5 Lecturer : Michel

Generating Functions Lecturer : Michel Goemans

We are going to discuss enumeration problems, and how to solve them using a powerful tool: generating functions. What is an enumeration problem? That’s trying to determine the number of objects of size n satisfying a certain definition. For instance, what is the number of permutations of {1, 2, . . . , n}? (answer: n!), or what is the number of binary sequences of length n? (answer: 2n). Ok, no...

Probability Theory Lecturer : Michel Goemans

To treat probability rigorously, we define a sample space S whose elements are the possible outcomes of some process or experiment. For example, the sample space might be the outcomes of the roll of a die, or flips of a coin. To each element x of the sample space, we assign a probability, which will be a non-negative number between 0 and 1, which we will denote by p(x). We require that p(x) = ...

What to Make of Michel Foucault's Perceptions of the Iranian Revolution

Lecturer : Michel

The point (0.5, 0.5, 0.5) ∈ P1, i.e. it satisfies the constraints above; however it is not in the convex hull of the matching vectors. The above example motivates the following family of additional constraints (introduced by Edmonds). Observe that for any matching M , the subgraph induced by M on any odd cardinality vertex subset U has at most (|U | − 1)/2 edges. Thus, without losing any of the...

Lectures 9 and 10 Lecturer : Michel

1 Semidefinite programming Let Sn×n be the set of n by n real symmetric matrices. Definition 1 A ∈ Sn×n is called positive semidefinite, denoted A 0, if xAx ≥ 0 for any x ∈ R. There are several well-known equivalent ways to state positive semidefiniteness. Proposition 1 The following are equivalent: (i) A is positive semidefinite. (ii) Every eigenvalue of A is nonnegative. (iii) There is a matr...