# Pre-sessional Advanced Mathematics

## EC9A0 course outline and notes

The course will be taught by Peter Hammond and Pablo Beker and will run from Tuesday 20th September to Monday 3rd October, Monday to Friday, excluding weekends. There will be four hours of teaching each day (10-11am; 11.30-12.30pm; 3-4pm and 4.30- 5.30pm). All teaching for the first two weeks will take place in room S2.79 (second floor of the Social Sciences building).

Greek alphabet (for mathematically inclined economists)

Click hereLink opens in a new window for Reading List.

**1. Real Analysis: **(Pablo Beker)

Metric and normed spaces, sequences, limits, open and closed sets, continuity, subsequences, compactness.

Readings: lecture notes, FMEA ch. 13.

Slides Real Analysis

Lecture Notes: Real Analysis

**2. Unconstrained optimization: **(Pablo Beker)

Concave and convex functions, Weierstrass' theorem, first- and second-order conditions, envelope theorems.

Readings: lecture notes; FMEA chs. 2, 3. Simon and Blume (ch. 18)

Slides: Unconstrained Optimisation

Lecture Notes: Unconstrained Optimisation

**3. Constrained Optimisation: **(Pablo Beker)

(a) Constraint qualifications,

(b) Karush/Kuhn/Tucker Theorem, first- and second-order conditions;

Readings: lecture notes; EMEA, Simon and Blume (chs. 19, 30) and Mas-Colell et al (Section M of Mathematical Appendix)

Slides: Constrained Optimisation with Equality Constraints.

Slides: Constrained Optimisation with Inequality Constraints.

Lecture Notes: Constrained Optimisation I

Lecture Notes: Constrained Optimisation II

**4. Matrix algebra:** (Peter J. Hammond)

A. Introduction to Vectors and Matrices

B. Special Matrices and Introduction to Determinants

C. Determinants and Gaussian elimination

D. Determinants and Rank

E. Quadratic Forms and Their Definiteness

F. Eigenvalues and Eigenvectors

Readings: EMEA6 chs. 13 and 14 (or EMEA5 chs. 15 and 16, plus FMEA ch.1)

Matrix algebra slides, Part A

Matrix algebra slides, Part B

Matrix algebra slides, Part C

Matrix algebra slides, Part D

Matrix algebra slides, Part D appendix

Matrix algebra slides, Part E

Matrix algebra slides, Part F

**5. Difference and differential equations: (**Peter J. Hammond)

Readings: FMEA chs. 11, 5, 6, 7. Difference equation slides, Part A

Difference equation slides, Part B

Difference equation slides, Part C

Differential equation slides

**6. Theorem of the Maximum and Envelope Theorem: **(Peter J. Hammond)

(a) Correspondences: upper and lower hemi-continuity

(b) Comparative statics and Berge's maximum theorem

Readings: chapter 3 in Stokey and Lucas.

Slides: Theorem of the Maximum and Envelope Theorem

Lecture Notes: Theorem of the Maximum and Envelope Theorem

**7. Fixed Point Theorems:** (Peter J. Hammond)(a) Contraction Mapping Theorem

(b) Brouwer's Fixed Point Theorem

(c) Kakutani's Fixed Point Theorem

Readings: chapter 3 in Stokey and Lucas, FMEA (ch. 14)

Slides: Fixed Point Theorems

Lecture Notes: Fixed Point Theorems

**8. Calculus of Variations and Optimal Control:** (Peter J. Hammond)

Readings: FMEA, chs. 8, 9,10.

Slides on calculus of variations

Slides on optimal control

**9. Probability:** (Peter J. Hammond)

Integral and measure, conditional probability and independence, random variables and moments, laws of large numbers, central limit theorem.

Readings: Notes.

Slides on Probability

**10.** **Deterministic and stochastic dynamic programming**: (Peter J. Hammond)

Readings: FMEA ch. 12, plus notes.

Slides on dynamic programming