On Monday the STEM scholars had our second professional development meeting with Dr. Embree. We spoke this time about possible careers in science and how to find the best one. After a brief discussion we all explored the myIDP website which includes quizzes on experience and interests to help match students to career paths. Although I found the suggestions helpful, I noticed that the quizzes were more asking how much math the student would be willing to do, and was less geared to students that wanted to go into math. I decided to look into other career tests that night. I found that there were plenty of options for people that liked math — perhaps too many. One thing I’m sure about in my interests is that I would like a job that involves math, is secure, and lets me travel. Beyond that I am open to almost any career path. I know that I don’t have to have it all figured out now, especially if I want to go to grad school, but I would like the security of having a more detailed plan for myself.
This week we spent most of our time fixing claims from last week and considering our own research questions. The claims we considered were:
Lemma 1: For any periodic orbit, the number of points that are skipped between two consecutive iterates is equal in any iteration.
Lemma 1 is used to prove our claims for modulo ordering. Although we based the lemma off of ideas from the paper, this claim was our own. The proof relies on the preservation of strong cyclic ordering, as opposed to weak, and is included in our Beamer.
Definition 1: We say that the points of are “cyclically ordered'” if .
In our beamer for the week we reiterated definitions for skip, pass, and hit and included this new definition and example (below) for cyclic ordering.
In this example are cyclically ordered.
Fact 1: If the vertices of a periodic orbit are cyclically ordered as , then we have that is the winding number if and only if . (1)
Note that this fact for modulo ordering does not match the one from last week, which is from (1). We found that the claim was clearer and stronger as it includes an “if and only if” statement, and we proved our claim in our Beamer.
Fact 2: For a periodic orbit of length and winding number , it must be that and are relatively prime. (1)
We decided to switch the ordering for facts 1 and 2 from what was included in the paper so that we could use modulo ordering to prove that and are relatively prime. The proof is in our Beamer.
Fact 3: If a periodic orbit of has length and winding number , with fully reduced, then .
This claim is part of a larger struggle to prove the claim “Suppose and contains a -swift point . Then, is periodic and the system has a single periodic orbit whose winding number and length satisfy .” Although we have proved claim 4 we have yet to prove that, given the original conditions from the paper, . We believe the claim and are trying to work out a proof through the division algorithm.
The Beamer presentation for this week is here!
Adamaszek, Michal, Henry Adams, and Francis Motta. “Random Cyclic Dynamical Systems.” Eprint ArXiv:1511.07832 (2015): n. pag. Web.