# Math 2325 Intro to Higher Math -- Assignments

## Dr. Duval

### Chapter 1, Linear Iteration

Report due Tuesday, September 21.
Revised report due Thursday, October 7.

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, The Euclidean Algorithm

Report due Thursday, September 30.
Revised report due Thursday, October 14.

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 at least Questions 1-4, and either Question 5 or Question 6.

Questions 1 and 2 provide some structure for experimentally analyzing the distribution of the number of steps, while Questions 3 and 4 provide some structure for experimentally analyzing the distribtuion of GCDs.

Question 5 provides hints for theoretically analyzing the distribution of GCDs. Question 6 asks you to investigate the relationship of Fibonacci numbers and the Euclidean algorithm, and to prove a few things that might be helpful for theoretical analysis of the questions of the two main goals.

### Chapter 6, Randomized Response Surveys

Report due Tuesday, October 19.
Revised report due Thursday, November 4.

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 Thursday, October 28.
Revised report due Thursday, November 11.

Your main goal is to find the chromatic number, and, especially, 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.

### Chapter 8, The p-adic Numbers

Report due Tuesday, November 16.
Revised report due Thursday, December 2.

Your main goal is to answer Question 5, to find the negative of a p-adic rational number. Intermediate questions (special cases) leading to that goal are Questions 3 and 4. Other warmups (examples) you might consider are Exercises 8, 9, and 11. Don't limit yourself to just the examples given these Exercises; instead use them as motivation to find more examples, until you feel ready to tackle Questions 3-5.

### Chapter 2, Cyclic Difference Sets

Option 1
Report due Tuesday, November 23 (actually, Wed. Nov. 24 11:30am)
Revised report due Friday, December 10.
Option 2
If you don't want the chance to officially revise your draft, simply submit your report by Friday, December 10.

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.

### Optional 7th Lab

Report due Friday, December 10.
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.)