References

Also, please gather a preliminary list of references and add them as a final section in the main document.

Get to Work!

Charlie's post looks like a big improvement. I'll see the three of you on April 9. It would be great if a little more progress could transpire by then. For starters,

* a reasonably polished draft of the 14 section by Mike;
* a good start on the 134 material by Hannah;
* insertion of the Notation into the main document (including the new rotation notation).

Also, first drafts of the figures would be great.

Have a nice break,
D.

2 things to do / keep in mind

1. Delete unnecessary "rotated" throughout the document
2. Include periods, commas and semicolons in displays

1 or 3 triangle business

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."

Six triangle picture

Here's a possible picture to illustrate the triangles.

pic

First draft of 1 or 3.

Here's 1 or 3 so far.

pdf
tex

Master document; proof of "no backfill" for exactly 1

Here is the LaTeX and PDF for the "master document" (i.e., the document containing all of the sections). I included section titles for 14 and 174 (M) and 13 and 134 (H) as placeholders.

The proof of "no backfill" for Exactly 1 is at the top of page 4. The other new thing is that I introduce the claim of "no backfill" in the third paragraph of Section 2: Exactly 1 (page 2).

A couple of things to consider for the "no backfill" proof for Exactly 1:

• I showed it for the rotated version; should we bother to show afterward that it also holds for the unrotated version?
• In the second paragraph of the proof (the "assume holds for n and show holds for n+1" part), I may need to say a bit more. For example, I may need to show that the sites outside B_N_(n+1) cannot affect sites in B_N_(n+1) because of Lemma 2 (analogous to the statement in the previous paragraph in which I show that sites outside B_3 do not affect sites in B_3 because of the boundary of empty cells formed by Lemma 2). Or is what I have written enough?