- Our next exam is two weeks away, Thursday, 11/18.
- Next assignment:
- Read Idea #26: Chaos (p. 104).
- For Tuesday, 11/9: revise your project idea (make sure you type it, or you will be losing points). Email it to me by Tuesday.
- Your questions about the voting readings are due today.
- For those who submitted project ideas: you should have received an
email concerning your project. Some general comments:
- To discuss your project with me, please hit "Reply" to my email to you. I want to keep my sanity while doing this, so I can't take stray emails that don't preserve the conversation history in them.
- Make sure that your project will be interesting to your peers! In the end you will be voting for favorites (with reasons), and your project will be closely evaluated by several of your classmates.
- You will need to go beyond Wikipedia -- find some real, solid references. Don't just grab whatever you can find on the internet. Poor references will result in a poor score....
- I hope that you've actually read some of the book, so that you have 50 models to work from. Look at the examples! Your example should look something like the author's examples.
- However, you do not want to repeat one of the other chapters: you are supposed to create the 51st great mathematical idea. Have you actually read some omre for a model?
- The project idea has been worth 2 of your 20 percentage points.
- For your "personal fractal", think of these things:
- how would you describe your 'process'?
- Can you "do it again, do it again, do it again, ...."?
- What's your personal connection? Some of you just grabbed something, perhaps off the internet....
- How did it go at the polls? Election results:
- Campbell County, by precinct (summary)
- Boone County, by precinct (summary)
- Kenton County, by precinct (summary)
- Your results are due by Tuesday of next week. Please get them earlier, so that I can begin to compile results.
Imagine an election in which 100 voters express the following preferences:
| # | 51 voters | 5 voters | 23 voters | 21 voters |
|---|---|---|---|---|
| 1st | Andrew | Catherine | Brian | David |
| 2nd | Catherine | Brian | Catherine | Catherine |
| 3rd | Brian | David | David | Brian |
| 4th | David | Andrew | Andrew | Andrew |
Three voting schemes: who wins in each case?
- Plurality voting:
the person with the most first place votes wins - "Vote for Two":
you get two votes, which are equal in weight (your top two). The person with the most votes wins. - Borda Count Method:
votes are weighted (e.g. 3 points for first, 2 points for second, 1 point for third). The person with the largest total wins. - What other voting methods do you know?
The upshots:
- the method of voting chosen may decide the election!
- Arrow says: no good democratic method for deciding between three candidates. [What if there are only two candidates?]
- Let's begin with Laplace's notion: that, given the positions and
velocities of all the particles in the universe, we should be able to
predict the future of "the system" (the universe) exactly for all
time. This is the "determinism" in "free will versus
determinism". Fortunately for free will, determinism is shown to fail.
- Chaos is roughly defined as "sensitive dependence on initial
conditions", but what this means is that if you start at two different
points in a system (for example, in weather you might be thinking of a
state of three variables, temperature, barometric pressure, and
wind speed), very close to each other but distinct, then eventually
the system will produce wildly different behaviors (even those the
starting conditions were extremely close).
- Some classic examples of chaos
- Hexstat Probability Generator -- out of chaos comes some predictability
- "The Swinger"
- the fractal diagram, showing which magnet will result if you start at a given point (each of the five magnets on the pentagon is associated with a color)
- the path of the pendulum
- Chaotic Dynamical Systems (and the laws that drive them)
- The Lorenz Equations
- A gorgeous
example of what are called "Julia Sets".
- Let's talk about the logistic map:
(our authors use this in their discussion of the Verhulst Model for Population growth). This will lead us back to fractals, as well as to the new idea of "cobwebbing". Here's a nice animation.
- Hexstat Probability Generator -- out of chaos comes some predictability
