Day 15, Mat385

Last time

Next time

Announcements
- The Zooms continue
  A menu of our past zooms...
- As of next week, masks are optional. Today is the last day I plan to wear one, but you're invited to do so or not as you see fit.
- Homework, 3.3: graded 9, 10 and 40
  - #9: you are to trace the execution for the specific case, $x=4$, $p(x)=2x^3-7x^2+5x-14$, and should have discovered that it required 9 operations (additions and multiplications).
    Generally: $C(n)=3n$ is the closed form of the operation count.
    Recursively: $C(n)=C(n-1)+3$, with $C(0)=0$.
  - #10: Same calculation, different algorithm:
    Generally: $H(n)=2n$ is the closed form of the operation count.
    Recursively: $H(n)=H(n-1)+2$, with $H(0)=0$.
    So you save $n$ ops by computing your polynomials in Horner form.
  - #40: Many of you went one step too far in your gcd calculations: you should have stopped at 2 = 2*1+0
    rather than doing
    
    2 = 1*1+1 1 = 1*1+0
    You were on a roll! -- the successive Fibonacci numbers lend themselves to that, because the quotients $q_i$ are 1, except for the last one.
Last time:
- We got pretty well through Section 4.1 (Sets).
- One focal point was the analogy between the binary operations of logic ($\land$ and $\lor$) and the binary operations of sets ($\cup$ and $\cap$).
- The power set was the primary tool we encountered, and we saw how it might be used in Venn diagrams to describe how many different subsets one can make with two sets in a universe.
  It is also the tool by which we come to the realization that infinity comes in various sizes....
Today:
1. We'll finish off Section 4.1 (Sets), with more on the various flavors of infinity.
  In particular, permit me to share with you the story of Motel $\infty$....
2. Then we'll get a start on Section 6.1 (Graphs).
Links:
- Graph Terminology Handout
- Recursive computation of Fibonacci numbers (in lisp)
- Recursive computation of rows from Pascal's triangle (in lisp)
- "String exponention" (in lisp)
- Here is the recursive selection sort algorithm, in lisp:
- Here is a recursive binary search algorithm in lisp:
- Closed-form computation of Fibonacci numbers (in lisp)
- Some examples of the Euclidean algorithm:
  1. Simple and clean recursive Euclidean Algorithm in lisp:
  2. Written as a while loop, using the recurrence relation:
- 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).

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