- We changed A_n to A_t to free up n so that we can abbreviate N = (2^n)-1.
- We clarified that B_N, D_N, and Q_N are regions of Z^2 and not occupied sets.
- A_N = tau^N A_0 is also equal to A_infinity intersect D_N (since there is no back fill)
- The set of occupied sites in the first quadrant is now A_N^1 (rather than Q_N as we had before)
- We also have some observations about how applying T conserves population and inserts a checkerboard of blank cells
***** There is a slight problem with our notation for A_N^1. The definition of A_N^1 includes sites on the axes. This is okay for the rotated version of Exactly 1 because there are no occupied cells on the axes. However, the axes in the regular orientation are occupied. This means that it is not the case that A_N^1 = (TA_N) intersect Q_N.
The reason I am encountering this problem is that when I list some initial data for a_N and a_N^1 at the beginning of the Exactly 1 writeup, the data for a_N^1 on the regular neighborhood is not quite accurate because the sites on the axes are double counted. In other words, it is not the case that a_N = 4*a_N^1 -3, like it is for the rotated scheme.
An easy fix would be to simply not give initial data for a_N^1 at the beginning of the Exactly 1 writeup, but instead just initial data for a_N (since this is indepedent of whether we are in the rotated or regular neighborhood).
2 comments:
2 quick comments, Charlie.
Instead of "since T does not ...", write "since T is one-to-one ..."
We really want to write a_n instead of a_N, so the sequence of initial data is a_1, a_2, ... instead of a_1, a_3, a_7, ...
I think this is good notation once we get used to it ...
Okay I fixed those two.
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