grandes-ecoles 2016 Q10

grandes-ecoles · France · x-ens-maths__pc Proof Computation of a Limit, Value, or Explicit Formula

Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.
Identify $\Sigma_{N}(f)$. Show that $J_{f,*} = \ln\left(\sum_{i=1}^{N} e^{f_{i}}\right)$.

Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.

Identify $\Sigma_{N}(f)$. Show that $J_{f,*} = \ln\left(\sum_{i=1}^{N} e^{f_{i}}\right)$.

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