grandes-ecoles 2015 QI.B.3

grandes-ecoles · France · centrale-maths2__psi Groups Decomposition and Basis Construction

a) Show that the set of harmonic polynomials is a vector subspace of $\mathcal{P}$.
b) For all $m \geqslant 2$, we denote by $\Delta_m$ the restriction of $\Delta$ to $\mathcal{P}_m$. Show that $\operatorname{dim}(\operatorname{ker} \Delta_m) \geqslant 2m+1$.
c) What can be deduced about the dimension of the vector space of harmonic polynomials?

a) Show that the set of harmonic polynomials is a vector subspace of $\mathcal{P}$.

b) For all $m \geqslant 2$, we denote by $\Delta_m$ the restriction of $\Delta$ to $\mathcal{P}_m$. Show that $\operatorname{dim}(\operatorname{ker} \Delta_m) \geqslant 2m+1$.

c) What can be deduced about the dimension of the vector space of harmonic polynomials?

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QI.B.3 QI.C.1 QI.C.2 QII.A QII.B.1 QII.B.2 QII.B.3 QII.C.1 QII.C.2 QII.D.1 QII.D.2 QII.D.3 QII.D.4 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C.1 QIV.C.2