okay okay
we seem to just have a big miscommunication.
the definition I was putting forth was
\( f^{\diamond n}(g^{\diamond n}(x)) = g^{\diamond n}(f^{\diamond n}(x)) = x \)
so that \( f^{\diamond n}(x) \) is the inverse of \( g^{\diamond n}(x) \)
\( f^{\diamond n}(x) = f^{\diamond n+1}(g^{\diamond n+1}(x) + 1) \)
which defines a sequence of super functions
therefore
\( f^{\diamond 1}(x) \) is the superfunction of \( f(x) \)
and
\( f^{\diamond 2}(x) \) is the superfunction of \( f^{\diamond 1}(x) \)
so on and so forth
this is all perfectly consistent.
Now I related that to the Ackermann function, which is self-evident, and sort of generalizes the question of operators inbetween addition and multiplication that are analytic, to functions inbetween a subfunction and a superfunction.
namely, what is \( f^{\diamond \frac{1}{2}}(x) \), which if made sense of puts us a lot closer to an analytic Ackermann function.
I wrote about the "diamond operator" which would have this relation:
\( (f^{\diamond m} \diamond f^{\diamond n})(x) = f^{\diamond m + n}(x) \)
just as a shot in the dark to try and see if we can come up with a formula that relates half-superfunctions \( f^{\diamond \frac{1}{2}}(x) \) and superfunctions \( f^{\diamond 1}(x) \)
I posed this operator not as if it would be true, but just as a "does this work?" Clearly it doesn't work, which frankly, doesn't surprise me.
The point of this thread was to try to come up with a formula that will relate half-superfunctions \( f^{\diamond \frac{1}{2}}(x) \) and superfunctions \( f^{\diamond 1}(x) \) in the same way that we have a relation between half-iterates \( f^{\circ \frac{1}{2}}(x) \) and a function itself \( f(x) \)
Until we can agree upon a proper relation between half-superfunctions and superfunctions (even if that relation is the fact that there is no relation) then we ultimately cannot solve the Ackermann function for fractional values.
I was merely pointing out that if we do not have a relation between half-superfunctions and superfunctions then it will be about 100 times harder and will give about a 100 times more solutions.
And I am well aware about fractional calculus operators. By fractional operators I was referring to operators inbetween addition and multiplication and exponentiation and tetration etc etc that form a smooth analytic transition for all arguments (save maybe a few holes or what have you).
In conclusion. I just wanted to ask the question
how can we make the transition from function to superfunction to super-superfunction etc etc analytic? And what qualifications should a half-superfunction have in terms of the function and its superfunction itself?
that's it. The fact that the diamond operator is inconsistent doesn't really matter. That was just thrown in there to try to come up with a solution. Frankly I'm a little embarrassed I didn't see the obvious fault with it, lol
I just got the feeling that you were implying \( f^{\diamond n}(x) \) is inconsistent seperate from the diamond operator; which is not true. It's very simple and nothing about it implies inconsistency, unless you think an infinite sequence of superfunctions is inconsistent.
we seem to just have a big miscommunication.
the definition I was putting forth was
\( f^{\diamond n}(g^{\diamond n}(x)) = g^{\diamond n}(f^{\diamond n}(x)) = x \)
so that \( f^{\diamond n}(x) \) is the inverse of \( g^{\diamond n}(x) \)
\( f^{\diamond n}(x) = f^{\diamond n+1}(g^{\diamond n+1}(x) + 1) \)
which defines a sequence of super functions
therefore
\( f^{\diamond 1}(x) \) is the superfunction of \( f(x) \)
and
\( f^{\diamond 2}(x) \) is the superfunction of \( f^{\diamond 1}(x) \)
so on and so forth
this is all perfectly consistent.
Now I related that to the Ackermann function, which is self-evident, and sort of generalizes the question of operators inbetween addition and multiplication that are analytic, to functions inbetween a subfunction and a superfunction.
namely, what is \( f^{\diamond \frac{1}{2}}(x) \), which if made sense of puts us a lot closer to an analytic Ackermann function.
I wrote about the "diamond operator" which would have this relation:
\( (f^{\diamond m} \diamond f^{\diamond n})(x) = f^{\diamond m + n}(x) \)
just as a shot in the dark to try and see if we can come up with a formula that relates half-superfunctions \( f^{\diamond \frac{1}{2}}(x) \) and superfunctions \( f^{\diamond 1}(x) \)
I posed this operator not as if it would be true, but just as a "does this work?" Clearly it doesn't work, which frankly, doesn't surprise me.
The point of this thread was to try to come up with a formula that will relate half-superfunctions \( f^{\diamond \frac{1}{2}}(x) \) and superfunctions \( f^{\diamond 1}(x) \) in the same way that we have a relation between half-iterates \( f^{\circ \frac{1}{2}}(x) \) and a function itself \( f(x) \)
Until we can agree upon a proper relation between half-superfunctions and superfunctions (even if that relation is the fact that there is no relation) then we ultimately cannot solve the Ackermann function for fractional values.
I was merely pointing out that if we do not have a relation between half-superfunctions and superfunctions then it will be about 100 times harder and will give about a 100 times more solutions.
And I am well aware about fractional calculus operators. By fractional operators I was referring to operators inbetween addition and multiplication and exponentiation and tetration etc etc that form a smooth analytic transition for all arguments (save maybe a few holes or what have you).
In conclusion. I just wanted to ask the question
how can we make the transition from function to superfunction to super-superfunction etc etc analytic? And what qualifications should a half-superfunction have in terms of the function and its superfunction itself?
that's it. The fact that the diamond operator is inconsistent doesn't really matter. That was just thrown in there to try to come up with a solution. Frankly I'm a little embarrassed I didn't see the obvious fault with it, lol
I just got the feeling that you were implying \( f^{\diamond n}(x) \) is inconsistent seperate from the diamond operator; which is not true. It's very simple and nothing about it implies inconsistency, unless you think an infinite sequence of superfunctions is inconsistent.