Conclusion

This summer I worked on a mathematical proof. I, working with Professor Johnson, proved that a partial TP (TN) matrix with a border pattern is TP (TN) completable. Though TP and TN are very similar properties, to show their completability require two different proofs. In TN case, we developed a Northwest principle and a (m,n) entry argument to show competability. In TP case, penultimate line insertion was used to show TP competability. The insertion is only possible with a previous theorem proved in S.M. Fallat et al. (2000).

However, our work does not end here and I will continue this research in my honor thesis. Sofar I plan to extend current border pattern to more general patterns. For example, what will happen if one side of the pattern is doubled (two columns or rows)? Will it still be TN or TP completable? Some of the questions can be easily answered with the same  methods. Some variations nevertheless require us to come up with new arguments.

My next steps include typing our findings and  think about possible extensions. This blog conclude my summer research and I am looking forward to meeting you in the showcase!