STEM Scholars Blog

Weekly Reflection 6/24/16

It’s the end of week three! Kayleigh and I had a conversation this week with Dr. Bayless about the difficulties of quoting or citing conversations with people in our blogs. The gist of what she said was that we can’t possibly remember conversations verbatim, and that could lead to offense or communication issues. Although she wasn’t offended by anything we had written, it would be very easy to post an interpretation of what she or someone else  had said that didn’t match what they meant. This sort of offense could be done without malicious intent, as everyone walks away from a conversation with their own perceptions. However, innocent intent doesn’t mean a lack of negative consequences. I also pointed out that these blogs are often written conversationally but read as factual analyses. In conversation one could “quote” another conversation that they had had by saying “my friend said…” or “my friend was like…” but there is often an implicit assumption that the speaker’s words don’t exactly match those of their friend. When writing blog posts I find myself describing conversations in this mindset. This is contrasted with the written word, which is given a higher degree of credibility and assumed to be more accurate. I’m not entirely sure what the solution to this disparity in expectations is. It seems impractical to record every conversation which I may one day wish to quote, and I will almost certainly want to use ideas that other people have voiced. On the one hand an entire lack of attribution where precise quotation is not available seems disrespectful and could border on plagiarism. On the other misquotation could lead to accidental slander on someone’s character. My solution to this, at least for now, is to use language that either implies that I am not directly quoting or that describes what I took away from a conversation rather than what was said.

This week’s Beamer was different from previous weeks in that it had a lot more of our individual research. Our combined work was mostly on Lemma 2.6. We had come to similar conclusions but Kayleigh had a more thorough proof and hers was the one that was included. So far the proof relies on the x that is q-swift being periodic, but we want to prove it for all q-swift points. I also noticed that we were using “mod” in functions in different ways, and suggested that we create some standardised notation. We decided that given a,b,c\in \mathds{Z} and a=kc+b for some k\in\mathds{Z} we use modulo as an equivalence relation with a\equiv b\bmod c (as in 5\equiv 1\bmod 2) and modulo as an operator where  a\bmod c=b  (as in 5\bmod2=1).

My work for the week, which is also in the beamer, revolves around creating a definition of q-swift points which allows for multiple orbits. I had the idea that points which imply n orbits with \ell p-wq=n exist when I noticed that figure 1, example (b) from (1) has two periodic orbits, with \ell=5, w=1, p=5, and q=23 so \ell p-wq=2. I’ve created a simpler motivating example with \ell=3, w=1, p=3, and q=7:

Rendered by QuickLaTeX.com

Note r=\frac{3}{7} and \frac{1}{3}<\frac{3}{7}<\frac{1}{2} so this orbit is valid and \ell p-wq=(3)(3)-(1)(7)=9-7=2.

I point out similarities and contrasts between this example and the definition for q-swift in the Beamer, which allowed me to arrive at a definition for nq-swift points. Although the original definition is the one included in the Beamer, I’ve since created this clearer one:

Definition: Let there be some point x\in X whose iterates are all in X'\subseteq X and let X' be cyclically ordered. For r=\frac{p}{q} and f^{q}_r(x)=x_{i} we say x is nq-swift if the following conditions are met:

  1. wn(x,f_r(x),...,f^q_r(x)) = p
  2. the open arc (x_{i+n-1},x)_{S^1} contains exactly n-1 points, all of which can’t be reached by iterating x and are periodic.

 

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.

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