Here is what I have done over the past week.

1. My TA duties.
  Last week is the midterm week for my TA. So there were a lot of works
  involved. Like review lectures/Office hours, grading and solutions.

2. I've also looked possible extensions to the first two theorems.
   Now, I've abstracted the properties from the third and forth
   theorem, which we can prove to be true, and tried to apply it
   to the first two. 
   Intuitively, the first two theorems can still hold by adding
   those additionaly properties. However, the mathematical prove
   is very difficult. I have not found a good way to prove them.
   
   So I guess I have to postpone the proof for awhile and see
   if I can find a solution later. I'll start to write down the
   proof for the third and forth theorem in Latex and re-write 
   those paragraph for our submission.

   I'll spend Wed. in preparing the guest lectures.