**Fall 2016**

This course is an introduction to mathematical techniques used to model phenomena studied in political science, with special attention to the analysis of individual action. Mathematical topics covered include: sets, functions, and graphs; matrix algebra; differential calculus and optimization; probability and risk; integral calculus; and sequences, series, and limits. All these topics are useful in many settings in political science, including game theory, dynamic modeling, and statistics.

This course website will be updated to reflect any changes in schedule, topics covered, or assignments, as well as to provide relevant links to materials associated with the course.

## Course Outline and Approximate Schedule

## 0. Optional pre-semester basic math review sessions

#### Tue 8/23 and Thu 8/25 2:00-3:00 pm in Siegle 205

Mainly, these sessions will be devoted to your questions on the basic math review problems, previously distributed. **You should try to do the problems in at least the first three Sections before you come to the first session**.

The Assessment problems are for my information, and those topics will be covered during the semester. **All students are to hand in their answers **(or "don't knows")** to the Assessment problems on the first day of class, Monday 8/29.**

Click here for another complete copy of Basic Review and Assessment problems. Solutions to the basic review are posted below, section-by-section.

- Solutions for Section 1, Algebra
- Solutions for Section 2, Sets
- Solutions for Section 3, Functions and Graphs
- Solutions for Section 4, Statements and Proofs

Also for the review sessions: you should try to read the following sections in the Pemberton and Rau text. I'll take any questions you may have on those chapters.

- Chapter 1, sections 1.1, 1.3,
- Chapter 2, sections 2.1
- Chapter 3, sections 3.1-3.4
- Chapter 4, sections 4.1-4.4
- section 31.1 "Rigour" (pp. 700-702)

Further relevant material you may find helpful in the supplementary texts

- on algebra: Gill sections 1.2-1.4 and 1.7; Moore & Siegel chapter 2
- on sets: Gill section 7.3; Moore & Siegel sections 1.1-1.2
- on functions and graphs: Gill section 1.5 and 2.1-2.2; Moore & Siegel section 3.1
- on statements and proofs: Moore & Siegel section 1.7

**Additional exercises or examples on equations involving exponentials and logarithms:**

- There are several such exercises at the end of Section 17 in Gill's textbook.
- This page and especially its following pages from the website "Purplemath" offers several examples and solution techniques.
- Try the examples and problems in Tutorial 45 and Tutorial 46 on the "Virtual Math Lab" site of W. Texas A&M University.

## 1. Sets and relations, with applications to choice theory

M 8/29 to W 8/31: Relations on sets; preference and choice

Text: [Lecture notes on Relations, Preference, and Choice]

- sets; sets of real numbers
- relations on sets
- application: choice theory
- partial orders
**Exercises due Wednesday 9/7**all those from the Lecture Notes on Relations, Preferences, and Choice.

**Special Friday software session 9/2: ****Mathematical Typesetting with LaTeX**

- How to Install LaTeX: Please try this prior to the session
- all subsequent homework should be typeset in LaTeX

M 9/12 Special software session: Programming with R

## 2. Sequences and series; limits and continuity

W 8/31 to W 9/21: Analysis

Text: P&R, selections (below) from chapters 5, 6, 31; for outside reading, Moore & Siegel ch. 3

**You should read sections 5.1 by Wednesday 8/31, the remainder of Chapter 5 by Wednesday 9/7**

- real numbers, sequences, limits (Pemberton and Rao 5.1, 31.2, 31.3)
- sums and series (P&R 5.2; see also Gill, pp. 259-266)
**application:**choices over time (P&R 5.3)- continuous functions (P&R 5.4; Section 31.4)
- Fixed-point theorems (recommended advanced reading: P&R section 32.4)

**Exercises****due Monday 9/12--Please hand in to Dave**- 5.1.3
- 5.2.1

**Exercises****due Monday 9/19**- 5.2.2, 5.2.3
- 5.3.2, 5.3.4
- 5.4.1, 5.4.2
- and this exercise using R:
- Use the "function" command to create a list consisting of the first 1000 elements of the sequence (u_n) defined by u_n = 1+ 1/n. Use the clipboard to copy your method and the sequence values from the console to a page in an editing program such as Microsoft Word.
- Plot the values of the sequence (vertical axis) against the values of n from 1 to 1000 using the "plot" or other graphing command.

## 3. Differential calculus and applications

Text: P&R chapters 6-8.

**You should read sections 6.1-6.3 for Monday, 9/19**

**Homework problems due Monday 9/26:**

- All problems listed below from Chapter 6
- An R-programming exercise:
- In a single graph, plot the function y = f(x) = x^3 - 2x^2 (where "^" indicates an exponent) and its tangent line at x = 1.62. (Let the graph cover x values from -3 to 3.)
- In a single graph, plot the same function y = f(x) and its derivative y = f'(x).

- Read by Monday 9/19: P&R sections 6.1-6.3.
- Read by Wednesday 9/21: sections 7.1, 7.2
- Exercises
- 6.1.1, 6.1.3
- 6.2.1, 6.2.3
- 6.3.1 through 6.3.3
- The latter problem refers to "the answer to Exercise 5.4.4." Do that also, as needed to answer 6.3.3. Exercise 5.4.4 requires you to apply the intermediate value theorem.

- Exercises
- Methods of differentiation
**Exercises due Monday 9/26**- 7.1.1 and 7.1.2
- 7.2.1 through 7.2.3

M 9/26: Monotone functions and inverse functions

- Read by Monday 9/26: P&R sections 7.3; 7.4 up to the "Application: inverse demand functions"
**Exercises NOT TO TURN IN: bring, along with any questions from this and previous material, on Wed. 9/28**- 7.4.1
- 7.3.1, 7.3.2

**First Exam 9/28 - 10/3**

- Wed. Sept. 28: Review session
- Wed. or Fri.: distribute exam
- Exam due at beginning of class, Mon. 10/3

#### M 10/3 Limits: More on epsilon, delta arguments

**Homework problems due Monday 10/10:**

- Give an epsilon, delta proof to show the following without using any of our theorems about limits (such as limit of sum equals sum of limits, etc.):
- as x approaches 3, 2x-6 approaches zero.
- as x approaches 2, x squared approaches 4. (Hint: x squared minus 4 = (x+2)(x-2).)

- (additional exercises from Chapter 8 of P&R shown below)

#### W 10/5 and M 10/10 Finding maxima and minima

Read by Wednesday 10/5: P&R Chapter 8, first two sections; by Monday 10/10 last two sections

**Homework problems due Monday 10/10:**

- (Exercises on epsilon-delta arguments, above)
- Exercises 8.1.2 and 8.1.3 in P&R
- 8.2.2 through 8.2.4 in P&R
- 8.3.1 and 8.3.2

**Homework problems due ****Wednesday 10/19 (due to Fall Break)**

- 8.4.1 (all eight parts)
- possible additional problems from chapter 9 (below)

#### W 10/12 Exponential and logarithmic functions

Read by Monday 10/10: P&R sections 9.1-9.2

**Homework due Wednesday 10/19 (due to Fall Break)**

- 9.1.1 (as R exercise--show code); 9.1.4
- 9.2.2 all parts; 9.2.4, 9.2.5

#### FALL BREAK -- no class Monday 10/17

#### W 10/19 Approximations

Read by Wednesday 10/19: P&R sections 10.1-10.3

**Homework due Monday 10/24**

- 10.1.1-10.1.3
- 10.2.1; 10.2.3 (three parts)

Examples and Illustrations

- Click here to download an Excel spreadsheet illustrating the example calculation using Newton's method in Pemberton & Rao, p. 184-85. You can plug in different starting values, which I label "x0".
- There are a number of nice graphic illustrations of Newton's method online, such as this one from Wikipedia. Here is one from the same article in which the Newton method fails to converge. In the latter, the derivative at x = 0 is -1, the derivative at x = 1 is 1, and both tangent lines take you back to the other point, so that xn = 0, 1, 0, 1, ..., although the only true root is somewhere else entirely.

Text: P&R chapters 11-13, plus (if time) some material from 25, 26.

**HW problems due Monday 10/31**

- all exercises listed below from Chapter 11
- all exercises from sections of Chapter 12 covered in class

Mon. 10/24: Vectors and Matrices

- read P&R, chapter 11
**Exercises to turn in Monday, 10/31:**- 11.1.4 (a thru f)
- 11.2.1, 11.2.3, 11.2.4, and 11.2.5
- 11.3.1, 11.3.2, and 11.3.3. In 11.3.2, what's called
**y**in the problem is labeled**w**back in equation (11.3); they're supposed to be the same thing. - and more below from chapter 12

Wed. 10/26: Systems of Linear Equations

- P&R, chapter 12
**Exercises to turn in Monday, 10/31:**- 12.1.2 (a thru e)
- 12.2.1 (a thru c); 12.2.3

**Exercises to turn in Monday, 11/7:**- 12.3.1, 12.3.2, 12.3.3, 12.3.4. In 12.3.1, think about what the inverse of
**P**must do to**Px**. - 12.4.1, 12.4.2, 12.4.3
- plus additional exercises from Chapters 13, 27 below

- 12.3.1, 12.3.2, 12.3.3, 12.3.4. In 12.3.1, think about what the inverse of

Mon. 10/31 and Wed. 11/2: Determinants and Quadratic Forms; Eigenvalues and Eigenvectors

- P&R, chapter 13
**Exercises to turn in Monday, 11/7:**- 13.1.1, 13.1.3
- 13.2.1, 13.2.2

**Exercises to turn in Monday, 11/14:**- 13.3.1, 13.3.2, 13.3.7
- 13.4.2, 13.4.5, 13.4.6, 13.4.7
- 27.1.1
- plus more below from chapters14-16

- Chapter 3: 3.1-3.5 (omit examples 3.21, 3.22)
- Chapter 4: 4.3, 4.4, 4.6, 4.7, 4.9 (except example) ).

## 5. Multivariate calculus

Text: P&R chapters 14-17

**Exercises to turn in Monday, 11/14:**

- 14.1.1 , 14.1.2
- 14.2.2, 14.2.3
- Exercise "15.A" below
- Exercises from chapter 16, TBA

Mon. 11/7: Multivariate functions and partial derivatives

- P&R, chapter 14.1, 14.2
- Exercises:
- 14.1.1 , 14.1.2
- 14.2.2, 14.2.3

Wed. 11/9 Implicit functions; optimization

- P&R, chapter 15.1; 16.1, 16.2
- Exercises :
- 15.1.1
- For material in Chapter 15.1, also do the following exercise:
- "15.A": Suppose a decision maker faces this problem: a given level x of product safety regulation provides social benefits b(x) and imposes economic costs c(x; t), where t > 0 is a parameter measuring competition in the relevant industry; costs are higher for higher t. The decision maker chooses x (nonnegative) to maximize b(x) - c(x ; t).
- a. Suppose in particular that b(x) = 4x and c(x ; t) = tx
^{2}. Solve the maximization problem and use derivatives to show how the decision maker's choice x will change when t increases. - b. Rather than use the specific functional forms in (a.), simply assume that benefits increase in x at a decreasing rate; and that costs increase in x at an increasing rate. Give these assumptions in the form of conditions on partial derivatives, and use implicit differentiation to answer the same comparative statics question as in part (a.).

- a. Suppose in particular that b(x) = 4x and c(x ; t) = tx

- "15.A": Suppose a decision maker faces this problem: a given level x of product safety regulation provides social benefits b(x) and imposes economic costs c(x; t), where t > 0 is a parameter measuring competition in the relevant industry; costs are higher for higher t. The decision maker chooses x (nonnegative) to maximize b(x) - c(x ; t).
- 16.1.2, 16.1.3
- 16.2.1

6. Integral calculus

Text: P&R sections 19.1, 19.2, 20.1.

#### M 11/14, M 11/21 Integral Calculus

- for Monday 11/14: sections chapter 19.1 and 19.2
- for Monday 11/21 (after exam): section 20.1

**Exercises to be done for discussion on Wednesday 11/16**

- 19.1.2 and 19.1.4
- 19.2.1, 19.2.2, and 19.2.4

**Second Exam**

To cover optimization with one variable, convexity and concavity, the exponential function, Newton's method of approximating roots, and L'Hopital's rule; all our material on matrix analysis; calculus of functions of several variables, including optimization, concavity, and convexity; and finally, basic integration (P&R chapter 19).

Wed. 11/16 Review

exam handed out Wed. 11/16, due at beginning of class, Mon. 11/21.

**Exercises to be turned in Monday 11/28**

- 20.1.1, 20.1.3, 20.1.4, 20.1.5
- additional exercises on Probability and Statistics, TBA below

## 7. Probability and Mathematical Statistics

#### MW 11/28-30, MW 12/5-7: Probability Theory and Expected Utility

For Monday 11/28: chapters 1, 2

**Coverage (Chapers and sections from Wasserman)**

- Monday 11/28: Chapter 1
- Wednesday 11/30: Chapter 2 (though 2.5)
**read and bring questions Monday 12/5: Sections 2.6-2.8, 3.1-3.5.**I will cover minimally but they are needed for what follows. Look at the exercises from Chapters 2 and 3 in the final batch below, but we'll discuss those at the help session later.- Monday 12/5: Section 2.9; 5.2, 5.3, 5.4
- Wednesday 12/7: Sections 6.3, 10.1, 10.2

NOTE Wasserman's textbook website with data and rough R code for some exercises and examples

**Exercises to be turned in Monday 11/28**

- Chapter 1: Exercises 5, 12, 13

**Exercises to be turned in Monday 12/5**

- Chapter 1: Exercises 15, 19; computer experiments 21, 22
- Chapter 2: Exercises 2, 6, 9

**Exercises to be brought to help session for discussion**

- Chapter 2: Exercise 17
- Chapter 3: Exercises 3, 4, 5, 11
- Chapter 6: Exercise 2, 3

Recommended for later:

Peter C. Fishburn, "The Axioms of Subjective Probability." *Statistical Science*, Vol. 1, No. 3 (Aug., 1986), pp. 335-345. Click here to obtain via JSTOR.

**Third Exam**

Covering techniques of integration and mathematical statistics.

Help session (tentatively) Mon. 12/12, 11:00 (location TBA)

Exam to be distributed 5:00 Monday 12/12

Answers due: 5:00 Friday 12/16--**Please turn in PDFs via email.**