05/22/2022, 12:17 AM
My apologies James.
But there are reasons why I want more.
I want to find " the " hyperoperator.
Secondly , I was
1) trying to find a way to solve your equations and I felt it was too general bringing me to
2) I wanted to have a uniqueness criterion and doubted your equation has a unique solution.
Which bring me to my questions to you :
How do you think about uniqueness ?
I think there is no uniqueness with your setting.
This might be problematic in the sense that the bundle of functions that satisfy this and do not have closed forms can be best described by the equations and nothing else ??
That is maybe way to pessimistic but I compare with differential equations in many variables with a large number of solutions , where no single solution has a closed form , it is hard to describe all solutions in terms of a given one and we describe the set of all solutions just with the differential equation itself.
***
Another thing what bothers or confuses me is this
x + y and x * y are commutative.
x^y is not.
Should operators between x + y and x*y be commutative or not ?
I guess you say not.
But going from commutative to noncomm and back to commutative and then noncomm again bothers me.
Maybe it is just me.
And maybe im too focused on superfunctions.
But that is all I know ( superfunctions ) when it comes to hyperoperators.
For me asking 2 <s> 2 = 4 seems like a superfunction interpretation too.
And without x <s> 1 = x I feel i have no starting point.
Im not sure what you want is a half-superfunction idea, I think not.
***
What happens when we simply interpolate ?
x <s> y = a * ( x <A> y) + b * ( x <B> y ) + c * ( x <C> y)
where a , b , c , A , B , C are functions of s ? preferably simple functions ?
Could that work ??
***
Finally im not even sure you still want x <3> y = x^^y now , which brings up the question ; how is this related to tetration or ackermann ? And if not , is it not suppose to ?
***
Im not trying to sound hostile to your ideas sorry.
regards
tommy1729
But there are reasons why I want more.
I want to find " the " hyperoperator.
Secondly , I was
1) trying to find a way to solve your equations and I felt it was too general bringing me to
2) I wanted to have a uniqueness criterion and doubted your equation has a unique solution.
Which bring me to my questions to you :
How do you think about uniqueness ?
I think there is no uniqueness with your setting.
This might be problematic in the sense that the bundle of functions that satisfy this and do not have closed forms can be best described by the equations and nothing else ??
That is maybe way to pessimistic but I compare with differential equations in many variables with a large number of solutions , where no single solution has a closed form , it is hard to describe all solutions in terms of a given one and we describe the set of all solutions just with the differential equation itself.
***
Another thing what bothers or confuses me is this
x + y and x * y are commutative.
x^y is not.
Should operators between x + y and x*y be commutative or not ?
I guess you say not.
But going from commutative to noncomm and back to commutative and then noncomm again bothers me.
Maybe it is just me.
And maybe im too focused on superfunctions.
But that is all I know ( superfunctions ) when it comes to hyperoperators.
For me asking 2 <s> 2 = 4 seems like a superfunction interpretation too.
And without x <s> 1 = x I feel i have no starting point.
Im not sure what you want is a half-superfunction idea, I think not.
***
What happens when we simply interpolate ?
x <s> y = a * ( x <A> y) + b * ( x <B> y ) + c * ( x <C> y)
where a , b , c , A , B , C are functions of s ? preferably simple functions ?
Could that work ??
***
Finally im not even sure you still want x <3> y = x^^y now , which brings up the question ; how is this related to tetration or ackermann ? And if not , is it not suppose to ?
***
Im not trying to sound hostile to your ideas sorry.
regards
tommy1729