Math 2325 Intro to Higher Math -- Assignments

Fall 2006

Dr. Duval

Chapter 1, Linear Iteration

Report due Tuesday, September 19.
Revised report due Tuesday, October 3.

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 3, Euclidean Algorithm

Report due Thursday, September 28.
Revised report due Thursday, October 12.

Your main goal is to investigate the following:

The distribution of the number of steps the Euclidean algorithm takes to compute the GCD of two numbers, as a function of the size of the numbers; and
the distribution of the GCD itself of two numbers, as a function of the size of the numbers.

In particular, address Questions 2 and 3, and either of the following:

Questions 4 and 5; or
Question 6.

Questions 4 and 5 guide you in further analyzing one interesting feature of the distribution of the GCD's, while Question 6 guides you in looking at how the Euclidean algorithm works on Fibonacci numbers.

Note that there is not much to say, theoretically, in Questions 2 and 3; here, you will simply need to analyze data very carefully. You can still make somewhat precise statements about what you can expect. Questions 4-6, however, guide you to proving some interesting things about the Euclidean algorithm.

Chapter 9, Parametric Curve Representation

Report due Tuesday, October 17.
Revised report due Thursday, November 2.

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 2, Cyclic Difference Sets

Report due Thursday, October 26.
Revised report due Thursday, November 9.

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 6, Randomized Response Surveys

Report due Tuesday, November 14.
Revised report Thursday, November 30.

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 Tuesday, November 21 (with an automatic extension until Wednesday, November 22, which is the day before Thanksgiving).
Revised report due Friday, December 8 (last day of finals).

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

The path on n vertices; and
The cycle on n vertices.

This is Questions 7 and 9 (and just a very few parts of Question 5 as warmups). Exercises 4-10 are also good practice with some of the tools you'll need, though these exercises do not directly answer our main questions.

Optional 7th Lab

Report due Friday, December 8.
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.)