We consider a function $f \in \mathscr { C } _ { b } ^ { 0 }$. We define for $( t , x ) \in \mathbb { R } ^ { 2 }$ $$\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$$ where $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$. Show that the function $\Phi _ { f } : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R }$ is well defined and continuous.

We assume that $k \geqslant 0$ and that $C$ is a matrix in $\Delta_{k+1}$. We set $P = I_n + C$. Show that $P$ is invertible and that $P^{-1} \in \bigoplus_{p=0}^{n-1} \Delta_{p(k+1)}$.

Show that the application $\Phi : \begin{aligned} \mathcal{P}(\mathbb{N}) &\rightarrow \{0,1\}^{\mathbb{N}} \\ A &\mapsto \mathbb{1}_A \end{aligned}$ is bijective.

Show that the application $$\Psi : \begin{aligned} \{0,1\}^{\mathbb{N}} &\rightarrow [0,1] \\ (x_n) &\mapsto \sum_{n=0}^{+\infty} \frac{x_n}{2^{n+1}} \end{aligned}$$ is well-defined and surjective. Is it injective?

We denote $D^{\star} = D \backslash \{0\}$. We set for all $(x_n) \in \{0,1\}^{\mathbb{N}}$ $$\Lambda\left((x_n)\right) = \begin{cases} \Psi\left((x_n)\right) & \text{if } \Psi\left((x_n)\right) \in [0,1[ \backslash D^{\star} \\ \frac{\Psi\left((x_n)\right)}{2} & \text{if } \Psi\left((x_n)\right) \in D \cup \{1\} \text{ and } (x_n) \text{ is eventually constant at } 1 \\ \frac{1 + \Psi\left((x_n)\right)}{2} & \text{if } \Psi\left((x_n)\right) \in D^{\star} \text{ and } (x_n) \text{ is eventually constant at } 0 \end{cases}$$
Show that $\Lambda$ realizes a bijection from $\{0,1\}^{\mathbb{N}}$ to $[0,1[$.

We denote $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Let $n \in \mathbb{N}^{\star}$. Show that the application $$\Psi_n : \begin{gathered} \{0,1\}^n \rightarrow D_n \\ (x_j)_{j \in \llbracket 1,n \rrbracket} \mapsto \sum_{j=1}^{n} \frac{x_j}{2^j} \end{gathered}$$ is bijective.

Show that the application $\Phi : \begin{aligned} \mathcal{P}(\mathbb{N}) &\rightarrow \{0,1\}^{\mathbb{N}} \\ A &\mapsto \mathbb{1}_A \end{aligned}$ is bijective.

Show that the application $$\Psi : \begin{aligned} \{0,1\}^{\mathbb{N}} &\rightarrow [0,1] \\ (x_n) &\mapsto \sum_{n=0}^{+\infty} \frac{x_n}{2^{n+1}} \end{aligned}$$ is well-defined and surjective. Is it injective?

We denote $D^{\star} = D \setminus \{0\}$ where $D = \bigcup_{n \in \mathbb{N}^{\star}} D_n$ and $D_n = \left\{ \sum_{j=1}^{n} \frac{x_j}{2^j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$. We set for all $(x_n) \in \{0,1\}^{\mathbb{N}}$ $$\Lambda\left((x_n)\right) = \begin{cases} \Psi\left((x_n)\right) & \text{if } \Psi\left((x_n)\right) \in [0,1[ \setminus D^{\star} \\ \frac{\Psi\left((x_n)\right)}{2} & \text{if } \Psi\left((x_n)\right) \in D \cup \{1\} \text{ and } (x_n) \text{ is eventually constant at } 1 \\ \frac{1 + \Psi\left((x_n)\right)}{2} & \text{if } \Psi\left((x_n)\right) \in D^{\star} \text{ and } (x_n) \text{ is eventually constant at } 0 \end{cases}$$ where $\Psi : \{0,1\}^{\mathbb{N}} \rightarrow [0,1]$, $(x_n) \mapsto \sum_{n=0}^{+\infty} \frac{x_n}{2^{n+1}}$.
Show that $\Lambda$ realizes a bijection from $\{0,1\}^{\mathbb{N}}$ to $[0,1[$.

We denote by $E$ the vector space of continuous functions defined on $[0,1]$ and taking values in $\mathbb{R}$ equipped with the inner product defined by $$\langle f, g \rangle = \int_0^1 f(t) g(t)\,\mathrm{d}t$$ We consider a function $A:[0,1] \times [0,1] \rightarrow \mathbb{R}$ continuous. We are interested in the application $T: E \rightarrow E$ defined by $$T(f)(x) = \int_0^1 A(x,t) f(t)\,\mathrm{d}t$$ We suppose that $\ker T$ is finite-dimensional. We denote by $S$ the inverse bijection of the isomorphism induced by $T$ from $(\ker T)^\perp$ onto $\operatorname{Im} T$. We define the inner product $\varphi$ on $\operatorname{Im} T$ by setting, for all $(f,g) \in (\operatorname{Im} T)^2$, $$\varphi(f,g) = \langle S(f), S(g) \rangle$$ We consider the application $K$ defined on $[0,1]^2$ by $$K(x,y) = \int_0^1 A(x,t) A(y,t)\,\mathrm{d}t$$ Show that $(\operatorname{Im} T, \varphi)$ is a reproducing kernel Hilbert space, with kernel $K$.

We denote by $E$ the vector space of continuous functions defined on $[0,1]$ and taking values in $\mathbb{R}$ equipped with the inner product defined by $$\langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ We consider a function $A : [0,1] \times [0,1] \rightarrow \mathbb{R}$ continuous. We are interested in the application $T : E \rightarrow E$ defined by $$T(f)(x) = \int_0^1 A(x,t) f(t) \, \mathrm{d}t$$ We suppose that $\ker T$ is finite-dimensional. We denote by $S$ the inverse bijection of the isomorphism induced by $T$ from $(\ker T)^\perp$ onto $\operatorname{Im} T$. We define the inner product $\varphi$ on $\operatorname{Im} T$ by setting, for all $(f,g) \in (\operatorname{Im} T)^2$, $$\varphi(f,g) = \langle S(f), S(g) \rangle$$ We consider the application $K$ defined on $[0,1]^2$ by $$K(x,y) = \int_0^1 A(x,t) A(y,t) \, \mathrm{d}t$$ Show that $(\operatorname{Im} T, \varphi)$ is a reproducing kernel Hilbert space, with kernel $K$.

Let $n \in \mathbb{N}^*$. For $k \in \mathbb{N}$, we denote $\mathcal{B}(n, k)$ the set of maps $\sigma \in \operatorname{MD}(n+1)$ such that $\sigma(2) - \sigma(1) = k+1$. For $k \in \mathbb{N}$ and $s \in \mathbb{N}$, we denote $\mathcal{C}(n, s, k)$ the set of elements $\sigma$ of $\operatorname{MD}(n+2)$ such that $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$.
1. For $m \geq 2$ verify that the map $\operatorname{Opp} : \Sigma_m \rightarrow \Sigma_m$, which to $\sigma \in \Sigma_m$ associates $\eta \in \Sigma_m$ defined by $$\eta(i) = m + 1 - \sigma(i)$$ is a bijection satisfying $\operatorname{Opp}(\operatorname{MD}(m)) = \operatorname{DM}(m)$ and $\operatorname{Opp}(\operatorname{DM}(m)) = \operatorname{MD}(m)$. Verify that if $\sigma \in \Sigma_m$ and if $i, j$ are elements of $\{1, \ldots, m\}$ satisfying $\sigma(j) > \sigma(i)$, $$\sigma(j) - \sigma(i) = 1 + \operatorname{Card}\{k \in \Delta_m \mid \sigma(i) < \sigma(k) < \sigma(j)\}$$

Let $n \in \mathbb{N}^*$. For $k \in \mathbb{N}$, we denote $\mathcal{B}(n, k)$ the set of maps $\sigma \in \operatorname{MD}(n+1)$ such that $\sigma(2) - \sigma(1) = k+1$. For $k \in \mathbb{N}$ and $s \in \mathbb{N}$, we denote $\mathcal{C}(n, s, k)$ the set of elements $\sigma$ of $\operatorname{MD}(n+2)$ such that $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$.
3. In this question and the next, we fix $n \geq 2$, $1 \leq k \leq n-1$ and $1 \leq s \leq n-k$. We propose to construct a bijection from $\mathcal{C}(n, s, k)$ to $\mathcal{B}(n, k)$. Let $\sigma \in \mathcal{C}(n, s, k)$. a. Verify that the number $m$ of integers $j \geq 4$ such that $\sigma(j) > \sigma(3)$ satisfies $m \geq k$. We denote $j_1, \ldots, j_m$ these integers, which we order in such a way that $\sigma(j_1) < \sigma(j_2) < \cdots < \sigma(j_m)$. b. Show that $\xi$ defined by $\xi(1) = \sigma(j_k) + \frac{1}{2}$, $\xi(2) = \sigma(3)$, $\xi(n+1) = \ldots$ satisfies $$\xi(p) > \xi(p+1) \text{ for } p \text{ odd}, \quad \xi(p) < \xi(p+1) \text{ for } p \text{ even}$$ and that the interval $]\xi(2), \xi(1)[$ contains exactly $k$ elements of $\{\xi(3), \ldots, \xi(n+1)\}$. c. We denote $A = \xi(\Delta_{n+1})$ and we set $\bar{\xi} = \beta_A \circ \xi$ (we recall that $\beta_A$ denotes the unique increasing bijection from $A$ to $\Delta_{n+1}$). Show that $\bar{\xi} \in \operatorname{DM}(n+1)$. d. Let $\eta = \operatorname{Opp}(\bar{\xi})$. Verify that $\eta \in \mathcal{B}(n, k)$.

Let $n \in \mathbb{N}^*$. For $k \in \mathbb{N}$, we denote $\mathcal{B}(n, k)$ the set of maps $\sigma \in \operatorname{MD}(n+1)$ such that $\sigma(2) - \sigma(1) = k+1$. For $k \in \mathbb{N}$ and $s \in \mathbb{N}$, we denote $\mathcal{C}(n, s, k)$ the set of elements $\sigma$ of $\operatorname{MD}(n+2)$ such that $\sigma(2) - \sigma(1) = s+1, \quad n+2-\sigma(2) = k$.
We denote by $\Psi_{n,s,k}$ the map from $\mathcal{C}(n,s,k)$ to $\mathcal{B}(n,k)$ defined by $\Psi_{n,s,k}(\sigma) = \eta$.
4. Let $\eta \in \mathcal{B}(n, k)$ and let $\xi = \operatorname{Opp}(\eta)$. a. Verify that the number $m$ of integers $j \geq 3$ such that $\xi(j) > \xi(2)$ satisfies $m \geq k$. We denote these integers by $j_1, \ldots, j_m$, with $\xi(j_1) > \xi(j_2) \cdots > \xi(j_m)$. b. We set $u_2 = \xi(j_k) - \frac{1}{2} > \xi(2)$. Show that the number $m'$ of integers $i \geq 2$ such that $\xi(i) < u_2$ satisfies $m' \geq s$. We denote them by $i_1, \ldots, i_{m'}$, with $\xi(i_1) > \cdots > \xi(i_{m'})$ and we set $u_1 = \xi(i_k) - \frac{1}{2}$. c. By considering the map $\theta$ defined by $$\theta(1) = u_1, \theta(2) = u_2, \theta(3) = \xi(2), \ldots, \theta(n+2) = \xi(n+1)$$ show the existence of $\sigma \in \mathcal{C}(n, s, k)$ satisfying $\Psi_{n,s,k}(\sigma) = \eta$. d. Show that $\Psi_{n,s,k}$ is bijective.

We define, for all $x \in \mathbb{R}^{+*}$, $$\Gamma(x) = \int_0^{+\infty} t^{x-1} \mathrm{e}^{-t} \, \mathrm{d}t$$ Show that the function $\Gamma$ is continuous and strictly positive on $\mathbb{R}^{+*}$.

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Verify that $\|\cdot\|_K$ is a norm on $\mathbb{C}[X]$.

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We define the function: $$\begin{aligned} \varphi : [0,+\infty[ &\rightarrow \mathbb{R} \\ p &\mapsto \begin{cases} \ln\left(M_p(Q)\right) & \text{if } p > 0 \\ 0 & \text{if } p = 0 \end{cases} \end{aligned}$$ where $M_p(Q) = \frac{1}{2\pi}\int_0^{2\pi}\left|Q(e^{i\theta})\right|^p d\theta$. Show that $\varphi$ is continuous on $[0,+\infty[$.

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We define the function: $$\begin{aligned} \varphi : [0,+\infty[ &\rightarrow \mathbb{R} \\ p &\mapsto \begin{cases} \ln\left(M_p(Q)\right) & \text{if } p > 0 \\ 0 & \text{if } p = 0 \end{cases} \end{aligned}$$ where $M_p(Q) = \frac{1}{2\pi}\int_0^{2\pi}\left|Q(e^{i\theta})\right|^p d\theta$. Show carefully that $\varphi$ is differentiable on $]0,+\infty[$ and compute its derivative on this interval.

Let $A \in \mathcal{S}_n(\mathbb{R})$ and $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. Prove that the application $\rho$ defines a norm on $\mathcal{S}_n(\mathbb{R})$.

In this question, we are given a real random variable $X$ following a geometric distribution with parameter $p \in ] 0,1 [$ arbitrary. We set $q = 1 - p$.
Show that for all $k \in \mathbf { N }$, the random variable $X ^ { k }$ has finite expectation. Show that $\Phi _ { X }$ is of class $\mathcal { C } ^ { \infty }$ on $\mathbf { R }$ and that $\Phi _ { X } ^ { ( k ) } ( 0 ) = i ^ { k } \mathbf { E } \left( X ^ { k } \right)$ for all $k \in \mathbf { N }$.

The function $q$ associates to any real $x$ the real number $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$, where $\lfloor x \rfloor$ denotes the integer part of $x$.
Show that $q$ is piecewise continuous on $\mathbf { R }$, that it is 1-periodic, and that the function $| q |$ is even.

In this part, we introduce the function $q$ which associates to any real $x$ the real number $q(x) = x - \lfloor x \rfloor - \frac{1}{2}$, where $\lfloor x \rfloor$ denotes the integer part of $x$.
Show that $q$ is piecewise continuous on $\mathbf{R}$, that it is 1-periodic and that the function $|q|$ is even.

For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \operatorname{det}(\operatorname{Gram}(e, u)).$$
(a) Show that for all $e \in E^p$, we have $\Omega_p(e) \in \mathscr{A}_p(E, \mathbb{R})$.
(b) Verify that for all $(e, u) \in E^p \times E^p$, we have $\Omega_p(e)(u) = \Omega_p(u)(e)$.
(c) Show that $\Omega_p \in \mathscr{A}_p(E, \mathscr{A}_p(E, \mathbb{R}))$.

For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \operatorname{det}(\operatorname{Gram}(e, u)).$$
(a) Let $M \in \mathcal{M}_p(\mathbb{R})$, $e = (e_1, \ldots, e_p)$ and $e^{\prime} = (e_1^{\prime}, \ldots, e_p^{\prime})$ in $E^p$ satisfying $e_i^{\prime} = \sum_{j=1}^p M_{ij} e_j$ for all $i \in \llbracket 1, p \rrbracket$. Show that $\Omega_p(e^{\prime}) = \operatorname{det}(M) \Omega_p(e)$.
(b) Let $e \in E^p$. Show that $\Omega_p(e) \neq 0$ if and only if $e$ is a free family.
(c) Verify that $\Omega_p(e)(e) \geqslant 0$ for all families $e \in E^p$.

For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \det(\operatorname{Gram}(e, u))$$
(a) Show that for all $e \in E^p$, we have $\Omega_p(e) \in \mathcal{A}_p(E, \mathbb{R})$.
(b) Verify that for all $(e, u) \in E^p \times E^p$, we have $\Omega_p(e)(u) = \Omega_p(u)(e)$.
(c) Show that $\Omega_p \in \mathcal{A}_p(E, \mathcal{A}_p(E, \mathbb{R}))$.