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exponential distributivity
#1
quickfur Wrote:I forgot to note, what I had in mind here is and .
Oh right, misinterpreted that.
Another idea would then be to investigate the distribution law for exponentiation, i.e. whether there is an operation # such that
(a^b)#c=(a#c)^(b#c).

As a side node I give a proof that there are only trivial operations # satisfying this property. Note however that on the just discussed hierarchy of operations {n}, there is always distributivity from {n+1} over {n}, but exponentiation does not belong to this hierarchy.

Now to the proof, if there would be such an operation then we can consider the function f(x)=x#c (fixed c) which satisfies f(a^b)=f(a)^f(b). Then:
f(a^(b^n))=f((...(a^b)^b...)^b)=f(a)^(f(b)^n) but also
f(a^(b^n))=f(a)^f(b^n)=f(a)^(f(b)^f(n)) hence

f(a)^(f(b)^f(n))=f(a)^(f(b)^n) then for f(a)!=1, f(a)>0
f(b)^f(n)=f(b)^n and again for f(b)!=1, f(b)>0
f(n)=n

f(a)^m=f(a^m)=f((a^(m/n))^n)=(f(a)^f(m/n))^n=f(a)^(f(m/n)n), then for f(a)!=1, f(a)>0:
m=f(m/n)n
f(m/n)=m/n

If we now suppose that f is continuos we extend this law from the fractional to the positive real numbers:
f(x)=x for all positive real x.

This was however under the assumption that there is an with . So the other possibility is:
f(x)=1 for all positive real x.

Summarizing
Proposition: Any operation # defined on the positive real numbers such that (a^b)#c=(a#c)^(b#c) (for each a,b,c) and such that f(x)=x#c is continuous (for each c), is either given by x#c=x or by x#c=1 for all x,c.
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#2
bo198214 Wrote:[...]
Another idea would then be to investigate the distribution law for exponentiation, i.e. whether there is an operation # such that
(a^b)#c=(a#c)^(b#c).

As a side node I give a proof that there are only trivial operations # satisfying this property. [...]
Now to the proof, if there would be such an operation then we can consider the function f(x)=x#c (fixed c) which satisfies f(a^b)=f(a)^f(b).
[...]
Summarizing
Proposition: Any operation # defined on the positive real numbers such that (a^b)#c=(a#c)^(b#c) (for each a,b,c) and such that f(x)=x#c is continuous (for each c), is either given by x#c=x or by x#c=1 for all x,c.
Hmm, interesting. I was going to ask what happens if we consider f(x)=c#x instead, but it doesn't change the conclusion because we assume the same operator #. So we established that a higher operation cannot distribute over exponentiation without becoming trivial.

I wonder what is the root cause of this... maybe it has something to do with the non-associativity of exponentiation?
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#3
quickfur Wrote:Hmm, interesting. I was going to ask what happens if we consider f(x)=c#x instead, but it doesn't change the conclusion because we assume the same operator #. So we established that a higher operation cannot distribute over exponentiation without becoming trivial.

I wonder what is the root cause of this... maybe it has something to do with the non-associativity of exponentiation?

Who knows, but the proof can be similarly done for other distributivity type functional equations, for example
with the only continuous solutions or
with the only continuous solutions .
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#4
That's interesting, because analysing f(x+y) = f(x)f(y) leads to the exponential function . Unfortunately, we don't seem to have such a handy property involving tetration that we can use, except for , which isn't very helpful for non-integer x.

Now that I think of it, I think the root of the problem is that exponentiation is non-associative, so there's no easy way to algebraically "access the top of the exponential tower", so to speak. In order to derive any useful relations, we need to have some way of "reaching the top of the tower". For example, given b[4]n, if we can somehow reach the top of the tower and add another tower of height m, then we can state the property that J(b[4]n, b[4]m) = b[4](n+m), where the hypothetical J operator attaches the second tower to the top of the first tower.

But I doubt that J is expressible as an algebraic operation (I'm not even sure if it can be consistently defined if b is not fixed!). And so even if we could make such a statement, it wouldn't be of the same utility as w.r.t. exponentiation.
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#5
(02/23/2008, 12:29 AM)quickfur Wrote: That's interesting, because analysing f(x+y) = f(x)f(y) leads to the exponential function . Unfortunately, we don't seem to have such a handy property involving tetration that we can use, except for , which isn't very helpful for non-integer x.

Now that I think of it, I think the root of the problem is that exponentiation is non-associative, so there's no easy way to algebraically "access the top of the exponential tower", so to speak. In order to derive any useful relations, we need to have some way of "reaching the top of the tower". For example, given b[4]n, if we can somehow reach the top of the tower and add another tower of height m, then we can state the property that J(b[4]n, b[4]m) = b[4](n+m), where the hypothetical J operator attaches the second tower to the top of the first tower.

But I doubt that J is expressible as an algebraic operation (I'm not even sure if it can be consistently defined if b is not fixed!). And so even if we could make such a statement, it wouldn't be of the same utility as w.r.t. exponentiation.

I'd like to respectfully disagree.

consider:




it's very easy to see that:


I think even, this operator may give some short cuts for algebra involving tetration.

I did a little research into this under the following thread:
http://math.eretrandre.org/tetrationforu...hp?tid=699


The only rule is if we create:





whereas when we create the same operator with exponentiation instead of tetration, the operators are equivalent and both are multiplication.
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