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Pre-sessional Advanced Mathematics

EC9A0 course outline and notes

The course will be taught by Peter Hammond and Pablo Beker and will run from 20th September- 1st 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. 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. 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: 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. 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

3. 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

4. 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)

5. 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. (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)

6. 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)

7. Calculus of Variations and Optimal Control: (Peter J. Hammond)
Readings: FMEA, chs. 8, 9,10.
Slides on calculus of variations
Slides on optimal control

8. 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

9. 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

10. Deterministic and stochastic dynamic programming: (Peter J. Hammond)
Readings: FMEA ch. 12, plus notes.
Slides on dynamic programming