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Pre-sessional Mathematics for MRes/PhD students



Week 1: Topics & Material

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)


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)