#### Fall 2018

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; linear algebra; differential calculus and optimization; probability, mathematical statistics, and decisions under risk; integral calculus; and sequences, series, and limits. All these topics are useful in many settings in political science, across formal theory 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. Some Preliminaries

##### Monday and Wednesday, 8/27-8/29

Sets and functions (Pemberton & Rao Chapter 3)

Intervals and real numbers (reference: P&R section 31.2)

A word on logic and proofs (P&R 31.1)

Sequences, convergence, limits, and continuity (P&R Chapter 5; for advanced reference, 31.3 & 31.4)

#### Exercises to discuss on Friday

- Try proving the other of DeMorgan's Laws.
- 5.1.3
- If we get to series on Wednesday: 5.2.2 and 5.2.3
- 5.4.1, 5.4.2, 5.4.3
- Additional exercises on sequences and convergence.
- If we get to Proposition 3 from p. 714 by Wednesday: 31.4.2. Erratum: Rather than using property
**SQ8**as the exercise says, they mean to invite you to use Proposition 3. If you do that, you won't need to invoke any epsilons or deltas or Ns to prove the claim in the exercise.

**Homework to turn in Wed. Sept. 5**

- As a LaTeX exercise: Exercise 5.4.3 (b)

#### Exercises to discuss on Friday Sept. 7

- left over from Chapter 5 and related material on functions and continuity, series
- 5.2.2 and 5.2.3
- 5.4.1, 5.4.2, 5.4.3 (including LaTeX for 5.4.3(b))
- 31.4.2. Erratum: Rather than using property
**SQ8**as the exercise says, they mean to invite you to use Proposition 3. If you do that, you won't need to invoke any epsilons or deltas or Ns to prove the claim in the exercise.

#### Homework to turn in Monday Sept. 10

- Do a new version of problem 5.2.2 and 5.2.3—call it part (e)—namely, a geometric progression with first term a = 1 and common ratio r = .95.

## 1. Calculus of one variable

### Newton's quotient and differentiation

Read Chapters 6 and 7 (excluding 6.4)

particularly including

- linear approximation using Newton's quotient
- composite functions
- monotone functions
- inverse functions

#### Exercises to discuss on Friday Sept. 14

- from Chapter 6 on differentiation
- 6.1.1
- 6.2.1: In each case, in addition to finding the approximate change in f(x) as instructed in the textbook, also find the exact change, show how much they differ, and compare that to the size of h, as suggested at the end of the first paragraph on p. 109.
- 6.2.3
**Additional exercise:**Let y = f(x) = x^{8}- x^{7}+ x^{6}- x^{5}+ x^{4}. Using linear approximation, find f(1.02). How far off is your approximation from the true value?- from Chapter 7 on methods of differentiation: any of the following that we get to by Wednesday in class, as announced
- 7.2.2 and 7.2.3, all parts

#### Homework to turn in Monday Sept. 17

- 6.1.1
- 7.2.2 (g) only

#### Exercises to discuss on Friday Sept. 21

- any remaining problems from Chapter 7 on methods of differentiation
- 7.3.1, all parts
- 7.4.1
- see below for additional exercises from Chapter 8

### Optimization and convexity

Chapter 8 Maxima and Minima -- read the whole chapter, even the economics application on pp. 150-51.

#### More exercises to discuss on Friday Sept. 21

- from Chapter 8 on maxima and minima
- 8.1.2, 8.1.3 critical points and curve-sketching
- 8.2.2, 8.2.3, 8.2.4 using second derivatives
- 8.3.1, 8.3.2, 8.3.4, 8.3.6 on global max and min
- 8.4.2, 8.4.3, 8.4.4 concave and convex functions

#### Homework to turn in Monday Sept. 24

- 7.4.1 with respect to the function in 7.3.1 (b) only
- 8.3.6

***** First Exam out about Wed. Sep. 26 *****

### The Exponential and Logarithmic Functions

(Review Sections 4.3, 4.4.) Read Sections 9.1, 9.2

**Exercises to discuss Friday**

- all exercises from Master List below from Chapter 8
- exercises from Chapter 9 from Master List covered in class (to be announced in class)
- any exercises from Chapter 10 from Master List covered in class (10.2.3)

### Mean value theorem, l'Hôpital's rule, Taylor's theorem

Sections 10.2 through 10.4

**Master List of exercises to be assigned from Chapters 8, 9, 10 (differential calculus)**

- 8.2.2, 8.2.3, 8.2.4 finding and assessing critical points, using the second derivative; and curve-sketching
- refers back to 8.1.2, 8.1.3
- 8.3.1, 8.3.2 global max & min
- 8.4.2, 8.4.3, 8.4.4 concavity, convexity.
- For optional extra practice do 8.4.1 (don't turn in).
- 9.1.4
- 9.2.2, 9.2.4, 9.2.5
- 10.2.3 all parts (a)-(c)
- 10.2.1, 10.3.2; 10.4.1 (a) and (b)

### Integration

Chapters 19, 20 (excluding 19.3)

**Exercises to discuss Friday**

- Exercises from chapter 10 (Master List, above) not yet done, namely: 10.2.1, 10.3.2; 10.4.1 (a) and (b)
- Exercise 19.1.3

**Master List of exercises to be assigned from Chapters 19, 20 (integral calculus)**

- area: 19.1.3
- integration: 19.2.3, 19.2.4 (all 3 parts); 19.2.6
- integration by parts & by substitution: 20.1.1, 20.1.3, 20.1.4, 20.1.5
- improper integrals: 20.2.1

## 2. Linear algebra

dates TBA

- vectors, linear dependence, length: Section 11.1, 13.3
- matrices as linear mappings; inverses: Section 11.2, 12.3
- linear dependence and rank: Section 12.4
- determinants and quadratic forms: Chapter 13
- (if time:) eigenvalues and eigenvectors: Chapter 27

**Exercises on linear algebra to discuss Friday**

- vector arithmetic: 11.1.4
- vector length; norm: 13.3.1, 13.3.2, 13.3.7
- matrices as linear mappings: 11.2.3

**Additional exercises on linear algebra: not to turn in, but will discuss in review sessions**

- matrix rank: 12.4.3 (refers to 11.1.4)
- determinant: 13.1.1
- quadratic forms: 13.4.7

**A nice supplemental FREE linear algebra textbook:** Pemberton and Rau occasionally make the connection between matrix operations and the corresponding properties of linear transformations, but their treatment of linear transformations is not complete. For your future reference, a nice-looking accessible supplementary treatment can be found online:

- David Cherney, Tom Denton, Rohit Thomas, and Andrew Watson,
*Linear Algebra*, online (1.4 megabytes, 460 pages), University of California, Davis (2013).

A succinct statement of the **test for positive and negative (semi-)definiteness of a symmetric matrix** based on its principal minors (by economist Nathan Barczi) can be found at MIT's website.

Another alternative, and interesting, definition of the determinant, in terms of permutations of the columns is given here.

## 3. Multivariate calculus

material from chapters 14, 15, 16

- partial derivatives: Read Section 14.1 up through p. 271 only.
- the chain rule: Read Section 14.2, especially begining from top of p. 276
- if time: implicit functions, section 15.1
- optimization: Read sections 16.1, 16.2

**Exercises on multivariate calculus: not to turn in, but will discuss in review sessions**

- critical points and second-order conditions: 16.1.2 (a, b, c), 16.1.3 (a, b, c), 16.1.5
- Hint for 16.1.3: consider movements along the line
*y*=*kx*for any constant*k*(whether positive or negative); then - in (a),
*f*(*x*,*y*) =*x*^{4}+*k*^{4}*x*^{4}= (1 +*k*^{4})*x*^{4}; - In (c), it becomes (1 -
*k*^{4})*x*^{4}, and whether*f*reaches a max or min at (0, 0) depends on whether*k*is greater than or less than 1. - global optima, concavity, and convexity: 16.2.1 (for a, b, c only); 16.2.2

***** Second Exam out about Wed. Oct. 31 *****

## 4. Probability and Statistical Inference

Text: Excerpts from Wasserman, *All of Statistics*, chapters 1-10

### Probability

##### dates TBA

**read: chapters 1 and 2.1-2.2**

Exercises to discuss Friday: Chapter 1, exercises 5, 12, 13, 15.

Exercises to discuss Friday:

- Chapter 1, Exercise 19; Computer Experiments 21, 22
- Chapter 2: Exercises 2, 6

Exercises to discuss Friday

- Chapter 2: Exercises 4, 7, 9, 18

### Expectations and Moments

##### dates TBA

Read: sections 3.1-3.5

Exercises to discuss Friday

- Chapter 2: Exercise 17
- Chapter 3: Exercises 3, 4, 5, 11, 13

### Convergence of Random Variables

##### dates TBA

Read: sections 5.1-5.4

**Exercises to discuss Friday**

- Chapter 5, "Problem A" and a version of Exercise 4, both as described below under "Exercises to be assigned."
- Exercises from Chapters 6 and 10, TBA.

### Statistical Inference and Hypothesis Testing

##### dates TBA

Read: sections 6.3, 10.1, 10.2, 13.1

- 6.3 estimation, confidence sets, and hypothesis testing
- linear regression as an application: Example 6.6 (p. 89)
- 10.1 the Wald test: read only through Remark 10.5, plus Example 10.8 (Comparing Two Means) and Theorem 10.10 and the Warning following it.
- 10.2
*p*-values - linear regression as an application: Section 13.1, especially Example 13.6 (p. 211-212)

**Exercises to discuss Friday**

- Chapter 5, "Problem A" and a version of Exercise 4, both as described below under "Exercises to be assigned."
- Chapter 6: Exercise 2, 3
- Chapter 10: Exercise 6 (see hint below)

**Exercises to be assigned**(subject to revision)

- Chapter 1: Exercises 5, 12, 13, 15, 19; Computer Experiments 21, 22
- Chapter 2: Exercises 2, 4, 6, 7, 9, 17, 18
- Chapter 3: Exercises 3, 4, 5, 11, 13
- Chapter 5:
- Do the following practice problem, which you can label "Problem A": For each
*n*= 1, 2, ..., let*X*have the uniform distribution on the interval [-1/_{n}*n*, 1 + 1/*n*]. Let*X*be distributed uniformly on [0, 1]. Show that*X*converges both in probability and in distribution to_{n}*X*. (You should prove convergence in distribution__directly__-- that is, without appealing to the fact that convergence in probability implies convergence in distribution.) - Do the version of Exercise 4 as we discussed in class: Don't examine convergence in "quadratic mean." Rather, in addition to showing that
*X*converges in probability to 0, show (_{n}__directly__, again) that it also converges in distribution to 0.

- Do the following practice problem, which you can label "Problem A": For each
- Chapter 6: Exercise 2, 3
- Chapter 10: Exercise 6 (Hint:
*n*= 1919 is definitely a "large" sample, so the binomial distribution suggested in the exercise is approximately normal. Use the Wald test.)

**Third Exam**

- review session Mon. Dec. 10
- Exam distributed by Tue. Dec. 11
- Exam due date Sun. Dec. 16