We assume that $A$ satisfies (P2). We consider $(f_n)_{n \in \mathbb{N}}$ a sequence of elements of $A$. Let $(x_p)_{p \geqslant 0}$ be a sequence of elements of $K$.
(a) Show that there exists a sequence $(\varphi_p)_{p \in \mathbb{N}}$ of strictly increasing functions from $\mathbb{N}$ to $\mathbb{N}$ such that for all $p \geqslant 0$, $f_{\psi_p(n)}(x_p)$ converges as $n$ tends to infinity with $\psi_0 = \varphi_0$ and $\psi_p = \psi_{p-1} \circ \varphi_p$ for $p \geqslant 1$.
(b) Show that for all $p \geqslant 0$, $f_{\psi_n(n)}(x_p)$ converges as $n$ tends to infinity.

We assume that $A$ satisfies (P2). We consider $(f_n)_{n \in \mathbb{N}}$ a sequence of elements of $A$.
(a) Show that we can extract from the sequence $(f_n)_{n \in \mathbb{N}}$ a subsequence that converges pointwise on $\mathbb{Q} \cap K$. We denote $(g_n)_{n \in \mathbb{N}}$ this extraction.
(b) For $x \in K$, show that $(g_n(x))_{n \in \mathbb{N}}$ admits a unique cluster value denoted $g(x)$ and conclude on the pointwise convergence of the sequence $(g_n)_{n \in \mathbb{N}}$ on $K$ to $g$.

We assume that $A$ satisfies (P2). We consider $(f_n)_{n \in \mathbb{N}}$ a sequence of elements of $A$, and $(g_n)_{n \in \mathbb{N}}$ a subsequence converging pointwise on $K$ to $g$.
(a) Show that $g$ is continuous on $K$.
(b) Show that the sequence $(g_n)_{n \in \mathbb{N}}$ converges uniformly to $g$ on $K$. (Hint: you may reason by contradiction.)
(c) Deduce that $(P2) \Rightarrow (P1)$.

We consider that the function $F$ is continuous and $y_{\text{init}} \in \Omega$. With the functions $\phi_N$ constructed in question III.2, show, using Theorem 1, that there exists a subsequence of $\phi_N$ that converges uniformly on $[0, T]$ to a continuous function $\phi$.

Let $A$ be a commutative ring. Show that if $A$ has property (F), then it has property (TF).

Show that, for all $\rho > 0$ and all $m , n \in \mathbb { N } ^ { * }$, the sets $\mathscr { D } _ { \rho } ( \mathbb { R } ) , \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ and $\mathscr { D } _ { \rho } \left( \mathscr { M } _ { m , n } ( \mathbb { R } ) \right)$ are closed under addition.

Let $\mathscr{P}$ be the set of row vectors of size $d$ with non-negative coefficients whose coordinate sum equals 1: $$\mathscr{P} = \left\{ u \in \mathscr{M}_{1,d}\left(\mathbb{R}_{+}\right) : \sum_{j=1}^{d} u_j = 1 \right\}.$$ We consider a square matrix $P \in \mathscr{M}_d\left(\mathbb{R}_{+}\right)$ such that for all $i \in \{1,\ldots,d\}$, $$\sum_{j=1}^{d} P_{i,j} = 1$$ We further assume that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$
Justify that $c \leqslant 1$.

We fix a Markov kernel $K$. Show that for all $n \in \mathbf{N}$, $K^n$ is a Markov kernel.

If $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, the support of $\varphi$ is defined by: $$\operatorname{Supp}(\varphi) = \overline{\{x \in \mathbb{R} : \varphi(x) \neq 0\}}$$ We say that $\varphi$ has compact support if $\operatorname{Supp}(\varphi)$ is a bounded subset of $\mathbb{R}$. We denote by $\mathcal{C}_{c}(\mathbb{R})$ the set of continuous functions with compact support on $\mathbb{R}$. If $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we set $$\|\varphi\|_{\infty} = \sup_{x \in \mathbb{R}} |\varphi(x)| \text{ and } \|\varphi\|_{1} = \int_{-\infty}^{+\infty} |\varphi(t)| dt$$
Show that $\|\cdot\|_{1} : \varphi \mapsto \|\varphi\|_{1}$ is a norm on $\mathcal{C}_{c}(\mathbb{R})$. One may admit without proof that $\|\cdot\|_{\infty}$ is also a norm.

Show that, for all $\rho > 0$ and all $n \in \mathbb { N } ^ { * }$, the sets $\mathscr { D } _ { \rho } ( \mathbb { R } )$ and $\mathscr { D } _ { \rho } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ are closed under multiplication.

If $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, the support of $\varphi$ is defined by: $$\operatorname{Supp}(\varphi) = \overline{\{x \in \mathbb{R} : \varphi(x) \neq 0\}}$$ We say that $\varphi$ has compact support if $\operatorname{Supp}(\varphi)$ is a bounded subset of $\mathbb{R}$. We denote by $\mathcal{C}_{c}(\mathbb{R})$ the set of continuous functions with compact support on $\mathbb{R}$. If $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we set $$\|\varphi\|_{\infty} = \sup_{x \in \mathbb{R}} |\varphi(x)| \text{ and } \|\varphi\|_{1} = \int_{-\infty}^{+\infty} |\varphi(t)| dt$$
Are the norms $\|\cdot\|_{\infty}$ and $\|\cdot\|_{1}$ equivalent?

Let $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$. Show that the map $\mathscr { D } _ { \rho } ( \mathbb { R } ) \rightarrow \mathscr { D } _ { r } ( \mathbb { R } )$ which associates to a function $f$ its restriction to $U _ { r }$ is injective.

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 $f : \mathbb{R} \rightarrow \mathbb{R}$ 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 $n \geqslant 0$, $f^{(n)}(x_{0}) = 0$.

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
$$f(x) = \int_{x_{0}}^{x} \frac{(x-t)^{n}}{n!} f^{(n+1)}(t)\, dt$$

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 $$\| P Q \| _ { r , s } \leqslant \| P \| _ { r , s } \cdot \| Q \| _ { r , s }$$

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 $Q$.

Show the inequality $$\forall (A, B) \in S_n^{++}(\mathrm{R})^2, \quad \operatorname{det}^{1/n}(A + B) \geq \operatorname{det}^{1/n}(A) + \operatorname{det}^{1/n}(B)$$

Show the inequality
$$\forall ( A , B ) \in S _ { n } ^ { + + } ( \mathrm { R } ) ^ { 2 } , \quad \operatorname { det } ^ { 1 / n } ( A + B ) \geq \operatorname { det } ^ { 1 / n } ( A ) + \operatorname { det } ^ { 1 / n } ( B )$$

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 $$J_\epsilon(\boldsymbol{q}) = \sum_{ij} q_{ij} C_{ij} + \epsilon \operatorname{KL}(\boldsymbol{q}, \boldsymbol{p})$$ (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 $\epsilon = 0$.

Let $(f^0, g^0) \in \mathbb{R}^{I \times J}$. For all $k \geq 0$, we consider $$g^{k+1} = g_*(f^k) \text{ and } f^{k+1} = f_*(g^{k+1})$$ Show that the sequence $(G(f^k, g^k))_{k \geq 0}$ is increasing.

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A_1, \ldots, A_n$ be as defined in Q13.
Justify that for all $p \in \llbracket 1 , n \rrbracket$, we have the inclusion $$A _ { p } \cap \left\{ \left| R _ { n } \right| < x \right\} \subset A _ { p } \cap \left\{ \left| R _ { n } - R _ { p } \right| > 2 x \right\} .$$

Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$. We recall that the degree of the zero polynomial is by convention equal to $-1$.
Show that there exists a natural number $n(T)$ such that, for every polynomial $p \in \mathbb{K}[X]$, $$\deg(Tp) = \max\{-1, \deg(p) - n(T)\}$$

In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part. Let $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ be the inner product on $\mathbf{R}^n$, and $q$ the associated quadratic form, i.e., $q(x) = b(x,x)$ for all $x \in \mathbf{R}^n$.
Prove then that: $$\forall x \in \mathbf { R } ^ { n } , \quad d q ( x ) ( a ( x ) ) = 2 b ( x , a ( x ) ) = - \| x \| ^ { 2 }$$

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by $$f_A(t) = \operatorname{det}(A + tM).$$ Show that there exists $\varepsilon_0 > 0$ such that, for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$.