STEM Scholars Blog

Weekly Reflection 6/17/16

Welcome to week two! This week I met with Dr. Bayless to discuss the proofs that  were to be included in the Beamer from 6/15. We started our meeting by discussing my plans going forward for this summer. When Kayleigh and I gave our first Beamer presentation on 6/8, I had mentioned to Dr. Bayless that I was interested in creating an applet that would plot graphs for assigned or randomised points. Having thought on it Dr. Bayless suggested that such an applet be done in JavaScript. However, there remains a question as to the worth of the endeavor. Dr. Bayless believes that by the end of the summer I could almost certainly create the app I want, but there is also the possibility of some combination of Dr. Bayless, Kayleigh, and myself creating a publishable paper. The question is: would the amount of time spent on an applet be worth the time sacrificed from paper writing? I think that having authorship on a paper would be much more significant than creating an applet, but having a tool to plot points for me would be very useful for when I want to experiment with different point configurations. I decided to start the Codecademy JavaScript course this week. My intention is to attempt to create the applet but if it becomes clear that writing the code will take too much time I can put it on hold. If I end up not finishing the app by the end of the summer, I will work on it at a later date so that I can use it if I decide to return to cyclic dynamical  systems.

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 x_0,...,x_n of X are “cyclically ordered'” if [x_0...x_n].

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.

Rendered by QuickLaTeX.com

In this example x_0,...,x_4 are cyclically ordered.

Fact 1: If the vertices of a periodic orbit are cyclically ordered as x_0, \dots, x_{\ell-1}, then we have that w is the winding number if and only if f_r(x_i)= x_{(i+w)\mod{\ell}} \forall i. (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 \ell and winding number w, it must be that \ell and w 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 \ell and w are relatively prime. The proof is in our Beamer.

Fact 3: If a periodic orbit of f_r has length \ell and winding number w, with r= \dfrac{p}{q} fully reduced, then \ell p -wq \geq 1.

This claim is part of a larger struggle to prove the claim “Suppose r = \dfrac{p}{q} and f_r: X \rightarrow X contains a q-swift point x. Then, f_r^q(x) is periodic and the system has a single periodic orbit whose winding number w and length \ell satisfy \ell p -wq=1.” Although we have proved claim 4 we have yet to prove that, given the original conditions from the paper, \ell p -wq =1. We believe the claim and are trying to work out a proof through the division algorithm.

 

The Beamer presentation for this week is here!

 

Works cited:

  1. Adamaszek, Michal, Henry Adams, and Francis Motta. “Random Cyclic Dynamical Systems.” Eprint ArXiv:1511.07832 (2015): n. pag. Web.

One Comment

  • Kathy stone

    Hi Brainiac. So fun to read your blog, even though I am clueless. I relish your accomplishments and curiosity. You go girl! This is my third attempt at posting…..here goes. Love you.

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