Arbitrary Order Transfer Equations JmsNxn Ultimate Fellow Posts: 1,176 Threads: 123 Joined: Dec 2010 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 Attached Files   Arbitrary_Difference_Equations.pdf (Size: 249.02 KB / Downloads: 192) « Next Oldest | Next Newest »

 Possibly Related Threads… Thread Author Replies Views Last Post New Quantum Algorithms (Carleman linearization) Finally Crack Nonlinear Equations Daniel 2 2,298 01/10/2021, 12:33 AM Last Post: marraco Moving between Abel's and Schroeder's Functional Equations Daniel 1 4,060 01/16/2020, 10:08 PM Last Post: sheldonison Taylor polynomial. System of equations for the coefficients. marraco 17 33,985 08/23/2016, 11:25 AM Last Post: Gottfried Totient equations tommy1729 0 3,846 05/08/2015, 11:20 PM Last Post: tommy1729 Bundle equations for bases > 2 tommy1729 0 3,976 04/18/2015, 12:24 PM Last Post: tommy1729 A system of functional equations for slog(x) ? tommy1729 3 9,489 07/28/2014, 09:16 PM Last Post: tommy1729 partial invariant equations ? tommy1729 0 3,659 03/16/2013, 12:32 AM Last Post: tommy1729 tetration base conversion, and sexp/slog limit equations sheldonison 44 107,440 02/27/2013, 07:05 PM Last Post: sheldonison Superfunctions in continu sum equations tommy1729 0 4,212 01/03/2013, 12:02 AM Last Post: tommy1729 Equations for Kneser sexp algorithm sheldonison 18 50,487 06/21/2011, 01:48 AM Last Post: sheldonison

Users browsing this thread: 1 Guest(s)