Matthew L. Wright
Assistant Professor, St. Olaf College

Modern Computational Math

Math 242 ⋅ Spring 2021

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Do the following before the first class:
Monday
February 15
Introduction; Mathematica basics
Do the following before the next class:
Wednesday
February 17
Computing the digits of $$\pi$$
Do the following before the next class:
• Watch this video, which explains why the sum of reciprocals of squares converges to $$\pi^2/6$$.
• Start the $$\pi$$ Project (due Monday Wednesday). Implement at least one of the methods for approximating digits of $$\pi$$ before Friday's class. Also look over the sample project report.
• Complete Mathematica Quiz 1 (on Moodle).
• Optional bonus: Watch this video to learn why the product formula from the Intro Mathematica assignment converges to $$\pi$$.
Friday
February 19
Computing the digits of $$\pi$$
Do the following before the next class:
Monday
February 22
Fibonacci numbers — meet in the classroom
Do the following before the next class:
• Finish the $$\pi$$ Project. Pay attention to the grading rubric in the assignment file and refer to the sample project report. Submit your notebook to the Pi Project on Moodle.
• Investigate $$F_n^2 - F_{n+1}F_{n-1}$$, where $$F_n$$ is the $$n$$th Fibonacci number. Evaluate this quantity for lots of values of $$n$$. What pattern do you observe?
Wednesday
February 24
Fibonacci identities
Do the following before the next class:
• Catalan's identity says $$F_n^2 - F_{n+r}F_{n-r} = (-1)^{n-r}F_r^2$$. Verify this for at least three values of $$r > 2$$. For each value of $$r$$, check at least 100 values of $$n$$.
• Vajda's identity says $$F_{n+i}F_{n+j} - F_nF_{n+i+j} = (-1)^n F_i F_j$$. Verify this for at least six pairs $$i,j$$. For each pair $$i,j$$, check at least 100 values of $$n$$.
• Submit a Mathematica notebook containing your verifications of Catalan's and Vajda's identities to the Fibonacci Assignment on Moodle. Please put your name at the top of your notebook. (Note that this is an Assignment, not a Project.)
Friday
February 26
Pell numbers
Do the following before the next class:
• Complete Mathematica Quiz 2 (on Moodle). This quiz covers lists, indexed variables, functions, and Modules.
• Take a look at this paper, which proves various identities involving the Pell numbers. Read through the Introduction, which gives some background about the Pell numbers. Note that Proposition 1 corresponds to our observations in class. Take a quick look at the other propositions and theorems that the authors prove.
• Begin the Pell Project, due Wednesday, March 3.
Monday
March 1
Iterated functions: Collatz conjecture
Do the following before the next class:
• Finish the Pell Project (due Wednesday). Upload your notebook to Moodle.
• Continue your investigation of sequences that arise when iterating the Collatz function or some other function. Bring observations and questions to class on Wednesday.

Extra credit opportunity: Attend either of Dr. Trachette Jackson's lectures on March 2 or 3 and answer these two questions on Moodle to earn two extra-credit points.

Wednesday
March 3
Iterated functions: logistic map and chaos
Do the following before the next class:
Friday
March 5
Iterated functions: fractals

Bonus video: Steven Strogatz — The science of sync

Do the following before the next class:
Monday
March 8
Primes
Do the following before the next class:
Wednesday
March 10
Prime sieves
Do the following before the next class:
• Finish implementing the Sieve of Eratosthenes in Mathematica.
Friday
March 12
Prime sieves

Bonus video: Yitang Zhang: An Unlikely Math Star Rises

Do the following before the next class:
• Finish implementing the Sieve of Sundaram in Mathematica.
• Take a look at the Primes Project, which is due March 22.
Monday
March 15
Primes powers
Wednesday
March 17
Rest Day — no class
Do the following before the next class:
Friday
March 19
Mathematics of RSA Cryptography
Do the following before the next class:
Monday
March 22
Encrypting text with RSA cryptography
Do the following before the next class:
Wednesday
March 24
Counting primes
Do the following before the next class:
Friday
March 26
Prime patterns and the Riemann zeta function
Do the following before the next class:
Monday
March 29
Introduction to Python
Do the following before the next class:
Wednesday
March 31
Yahtzee in Mathematica and Python
Do the following before the next class:
• Finish four of the six exercises in the Intro to Python notebook and submit your notebook link to the Intro Python assignment on Moodle.
• Finish implementing the Yahtzee simulation in Python. You don't have to submit it for a grade, but we will use it for further investigation in Friday's class.
Friday
April 2
Yahtzee investigation and plotting with Matplotlib
Do the following before the next class:
• Complete Python Quiz 1 (topics are variables, lists, if statements, and functions).
• Use simulation and make plots to answer the three questions in the Yahtzee Investigation notebook. Give this your best shot before class on Monday, and bring questions to class.
Monday
April 5
Trouble simulation
Wednesday
April 7
Rest Day — no class
Do the following before the next class:
• Finish the Yahtzee Investigation, if you haven't done so already, and submit a link to your notebook to the Yahtzee Investigation assignment on Moodle.
• Work on the Trouble Investigation. Bring questions to class on Friday.
Friday
April 9
Trouble Investigation; Intro to Random Walks
Do the following before the next class:
• Finish the Trouble Investigation and submit a link to your notebook to Moodle.
• Investigate the questions in the Random Walk Notebook.
Monday
April 12
1D and 2D Random Walks
Do the following before the next class:
Wednesday
April 14
2D Random walks
Do the following before the next class:
• Work on the Random Walk Project, due Monday Wednesday.
• Think about these questions: Do all 2D random walks return to the origin? What does your computational investigation show? How would the 1D proof from the video adapt to 2D?
Friday
April 16
2D and 3D Random Walks
Do the following before the next class:
• Work on the Random Walk Project, due Monday Wednesday.
• Continue thinking about these questions: Do all 2D symmetric random walks return to the origin? How about 3D random walks? What does your computational investigation show? How would the 1D proof from the video relate to higher dimensional random walks?
Monday
April 19
Percolation
Do the following before the next class:
• Finish the Random Walk Project, due Wednesday Friday. Submit your notebook link to Moodle.
• Think about how to write an algorithm that determines whether or not percolation occurs in a grid. Bring ideas to class on Wednesday Friday.
Wednesday
April 21
No class — Day of Healing
Take time for reflection and conversation.
Do the following before the next class:
• Finish the Random Walk Project, due Wednesday Friday. Submit your notebook link to Moodle.
• Think about how to write an algorithm that determines whether or not percolation occurs in a grid. Bring ideas to class on Wednesday Friday.
Friday
April 23
Percolation
Do the following before the next class:
• Implement the query() function from class. Can you use it to estimate the probability of percolation for given values of $$n$$ and $$p$$?
• Take a look at the Percolation Project, due next Friday.
Monday
April 26
Percolation

Bonus: Susan D'Agostino book and interview

Enjoy a rest day!
Wednesday
April 28
Monte Carlo optimization (simulated annealing)
Do the following before the next class:
• Finish the Percolation Project and submit your notebook link to Moodle.
• Experiment with the simulated annealing method from class. How are the results affected if you change the decrease factor, the number of steps, or the possible moves?
Friday
April 30
Monte Carlo optimization (simulated annealing)
Do the following before the next class:
• Experiment with the simulated annealing method from class. How are the results affected if you change the decrease factor, the number of steps, or the possible moves?
• Read the Final Project Information. Think about a topic (and optionally a partner) for the final project.
Monday
May 3
Magic squares
Do the following before the next class:
Wednesday
May 5
Magic squares

Bonus: Satyan Devadoss Mage Merlin's Unsolved Mathematical Mysteries

Do the following before the next class:
Friday
May 7
Traveling salesperson problem
Do the following before the next class:
• Work on implementing a simulated annealing algorithm to find approximate solutions to the traveling salesperson problem. Take a look at the TSP Project.
Monday
May 10
Traveling salesperson problem
Do the following before the next class:
Wednesday
May 12
TSP questions; mathematics of juggling
Do the following before the next class:
• Finish the TSP Project and submit your work to Moodle.
• Work on your final project.
• Take a look at the bonus videos (below) on areas of computational mathematics that we didn't study this semester.
Friday
May 14
Do the following before the next class:
• Work on your final project.
• Email the professor a brief update (1–2 paragraphs) on your final project. Describe what you have accomplished, what remains to do, and what questions you have.
• Take a look at the bonus videos on areas of computational mathematics that we didn't study this semester.