Show that for all functions $f, g \in C^{2}(\mathbf{R})$ such that the functions $f, f^{\prime}, f^{\prime\prime}$ and $g$ have slow growth, we have
$$\int_{-\infty}^{+\infty} L(f)(x) g(x) \varphi(x) \mathrm{d}x = -\int_{-\infty}^{+\infty} f^{\prime}(x) g^{\prime}(x) \varphi(x) \mathrm{d}x$$
where $\forall x \in \mathbf{R}, \quad L(f)(x) = f^{\prime\prime}(x) - x f^{\prime}(x).$

Let $g$ be the function defined by
$$\begin{aligned} g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Deduce that
$$\int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t } \mathrm {~d} t = \frac { \pi } { \sin ( \pi x ) }$$

Show that for all functions $f, g \in C^2(\mathbf{R})$ such that the functions $f, f', f''$ and $g$ have slow growth, we have $$\int_{-\infty}^{+\infty} L(f)(x)\,g(x)\,\varphi(x)\,\mathrm{d}x = -\int_{-\infty}^{+\infty} f'(x)\,g'(x)\,\varphi(x)\,\mathrm{d}x,$$ where $L(f)(x) = f''(x) - x f'(x)$ and $\varphi(x) = \frac{1}{\sqrt{2\pi}}\mathrm{e}^{-x^2/2}$.

Deduce the characteristic polynomial of a graph with $n$ vertices whose non-isolated vertices form a star with $d$ branches with $1 \leq d \leq n - 1$. Then determine the eigenvalues and eigenvectors of an adjacency matrix of this graph.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ We denote $W = \left\{y \in E \left\lvert\, \|y - f(a)\| < \frac{r}{4}\right.\right\}$ and $V = f^{-1}(W) \cap B(a,r)$.
Justify that $V$ and $W$ are open sets of $E$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ We denote $W = \left\{y \in E \left\lvert\, \|y - f(a)\| < \frac{r}{4}\right.\right\}$ and $V = f^{-1}(W) \cap B(a,r)$.
Show that $$\begin{array}{clcc} f_{\mid V} : & V & \longrightarrow & W \\ & x & \longmapsto & f(x) \end{array}$$ is a continuous bijection from $V$ to $W$ whose inverse is a continuous function on $W$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ We denote by $W = \left\{y \in E \left\lvert\, \|y - f(a)\| < \frac{r}{4}\right.\right\}$ and $V = f^{-1}(W) \cap B(a,r)$.
Justify that $V$ and $W$ are open sets of $E$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ We denote by $W = \left\{y \in E \left\lvert\, \|y - f(a)\| < \frac{r}{4}\right.\right\}$ and $V = f^{-1}(W) \cap B(a,r)$.
Show that $$f_{\mid V} : \begin{array}{lcc} V & \longrightarrow & W \\ x & \longmapsto & f(x) \end{array}$$ is a continuous bijection from $V$ to $W$ whose inverse is a continuous function on $W$.

For all $g = (\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$, we denote by $\phi_{g} : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ the map defined by $\phi_{g}(x) = Rx + \tau$. Let $g \in \operatorname{Dep}(\mathbb{R}^{d})$.

• [(a)] Show that $\phi_{g}$ is bijective. We denote by $\phi_{g}^{-1}$ its inverse map.
• [(b)] Show that there exists a unique $g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$, which we will express in terms of $g$, such that $\phi_{g^{\prime}} = \phi_{g}^{-1}$. We denote $g^{\prime} = g^{-1}$.
• [(c)] Verify that $ge = eg = g$ and then that $gg^{-1} = g^{-1}g = e$.

Show that if $f \in C^{1}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime} \in CL(\mathbf{R})$ and $x \in \mathbf{R}$, then $t \in \mathbb{R}_{+} \mapsto P_{t}(f)(x)$ is of class $C^{1}$ on $\mathbb{R}_{+}$ and show that for all $t > 0$, we have
$$\frac{\partial P_{t}(f)(x)}{\partial t} = \int_{-\infty}^{+\infty} \left(-x \mathrm{e}^{-t} + \frac{\mathrm{e}^{-2t}}{\sqrt{1 - \mathrm{e}^{-2t}}} y\right) f^{\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y$$

Show that if $f \in C^1(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f' \in CL(\mathbf{R})$ and $x \in \mathbf{R}$, then $t \in \mathbf{R}_+ \mapsto P_t(f)(x)$ is of class $C^1$ on $\mathbf{R}_+^*$ and show that for all $t > 0$, we have $$\frac{\partial P_t(f)(x)}{\partial t} = \int_{-\infty}^{+\infty} \left(-x\mathrm{e}^{-t} + \frac{\mathrm{e}^{-2t}}{\sqrt{1-\mathrm{e}^{-2t}}}\,y\right) f'\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y.$$

Let $G _ { 1 } = \left( S _ { 1 } , A _ { 1 } \right)$ and $G _ { 2 } = \left( S _ { 2 } , A _ { 2 } \right)$ be two non-empty graphs such that $S _ { 1 }$ and $S _ { 2 }$ are disjoint, that is, such that $S _ { 1 } \cap S _ { 2 } = \varnothing$. Let $s _ { 1 } \in S _ { 1 }$ and let $s _ { 2 } \in S _ { 2 }$.
We define the graph $G = ( S , A )$ with $S = S _ { 1 } \cup S _ { 2 }$ and $A = A _ { 1 } \cup A _ { 2 } \cup \left\{ \left\{ s _ { 1 } , s _ { 2 } \right\} \right\}$.
Show that : $$\chi _ { G } = \chi _ { G _ { 1 } } \times \chi _ { G _ { 2 } } - \chi _ { G _ { 1 } \backslash s _ { 1 } } \times \chi _ { G _ { 2 } \backslash s _ { 2 } }$$

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 $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Justify that $(\mathbb{C}[A])^*$ is an abelian subgroup of $\mathrm{GL}_n(\mathbb{C})$.

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 $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Show that $(\mathbb{C}[A])^* = \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C})$.

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 $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Justify that $\mathbb{C}[A]^*$ is an abelian subgroup of $\mathrm{GL}_n(\mathbb{C})$.

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 $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Show that $(\mathbb{C}[A])^* = \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C})$.

Let $f \in C^{2}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime}$ and $f^{\prime\prime}$ have slow growth and $t \in \mathbf{R}_{+}$.
Show that $x \in \mathbb{R} \mapsto P_{t}(f)(x)$ is of class $C^{2}$ on $\mathbf{R}$. Also show that
$$\forall x \in \mathbf{R}, \quad P_{t}(f)^{\prime}(x) = \mathrm{e}^{-t} \int_{-\infty}^{+\infty} f^{\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y$$
and
$$\forall x \in \mathbf{R}, \quad P_{t}(f)^{\prime\prime}(x) = \mathrm{e}^{-2t} \int_{-\infty}^{+\infty} f^{\prime\prime}\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y.$$

Let $f \in C^2(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f'$ and $f''$ have slow growth and $t \in \mathbf{R}_+$. Show that $x \in \mathbf{R} \mapsto P_t(f)(x)$ is of class $C^2$ on $\mathbf{R}$. Also show that $$\forall x \in \mathbf{R}, \quad P_t(f)'(x) = \mathrm{e}^{-t}\int_{-\infty}^{+\infty} f'\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y$$ and $$\forall x \in \mathbf{R}, \quad P_t(f)''(x) = \mathrm{e}^{-2t}\int_{-\infty}^{+\infty} f''\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y.$$

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Let $$m = \min\{k \in \mathbb{N} \mid \exists P \in \mathscr{V}(A) \text{ with } \deg(P) = k\}$$ Show that there exists a unique polynomial $p \in \mathbb{C}[X]$ satisfying the three conditions
(i) $p \in \mathscr{V}(A)$,
(ii) $\deg(p) = m$,
(iii) $p$ is monic.

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 $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Show that $\exp(\mathbb{C}[A]) \subset (\mathbb{C}[A])^*$.

We consider $n$ a strictly positive integer and $\mathscr{E}_{d}^{n}(\mathbb{R}) = \{ \boldsymbol{z} = (\boldsymbol{z}_{i})_{1 \leqslant i \leqslant n} \mid \boldsymbol{z}_{i} \in \mathbb{R}^{d}, 1 \leqslant i \leqslant n \}$ the vector space of families of $n$ points in $\mathbb{R}^{d}$ equipped with the norm $\|\boldsymbol{z}\| = \sqrt{\sum_{i=1}^{n} |\boldsymbol{z}_{i}|^{2}}$. For all $g \in \operatorname{Dep}(\mathbb{R}^{d})$ and $\boldsymbol{z} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ we denote $g \cdot \boldsymbol{z} = (\phi_{g}(\boldsymbol{z}_{i}))_{1 \leqslant i \leqslant n}$.

• [(a)] Show that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$ and $\boldsymbol{z} \in \mathscr{E}_{d}^{n}(\mathbb{R})$, we have $g \cdot (g^{\prime} \cdot \boldsymbol{z}) = (gg^{\prime}) \cdot \boldsymbol{z}$.
• [(b)] Show that for all $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ and all $g \in \operatorname{Dep}(\mathbb{R}^{d})$, if $\boldsymbol{x} = g \cdot \boldsymbol{y}$ then $\boldsymbol{y} = g^{-1} \cdot \boldsymbol{x}$.

Deduce that for $f \in C^{2}(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f^{\prime}$ and $f^{\prime\prime}$ have slow growth, we have
$$\forall t \in \mathbf{R}_{+}^{*}, \forall x \in \mathbf{R}, \quad \frac{\partial P_{t}(f)(x)}{\partial t} = L\left(P_{t}(f)\right)(x)$$
where $\forall x \in \mathbf{R}, \quad L(f)(x) = f^{\prime\prime}(x) - x f^{\prime}(x).$

Deduce that for $f \in C^2(\mathbf{R}) \cap CL(\mathbf{R})$ such that $f'$ and $f''$ have slow growth, we have $$\forall t \in \mathbf{R}_+^*, \forall x \in \mathbf{R}, \quad \frac{\partial P_t(f)(x)}{\partial t} = L\!\left(P_t(f)\right)(x),$$ where $L(f)(x) = f''(x) - xf'(x)$.

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 $\omega(\sigma) = k$.
Specify $s(n,n)$ and $s(n,1)$ then show that, for $2 \leqslant k \leqslant n-1$, we have $$s(n,k) = s(n-1, k-1) + (n-1) s(n-1, k)$$ For $\sigma \in \mathfrak{S}_{n}$, one may distinguish the cases $\sigma(1) = 1$ and $\sigma(1) \neq 1$.

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 $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t) \end{array}$$ Show that the application $$\begin{array}{rcc} ]0,1[ \times \mathbb{R} & \longrightarrow & \mathbb{C} \\ (t,a) & \longmapsto & Z_a(t) \end{array}$$ is injective.