I think I got the triangle book-keeping down in a relatively neat fashion. It required some new notation (P_I for the population in region I, P_II for region II, etc.), but I think it's pretty slick in the end.
Master document: LaTeX, PDF
Suggested change: instead of
"we examine the six triangular regions and one square (see Fig. **) bounded by"
how about we delete the "bounded by" (it's awkward because we give region 7 as a Cartesian product) and change all the ='s to >='s in the description of each region.
Also, we need to add a paragraph to this section proving "no backfill."
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