grandes-ecoles 2014 QIII.C.1

grandes-ecoles · France · centrale-maths1__psi Not Maths

We assume $\alpha = 1$. We denote $\|\cdot\|$ the norm associated with $S_1$, and $$\varphi_1(y) : t \mapsto \left(1-t^2\right)y''(t) - 3t\,y'(t)$$ Justify that, for all $k \in \mathbb{N}$, there exists a unique polynomial eigenvector of $\varphi_1$ of degree $k$, of norm 1 and with positive leading coefficient. We denote it $T_k$.

We assume $\alpha = 1$. We denote $\|\cdot\|$ the norm associated with $S_1$, and
$$\varphi_1(y) : t \mapsto \left(1-t^2\right)y''(t) - 3t\,y'(t)$$
Justify that, for all $k \in \mathbb{N}$, there exists a unique polynomial eigenvector of $\varphi_1$ of degree $k$, of norm 1 and with positive leading coefficient. We denote it $T_k$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.A.6 QI.A.7 QI.B.1 QI.B.2 QI.B.3 QI.B.4 QII.A.1 QII.A.2 QII.B.1 QII.B.2 QII.C.1 QII.C.2 QII.C.3 QII.C.4 QII.C.5 QII.C.6 QII.C.7 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.B.5 QIII.C.1 QIII.C.2 QIII.C.3 QIII.C.4 QIII.C.5 QIII.C.6