Poll: We need a new subsection for Hyperoperations and related studies.
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YES, let's add a new subsection.
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[Poll] Add a new subsection to the forum
#1
I realized that I posted this in the wrong place so here's the fix. This can be seen as a followup of this post.


Since 2007 lot of new threads and ideas appeared on the forum.
I still think that the forum categorization should be minimal: the General Discussion/Computation partition should be conserved. It is the division between theoretic arguments and real implementation.

Now, in 2022, I believe that the General discussion section (theory) should have at least one subsection. I feel like a post here should be by default about tetration or iteration: so no need for a subsection specific for Tetration or Iteration theory. But a subsection for generalized hyperoperations can be useful, I think.
Here some example of the thread that should be moved to a new "Generalized Hyperoperations" subsection
  • All the threads about Zeration, and sub-addition operators;
  • Everything about Knuth's , Conway's, Nambiar's, Reihenalgebra... and other notations including notations for large numbers that use ranks bigger than 4;
  • All the threads about balanced and Bennet/commutative hyperoperations;
  • All the threads discussing Goodstein and Ackermann function;
  • Everything about Trapmann's Arborescent Numbers;
  • Extension of Ackermann or of Goodstein Hyperoperations to non-integer ranks (mainly Jmsn's);
  • All the discussions about tetration and higher ops. over transifinite ordinals/cardinals;
  • All the Tommy's threads about various exotic functional equations for alternative H.Os. sequences;
  • Everything about iteration of iteration, and higher form of iteration, kind of stuff;

What should NOT end up there:
  • Integer tetration, integer pentation;
  • Discussion about alternative to tetration if the do not contain explicit discussion of extending to higher ranks;
  • The study of the dynamics and analytic properties of exponentiation, and tetration functions;
  • The problem about half, rational and real/complex iteration;
  • Fractals;
  • General theory of superfunctions (Kouznetsov's);
  • Operators iteration, matrix methods and fractional calculus (JmsNxn's when not applied directly to HOs, Woon's stuff, Gotfried's matrices stuff);

To summarize: everything that talks about ranks, and has variables concerning ranks (going up or down) should be in this section.


I hope this will make easier for tetrations-only folks to browse threads, and makes ranks-lovers, like me, to find everything they need in one place.
No need to add that a section like the one will be the richest thematic resource on the whole internet.
If this idea will be accepted I can help sorting the threads.

MathStackExchange account:MphLee

Fundamental Law
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#2
Sounds reasonable to me, any suggestions for the name?
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#3
Generalized Iterations... maybe?

I feel it's a little redundant to used Generalized again, but I can't think of a better name. Maybe Greater Iteration Theory? As Tetration would fall under a subset of all the topics Mphlee described for the new subsection. Hyper-operational iteration theory...? Lol, that one's too much of a mouthful.
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#4
Maybe we could just use one of these:
  • General theory of Hyperoperations
  • Hyperoperations and related studies
  • Hyperoperations and Ackermann-like functions

I feel the first is too rigid and demanding, the third too limiting and the second to be the perfect balance between informal/informative/broad but precise..

ps: I like greater iteration theory a lot ...but the problem is... there is not such a theory yet... if not inside my mind and of few others: I'm much more near having a general theory of hyperoperation then having an even vague understanding of how greater or higher iteration theory can be defined.

pps: for the description I'd go for

Hyperoperations and related studies
"Discuss (the mathematics of) hyperoperations families, how to extend and generalize them."

Where with hyperoperations I'd stick with the Robbins' general definition: I used that to write the MathStackExchange tag for the topic, and it resisted 8 years without being modified, so I guess it's a good one

Hyperoperation is a field of mathematics which studies indexed families of binary operations, Hyperoperations families, that generalize and extend the standard sequence of the basic arithmetic operations of addition, multiplication and exponentiation.

pps: feel free to fix the English, if needed.

MathStackExchange account:MphLee

Fundamental Law
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#5
Lol, my mom's an editor, my mom reads and criticizes all the time, and I'm being my mom right now; where she'd say "you're using 'general' too many times." I apologize. Just, we shouldn't use the word generalized, in a section called general discussion.

Greater Studies of Iteration?

Where we assume this subsection \(A \supset B\), where \(B\) is the specification of tetration/iterated exponentials, and more pentation, etc..

Then \(A\), is the greater study of iteration. I don't mean to say great as in alexander the great. I mean, a larger set of questions. The greater study of iteration.
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#6
Idk. If we go the way you suggest, I fear we should re-categorize all the structure of "general discussions".
The proposed subdivision was not meant to be the real logical subdivision but the organization suggested by the users activity. It seems to me that most viewers visit the threads on tetration, iterated exponentials and maybe classical iteration so I wanted to clean that from hyperoperations-related stuff. That would help the average user, and help me too and everyone focused on hyperoperations.

ps: I hope you'll never let your mother read my posts xd ahahah, jk.
pps: I guess that she's a good influence on your style. I've read many, many math papers and yours are among the most warm, vibrant and friendly ones.

MathStackExchange account:MphLee

Fundamental Law
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#7
(05/07/2022, 01:20 PM)MphLee Wrote: Idk. If we go the way you suggest, I fear we should re-categorize all the structure of "general discussions".
The proposed subdivision was not meant to be the real logical subdivision but the organization suggested by the users activity. It seems to me that most viewers visit the threads on tetration, iterated exponentials and maybe classical iteration so I wanted to clean that from hyperoperations-related stuff. That would help the average user, and help me too and everyone focused on hyperoperations.

ps: I hope you'll never let your mother read my posts xd ahahah, jk.
pps: I guess that she's a good influence on your style. I've read many, many math papers and yours are among the most warm, vibrant and friendly ones.

For sure, I see your point!

Generalized Iteration Theory is great, no need to worry.

ps: Thanks, love my mom, she doesn't edit my work anymore (that was more when I would write fiction/essays). But she's a cold editor tbh--zero warmth. She would read my stuff and be a harsh harsh critic, even when I was 12 years old. I remember I made a card for her, but I wrote 'greatful' instead of 'grateful', and that's all she took away from it, lmaoo. "You spelt grateful wrong..."
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#8
Again, idk If I'm gettng your point... I do not think generalized iteration is good for the section I have in mind, I don't think I have proposed it.
For me, generalized iteration is about widening the field of application of real complex dynamics of functions beyond the realm of real/complex times. I'd say also omega notation and iteration integral is part of that. Instead ranks are not directly about iterations, more like "iteration of recursive constructions", a kind of meta-iteration height: and all the hyperoperations and related studies have their focus on isolating and iterating these recursive constructions.

My proposal is this

Quote:Hyperoperations and related studies
"Discuss (the mathematics of) hyperoperations families, how to extend and generalize them."

I'm not sure to agree when you say

Quote:As Tetration would fall under a subset of all the topics Mphlee described for the new subsection.

I do not think tetration falls under this. Probably Hyperoperations fall under what we could call "iteration of iteration" and variants of.. So maybe... meta-iteration theory? But to be honest I'd like to avoid technical terms to name subsection.

MathStackExchange account:MphLee

Fundamental Law
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