# Math 2325 Intro to Higher Math -- Assignments

## Dr. Duval

### Chapter 1, Linear Iteration

Report due Wednesday, February 9.
Revised report due Friday, February 25.

Your main goal is to address Question 5 (you may want to, but do not have to, organize your answer along the lines suggested by Question 6).

Also address one of the following three:

• Questions 7 and 8; or
• Question 9 (be sure to read "Visualizing Iteration" after Question 9); or
• Question 10 [note that "Question 10" on page 11 has a typographical error, and is printed as "Question 1: 0"].

### Chapter 2, Cyclic Difference Sets

Report due Friday, February 18.
Revised report due Friday, March 4.

Your main goal is to answer the question "For which m does the set of non-zero squares (mod m) form a cyclic difference set with (m-1)/2 elements?". Questions 1-3 help you discover this experimentally. Theoretically, Questions 4-8 help you answer for which m are there (m-1)/2 distinct non-zero squares, and Questions 9-10 help you answer for which of those do we get cyclic difference sets.

### Chapter 9, Parametric Curve Representation Sets

Report due Wenesday, March 9.
Revised report due Friday, April 1.

Your main goal is to answer the question "Which properties of the parametersp, q, r, s determine the symmetry of the parametric curves

x(t) = sin(pt) + cos(qt)
y(t) = sin(rt) + cos(st)?".
This is, essentially, the last part of Question 2. Question 1 and the first part of Question 2 are good warmups, so do those first; they can also be your first data points.

### Chapter 4, Prime Numbers

Report due Friday, March 18.
Revised report due Friday, April 8.

• Address the issues of primes (mod 4) in Exercises 15 and 16. (Exercises 13 and 14 are good warmups for this. You may not be able to prove everything you observe in Exericse 15.) AND
• One of the following:
• Address the issues of Euler's simple method for finding primes in Exercises 9, 11, and 12. (Again, you may not be able to prove everything you observe, though you can prove at least a few simple observations.) OR
• Address the issues of the density of primes in Exercises 19 and 20. (Again, you may not be able to prove everything you observe, for instance a formula for the density of primes, but you can check your answer for reasonableness.)

### Chapter 6, Randomized Response Surveys

Report due Wednesday, April 13.
Revised report due Wednesday, April 27 (with an automatic extension until Friday, April 29, which is Dead Day) .

Your main goal is to answer the question: "What should the survey-taker do with the results?" In other words, what is your estimate of the proportion of True Yesses as a function of the proportion of reported yesses? Answer this in the most general setting, where the probabilities of answering the real question (dime lands heads) and the answer to the decoy question being yes (penny lands heads) are variables.

This is some mixture of Questions 1-3, 8, and 11.

### Chapter 5, The Coloring of Graphs

Report due Friday, April 22.
Revised report due Friday, May 6 (last day of finals).

Your main goal is to find the chromatic polynomial of the following three kinds of graphs:

• The complete graph on n vertices;
• The path on n vertices; and
• The cycle on n vertices.

Structure for these investigations is provided by Questions 1-3, parts of Question 5, and Questions 6-9.

### Optional 7th Lab

Report due Friday, May 6.
No official revisions, but I encourage you to consult with me as you write your report.

Talk to me in advance so that we can set up a reasonable main goal for you to pursue.

Also note that, instead of an optional 7th lab, you may turn in a re-revision of any of the first six reports, or you may turn in nothing at all. (See syllabus for how all this affects your grades.)