# Pre-sessional Advanced Mathematics

## EC9A0 course outline and notes

The course will be taught by Peter Hammond and Pablo Beker and will run from Monday 16th September to Friday 27th September, Monday to Friday, excluding weekends. There will be four hours of teaching each day. All teaching for the first week will take place in room S2.79 (second floor of the Social Sciences building), and the second week in S0.10Link opens in a new window (ground floor of the Social Sciences Building).

Greek alphabet (for mathematically inclined economists)

Click hereLink opens in a new window for Reading List.

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

Matrix algebra slides, Part ALink opens in a new window

Matrix algebra slides, Part BLink opens in a new window

Matrix algebra slides, Part CLink opens in a new window

Matrix algebra slides, Part DLink opens in a new window

Matrix algebra slides, Part D appendixLink opens in a new window

Matrix algebra slides, Part ELink opens in a new window

Matrix algebra slides, Part FLink opens in a new window

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

Readings: FMEA chs. 11, 5, 6, 7.Part ALink opens in a new window

Part BLink opens in a new window

Part CLink opens in a new window

Part DLink opens in a new window

**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 AnalysisLink opens in a new window (updated 19/9/24)

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 OptimisationLink opens in a new window (updated 22/9/23)

Lecture Notes: Unconstrained Optimisation

**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 ConstraintsLink opens in a new window. (updated 22/9/23)

Slides: Constrained Optimisation with Inequality Constraints</a>.Link opens in a new window

Lecture Notes: Constrained Optimisation I

Lecture Notes: Constrained Optimisation II

**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 TheoremLink opens in a new window

Lecture Notes: Theorem of the Maximum and Envelope Theorem

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

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

Slides on calculus of variationsLink opens in a new window

Slides on optimal controlLink opens in a new window

**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.

Part ALink opens in a new window

Part BLink opens in a new window

Part CLink opens in a new window

**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 TheoremsLink opens in a new window

Lecture Notes: Fixed Point Theorems

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

Readings: FMEA ch. 12, plus notes.

Slides on dynamic programmingLink opens in a new window