Preparatory Math Course for the Intake 2013

Guest Instructor: 
Anton Velinov
Christopher Boortz
Time I: 
Monday, 09:00am to 04:00pm
Venue I: 
HU Berlin, Hegelplatz 1, Seminargebäude, Room 1.403

Starting Date: October 7

Preliminary Outline:

--> Logic and Sets:
ˆ Logic: methods of proof
-Constructive proof
-Contrapositive proof
-Proof by contradiction
ˆ De Morgan's laws
ˆ Convexity

--> Functions:
ˆ Mappings: surjective, injective, bijective
ˆ correspondence: Upper and lower hemicontinuity
ˆ Continuity and dierentiability
ˆ Integration
ˆ Concavity and convexity (for multidimensional functions)
ˆ Quasiconcavity
ˆ Taylor's Theorem
ˆ Implicit Function Theorem
ˆ Homogeneous functions and Euler's formula

--> Sequences:
ˆ Metric and normed spaces
ˆ Convergence of sequences in metric spaces
ˆ Bounded and Compact sets
ˆ Limits of functions
ˆ Contraction Mapping Theorem
ˆ Extreme Value Theorem
ˆ Separating Hyperplane Theorem
ˆ Fixed Point Theorems

--> Matrix Algebra:
ˆ Addition, subtraction, multiplication
ˆ Trace, determinant, rank
ˆ Inversion
ˆ Positive deniteness
ˆ Eigenvalues, Eigenvectors
ˆ Diagonalization, Jordan decomposition, Choleski decomposition
ˆ Matrix dierentiation, Kronecker products, vectorization

--> Complex numbers:
ˆ sin, cos functions
ˆ De Moivre's Theorem

--> Additional Topics:
ˆ Unconstrained optimization
ˆ Constrained optimization
ˆ Linear programming
ˆ Dynamic programming

Fall 2013
Humboldt-Universität zu Berlin