Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. If $i$, $j$ and $k$ are three integers in $\llbracket 1, n \rrbracket$, the second partial derivative of $f_k$ at $x$ with respect to the variables $x_i$ and $x_j$ is denoted $f_{i,j,k}(x)$.
We assume that the Jacobian matrix $J_f(x)$ is antisymmetric for all $x$ in $\mathbb{R}^n$.
Show that there exist a real square matrix $A$ of size $n$ and an element $b$ of $\mathbb{R}^n$ such that for all $x$ in $\mathbb{R}^n$, $f(x) = Ax + b$. Justify that $A$ is antisymmetric.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. What is the necessary and sufficient condition on $f$ for the Jacobian matrix $J_f(x)$ to be antisymmetric for all $x$ in $\mathbb{R}^n$?

Now $f$ is a function of class $C^1$ from $\mathbb{R}^n$ to itself.
Show that the Jacobian matrix $J_f(x)$ is symmetric for all $x$ in $\mathbb{R}^n$ if and only if there exists $g$ of class $C^2$ on $\mathbb{R}^n$ with values in $\mathbb{R}$ such that $$\forall x \in \mathbb{R}^n, \forall i \in \llbracket 1, n \rrbracket, \quad f_i(x) = \mathrm{D}_i g(x)$$
One may consider the map $g$ defined by $g(x) = \sum_{i=1}^n x_i \int_0^1 f_i(tx)\, \mathrm{d}t$ and express $\mathrm{D}_i g(x)$ as a single integral.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal.
For $x$ in $\mathbb{R}^n$ and $i$, $j$, $k$ in $\llbracket 1, n \rrbracket$, we denote $$\alpha_{i,j,k}(x) = \sum_{p=1}^n \frac{\partial f_p}{\partial x_i}(x) \cdot \frac{\partial^2 f_p}{\partial x_j \partial x_k}(x)$$
We assume $(\mathcal{P})$. Show that for all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, $\alpha_{i,j,k} = \alpha_{i,k,j} = -\alpha_{k,j,i}$.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal.
For $x$ in $\mathbb{R}^n$ and $i$, $j$, $k$ in $\llbracket 1, n \rrbracket$, we denote $$\alpha_{i,j,k}(x) = \sum_{p=1}^n \frac{\partial f_p}{\partial x_i}(x) \cdot \frac{\partial^2 f_p}{\partial x_j \partial x_k}(x)$$
We assume $(\mathcal{P})$. Deduce that for all $i$, $j$ and $k$ in $\llbracket 1, n \rrbracket$, $\alpha_{i,j,k} = 0$.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal.
For $x$ in $\mathbb{R}^n$ and $i$, $j$, $k$ in $\llbracket 1, n \rrbracket$, we denote $$\alpha_{i,j,k}(x) = \sum_{p=1}^n \frac{\partial f_p}{\partial x_i}(x) \cdot \frac{\partial^2 f_p}{\partial x_j \partial x_k}(x)$$
We assume $(\mathcal{P})$. Show that there exist an orthogonal matrix $A$ and an element $b$ of $\mathbb{R}^n$ such that, for all $x$ in $\mathbb{R}^n$, $f(x) = Ax + b$.
One may interpret the relations $\alpha_{i,j,k} = 0$ using matrix products.

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal.
What is the necessary and sufficient condition on $f$ for proposition $(\mathcal{P})$ to hold?

Let $f$ be a function of class $C^2$ from $\mathbb{R}^n$ to itself. We consider the proposition $(\mathcal{P})$: for all $x$ in $\mathbb{R}^n$, the Jacobian matrix $J_f(x)$ of $f$ is orthogonal.
If $g$ is a function of class $C^2$ from $\mathbb{R}^n$ to $\mathbb{R}$, we denote $\Delta_g(x) = \sum_{i=1}^n \frac{\partial^2 g}{\partial x_i^2}(x)$ (Laplacian of $g$ at $x$). Show that $(\mathcal{P})$ is equivalent to the proposition $$\text{For every function } g \text{ of class } C^2 \text{ from } \mathbb{R}^n \text{ to } \mathbb{R},\quad \Delta_{g \circ f} = (\Delta_g) \circ f.$$

We denote $\alpha$ a real number such that $\alpha > -1/2$, $E$ the $\mathbb{R}$-vector space of functions of class $\mathcal{C}^\infty$ on $[-1,1]$ with real values, and $$S_\alpha(f,g) = \int_{-1}^{1} f(t)g(t)\left(1-t^2\right)^{\alpha - \frac{1}{2}} \mathrm{~d}t$$ Verify that $S_\alpha$ is an inner product on $E$.

We denote $\alpha > -1/2$, $E$ the $\mathbb{R}$-vector space of functions of class $\mathcal{C}^\infty$ on $[-1,1]$ with real values, $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ and $$S_\alpha(f,g) = \int_{-1}^{1} f(t)g(t)\left(1-t^2\right)^{\alpha - \frac{1}{2}} \mathrm{~d}t$$ Show that $$\forall (f,g) \in E^2, \quad S_\alpha\left(\varphi_\alpha(f), g\right) = S_\alpha\left(f, \varphi_\alpha(g)\right)$$ One may calculate the derivative of $t \mapsto \left(1-t^2\right)^{\alpha+\frac{1}{2}} f'(t)$.

Show that the set $O ( 1 , p )$ is a subgroup of $G L _ { p + 1 } ( \mathbb { R } )$ and that $O ^ { + } ( 1 , p )$ is a subgroup of $O ( 1 , p )$.

In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$.
V.A.1) Show that there exists a unique inner product on $E$ for which $\mathcal{B}$ is orthonormal.
This inner product is denoted in the usual way by $\langle u, v \rangle$ or more simply $u \cdot v$ for all $(u, v)$ in $E^2$.
V.A.2) If $u$ and $v$ are represented by the respective column matrices $U$ and $V$ in the basis $\mathcal{B}$, what simple relation exists between $u \cdot v$ and the matrix product ${}^t U V$ (where ${}^t U$ is the transpose of $U$)?

In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$. There exists a unique inner product on $E$ for which $\mathcal{B}$ is orthonormal, denoted $\langle u, v \rangle$ or $u \cdot v$.
Let $H$ be a hyperplane of $E$ and $D$ its orthogonal complement. If $(u)$ is a basis of $D$ and if $U$ is the column matrix of $u$ in $\mathcal{B}$, show that $H$ is stable by $f$ if and only if $U$ is an eigenvector of the transpose of $A$.

In this question, $E$ is a real vector space of dimension $n$ and $f$ is an endomorphism of $E$.
V.D.1) Show that if $f$ is diagonalisable then there exist $n$ hyperplanes of $E$, $(H_i)_{1 \leqslant i \leqslant n}$, all stable by $f$ such that $\bigcap_{i=1}^{n} H_i = \{0\}$.
V.D.2) Is an endomorphism $f$ of $E$ for which there exist $n$ hyperplanes of $E$ stable by $f$ and with intersection reduced to the zero vector necessarily diagonalisable?

We say that a real function $f$ of class $\mathcal{C}^{\infty}$ on $\mathbb{R}$ has rapid decay if $$\forall (n,m) \in \mathbb{N}^2, \lim_{x \rightarrow +\infty} x^m f^{(n)}(x) = \lim_{x \rightarrow -\infty} x^m f^{(n)}(x) = 0$$ We denote $\mathcal{S}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ with rapid decay.
Show that if $P$ is a polynomial function and if $f$ is in $\mathcal{S}$, then $Pf$ belongs to $\mathcal{S}$.

Explicitly determine a basis of $\mathcal{H}_3$.

Let $\mathcal{U}$ and $\mathcal{V}$ be two vector subspaces of $\mathbb{R}^{n}$ such that $$\operatorname{dim} \mathcal{U} + \operatorname{dim} \mathcal{V} > n.$$ Show that $\mathcal{U} \cap \mathcal{V}$ is not reduced to $\{0\}$.

Let $\mathcal{U}, \mathcal{V}$ and $\mathcal{W}$ be three vector subspaces of $\mathbb{R}^{n}$ such that $$\operatorname{dim} \mathcal{U} + \operatorname{dim} \mathcal{V} + \operatorname{dim} \mathcal{W} > 2n.$$ Show that $\mathcal{U} \cap \mathcal{V} \cap \mathcal{W}$ is not reduced to $\{0\}$.

Let $j$ and $k$ be non-negative integers such that $1 \leqslant j < k$ and $s_{1} \geqslant s_{2} \geqslant \cdots \geqslant s_{k}$ be real numbers. We define $\mathcal{D}_{j,k} = \left\{\left(t_{1}, \ldots, t_{k}\right) \in [0,1]^{k} \mid t_{1} + \cdots + t_{k} = j\right\}$ and $f$ the function from $\mathcal{D}_{j,k}$ to $\mathbb{R}$ defined by $$f\left(t_{1}, \ldots, t_{k}\right) = \sum_{i=1}^{k} s_{i} t_{i}.$$ Prove that for every $\left(t_{1}, \ldots, t_{k}\right) \in \mathcal{D}_{j,k}$, $$\sum_{i=1}^{j} s_{i} - f\left(t_{1}, \ldots, t_{k}\right) \geqslant \sum_{i=1}^{j} \left(s_{i} - s_{j}\right)\left(1 - t_{i}\right).$$ Deduce that $$\sup_{\mathcal{D}_{j,k}} f = \sum_{i=1}^{j} s_{i}.$$

In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $a_{1} \geqslant a_{2}$ and $b_{1} \geqslant b_{2}$, four real numbers satisfying the relation $a_{1} - a_{2} \geqslant b_{1} - b_{2}$, and $$\Sigma = \left\{s^{\downarrow}(A + B) \mid A \in S\left(a_{1}, a_{2}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$
Show that $$\Sigma = \left\{s^{\downarrow}(A + B) \,\middle|\, A = \left(\begin{array}{cc} a_{1} & 0 \\ 0 & a_{2} \end{array}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$

In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $a_{1} \geqslant a_{2}$ and $b_{1} \geqslant b_{2}$, four real numbers satisfying the relation $a_{1} - a_{2} \geqslant b_{1} - b_{2}$, and $$\Sigma = \left\{s^{\downarrow}(A + B) \mid A \in S\left(a_{1}, a_{2}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$ Let $L$ be the line segment identified in question 9.
Show that $\Sigma = L$.

Prove that $\mathcal{Y}_n$ is a convex and compact subset of $\mathcal{M}_n(\mathbb{R})$.

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ Let $P \in \mathbb{R}_{n-1}[X]$. Show that $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} P(j) = 0$$

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ In this question, we propose to show that there does not exist a linear application $u : \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X]$ such that $u \circ u = \delta$. We suppose, by contradiction, that such an application $u$ exists.
a) Show that $u$ and $\delta^2$ commute.
b) Deduce that $\mathbb{R}_1[X]$ is stable under the application $u$.
c) Show that there does not exist a matrix $A \in \mathcal{M}_2(\mathbb{R})$ such that $$A^2 = \left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right)$$
d) Conclude.

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ In this question, we seek all vector subspaces of $\mathbb{R}_n[X]$ stable under the application $\delta$.
a) For a nonzero polynomial $P$ of degree $d \leqslant n$, show that the family $(P, \delta(P), \ldots, \delta^d(P))$ is free. What is the vector space spanned by this family?
b) Deduce that if $V$ is a vector subspace of $\mathbb{R}_n[X]$ stable under $\delta$ and not reduced to $\{0\}$, there exists an integer $d \in \llbracket 0, n \rrbracket$ such that $V = \mathbb{R}_d[X]$.