All Maps Have Flows & All Hyperoperators Operate on Matrices - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: All Maps Have Flows & All Hyperoperators Operate on Matrices (/showthread.php?tid=1261) All Maps Have Flows & All Hyperoperators Operate on Matrices - Daniel - 03/14/2020 In 1987 Stephen Wolfram introduced me to the question of whether all maps are flows. Given the fifteen-year-old mathematics on Tetration.org, I have a simple proof that all maps are flows, that they are two different views of the same thing. Consider the Taylor series of an arbitrary smooth iterated function and it's representation as the combinatorial structure total partitions, the recursive version of set partitions. Each enumerated combinatorial structure has a symmetry associated with it. Let's say we want to consider $S_2$, just remove all combinatorial structures inconsistent with $S_2$. Because I can define $GL(n)$ as the domain and the iterant, through representation theory, that if I can compute with matrices, I can compute within any symmetry. Just as the exponential function of invertible matrices can be computed, all hyperoperations can be defined with invertible matrices. Daniel