10/09/2009, 08:38 PM
This is only a sketch, yet; I didn't compute values. Also there is one conceptional problem.
Consider the binary operations add(x,b) = x+b , mult(x,b)=x*b , pow(x,b) = x^b (the version with left associativity) and their iterable notations add(x,b,h)=x+b*h, mult(x,b,h) = x*b^h , pow(x,b,h) = x^b^h
It may look overartificial to find the schröder-functions for fractional iterates of add(), mult() and pow(), since it is trivial to insert fractional heights h into the formulae.
But for more intense study let's try to express this with the use of schröder-functions.
Always we have the general form of composition
g ° b^h ° f
or
g( b^h * f(x))
where f is the appropriate schröder-function and g its inverse.
add(x,b,h) can be expressed as
logb ( b^(b*h) * (b^x))
where the schröder-function is b^x and its inverse the logb (=log to base b)
Explicitely we have by this
logb( b^(b*h) * (b^x)) = logb( b^(x+b*h)) = x + b*h
Now we look at the mult-function. The schröder-function is simply the id-function
id(b^h*id(x)) = x*b^h
The pow-function has the logb-function as schröder-function:
b^( b^h * logb(x)) = b^(logb( x^b^h)) = x^(b^h)
Now we see, that the schröder-functions for the three basic operations are the iterates of the exp-function:
add() --> exp_b°1(x) = b^x
mul() --> exp_b°0(x) = id(x)
pow() --> exp_b°-1(x) = logb(x)
where the iteration-heights of the exp-function may serve as integer index with an unlucky offset. But let's ignore that unusual offset.
What we immediately see is, that we could study which sense it makes to define operations fractionally indexed between add() and mul() based on fractional iterates of the exp-/log-function.
However, for one inconsistency I didn't find a smooth workaround yet: for add() the cofactor to the schröder-function is b^(b*h), while for mult() and pow() it is b^h only.
I also have not yet numerical examples, and think I'll need some more time for this (I'm short, courses begin next week), but thought, perhaps someone other could already look at this.
What do you think?
Consider the binary operations add(x,b) = x+b , mult(x,b)=x*b , pow(x,b) = x^b (the version with left associativity) and their iterable notations add(x,b,h)=x+b*h, mult(x,b,h) = x*b^h , pow(x,b,h) = x^b^h
It may look overartificial to find the schröder-functions for fractional iterates of add(), mult() and pow(), since it is trivial to insert fractional heights h into the formulae.
But for more intense study let's try to express this with the use of schröder-functions.
Always we have the general form of composition
g ° b^h ° f
or
g( b^h * f(x))
where f is the appropriate schröder-function and g its inverse.
add(x,b,h) can be expressed as
logb ( b^(b*h) * (b^x))
where the schröder-function is b^x and its inverse the logb (=log to base b)
Explicitely we have by this
logb( b^(b*h) * (b^x)) = logb( b^(x+b*h)) = x + b*h
Now we look at the mult-function. The schröder-function is simply the id-function
id(b^h*id(x)) = x*b^h
The pow-function has the logb-function as schröder-function:
b^( b^h * logb(x)) = b^(logb( x^b^h)) = x^(b^h)
Now we see, that the schröder-functions for the three basic operations are the iterates of the exp-function:
add() --> exp_b°1(x) = b^x
mul() --> exp_b°0(x) = id(x)
pow() --> exp_b°-1(x) = logb(x)
where the iteration-heights of the exp-function may serve as integer index with an unlucky offset. But let's ignore that unusual offset.
What we immediately see is, that we could study which sense it makes to define operations fractionally indexed between add() and mul() based on fractional iterates of the exp-/log-function.
However, for one inconsistency I didn't find a smooth workaround yet: for add() the cofactor to the schröder-function is b^(b*h), while for mult() and pow() it is b^h only.
I also have not yet numerical examples, and think I'll need some more time for this (I'm short, courses begin next week), but thought, perhaps someone other could already look at this.
What do you think?
Gottfried Helms, Kassel
