Study the variations of the function $t \mapsto t \ln(t)$ on $\mathbf{R}_{+}^{*}$. Verify that the function can be extended by continuity at 0 and verify that the quantity $\operatorname{Ent}_{\varphi}(g)$ is well defined for all $g \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ with strictly positive values such that $\int_{-\infty}^{+\infty} g(x) \varphi(x) \mathrm{d}x = 1$.
Recall that for $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ with strictly positive values such that $\int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x = 1$, the entropy of $f$ with respect to $\varphi$ is defined by: $$\operatorname{Ent}_{\varphi}(f) = \int_{-\infty}^{+\infty} \ln(f(x)) f(x) \varphi(x) \mathrm{d}x$$

Study the variations of the function $t \mapsto t\ln(t)$ on $\mathbf{R}_+^*$. Verify that we can extend the function by continuity at $0$.

We denote by $\mathbf{e}$ the matrix of $\mathcal{M}_{n,1}(\mathbb{R})$ whose coefficients are all equal to 1, and $P = I_n - \frac{1}{n} \mathbf{e} \cdot \mathbf{e}^T$. We denote by $\Delta_n$ the set of EDM of order $n$ and $\Omega_n$ the set of symmetric positive matrices of order $n$ such that $M \cdot \mathbf{e} = 0$. We denote by $T$ the application from $\Delta_n$ to $\mathcal{M}_n(\mathbb{R})$ which associates to $D$ $$T(D) = -\frac{1}{2} P D P.$$
Let $D \in \Delta_n$. Let $A_1, \ldots, A_n$ be points whose matrix $D$ is the Euclidean distance matrix. We denote by $x_i$ the coordinate vectors of the $A_i$ and $M_A$ the matrix whose columns are the $x_i$ and $C$ the column formed by the $\|x_i\|^2$. Write $D$ as a linear combination of $C\mathbf{e}^T$, $\mathbf{e}C^T$ and $M_A^T \cdot M_A$. Deduce that for every matrix $D$ of $\Delta_n$ we have $T(D) \in \Omega_n$.

Let $n$ be a non-zero natural integer. For any permutation $\sigma \in \mathfrak{S}_{n}$, we recall that there exists, up to order, a unique decomposition $\sigma = c_{1} c_{2} \cdots c_{\omega(\sigma)}$, where $\omega(\sigma) \in \mathbb{N}^{*}$ where $c_{1}, \ldots, c_{\omega(\sigma)}$ are cycles with disjoint supports of respective lengths $\ell_{1} \leqslant \ell_{2} \leqslant \cdots \leqslant \ell_{\omega(\sigma)}$ and $\ell_{1} + \ell_{2} + \cdots + \ell_{\omega(\sigma)} = n$. For an integer $k$ at most $n$, we denote by $s(n,k)$ the number of permutations of $\mathfrak{S}_{n}$ such that $\omega(\sigma) = k$.
Establish that, for any real $x$, $\prod_{i=0}^{n-1}(x+i) = \sum_{k=1}^{n} s(n,k) x^{k}$.

Justify that the quantity $\operatorname{Ent}_{\varphi}(g)$ is well defined for all $g \in C^0(\mathbf{R}) \cap CL(\mathbf{R})$ with strictly positive values such that $\int_{-\infty}^{+\infty} g(x)\varphi(x)\,\mathrm{d}x = 1$, where $$\operatorname{Ent}_{\varphi}(g) = \int_{-\infty}^{+\infty} \ln(g(x))\,g(x)\,\varphi(x)\,\mathrm{d}x.$$
Hint: You may use question 11.

For $t \in \mathbf{R}_{+}$, we set $S(t) = \operatorname{Ent}_{\varphi}\left(P_{t}(f)\right)$. Justify that $S(t)$ is well defined.
Here $f$ is an element of $C^{2}(\mathbf{R})$ with strictly positive values such that the functions $f, f^{\prime}, f^{\prime\prime}$ and $\frac{f^{\prime 2}}{f}$ have slow growth, and $\int_{-\infty}^{+\infty} f(x) \varphi(x) \mathrm{d}x = 1$.

For $t \in \mathbf{R}_+$, we set $S(t) = \operatorname{Ent}_{\varphi}(P_t(f))$. Justify that $S(t)$ is well defined.

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ We want to show that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$. We suppose that $\exp(\mathbb{C}[A]) \neq (\mathbb{C}[A])^*$ and we fix $M_1, M_2 \in (\mathbb{C}[A])^*$ such that $M_1 \in \exp(\mathbb{C}[A])$ and $M_2 \notin \exp(\mathbb{C}[A])$.
Show that there exists a continuous map $f$ from $(\mathbb{C}[A])^*$ to $\{0,1\}$ such that $f(M_1) = 0$ and $f(M_2) = 1$.

Show that $S$ is continuous on $\mathbf{R}_+$.
Hint: You may first show that, if $x \in \mathbf{R}$, $t \mapsto P_t(f)(x)$ is continuous on $\mathbf{R}_+$.

Verify that we have $S(0) = \operatorname{Ent}_{\varphi}(f)$ and $\lim_{t \rightarrow +\infty} S(t) = 0$.

We admit that $S$ is of class $C^1$ on $\mathbf{R}_+^*$ and that $$\forall t \in \mathbf{R}_+^*, \quad S'(t) = \int_{-\infty}^{+\infty} \frac{\partial P_t(f)(x)}{\partial t}\left(1 + \ln\!\left(P_t(f)(x)\right)\right)\varphi(x)\,\mathrm{d}x.$$ Show that $$\forall t \in \mathbf{R}_+^*, \quad S'(t) = \int_{-\infty}^{+\infty} L\!\left(P_t(f)\right)(x)\left(1 + \ln\!\left(P_t(f)(x)\right)\right)\varphi(x)\,\mathrm{d}x.$$

By admitting that the result of question 7 is valid for the functions $P_t(f)$ and $1 + \ln(P_t(f))$, show that $$\forall t \in \mathbf{R}_+^*, \quad -S'(t) = \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} \frac{P_t(f')(x)^2}{P_t(f)(x)}\,\varphi(x)\,\mathrm{d}x.$$

By using the Cauchy-Schwarz inequality, show that $$\forall t \in \mathbf{R}_+^*, \quad -S'(t) \leq \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} P_t\!\left(\frac{f'^2}{f}\right)(x)\,\varphi(x)\,\mathrm{d}x.$$

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. Its closure, denoted $\overline { U _ { n } }$, is $\left( \mathbb { R } _ { + } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $M _ { n }$ the maximum of $F _ { n }$ on $\overline { U _ { n } } \cap H _ { n }$ and we denote by $( a _ { 1 } , \ldots , a _ { n } )$ a point of $U _ { n } \cap H _ { n }$ at which it is attained. For $k$ between 1 and $n$, we denote $\gamma _ { k } = \left( a _ { 1 } a _ { 2 } \cdots a _ { k } \right) ^ { 1 / k }$. There exists a real number $\lambda > 0$ satisfying the system in Q17.
Deduce that:
a) $\lambda = \gamma _ { 1 } + \gamma _ { 2 } + \cdots + \gamma _ { n } = M _ { n }$;
b) for all $k$ in $\llbracket 1 , n \rrbracket , \gamma _ { k } = \lambda \omega _ { k } a _ { k }$, where
$$\left\{ \begin{array} { l } \omega _ { k } = k \left( 1 - \frac { a _ { k + 1 } } { a _ { k } } \right) \text { if } k \in \llbracket 1 , n - 1 \rrbracket \\ \omega _ { n } = n \end{array} \right.$$

Deduce that we have: $$\forall t \in \mathbf{R}_+^*, \quad -S'(t) \leq \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} \frac{f'^2(x)}{f(x)}\,\varphi(x)\,\mathrm{d}x.$$

Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. Let $X _ { n } ^ { 0 }$ be the discrete real random variable defined on $\mathcal{E}_n$ such that for $G \in \Omega_n$, the integer $X_n^0(G)$ equals the number of copies of $G_0$ contained in $G$. Express $X _ { n } ^ { 0 }$ using random variables of the type $X _ { H }$, and show that : $$\mathbf { E } \left( X _ { n } ^ { 0 } \right) = \sum _ { H \in \mathcal { C } _ { 0 } } \mathbf { P } ( H \subset G ) \leq n ^ { s _ { 0 } } p _ { n } ^ { a _ { 0 } } .$$

The objective of this question is part of proving that $\lambda \leqslant \mathrm { e }$. We assume by contradiction that $\lambda > \mathrm { e }$.
Verify that, for all $k$ in $\mathbb { N } , \frac { 1 } { \mathrm { e } } \leqslant \left( \frac { k + 1 } { k + 2 } \right) ^ { k + 1 }$.

Establish the following inequality $$\operatorname{Ent}_{\varphi}(f) \leq \frac{1}{2}\int_{-\infty}^{+\infty} \frac{f'^2(x)}{f(x)}\,\varphi(x)\,\mathrm{d}x.$$

Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. Let $X _ { n } ^ { 0 }$ be the discrete real random variable counting the number of copies of $G_0$ contained in $G \in \Omega_n$, and let $$\omega _ { 0 } = \min _ { \substack { H \subset G _ { 0 } \\ a _ { H } \geq 1 } } \frac { s _ { H } } { a _ { H } }$$ Deduce that if $p _ { n } = \mathrm { o } \left( n ^ { - \omega _ { 0 } } \right)$, then $\lim _ { n \rightarrow + \infty } \mathbf { P } \left( X _ { n } ^ { 0 } > 0 \right) = 0$. Hint: one may introduce $H _ { 0 } \subset G _ { 0 }$ achieving the minimum giving $\omega _ { 0 }$.

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $M _ { n }$ the maximum of $F _ { n }$ on $\overline { U _ { n } } \cap H _ { n }$ and we denote by $( a _ { 1 } , \ldots , a _ { n } )$ a point of $U _ { n } \cap H _ { n }$ at which it is attained. For $k$ between 1 and $n$, we denote $\gamma _ { k } = \left( a _ { 1 } a _ { 2 } \cdots a _ { k } \right) ^ { 1 / k }$. We assume by contradiction that $\lambda > \mathrm { e }$, where $\lambda$ is the real number from Q17, and $\omega_k$ is as defined in Q18b.
Prove that $\omega _ { 1 } \leqslant \frac { 1 } { \mathrm { e } }$ and that, for all $k$ in $\llbracket 1 , n \rrbracket , \omega _ { k } \leqslant \frac { k } { k + 1 }$.
You may prove, for $k \in \llbracket 1 , n - 1 \rrbracket$, that $\omega _ { k + 1 } ^ { k + 1 } = \frac { 1 } { \lambda } \omega _ { k } ^ { k } \left( 1 - \frac { \omega _ { k } } { k } \right) ^ { - k }$.

Let $H$ be a Hadamard matrix of order $n$ with first row constant equal to 1. Let $\lambda_1, \ldots, \lambda_n$ be real numbers such that $$\lambda_1 > 0 \geq \lambda_2 \geq \ldots \geq \lambda_n$$ and $$\sum_{i=1}^{n} \lambda_i = 0.$$ We denote by $U$ the matrix $\frac{1}{\sqrt{n}} H$ and $\Lambda$ the diagonal matrix whose diagonal coefficients are the $\lambda_i$. We finally denote by $D = U^T \Lambda U$.
Give a Euclidean distance matrix of order 4 such that its spectrum is $\{5, -1, -2, -2\}$.

Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. We now assume that $\lim _ { n \rightarrow + \infty } \left( n ^ { \omega _ { 0 } } p _ { n } \right) = + \infty$. Show that the expectation $\mathbf { E } \left( \left( X _ { n } ^ { 0 } \right) ^ { 2 } \right)$ satisfies : $$\mathbf { E } \left( \left( X _ { n } ^ { 0 } \right) ^ { 2 } \right) = \sum _ { \left( H , H ^ { \prime } \right) \in C _ { 0 } ^ { 2 } } \mathbf { P } \left( H \cup H ^ { \prime } \subset G \right) = \sum _ { \left( H , H ^ { \prime } \right) \in C _ { 0 } ^ { 2 } } p _ { n } ^ { 2 a _ { 0 } - a _ { H \cap H ^ { \prime } } } .$$

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $M _ { n }$ the maximum of $F _ { n }$ on $\overline { U _ { n } } \cap H _ { n }$ and we denote by $( a _ { 1 } , \ldots , a _ { n } )$ a point of $U _ { n } \cap H _ { n }$ at which it is attained. For $k$ between 1 and $n$, we denote $\gamma _ { k } = \left( a _ { 1 } a _ { 2 } \cdots a _ { k } \right) ^ { 1 / k }$. We assume by contradiction that $\lambda > \mathrm { e }$, where $\lambda$ is the real number from Q17, and $\omega_k$ is as defined in Q18b.
Reach a contradiction on $\omega _ { n }$. Deduce that, for all $n$ in $\mathbb { N } ^ { * }$, for all $\left( x _ { 1 } , \ldots , x _ { n } \right) \in \left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$ such that $x _ { 1 } + \cdots + x _ { n } = 1$,
$$\sum _ { k = 1 } ^ { n } \left( x _ { 1 } x _ { 2 } \cdots x _ { k } \right) ^ { 1 / k } \leqslant \mathrm { e }$$

Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix from the normal form $M(t) = Q(t)\exp(tB)$. Let the differential system $$X'(t) = A(t) X(t) + b(t) \tag{3}$$ where $b : \mathbb{R} \rightarrow \mathbb{C}^n$ is a continuous function on $\mathbb{R}$ and $T$-periodic.
We assume that $1$ is not an eigenvalue of $\exp(TB)$. Show that (3) possesses a unique $T$-periodic solution.

Let $G _ { 0 } = \left( S _ { 0 } , A _ { 0 } \right)$ be a particular fixed graph with $s _ { 0 } = s _ { G _ { 0 } }$, $a _ { 0 } = a _ { G _ { 0 } }$, $s_0 \geq 2$, $a_0 \geq 1$. We now assume that $\lim _ { n \rightarrow + \infty } \left( n ^ { \omega _ { 0 } } p _ { n } \right) = + \infty$. For $k \in \llbracket 0 , s _ { 0 } \rrbracket$, we denote: $$\Sigma _ { k } = \sum _ { \substack { \left( H , H ^ { \prime } \right) \in C _ { 0 } ^ { 2 } \\ s _ { H \cap H ^ { \prime } } = k } } \mathbf { P } \left( H \cup H ^ { \prime } \subset G \right)$$ Show that $\Sigma _ { 0 } \leq \left( \mathbf{E} \left( X _ { n } ^ { 0 } \right) \right) ^ { 2 }$.