Day 18: mat385

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Today:

• Announcements:
  • Your Section 4.1 homework is graded. I graded four problems, two from each assignment. I did, however, avoid too many comments on your homeworks. Please just see the "key" that I've posted with them:
    • 4.1: 6 and 13; 39 and 100

    • A couple of things:
      1. Remember that you need to pay attention to the domain.
      2. #13: Please give reasons for things. Show work.
      3. #39: some of you were obviously paying too much attention to the back of the book. The author is incorrect: $0 \notin \mathbb{N}$. The natural numbers begin from 1.

         There's nothing natural about 0. It was the last "reasonable" number discovered, because most civilizations had a hard time coming to terms with the need for a symbol for nothing. It seemed like nothing itself would do....

      4. #100: It did say "prove", so I expected a little more than "voila"....

         See my mapping, which shows that there's plenty of room at the Motel $\infty$ for the set of finite length strings. That means that the set is denumerable, since the naturals have it covered.....

  • Your project proposals have been approved, or I'm still looking for input from you on that.

    If you didn't get a 10/10 on your proposal, then I'm still waiting to hear from you, so that we can get your topic straightened out.

    Then you'll get your 10!:)

  • I've created a wiki account for everyone today. The invitations are generally flagged by NKU's email server and sent off to spam; you may find the invite on webmail.nku.edu, in the quarantine.

    If you have any trouble let me know -- maybe you'll even visit during office hours!

• Today: a few musings on the importance of recursion, and graphs.
  • Simple Graphs. Five vertices. Six vertices? Here's some python code to generate all simple graphs, no matter what the number of vertices. Here are the sixes... they grow fast, according to The On-Line Encyclopedia of Integer Sequences.

  • Calculus is actually our friend.... Fibonacci numbers.

  • Recordings for today:
    1. Part I: Intro and Simple Graphs
    2. Part II: Closed-form Fibonacci Numbers

• Links:
  • If you check out this code, you'll see where I compute the power set ("all-subsets") of a set, to cover all the bases. I think that the end result is pretty beautiful, too!

  • Recursive in-order, post-order, pre-order in lisp
  • Here is a recursive bubbleSort algorithm in lisp
    (coded up following our author's algorithm, in section 3.3, p. 214, as closely as possible).

    Since I had to grade a bubbleSort problem, I couldn't resist coding it up.

  • I've discovered a really interesting fellow, James Tanton, who makes a fascinating presentation on Exploding Dots!

    James shows us how to conceive of different bases, algebra as arithmetic, the geometric series as infinite algebra, and the Fibonacci numbers as an infinite sequence of dots -- all this, via the magic of exploding dots! Very entertaining.

    We'll probably try to invite this guy to campus -- hopefully this fall. I'm working on that!