Day 38, MAT115

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Announcements:
- Your homework:
  1. Draw all simple graph on four distinguished, or labelled, vertices. Imagine that each one is a different person on Facebook.
    solution
  2. Draw your family tree (as a directed graph) with one of your grand parents as the root of the tree (include only those actually descended from that grandparent). Label each vertex (that is, each person!).
    - Why do your "directed graphs" have no arrows?
    - What does an arrow mean?
    - I was surprised that many of you don't seem to know what a "family tree" is! First of all, as a graph, it's a tree -- there shouldn't be any cycles.
  3. Identify the knots (or links?) in this "story", which I call A Knotty Tale. You might use a piece of string to help you decide. Just put a name next to each one.
    Solution
    - You can't say "torus" -- that's a family
    - You can't say "twist" -- that's a family
    - You can't say "five-knot" -- there are two different five knots (3-twist and cinquefoil)
    - You can't say "link" -- that's a family -- you should know Solomon's and Borromean rings (the other is Hopf)
    - There's no difference between "trefoil twist" and "trefoil torus" (except in how it appears). You can interchange them -- make each look like the other.
    - If you can only see 3 crossings (e.g. "#5"), then it can't be a 4 knot (the figure eight). It can be a link, a trefoil, or the unknot -- that's it.
  4. You have a final assignment, due Wednesday.
  5. Logo day is coming up: Wednesday, 4/29. You will submit your logo and logo paper to me at the time of class. You may not hand that in late, so get cracking!
  6. Reading for next time: Commuting (p. 23)
Question of the Day:

Loves me, loves me not: the dynamics of love and life
Last time we talked about The Loneliest Numbers: primes
- One is the loneliest number (one is not prime, but can't be expressed as a product of primes in a unique way -- rats!).
- The sieve of Erotasthenes -- finding primes
- Twin primes: A love poem for lonely prime numbers
- The prime factorization theorem: every natural number (other than 1) is either prime, or can be expressed as a product of primes in a unique way.
- In a way most series get lonelier and lonelier as the get farther from zero. Think of these:
  - Power of 2
  - Fibonacci numbers
  - So why not the primes? But it's that they're getting lonelier in a kind of regular way that was so interesting... to Gauss, anyway!:)
Today we talk about dynamics: those of love, hate, the weather, etc.
- Here's what one college in a university is proposing as a budget strategy:
  - Departments within the college will be rewarded with more money for passing more students.
  - Departments within the college will be rewarded with more money for retaining more students.
  The administration suggests that this will lead to more student success (in fact, I'm sure that they would tell you that it will lead to "more success all around!").
  You tell me what you think will happen (that is, how will the system -- departments, college, students -- change through time -- this is a dynamical system, changing dynamically in time).
- Let's now turn to the situation that Strogatz describes, of his "Romeo and Juliet" situation from college. I'll share another way to think about that, in what I call "Ardor space".
- Let's look at a more realistic situation: what's called predator-prey (rabbits and foxes).
  We'll also look at the Lorenz system, which demonstrates chaos.
- What if we had a situation of two enemies. What would the situation look like?

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