A proof of the irrationality of \(\zeta(2)=\sum_{n=1}^{+\infty} \frac{1}{n^2}\).

The main topic of this problem set is the optimization of multivariable functions.

The first part of the problem is about polynomial osculation. The second is linear algebra in \(\mathbb{R}_n[X]\) fitted with an inner product. The third is about the power series development of \(r\mapsto (1-2rx+r^2)^{-\lambda}\).In the last part, we introduce positive-type functions in dimension \(N\).

This year's A problem set was pure linear algebra. With lots of matrices. Jordan decomposition was the main theme, with an emphasis on nilpotent endomorphisms. Please note that, although complete, this document is temporary and still needs careful proofr…

We've seen that a qubit lives in a complex vector space of dimension 2, whose basis is given by \((\ket0, \ket1)\). Let's call this vector space \(E\). Now, how do we describe the state of 2 qubits? We introduce the tensor product of \(E\) with itself, deno…