# Pre-sessional Mathematics for MRes/PhD students

**1. Matrix algebra:**

A. Linear systems: Gaussian elimination, linear dependence, rank;

B. Determinants, Cramer's rule, inverses;

C. Quadratic forms

Readings: EMEA chs. 15, 16; FMEA ch.1; Gilbert Strang, Introduction to Linear Algebra

Slides: Matrix algebra

**2. Real Analysis:**

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

Readings: lecture notes, FMEA ch. 13.

**3. Unconstrained optimization:**

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)

**4. Constrained Optimisation:**

(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)

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

**5. Theorem of the Maximum and Envelope Theorem**

(a) Correspondences: upper and lower hemi-continuity

(b) Comparative statics and Berge's maximum theorem;

Readings: chapter 3 in Stokey and Lucas.

**6. Fixed Point Theorems**

(a) Contraction Mapping Theorem

(b) Brouwer's Fixed Point Theorem

(c) Kakutani's Fixed Point Theorem

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