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.

5. Give a procedure for computing $\operatorname{Card} \operatorname{MD}(n)$ by recursion.

We denote by $\widehat{S}$ the set of $f \in S_*$ satisfying $\lim_{x \rightarrow -\infty} f(x) = -\infty$ and $\lim_{x \rightarrow +\infty} f(x) = +\infty$. We denote by $\operatorname{Mi}(f)$ the set of minima of $f$ and by $\operatorname{Ma}(f)$ the set of maxima of $f$, so $E(f) = \operatorname{Mi}(f) \cup \operatorname{Ma}(f)$.
1. Let $f \in \widehat{S}$. a. Verify that $\operatorname{Card} \operatorname{Mi}(f) = \operatorname{Card} \operatorname{Ma}(f)$ and that for $y \in \mathbb{R}$, $f^{-1}([-\infty, y])$ is the union of non-empty open intervals that are pairwise disjoint. We denote by $\mathscr{I}(y)$ their set. b. Show that for every element $M$ of $\operatorname{Ma}(f)$, there exists a unique element $m$ of $\operatorname{Mi}(f)$ such that $f(m) < f(M)$ and $m > M$.

We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$.
Given $k \in \mathbf{N}$, give a simplified expression for $\operatorname{tr}(f_{1}^{(k+1)})$, and deduce from this the validity of Lemma A: for $u, v \in \mathcal{V}$, $\operatorname{tr}(u^{k} v) = 0$ for every natural integer $k$.

We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. We introduce the subset $\mathcal{V}^{\bullet}$ of $E$ formed by vectors belonging to at least one of the sets $\operatorname{Im} u^{p-1}$ for $u$ in $\mathcal{V}$, and the vector subspace $K(\mathcal{V}) := \operatorname{Vect}(\mathcal{V}^{\bullet})$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$.
Let $y \in E$. Prove that $f_{1}^{(p-1)}(y) \in K(\mathcal{V})$. Using a relation between $u(f_{1}^{(p-1)}(y))$ and $v(u^{p-1}(y))$, deduce that $v(x) \in u(K(\mathcal{V}))$ for every $x \in \operatorname{Im} u^{p-1}$.

In the rest of the problem, we assume that $\lambda$ is a real number distinct from 1 and we set $w = \frac{1}{\lambda - 1}$. We further set $\mathbf{f} = (1 + w)\delta - w\mathbf{1}$.
Show that $\mathbf{f} * \mathbf{b} = \delta$.

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 $K$.

We have two finite sequences of real numbers $0 = t_1 < t_2 < \cdots < t_K$ ($K \geqslant 2$) and $x_1 < x_2 < \cdots < x_L$ ($L \geqslant 2$). The formula from question 32 applied at $t_i$ with $n$ sufficiently large allows us to estimate $\lambda(t_i)$ by an approximate value $\hat{\lambda}(t_i)$. Justify that for all $i \in \{1, \ldots, L\}$, $$\hat{\lambda}^*\left(x_i\right) = \max_{1 \leqslant j \leqslant K} \left(t_j x_i - \hat{\lambda}\left(t_j\right)\right)$$ constitutes a reasonable approximate value of $\lambda^*\left(x_i\right)$.

Let $X : \Omega \rightarrow \mathbb { R }$ be a real-valued random variable. We assume that $X ( \Omega )$ is countable and we use the notation from question 2: $X ( \Omega ) = \left\{ x _ { n } , n \in \mathbb { N } \right\}$ with $a _ { n } = \mathbb { P } \left( X = x _ { n } \right)$. We also assume that, for all integer $n \in \mathbb { N } , X$ admits a moment of order $n$ and that there exists a real $R > 0$ such that $$\mathbb { E } \left( | X | ^ { n } \right) = O \left( \frac { n ^ { n } } { R ^ { n } } \right) \quad \text { when } n \rightarrow + \infty$$ Show that for all $n \in \mathbb { N }$ and all $y \in \mathbb { R } , \left| \mathrm { e } ^ { \mathrm { i } y } - \sum _ { k = 0 } ^ { n } \frac { ( \mathrm { i } y ) ^ { k } } { k ! } \right| \leqslant \frac { | y | ^ { n + 1 } } { ( n + 1 ) ! }$.

Using Table 1 below, give an approximate bound for the value of $m$ and the value of a real number $h > 0$ such that there exists a rank $n_0 \in \mathbb{N}^*$ satisfying for all $n \geqslant n_0$, $$P\left(S_n > 1{,}1 \times nm\right) \leqslant \mathrm{e}^{-nh}.$$

$x_i$	4,50	4,55	4,60	4,65	4,70
$\hat{\lambda}^*(x_i)$	$4{,}1 \times 10^{-12}$	$4{,}1 \times 10^{-12}$	$4{,}1 \times 10^{-12}$	$4{,}1 \times 10^{-12}$	$4{,}1 \times 10^{-12}$
$x_i$	4,75	4,80	4,85	4,90	4,95
$\hat{\lambda}^*(x_i)$	$5{,}1 \times 10^{-4}$	$5{,}5 \times 10^{-3}$	$1{,}1 \times 10^{-2}$	$1{,}6 \times 10^{-2}$	$2{,}1 \times 10^{-2}$
$x_i$	5,00	5,05	5,10	5,15	5,20
$\hat{\lambda}^*(x_i)$	$2{,}6 \times 10^{-2}$	$3{,}1 \times 10^{-2}$	$3{,}6 \times 10^{-2}$	$4{,}1 \times 10^{-2}$	$4{,}6 \times 10^{-2}$
$x_i$	5,25	5,30	5,35	5,40	5,45
$\hat{\lambda}^*(x_i)$	$5{,}1 \times 10^{-2}$	$5{,}6 \times 10^{-2}$	$6{,}1 \times 10^{-2}$	$6{,}6 \times 10^{-2}$	$7{,}1 \times 10^{-2}$

Justify that, for every natural integer $k$, $p _ { 1 } ^ { ( k ) } + \cdots + p _ { n } ^ { ( k ) } = 1$.

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Show that $\|P\|_K$ belongs to $\mathbb{R}$ for all $P \in \mathbb{C}[X]$.

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

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Let $Q$ and $R$ be two non-zero polynomials in $\mathbb{C}[X]$. Show that: $$\|Q\|_K \|R\|_K \geq \|QR\|_K.$$

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. Show that, if the inequality $\|Q\|_K \|R\|_K \geq \|QR\|_K$ is an equality, then there exists $z_0 \in K$ such that: $$\left|Q\left(z_0\right)\right| = \|Q\|_K, \quad \left|R\left(z_0\right)\right| = \|R\|_K \quad \text{and} \quad \left|Q\left(z_0\right)R\left(z_0\right)\right| = \|QR\|_K.$$

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. We fix two non-zero natural integers $n$ and $m$, and we set: $$C_{n,m}^K = \sup\left\{\left.\frac{\|Q\|_K \|R\|_K}{\|QR\|_K}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$ Show that $C_{n,m}^K > 1$.
To do this, one may choose two distinct elements $a$ and $b$ in $K$ and verify that, for $\rho \in \mathbb{R}$ sufficiently large, we have $\left\|Q_\rho R_\rho\right\|_K < \left\|Q_\rho\right\|_K \left\|R_\rho\right\|_K$ with $Q_\rho(X) = X - (\rho(b-a)+a)$ and $R_\rho(X) = X - (\rho(a-b)+b)$.

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. We fix two non-zero natural integers $n$ and $m$. We introduce the $\mathbb{C}$-vector space $V = \mathbb{C}_n[X] \times \mathbb{C}_m[X]$ as well as the set: $$E = \left\{(Q,R) \in V \mid \|Q\|_K = \|R\|_K = 1\right\}.$$ Show that there exists a pair $\left(Q_0, R_0\right) \in E$ such that: $$\left\|Q_0 R_0\right\|_K = \inf\left\{\|QR\|_K \mid (Q,R) \in E\right\}.$$ To do this, one may equip $V$ with the norm defined by $$\|(Q,R)\| = \|Q\|_K + \|R\|_K$$ for $(Q,R) \in V$, then study the application $$\begin{aligned} f : E &\rightarrow \mathbb{R} \\ (Q,R) &\mapsto \|QR\|_K. \end{aligned}$$

Let $K$ be a closed, bounded and infinite subset of $\mathbb{C}$. We fix two non-zero natural integers $n$ and $m$, and we set: $$C_{n,m}^K = \sup\left\{\left.\frac{\|Q\|_K \|R\|_K}{\|QR\|_K}\right\rvert\, Q \in \mathbb{C}_n[X]\backslash\{0\}, R \in \mathbb{C}_m[X]\backslash\{0\}\right\} \in \mathbb{R} \cup \{+\infty\}$$ Deduce that there exist two monic polynomials $Q_1 \in \mathbb{C}_n[X]$ and $R_1 \in \mathbb{C}_m[X]$ such that: $$\frac{\left\|Q_1\right\|_K \left\|R_1\right\|_K}{\left\|Q_1 R_1\right\|_K} = C_{n,m}^K.$$

Show that, for all $n \in \mathbb{N}$ and $\theta \in \mathbb{R}$, $T_n(\cos\theta) = \cos(n\theta)$.
The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.

Show that, for all $n \in \mathbb{N}$ and $\theta \in \mathbb{R}$, $T_n(\cos\theta) = \cos(n\theta)$.
The sequence of polynomials $\left(T_n\right)_{n \in \mathbb{N}}$ is defined by $T_0 = 1, T_1 = X$ and $\forall n \in \mathbb{N}, T_{n+2} = 2X T_{n+1} - T_n$.

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. Verify that the integral: $$\int_0^{2\pi} \ln\left|Q\left(e^{i\theta}\right)\right| d\theta$$ converges absolutely in the sense of Definition 1.
One may use the d'Alembert-Gauss theorem.

Let $Q \in \mathbb{C}[X]$ be a non-zero polynomial. We set for $p > 0$: $$M_p(Q) = \frac{1}{2\pi} \int_0^{2\pi} \left|Q\left(e^{i\theta}\right)\right|^p d\theta$$ Explain why $M_p(Q)$ is strictly positive for all $p > 0$.

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 $[0,+\infty[$.