DRE7007 Preparatory course in Mathematics, PhD - 2012/13

Reading

Compulsory reading: There is a certain overlap between [FMEA] and [ME]. Some advanced topics (in particular optimal control theory) are only covered in [FMEA], while [ME] is better and more complete for many of the more basic themes, and with many more examples and exercises. The following books are also useful (the book by Sundaram is especially recommended; the others are more difficult to read):

Lecture Plan

Date: Topic: Reading:
Aug 23
1200-1400 : A2-050 		Lecture 1 - [LSGE] Notes on Gaussian elimination:
Matrices. Eigenvalues and eigenvectors. Quadratic forms and definiteness. 		[FMEA] 1.1 - 1.7 [ME] 6 - 9, 23
[S] 1.3, 1.5
Aug 24
1200-1400 : A2-050 		Lecture 2 - Notes on Euclidean Spaces, sequences and topology
Euclidean spaces. Sequences. Topology. 		 [FMEA] A.1 - A.3, 13.1 -13.2
[ME] A1, 10, 12, 29 [S] A, C, 1.1 - 1.2
Aug 27
1200-1400 : A2-005 		Lecture 3
Functions. Continuity. Derivatives.
 [FMEA] 2.9, 13.3 [ME] 13.4, 14
[S] 1.4, 3
Aug 28
1200-1400 : C2-095 		Lecture 4
Convex sets. Separation theorems. (Quasi)Convex/concave functions.
 [FMEA] 2.2 - 2.3, 2.5, 13.5 - 13.6 [ME]
21.1 - 21.2 [S] 1.2, 1.6, 7.1 - 7.2, 8.1 - 8.3 
Aug 30
1200-1400 : A2-050 		Lecture 5
Optimization problems. Unconstrained optimization
 [FMEA] 3.1 - 3.2 [ME] 17
[S] 2, 4, 7.3 - 7.6, 8.4 - 8.7
Aug 31
1200-1400 : A2-050 		Lecture 6
Constrained optimization. Lagrange and Kuhn-Tucker problems.		 [FMEA] 3.3 - 3.6 [ME] 18 - 19
[S] 5 - 6, 7.7, 8.8
Sep 03
1200-1400 : A2-050 		Lecture 7
Differential equations. Systems of differential equations. Linearization. 
[FMEA] 5 - 7 [ME] 24 - 25
  
Sep 04
1200-1400 : A2-050 		Lecture 8
Optimal control theory: Continuous case, Pontryagin's maximum principle 		[FMEA] 9 (8, 10)
 
Sep 06
1400-1600 : C2-095 		Lecture 9
Optimal control theory: Discret case, Bellman's equation		 [FMEA] 12 [S] 11- 12
 
Sep 07
1400-1600 : C2-095 		Lecture 10 
Fixed points and fixed point theorems. Correspondences.
[FMEA] 14 [S] 9, 12
  
Sep 17 at 1200
  		Deadline: Final submission of portfolio
There will be 10 assignments, and at least 8 have to be passed to complete the course.

Problem Sets for the course Portfolio

Lecture: Problems:
Lecture 1 Problem Set 1 - Solutions Problem Set 1
Lecture 2 Problem Set 2 - Solutions Problem Set 2
Lecture 3 Problem Set 3 - Solutions Problem Set 3
Lecture 4 Problem Set 4 - Solutions Problem Set 4
Lecture 5 Problem Set 5 - Solutions Problem Set 5
Lecture 6
Problem Set 6 - Solutions Problem Set 6
Lecture 7 Problem Set 7 - Solutions Problem Set 7
Lecture 8 Problem Set 8 - Solutions Problem Set 8
Lecture 9 Problem Set 9 - Solutions Problem Set 9
Lecture 10 Problem Set 10 - Solutions Problem Set 10