STEM Scholars Blog

Weekly Reflection 7/08/16

I was able this week to start putting my new programming knowledge to work! Dr. Bayless and I met with Matt Ruby, the Senior Network Analyst at Agnes Scott College, on Friday to discuss how I might approach the program. I gave him an overview of the function and how I would like the user interface to look. He then pointed me to some relevant coding and a few text editors that I could try. I used NetBeans at first, but switched when Matt suggested using Cloud9. As the program I’m writing should be relatively small, the faster and more easily sharable style of Cloud9 is more suitable than NetBeans. Matt also shared code with me for drawing circles with a canvas environment. I edited the circle to the right size and worked out how to draw points on it. So far the program has a circle and randomly generated points, but there are no connections between the points and the user has to type the amount of points that they want directly into the text. My goals for the code are to:

  • enter user-supplied points and/or have the user determine the number of random points;
  • draw the full mapping from the points and have the line color be determined by periodicity;
  • signify when a point is q-swift, fixed, periodic, etc. with color.

Kayleigh and I are continuing to work on our summary paper, and sent a draft to Dr. Bayless this morning. We are also beginning work on our final presentation for STEM scholars. A lot of the material is from our chalk talk last week. I think our points are a lot clearer when we’ve got well drawn computer generated images, but we’re still planning on answering any questions with demonstrations on the board. Kayleigh and I are also both independently beginning work on presentations that can be used if we get the opportunity to speak at a conference. I’m aiming at having mine be around 20 minutes. It will cover definitions, Lemmas, and Theorems Kayleigh and I created as well as relevant work from the paper.

This week Adamaszek, Adams, and Motta surprised us with a new version of their paper. We were worried that what they released would be similar to what we were doing, but luckily nothing was too close. The new edition included a proof for why a (q,i)-swift point implies \ell p-wq=1. In the proof the authors used “regular” cyclic dynamical systems, which have 1 or more orbits with equally spaced points. The isomorphism between regular systems and ones with unequal spacing then is used to imply \ell p - wq=1. I found that it was relatively easy to adapt this proof to one for nq-swift points implying \ell p-wq=n. The next step now for me is to decide if I want to keep working on nq-swift points and create a definition for (nq,i) swift points from (q,i)-swift points, or if I want to begin another short project

In the Beamer for the week I pulled multiple Lemmas out of my original proof for an nq-swift point implying x_{i+n-1} being periodic and the existence of n orbits. I think this made the proof more digestible, although I need to implement a system for numbering and referencing Lemmas within Beamers. The only major issue with my proof was an implication which was unclear. I had some issue convincing Kayleigh and Dr. Bayless that x_{i+n-1}\neq c_{j+n-1} implies x_0\neq c_0 (line 12 from Lemma 2.6 for nq-swift points). I haven’t fixed that for this week’s Beamer but it is updated in our summary paper and I will include the revision next week as well as a proof for an nq-swift point implying \ell p -wq=n.

 

The Beamer presentation for this week is here!

Works cited:

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

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