Day 14, Mat385

Last time

Next time

Announcements
- The Zooms continue
  A menu of our past zooms...
- I hope that you all had a nice spring break! I wish that it had been longer, but alas.
- Your project proposal is due today. I've made a slot on Canvas for you to submit your proposal.
  Your proposal is just a one-page description of what you'd like to do. Then we'll no doubt iterate!:)
- This from NKU's President, RE Covid:
  Effective Tuesday, March 8, masks will be optional on campus, with these exceptions:
  - Masks are required during class meetings in classrooms and lab spaces, through Monday, March 21.
  - Masks are required in NKU Health, Counseling and Student Wellness facilities.
  - Masks are required for at least five days following diagnosis of or exposure to COVID-19, per CDC policy.
  - Individual employees may continue to require masks in individual private offices where they meet with students, other employees or visitors.
  So one more week we wear the masks; and then I think that I will be maskless in class. You're welcome to come masked or unmasked.
- Homework, 3.2: graded 6, 23, 48
  - #6: you were really expected to try out the closed form solution, rather than heading back to induction.... We used induction to get the closed form solution formula -- now it's time to use it! Nonetheless, kudos to those who properly used the expand, guess, check method.
  - #23: Here you were to first identify the recurrence relation, and then solve it -- again, using the linear, first order, constant coefficient, non-homogeneous formula!
  - #48: here you were expected to use the closed form solution for a divide and conquer problem. Once we have that formula, we just need to identify the various parts of the recurrence relation, and away you go.
- By the way, speaking of homework: I noticed that five of my MAT385 students haven't submitted a single homework for grading. That seems a little lax. Do the math:
  Homework is 25% of your grade. If you pitch 25% of your grade, the best you can do is a C, and that seems unlikely....
Last time:
- We wrapped up Section 3.3 (Analysis of Algorithms), spending a lot of time with the Euclidean algorithm.
  It's a little mysterious until you see the geometry of the method -- recognizing that it can be considered a rectangle tiling problem -- and understand "the recursive idea". In the case of the Euclidean algorithm, it's about "casting out squares" to get to a smaller rectangle; then do it again.
  It also led us to understand the worst-case scenario -- casting out a single square each time. Working backwards from a single unit square, we could see that it is successive Fibonacci numbers which case the algorithm the most agonizingly slow convergence to its solution.
- The recursive idea: reducing a problem to an identical smaller problem, with a definite ending point.
  As another example, with a twist, consider the homework problem #41 about the number of binary strings of length $n$ with no two successive zeros. Prove that there $F(n+2)$ of them.
  1. How do you get the strings of length $n$ that end in 1?
  2. How do you get the strings of length $n$ that end in 0?
  We know what we're looking for, so let's see if each comes from one of the preceeding terms....
- Recursive computation of binary strings (no two successive 0s):
Today:
1. We'll see how far we can get on Section 4.1 (Sets).
2. On a related note I found this article today (a little after $\pi$ day:
  Pi Day: How One Irrational Number Made Us Modern: The famous mathematical ratio, estimated to more than 22 trillion digits (and counting), is the perfect symbol for our species's long effort to tame infinity.
Links:
- 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).

Website maintained by Andy Long. Comments appreciated.