Let $r \in \mathbb { R } _ { + } ^ { * }$ such that $r < \rho$. Let $\left( f _ { n } \right) _ { n \geqslant 0 }$ be a sequence of elements of $\mathscr { D } _ { \rho } ( \mathbb { R } )$. We assume that $\sum _ { n \geqslant 0 } \left\| f _ { n } \right\| _ { r }$ converges. Show that $\sum _ { n \geqslant 0 } f _ { n }$ converges normally on $U _ { r }$ to a function $f \in \mathscr { D } _ { r } ( \mathbb { R } )$. Show that $\sum _ { n \geqslant 0 } f _ { n }$ also converges to $f$ for the norm $\| \cdot \| _ { r }$.

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$.
Show that, for $i \in \llbracket 1 ; r \rrbracket$, $E _ { i }$ is stable under $a$.

Let $M \in S_n^+(\mathbf{R})$ be a non-zero matrix. Express $\|M\|_2$ in terms of the eigenvalues of $M$.

Let $M \in S _ { n } ^ { + } ( \mathbf { R } )$ be a non-zero matrix. Express $\| M \| _ { 2 }$ in terms of the eigenvalues of $M$.

We consider the matrix $K \in \mathscr{M}_N(\mathbf{R})$ defined by $$\forall (i,j) \in \llbracket 1;N \rrbracket^2, K[i,j] = p_{ij}$$ where $p_{ij}$ is the probability of moving from state $i$ to state $j$ at each impulse. Justify that $K$ is a Markov kernel.

Let $\varphi_{0}$ be the function defined on $\mathbb{R}$ by
$$\left\{ \begin{array}{l} \varphi_{0}(x) = e^{-1/x^{2}} \text{ if } x \neq 0 \\ \varphi_{0}(0) = 0 \end{array} \right.$$
Using $\varphi_{0}$, show that there exists a function $\varphi_{1}$ of class $C^{\infty}$ on $\mathbb{R}$, whose support is $[0, \infty[$. Deduce that there exists a function $\varphi_{2}$ of class $C^{\infty}$ on $\mathbb{R}$ such that $\operatorname{Supp}(\varphi_{2}) = [-1, 1]$.

Let $q = (2^{-|c(x)|})_{x \in X}$ and $X$ a random variable taking values in $X$ with distribution $p$.
(a) Verify that $\ln(2) E(|c(X)|) = -\sum_{x \in X} p_x \ln(q_x)$.
(b) Deduce that $E(|c(X)|) \geq \frac{H(p)}{\ln(2)}$. (Hint: One may try to express $\ln(2) E(|c(X)|)$ in terms of $H(p)$ and $\mathrm{KL}(p, q)$)

Let $f \in \mathscr { D } _ { \rho } ( \mathbb { R } )$ such that $f ( 0 ) \neq 0$. The purpose of this question is to show that there exists $r \in \mathbb { R } _ { + } ^ { * } , r \leqslant \rho$ such that $\frac { 1 } { f } \in \mathscr { D } _ { r } ( \mathbb { R } )$.
6a. Show that we can assume without loss of generality that $f ( 0 ) = 1$.
We now write $f ( t ) = \sum _ { i = 0 } ^ { \infty } a _ { i } t ^ { i }$ and we assume that $a _ { 0 } = 1$.
6b. Only in this sub-question, we assume that there exists $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$ and $g \in \mathscr { D } _ { r } ( \mathbb { R } )$ such that $f ( t ) g ( t ) = 1$ for all $t \in U _ { r }$. We write $g ( t ) = \sum _ { i = 0 } ^ { \infty } b _ { i } t ^ { i }$. Show that: $$\left\{ \begin{aligned} & b _ { 0 } = 1 \\ & \text { for } n \geqslant 1 , b _ { n } = - \left( b _ { 0 } a _ { n } + \ldots + b _ { n - 1 } a _ { 1 } \right) \end{aligned} \right.$$
We now define the sequence $\left( b _ { n } \right) _ { n \geqslant 0 }$ by the above recurrence formula.
6c. Show that there exists $c \in \mathbb { R } _ { + } ^ { * }$ such that $\left| a _ { n } \right| \leqslant c ^ { n }$ for all $n \in \mathbb { N }$.
6d. Show that $\left| b _ { n } \right| \leqslant ( 2 c ) ^ { n }$ for all $n \in \mathbb { N }$.
6e. Conclude.

We consider the matrix $K \in \mathscr{M}_N(\mathbf{R})$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, K[i,j] = p_{ij}$, and the random variable $Z_k$ representing the state of the system after $k$ impulses, with $Z_0$ being the certain variable with value 1. Let $n \in \mathbf{N}$. Let $j \in \llbracket 1;N \rrbracket$, show that $P(Z_n = j) = K^n[1,j]$. One may proceed by induction.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function of class $C^{\infty}$ with compact support. For all $n \in \mathbb{N}$, we set $$M_{n} = \sup_{x \in \mathbb{R}} \left|f^{(n)}(x)\right| = \left\|f^{(n)}\right\|_{\infty}$$ and we seek properties satisfied by the sequence $(M_{n})$. In this part we assume that $f$ is not identically zero.
Show that there exists $x_{0} \in \operatorname{Supp}(f)$ such that for all integers $n \geqslant 0$, $f^{(n)}(x_{0}) = 0$.

We consider $\alpha = (\alpha_i)_{i \in I} \in (\mathbb{R}_+^*)^I$ and $\beta = (\beta_j)_{j \in J} \in (\mathbb{R}_+^*)^J$ such that $\sum_{i \in I} \alpha_i = \sum_{j \in J} \beta_j = 1$. We denote $$Q = \left\{(q_{ij})_{(i,j) \in I \times J} \in \mathbb{R}^{I \times J} \mid q_{ij} \geq 0 \text{ for all } (i,j) \in I \times J\right\}$$ and $$F(\alpha, \beta) = \left\{q \in Q \mid \sum_{j' \in J} q_{ij'} = \alpha_i \text{ and } \sum_{i' \in I} q_{i'j} = \beta_j \text{ for all } (i,j) \in I \times J\right\}.$$ Verify that $F(\alpha, \beta)$ is a convex set of the vector space $E = \mathbb{R}^{I \times J}$.

Show that $\mathscr { D } _ { \rho } ( \mathbb { R } )$ is an integral domain.

Let $A \in S_n^{++}(\mathbf{R})$ and $B \in S_n(\mathbf{R})$. Show that there exists a diagonal matrix $D \in M_n(\mathbf{R})$ and $Q \in GL_n(\mathbf{R})$ such that $B = QDQ^\top$ and $A = QQ^\top$. What can be said about the diagonal elements of $D$ if $B \in S_n^{++}(\mathbf{R})$?
Hint: You may use question 3.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $B \in S _ { n } ( \mathbf { R } )$. Show that there exists a diagonal matrix $D \in M _ { n } ( \mathbf { R } )$ and $Q \in G L _ { n } ( \mathbf { R } )$ such that $B = Q D Q ^ { \top }$ and $A = Q Q ^ { \top }$. What can be said about the diagonal elements of $D$ if $B \in S _ { n } ^ { + + } ( \mathbf { R } )$ ?
Hint: You may use question 3.

We consider the matrix $K \in \mathscr{M}_N(\mathbf{R})$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, K[i,j] = p_{ij}$, and the matrix $H_t$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, H_t[i,j] = e^{-t} \sum_{n=0}^{+\infty} \frac{t^n K^n[i,j]}{n!}$. Let $t \in \mathbf{R}_+$. We assume that the number of impulses after time $t$ is given by a random variable $Y_t$ following a Poisson distribution with parameter $t$. For all $j \in \llbracket 1;N \rrbracket$ we denote by $A_{t,j}$ the event ``the system is in state $j$ after time $t$''. Justify that $P(A_{t,j}) = H_t[1,j]$.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function of class $C^{\infty}$ with compact support. For all $n \in \mathbb{N}$, we set $$M_{n} = \sup_{x \in \mathbb{R}} \left|f^{(n)}(x)\right| = \left\|f^{(n)}\right\|_{\infty}$$ In this part we assume that $f$ is not identically zero, and $x_{0} \in \operatorname{Supp}(f)$ is such that for all integers $n \geqslant 0$, $f^{(n)}(x_{0}) = 0$.
Show that for all $x \in \mathbb{R}$ and all $n \in \mathbb{N}$, we have
$$f(x) = \int_{x_{0}}^{x} \frac{(x-t)^{n}}{n!} f^{(n+1)}(t)\, dt$$

We consider $\alpha = (\alpha_i)_{i \in I} \in (\mathbb{R}_+^*)^I$ and $\beta = (\beta_j)_{j \in J} \in (\mathbb{R}_+^*)^J$ such that $\sum_{i \in I} \alpha_i = \sum_{j \in J} \beta_j = 1$. We denote $$F(\alpha, \beta) = \left\{q \in Q \mid \sum_{j' \in J} q_{ij'} = \alpha_i \text{ and } \sum_{i' \in I} q_{i'j} = \beta_j \text{ for all } (i,j) \in I \times J\right\}.$$ We denote by $\boldsymbol{p}$ the element of $F(\alpha, \beta)$ defined by $p_{ij} = \alpha_i \beta_j > 0$ for all $(i,j) \in I \times J$. Let $X_1$ and $X_2$ be two random variables such that $X_1$ takes values in $I$ and $X_2$ takes values in $J$.
(a) Verify that if $\boldsymbol{q} \in F(\alpha, \beta)$, then $\sum_{i \in I} \sum_{j \in J} q_{ij} = 1$.
(b) Assume that $P(X_1 = i, X_2 = j) = q_{ij}$ with $q \in F(\alpha, \beta)$. Calculate the distribution of $X_1$ and that of $X_2$ in terms of $\alpha$ and $\beta$.
(c) What can we say about $X_1$ and $X_2$ when $\boldsymbol{q} = \boldsymbol{p}$?

Let $n , m \in \mathbb { N }$ and let $r , s \in \mathbb { R } _ { + } ^ { * } , r < \rho$.
8a. Show that $\| \cdot \| _ { r , s }$ is a norm on $\mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$.
8b. Show that if $P \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ and $Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { m } [ X ] \right)$, then $P Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n + m } [ X ] \right)$ and $$\| P Q \| _ { r , s } \leqslant \| P \| _ { r , s } \cdot \| Q \| _ { r , s }$$

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function of class $C^{\infty}$ with compact support. For all $n \in \mathbb{N}$, we set $$M_{n} = \sup_{x \in \mathbb{R}} \left|f^{(n)}(x)\right| = \left\|f^{(n)}\right\|_{\infty}$$ In this part we assume that $f$ is not identically zero, and $x_{0} \in \operatorname{Supp}(f)$ is such that for all integers $n \geqslant 0$, $f^{(n)}(x_{0}) = 0$.
Show that if there exist constants $A > 0$ and $B > 0$, and a subsequence $(n_{j})_{j \geqslant 1}$ such that $M_{n_{j}} \leqslant A B^{n_{j}} (n_{j})!$, then $f$ is identically zero on the interval $]x_{0} - 1/B,\, x_{0} + 1/B[$.

We consider $\alpha = (\alpha_i)_{i \in I} \in (\mathbb{R}_+^*)^I$ and $\beta = (\beta_j)_{j \in J} \in (\mathbb{R}_+^*)^J$ such that $\sum_{i \in I} \alpha_i = \sum_{j \in J} \beta_j = 1$. We denote by $\boldsymbol{p}$ the element of $F(\alpha, \beta)$ defined by $p_{ij} = \alpha_i \beta_j > 0$ for all $(i,j) \in I \times J$. Let $C = (C_{ij})_{(i,j) \in I \times J} \in \mathbb{R}_+^{I \times J}$ and $\epsilon > 0$. We consider $J_\epsilon : Q \rightarrow \mathbb{R}$ defined by $$J_\epsilon(\boldsymbol{q}) = \sum_{ij} q_{ij} C_{ij} + \epsilon \operatorname{KL}(\boldsymbol{q}, \boldsymbol{p})$$ where $\mathrm{KL}(\boldsymbol{q}, \boldsymbol{p})$ is defined by taking $X = I \times J$. Show that $J_\epsilon$ is strictly convex on $Q$.

Let $A , B \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$. We assume that $B$ is monic of degree $d \leqslant n$.
9a. Show that there exist elements $Q \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n - d } [ X ] \right)$ and $R \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { d - 1 } [ X ] \right)$ uniquely determined such that $A = B Q + R$.
The elements $Q$ and $R$ are called respectively the quotient and the remainder of the Euclidean division of $A$ by $B$.
9b. Let furthermore $r , s \in \mathbb { R } _ { + } ^ { * }$ with $r < \rho$. Show that, if $\left\| B - X ^ { d } \right\| _ { r , s } < s ^ { d }$, then $$\| Q \| _ { r , s } \leqslant \frac { \| A \| _ { r , s } } { s ^ { d } - \left\| B - X ^ { d } \right\| _ { r , s } } \quad \text { and } \quad \| R \| _ { r , s } \leqslant \frac { s ^ { d } \cdot \| A \| _ { r , s } } { s ^ { d } - \left\| B - X ^ { d } \right\| _ { r , s } }$$ (One may start by treating the case where $B = X ^ { d }$.)

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a function of class $C^{\infty}$ with compact support. For all $n \in \mathbb{N}$, we set $$M_{n} = \sup_{x \in \mathbb{R}} \left|f^{(n)}(x)\right| = \left\|f^{(n)}\right\|_{\infty}$$ In this part we assume that $f$ is not identically zero.
Deduce that for all $B > 0$ we have
$$\frac{M_{n}}{B^{n} n!} \underset{n \rightarrow \infty}{\longrightarrow} +\infty$$

We consider $\alpha = (\alpha_i)_{i \in I} \in (\mathbb{R}_+^*)^I$ and $\beta = (\beta_j)_{j \in J} \in (\mathbb{R}_+^*)^J$ such that $\sum_{i \in I} \alpha_i = \sum_{j \in J} \beta_j = 1$. We denote by $\boldsymbol{p}$ the element of $F(\alpha, \beta)$ defined by $p_{ij} = \alpha_i \beta_j > 0$ for all $(i,j) \in I \times J$. Let $C = (C_{ij})_{(i,j) \in I \times J} \in \mathbb{R}_+^{I \times J}$ and $\epsilon > 0$. We consider $J_\epsilon : Q \rightarrow \mathbb{R}$ defined by $$J_\epsilon(\boldsymbol{q}) = \sum_{ij} q_{ij} C_{ij} + \epsilon \operatorname{KL}(\boldsymbol{q}, \boldsymbol{p})$$ (a) Verify that $F(\alpha, \beta)$ is a closed bounded set of $\mathbb{R}^{I \times J}$.
(b) Show that there exists a unique $\boldsymbol{q}(\epsilon) \in Q$ minimizing $J_\epsilon$ on $F(\alpha, \beta)$.
(c) By considering a simple counterexample, show that uniqueness is no longer true if we assume that $\epsilon = 0$.

We consider $P = f _ { 0 } + f _ { 1 } X + \cdots + f _ { n } X ^ { n } \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$, with $\lambda = 0$, $f _ { 0 } ( 0 ) = \cdots = f _ { d - 1 } ( 0 ) = 0$ and $f _ { d }$ the constant function equal to 1.
Let $F \in \mathscr { D } _ { \rho } \left( \mathbb { R } _ { d } [ X ] \right)$ be monic and such that $F _ { \mid t = 0 } = X ^ { d }$. Let $R$ be the remainder of the Euclidean division of $P$ by $F$. Show that $F + R$ is monic of degree $d$ and that $( F + R ) _ { \mid t = 0 } = X ^ { d }$.

For $\mu > 0$ and $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we define $T_{\mu} : \varphi \mapsto T_{\mu}\varphi$, where for all $x \in \mathbb{R}$,
$$T_{\mu}\varphi(x) = \frac{1}{2\mu} \int_{x-\mu}^{x+\mu} \varphi(t)\, dt$$
Show that $T_{\mu}$ is a linear map, which sends the space $\mathcal{C}_{c}(\mathbb{R})$ into itself, and that for all $\varphi \in \mathcal{C}_{c}(\mathbb{R})$ we have $\|T_{\mu}\varphi\|_{\infty} \leqslant \|\varphi\|_{\infty}$.