by doing that we set going from x <s> y to x <s+1> y as a superfunction operator.
This gives us an opportunity to get analytic hyperoperators.
[...]
Now we only need to understand x < s > y for s between 0 and 1 but analytic at 0 and 1.
This way to frame it is really interesting Tommy, it's the same as Trapmann's 2008 proposal (see link) to define ranks as the iteration height of the "superfunction operator" on analytic functions (later reworked by James and Tommy aswell).
This seems very fruitful, and I'm investigating this approach since then. The problem is... it is not technically an operator in the strict sense imho, we lack the right linear structure. So it nonlinear. We need exotic and advanced methods to treat non-linear operators, and I'm ignorant af. James did some research on this under his differ-integral program. Basically he tried to conjugate the superfunction to the differential operator in order to turn fractional differentiation to fractional iteration of the superfunction operator.
So here James is trying to define an good interpolation for \(s\) between \(0\) and \(2\) and then do a piecewise extension, in order to avoid the iteration of non-linear operators.
Regards.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)