03/09/2021, 10:27 PM
Just for clarification, you mean taking invertible-matrices \( A \) and producing a matrix \( e \uparrow^n A \)? Is this done using a Taylor Series approach \( f(A) \) where \( f \) is a formal power-series? Is it possible yet to produce a proof that the formal Taylor series produced by the Matrix method is convergent? I assume you are doing the same process to define a formal power-series for hyper-operations, right? I only say "formally" because I'm curious to see a proof showing the radius of convergence is one (though I have no doubt it definitely is). That sounds really fascinating though.
Forgive me for asking but what does the functional equation look like? Does it look something like this,
\(
e \uparrow^{n-1} (e \uparrow^n A) = e \uparrow^n TA\\
\)
For some matrix/operator \( T \), essentially equivalent to the transfer operator for scalars \( Tx = x+1 \)--but for matrices? Or is it something stranger? Or is the functional equation essentially nulled; and only considered as a plug in play with a flow?
Sorry for so many questions, I'm just really curious. It would be very fascinating to have a built in hyper-operations code for Wolfram, I must say. I want to know more about plugging in matrices though and their structure! Is there any other literature you have on the subject?
Forgive me for asking but what does the functional equation look like? Does it look something like this,
\(
e \uparrow^{n-1} (e \uparrow^n A) = e \uparrow^n TA\\
\)
For some matrix/operator \( T \), essentially equivalent to the transfer operator for scalars \( Tx = x+1 \)--but for matrices? Or is it something stranger? Or is the functional equation essentially nulled; and only considered as a plug in play with a flow?
Sorry for so many questions, I'm just really curious. It would be very fascinating to have a built in hyper-operations code for Wolfram, I must say. I want to know more about plugging in matrices though and their structure! Is there any other literature you have on the subject?