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May 14th, 2013, 11:42 AM   #1
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Notations for tetration

Okay, since we have a lot of controversies on the notation of tetration among us, let's make a general notation which will generally be used at MMF.

First, the regular superexponentiation : . Which means x^(x^(x^(x^(x..)))..) y times.

Kneser's analytic continuation for the tetration (or superexponentiation) operator : .

And at last, the superexponentiation with integer arguments, which will be mostly used in NT purposes : for integer x and y.

Any comments and suggestions OR approval from CRGreathouse who initially shown his disagreement for for obvious reasons are welcome!

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May 14th, 2013, 12:10 PM   #2
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Re: Notations for tetration

I think the most common notations are x ^^ y (sometimes written more nicely as or using indexed notation as ) and (I do think the latter is unfortunate.)

I don't think Kneser's continuation is common at all. If I were writing a summary on tetration I might not have even mentioned it, except as a historical note. It's been largely surpassed by the work of Galidakis and others, or at least that was my impression the last time I seriously looked into the subject.

I've never seen either "sexp" or "ksexp" used by anyone other than Balarka.
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May 14th, 2013, 06:15 PM   #3
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Re: Notations for tetration

Oh, now I finally know what the topic was in that thread where CRG first criticized the subscript notation!

Hang around long enough and who knows what all you can learn around here!
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May 14th, 2013, 08:49 PM   #4
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Re: Notations for tetration

Quote:
Originally Posted by CRGreathouse
I don't think Kneser's continuation is common at all. If I were writing a summary on tetration I might not have even mentioned it, except as a historical note. It's been largely surpassed by the work of Galidakis and others, or at least that was my impression the last time I seriously looked into the subject.
Interesting enough, this is not the case as I see it. The researchers on this subject usually choses the recursion condition which Galidakis's works doesn't. There are currently four methods, Kneser's, Robbin's, The matrix method (probably by Gottfried) and the asymptotic sinh method which is not even known to be analytic. All of them results the same, just the difference that Kneser's method converges on a large domain.

Quote:
Originally Posted by CRGreathouse
I've never seen either "sexp" or "ksexp" used by anyone other than Balarka.
Everyone uses it on Tetration forum.

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May 14th, 2013, 10:07 PM   #5
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Re: Notations for tetration

Quote:
Originally Posted by mathbalarka
Quote:
Originally Posted by CRGreathouse
I've never seen either "sexp" or "ksexp" used by anyone other than Balarka.
Everyone uses it on Tetration forum.
I don't post there, clearly. I've gathered my own research on the topic over the years.
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May 14th, 2013, 11:59 PM   #6
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Re: Notations for tetration

Well, as a matter of fact, the Kneser's solution at least makes sense in the sense that at least 3 of the conditions gets satisfies by the method whereas Ioannis's method does't seem to be the correct continuation for superexponentiation at all since we can't even satisfy the definition of tetration. And even if we force it to satisfy the definition, the continuity requirement gets violated. Either way, Kneser's is far better than Galidakis's method which is analytic, continues and somewhat unique(!).

Although there are many positive results on applying Kneser, one also gets some difficulties like branch point at cheta and all the negative integers. We cannot tell whether Kneser is the *right* continuation since we still don't know whether similar arguments can be use to extend tetration to complex bases although we can use it to work for a few bases on Im(z)=1.

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May 15th, 2013, 04:54 AM   #7
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Re: Notations for tetration

Quote:
Originally Posted by mathbalarka
Well, as a matter of fact, the Kneser's solution at least makes sense in the sense that at least 3 of the conditions gets satisfies by the method whereas Ioannis's method does't seem to be the correct continuation for superexponentiation at all since we can't even satisfy the definition of tetration. And even if we force it to satisfy the definition, the continuity requirement gets violated. Either way, Kneser's is far better than Galidakis's method which is analytic, continues and somewhat unique(!).
Which method are you talking about? I've only read one of Galidakis' papers on tetration but it had two methods, and he may have others beside.
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May 15th, 2013, 05:12 AM   #8
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Re: Notations for tetration

Quote:
Originally Posted by CRGreathouse
Which method are you talking about?
The methods given in "On Extending hyper4 and Knuth's Up-arrow Notation to the Reals". Neither of them satisfies the recursion property.

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May 15th, 2013, 06:33 AM   #9
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Re: Notations for tetration

I can't find a copy online. I can check to see if I have hardcopy at home.
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May 15th, 2013, 06:57 AM   #10
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Re: Notations for tetration

Quote:
Originally Posted by CRGreathouse
I can't find a copy online.
It probably isn't available online. None of the Galidakis's papers are.
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