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Additional super exponential condition
#1
I was just thinking about the following for an arbitrary super exponential :
We surely have for natural numbers m and n that

So why not demand this rule also for the super exponential extended to the reals?

For a super logarithm the rule would be:


Note that this rule is not applicable to the left-bracketed super exponentials.
Because from the rule it follows already that:
which is not valid for left bracketed super exponentials because they grow more slowly.

I didnt verify the rule yet for our known tetration extensions. Do you think it will be valid?

However I dont think that this condition suffice as a uniqueness criterion. But at least it would reduce the set of valid candidates.
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#2
bo198214 Wrote:For a super logarithm the rule would be:

This is certainly consistent. For example:




which is true.

Andrew Robbins
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#3
Ansus Wrote:By the way, I had an idea to extend hyper-operator based on the sequence of mean values:
And how? I.e. what is ?
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#4
Ansus Wrote:

Ya of course, but what is ? You said you have an idea how to extend it to real via those means.
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#5
I doubt this is an option to extend the mean value operations.
In a given set of data (say a1, a2, ...), ordering is irrelevant for calculating a mean value. But a1^a2(^a3...an) is different from a2^a1(^a3...). Well, at least most of the time.
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