Octonion Hyper-Operations
#1
Question 
How does tetration work for quaternions? What about for octonions? How do pentation and higher hyper-operations work for those?
Please remember to stay hydrated.
ฅ(ミ⚈ ﻌ ⚈ミ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
Reply
#2
We try to walk before we can run Smile

We do not even understand tetration well for octonions ( or even complex numbers lol )

And there are many tetrations and many hyper-operators.

So I assume it might be case specific.

I like the question , but it seems very very general.

Wish I could say more ... sorry.


regards

tommy1729
Reply
#3
I think most (known) methods rely on properties such as commutative associative etc.

So that might imply they all fail for most non-complex numbers.

Note that it is also problematic to define derivatives for non-complex functions. ( zero-divisors , depending on the type of infinitesimal , ... ) 
so that makes formulas that use calculus concepts * uncomfortable *. 

Functional composition is associative ... but if we plug in non-associative numbers as parameters ... feels * delicate *.

Also we might get extra fixpoints in say the quaternions and such that harrass our convergeance domains and such.
Or as we learned before ; functions do no agree on the fixpoint. ( see the bummer thread )

There are also issues with combinatorial interpretations/methods but those are complicated.

Best I hope for is that the taylor series expansions of the solutions we have can be used ;

plug in a power-associative number x instead of the complex z and just compute the values.
... It should agree with the complex solution and by extension also the field/ring/whatever of x.

hopefully.



regards

tommy1729
Reply
#4
Question 
How do they work for matrices?
Please remember to stay hydrated.
ฅ(ミ⚈ ﻌ ⚈ミ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
Reply


Possibly Related Threads…
Thread Author Replies Views Last Post
Question Base Pi Hyper-Operations Catullus 3 749 11/08/2022, 06:51 AM
Last Post: Catullus
  How could we define negative hyper operators? Shanghai46 1 130 10/23/2022, 11:12 AM
Last Post: MphLee
Question Hyper-Operational Salad Numbers Catullus 9 1,105 09/17/2022, 01:15 AM
Last Post: Catullus
Question Rank-Wise Approximations of Hyper-Operations Catullus 48 9,576 09/08/2022, 02:52 AM
Last Post: JmsNxn
  Thoughts on hyper-operations of rational but non-integer orders? VSO 4 5,284 06/30/2022, 11:41 PM
Last Post: MphLee
Question Communitive Octonion Multiplication Catullus 1 500 06/24/2022, 08:44 AM
Last Post: JmsNxn
Question Weak Hyper-Operational Etas and Euler Numbers Catullus 0 382 06/17/2022, 09:45 AM
Last Post: Catullus
  On my old fractional calculus approach to hyper-operations JmsNxn 14 7,127 07/07/2021, 07:35 AM
Last Post: JmsNxn
  hyper 0 dantheman163 2 6,363 03/09/2021, 10:28 PM
Last Post: MphLee
  On to C^\infty--and attempts at C^\infty hyper-operations JmsNxn 11 7,101 03/02/2021, 09:55 PM
Last Post: JmsNxn



Users browsing this thread: 1 Guest(s)