deletions | additions       diff --git a/main.tex b/main.tex  index 89e5e41..7420b09 100644  --- a/main.tex  +++ b/main.tex 
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\end{equation}  Which rearranges to $m=2(r-1)/\beta$, which is the form of the first of the equations! equations (with $1$ relaced by $a_0$.  So it is justified. Long numbers here are based on the bifurcation points $a_n$ and will be replaced with appropriate symbols soon. We have \begin{equation}  \begin{array}{|c|c|}