12/06/2022, 12:41 AM
I have been talking alot lately about the semi-group iso.
However the problem is for tetration that there is no real solution.
Consider the fixpoint expansion at the primary fixpoint or equivalantly the kneser method BEFORE the riemann mapping.
This solution HAS THE CORRECT 1-periodic mapping satisfying the semi-group iso property.
And by uniqueness of the 1-periodic mapping this means it is the ONLY solution that has the semi-group iso property.
But it does not map any large real interval to the reals.
It maps the reals to complex numbers almost everywhere.
So that is not the type of tetration we want.
Im not sure how it works with other fixpoints but those also do not map to the reals.
This also implies that any method for REAL tetration can not have the semi-group iso property.
NOT EVEN the carleman matrices , at least not converging to anything meaningful ( like a taylor with nonzero radius ).
I was away for a while because I did not know how to handle it.
It is a big bummer and makes one wonder what it means to do continu iterations !
This problem does not only occur for tetration but for a very large class of problematic functions !!
---
IT seems the semi-group iso only works in 2 non-trivial cases ;
1)
f(z) has a fixpoint.
or
2)
f(z) is of the form g( k + inverse_g(z) ).
For instance f(z) = ln( k + exp(z) ) for k a real above or equal to 1.
---
Another thing is that this has also SEVERE implications to the whole ackermann .. hyperoperator .. pentation .. etc type topics.
So stuff like 3 <s> ( 3 <s+1> y ) = 3 <s+1> (y+1) and similar equation are also hard to consistantly define.
---
So i considered going to older ideas again.
somewhat against my will.
sigh.
---
So it was back to square 1 for formal uniqueness conditions that made sense and not completely arbitrary but rather logical.
So i started to focus again on the derivatives of the semi-exp and the extensions.
I will thus probably delve into old ideas but I feel a bit forced.
---
lets focus on the reals now.
LET f(x) be the semi-exp.
We want for all real x :
f '(x) > f '' (x) > f ''' (x) > 0.
This might not be completely a uniqueness critertion but puts strong conditions and all solutions visually look similar IF they agree on specific values
( f ( -oo ) in particular )
Another intresting idea is the idea of induction : if this holds on a given interval it holds everywhere.
For instance something such as : if f '(x) > f '' (x) > 0 for x < 0 then actually it holds for all real x.
proofs by inductions and infinite descent for partitions of the reals into intervals seems logical.
playing around with that seems fun.
---
perhaps more later
---
sorry for my absense , i had to review alot of things.
regards
tommy1729
However the problem is for tetration that there is no real solution.
Consider the fixpoint expansion at the primary fixpoint or equivalantly the kneser method BEFORE the riemann mapping.
This solution HAS THE CORRECT 1-periodic mapping satisfying the semi-group iso property.
And by uniqueness of the 1-periodic mapping this means it is the ONLY solution that has the semi-group iso property.
But it does not map any large real interval to the reals.
It maps the reals to complex numbers almost everywhere.
So that is not the type of tetration we want.
Im not sure how it works with other fixpoints but those also do not map to the reals.
This also implies that any method for REAL tetration can not have the semi-group iso property.
NOT EVEN the carleman matrices , at least not converging to anything meaningful ( like a taylor with nonzero radius ).
I was away for a while because I did not know how to handle it.
It is a big bummer and makes one wonder what it means to do continu iterations !
This problem does not only occur for tetration but for a very large class of problematic functions !!
---
IT seems the semi-group iso only works in 2 non-trivial cases ;
1)
f(z) has a fixpoint.
or
2)
f(z) is of the form g( k + inverse_g(z) ).
For instance f(z) = ln( k + exp(z) ) for k a real above or equal to 1.
---
Another thing is that this has also SEVERE implications to the whole ackermann .. hyperoperator .. pentation .. etc type topics.
So stuff like 3 <s> ( 3 <s+1> y ) = 3 <s+1> (y+1) and similar equation are also hard to consistantly define.
---
So i considered going to older ideas again.
somewhat against my will.
sigh.
---
So it was back to square 1 for formal uniqueness conditions that made sense and not completely arbitrary but rather logical.
So i started to focus again on the derivatives of the semi-exp and the extensions.
I will thus probably delve into old ideas but I feel a bit forced.
---
lets focus on the reals now.
LET f(x) be the semi-exp.
We want for all real x :
f '(x) > f '' (x) > f ''' (x) > 0.
This might not be completely a uniqueness critertion but puts strong conditions and all solutions visually look similar IF they agree on specific values
( f ( -oo ) in particular )
Another intresting idea is the idea of induction : if this holds on a given interval it holds everywhere.
For instance something such as : if f '(x) > f '' (x) > 0 for x < 0 then actually it holds for all real x.
proofs by inductions and infinite descent for partitions of the reals into intervals seems logical.
playing around with that seems fun.
---
perhaps more later
---
sorry for my absense , i had to review alot of things.
regards
tommy1729