03/16/2021, 08:45 PM
Hey, everyone!
This is slightly off-topic from tetration, but still in the realm of super-functions/recurrence relations, but I've finished another paper on transfer equations. Rather than looking at first order transfer equations, I look at at \( k \)'th order transfer equations. First order transfer equations look like,
\(
y(s+1) = F(s,y(s))\\
\)
Of which I spent a lot of time solving in multiple manners. Of which the \( \phi \) function is one type of these functions. One benefit of these types of equations is that they can be used to solve more complicated types of transfer equations. Thereby, in the attached paper, I solve equations of the form,
\(
u(s+k) = F(s,u(s),u(s+1),...,u(s+k-1))\\
\)
It's a relatively short paper, and relatively simple. If anything, it may help persons grasp the benefit of infinite compositions and the Omega notation. The general method of this paper is to infinitely compose a function, then infinitely compose that function, then infinitely compose that function, etc etc... Which is sequentially approaching the solution \( u \). Which is to say, we do infinite compositions infinite times. Which sounds silly, but is pretty simple--if we just think of it as a sequence of functions approaching our solution.
Anyway, here's the paper, it's pretty short; but I think fairly enlightening. Thanks to anyone who takes the time to read it.
Regards, James
This is slightly off-topic from tetration, but still in the realm of super-functions/recurrence relations, but I've finished another paper on transfer equations. Rather than looking at first order transfer equations, I look at at \( k \)'th order transfer equations. First order transfer equations look like,
\(
y(s+1) = F(s,y(s))\\
\)
Of which I spent a lot of time solving in multiple manners. Of which the \( \phi \) function is one type of these functions. One benefit of these types of equations is that they can be used to solve more complicated types of transfer equations. Thereby, in the attached paper, I solve equations of the form,
\(
u(s+k) = F(s,u(s),u(s+1),...,u(s+k-1))\\
\)
It's a relatively short paper, and relatively simple. If anything, it may help persons grasp the benefit of infinite compositions and the Omega notation. The general method of this paper is to infinitely compose a function, then infinitely compose that function, then infinitely compose that function, etc etc... Which is sequentially approaching the solution \( u \). Which is to say, we do infinite compositions infinite times. Which sounds silly, but is pretty simple--if we just think of it as a sequence of functions approaching our solution.
Anyway, here's the paper, it's pretty short; but I think fairly enlightening. Thanks to anyone who takes the time to read it.
Regards, James