# Pre-sessional Advanced Mathematics

## EC9A0 course outline and notes

The course will be taught by Peter Hammond and Pablo Beker and will run from 21st September- 2nd 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 delivered through a combination of face to face and on-line teaching (with all teaching sessions recorded for those unable to participate live). For those able to attend in person, all teaching will take place in room S2.79 (second floor of the Social Sciences building).

Greek alphabet (for mathematically inclined economists)

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

A. Linear systems: Gaussian elimination, echelon form, linear dependence, rank

B. Determinants, Cramer's rule, inverses

C. Quadratic forms: Eigenvalues, diagonalization, linear constraints

D. Ordinary least squares estimation.

Readings: EMEA chs. 15, 16; FMEA ch.1

Matrix algebra slides, Part A

Matrix algebra slides, Part B

Matrix algebra slides, Part C

Matrix algebra slides, Part D

**2. 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 (updated 21/09/20)

Lecture Notes: Real Analysis

**3. 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 (updated 25/09/20)

Lecture Notes: Unconstrained Optimisation (updated 25/09/20)

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

(a) Constraint qualifications,

(b) 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. (updated 26/09/19)

Lecture Notes: Constrained Optimisation I (updated 7/10/20, typo in Theorem 2 fixed)

Lecture Notes: Constrained Optimisation II (updated 26/09/19)

**5. Theorem of the Maximum and Envelope Theorem: **(Pablo Beker)

(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 (updated 26/09/19)

Lecture Notes: Theorem of the Maximum and Envelope Theorem (updated 26/09/19)

**6. Fixed Point Theorems:**(Pablo Beker)**
**(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

**7. 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

**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