Show that for $(t,s) \in \mathbf{R}_+^2$, $H_{t+s} = H_t H_s$. One may use a Cauchy product.

Let $U, V \in \mathbb{H}^{\mathrm{im}}$. a) Show that $U$ and $V$ are orthogonal if and only if $UV + VU = 0$. In this case show that $UV \in \mathbb{H}^{\mathrm{im}}$ and that the determinant of the family $(U, V, UV)$ in the basis $(I, J, K)$ of $\mathbb{H}^{\mathrm{im}}$ is non-negative. b) Show that if $(U, V)$ is an orthonormal family in $\mathbb{H}^{\mathrm{im}}$, then $(U, V, UV)$ is a direct orthonormal basis of $\mathbb{H}^{\mathrm{im}}$.

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 $\| \cdot \| _ { r }$.

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 $i,j \in \{1,\ldots,d\}$, $$P_{i,j} \geqslant c\nu_j.$$ Let $(x_n)_n \in \mathscr{P}^{\mathbb{N}}$ be defined by recursion by $x_0 \in \mathscr{P}$ and $x_{n+1} = x_n P$.
Deduce that $(x_n)_n$ converges to an element of $\mathscr{P}$.

Let $a \in \mathbb{K}$. Verify that the endomorphisms $I$ and $D$ are shift-invariant, as well as the endomorphisms $E_a$, $J$ and $L$ defined in part I. Are they delta endomorphisms?
Recall: $T$ is shift-invariant if for all $a \in \mathbb{K}$, $E_a \circ T = T \circ E_a$. $T$ is a delta endomorphism if $T$ is shift-invariant and $TX \in \mathbb{K}^*$.

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 $q = (2^{-|c(x)|})_{x \in X}$ 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 $\mathrm{KL}(p, q)$)

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 $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ and $q(x) = b(x,x)$. Let $x_0 \in \mathbf{R}^n$ and $f_{x_0}$ the solution of $y' = \varphi(y),\ y(0) = x_0$. For any function $y$ defined on $\mathbf{R}_+$, define $\varepsilon(y)(t) = \varphi(y(t)) - a(y(t))$.
Prove the existence of two strictly positive real numbers $\alpha$ and $\beta$ such that, for all $t \in \mathbf { R } _ { + }$, we have: $$q \left( f _ { x _ { 0 } } ( t ) \right) \leqslant \alpha \Rightarrow - \left\| f _ { x _ { 0 } } ( t ) \right\| ^ { 2 } + 2 b \left( f _ { x _ { 0 } } ( t ) , \varepsilon \left( f _ { x _ { 0 } } \right) ( t ) \right) \leqslant - \beta q \left( f _ { x _ { 0 } } ( t ) \right)$$

We use the notations of the previous parts. We assume that there exists an eigenvalue $\lambda > 0$ and an associated eigenvector column $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$: $$Mh = \lambda h,$$ and that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$M_{i,j} \geqslant c\nu_j.$$
We assume, in this question only, that $\lambda \in ]0,1[$. Show then that $\mathbb{E}\left(\|X_n\|_1\right)$ tends to $0$ as $n$ tends to infinity and $\mathbb{P}\left(\exists n \geqslant 0 : X_n = 0\right) = 1$. We say that the population becomes extinct almost surely in finite time.

Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$ that is invertible. Show that $T^{-1}$ is still a shift-invariant endomorphism.

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 $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ and $q(x) = b(x,x)$. Let $x_0 \in \mathbf{R}^n$ and $f_{x_0}$ the solution of $y' = \varphi(y),\ y(0) = x_0$. We fix strictly positive real numbers $\alpha$ and $\beta$ such that for all $t \in \mathbf{R}_+$: $$q(f_{x_0}(t)) \leqslant \alpha \Rightarrow -\|f_{x_0}(t)\|^2 + 2b(f_{x_0}(t), \varepsilon(f_{x_0})(t)) \leqslant -\beta q(f_{x_0}(t))$$
Show then that: $$q \left( x _ { 0 } \right) < \alpha \quad \Rightarrow \quad \forall t \geqslant 0 , q \left( f _ { x _ { 0 } } \right) ( t ) \leqslant e ^ { - \beta t } q \left( x _ { 0 } \right) .$$

We use the notations of the previous parts. For $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we denote $T(u) = (T_i(u))_{1 \leqslant i \leqslant d} \in \mathbb{R}^d$ the vector defined by $$T_i(u) = \operatorname{Var}\left(\langle L_i, u \rangle\right) \quad \text{for } i \in \{1,\ldots,d\}.$$
(a) Show that there exists $c_0 \geqslant 0$ such that for all $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we have $\|T(u)\|_1 \leqslant c_0 \|u\|_2^2$.
(b) Deduce the existence of $c_1 \geqslant 0$ such that for all $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we have $\|T(u)\|_1 \leqslant c_1 \|u\|_1^2$.

Let $T$ be a delta endomorphism of $\mathbb{K}[X]$.
Show that there exists a sequence of scalars $\left(\alpha_k\right)_{k \in \mathbb{N}}$ satisfying $\alpha_0 = 0$, $\alpha_1 \neq 0$ and $T = \sum_{k=1}^{+\infty} \alpha_k D^k$.

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 $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ and $q(x) = b(x,x)$. Let $x_0 \in \mathbf{R}^n$ and $f_{x_0}$ the solution of $y' = \varphi(y),\ y(0) = x_0$. We have established that $q(x_0) < \alpha \Rightarrow \forall t \geqslant 0,\ q(f_{x_0})(t) \leqslant e^{-\beta t} q(x_0)$.
Deduce the existence of three strictly positive constants $\tilde { \alpha } , C$ and $\beta$ such that: $$\forall x _ { 0 } \in B ( 0 , \tilde { \alpha } ) , \quad \forall t \in \mathbf { R } _ { + } , \quad \left\| f _ { x _ { 0 } } ( t ) \right\| \leqslant C e ^ { - \beta t } \left\| x _ { 0 } \right\| ,$$ where $B ( 0 , \tilde { \alpha } )$ denotes the open ball, for the norm $\|.\|$, with center 0 and radius $\tilde { \alpha }$.

Show that for any non-zero natural integer $n$, $$D_n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}.$$

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Let $y_0 \in E$ such that $\left\|y_0 - f(a)\right\| \leqslant \frac{r}{4}$.
Show that the application $$\begin{array}{ccc} \overline{B(a,r)} & \longrightarrow & \mathbb{R} \\ x & \longmapsto & \left\|y_0 - f(x)\right\|^2 \end{array}$$ admits a minimum attained at a point $x_0$ of $B(a,r)$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Let $y_0 \in E$ such that $\left\|y_0 - f(a)\right\| \leqslant \frac{r}{4}$. Let $x_0 \in B(a,r)$ be the point at which the minimum of $x \mapsto \|y_0 - f(x)\|^2$ is attained.
Show that $f(x_0) = y_0$.

For all $g = (\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$, we denote by $\phi_{g} : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ the map defined by $\phi_{g}(x) = Rx + \tau$.

• [(a)] Verify that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$, there exists a unique $g^{\prime\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$ such that $\phi_{g^{\prime\prime}} = \phi_{g^{\prime}} \circ \phi_{g}$. We denote this element by $g^{\prime}g$ in the following.
• [(b)] Verify that for all $g_{1}, g_{2}$ and $g_{3}$ in $\operatorname{Dep}(\mathbb{R}^{d})$ we have $g_{1}(g_{2}g_{3}) = (g_{1}g_{2})g_{3}$.

We assume that $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ is a decreasing sequence of strictly positive real numbers. We denote by $f$ the step function which, for all $k \in \mathbb { N } ^ { * }$, equals $a _ { k }$ on the interval $[ k - 1 , k [$.
Let $k$ be in $\mathbb { N } ^ { * }$. Prove that the function $v _ { k }$ defined on $[ k - 1 , k ]$ by
$$\left\{ \begin{array} { l } v _ { k } ( x ) = \frac { 1 } { x } \sum _ { i = 1 } ^ { k - 1 } \ln \left( a _ { i } \right) + \frac { 1 } { x } ( x - k + 1 ) \ln \left( a _ { k } \right) \quad \text { if } k \geqslant 2 \\ v _ { 1 } ( x ) = \ln \left( a _ { 1 } \right) \end{array} \right.$$
is minimal for $x = k$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ We denote $W = \left\{y \in E \left\lvert\, \|y - f(a)\| < \frac{r}{4}\right.\right\}$ and $V = f^{-1}(W) \cap B(a,r)$.
Justify that $V$ and $W$ are open sets of $E$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ We denote $W = \left\{y \in E \left\lvert\, \|y - f(a)\| < \frac{r}{4}\right.\right\}$ and $V = f^{-1}(W) \cap B(a,r)$.
Show that $$\begin{array}{clcc} f_{\mid V} : & V & \longrightarrow & W \\ & x & \longmapsto & f(x) \end{array}$$ is a continuous bijection from $V$ to $W$ whose inverse is a continuous function on $W$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ We denote by $W = \left\{y \in E \left\lvert\, \|y - f(a)\| < \frac{r}{4}\right.\right\}$ and $V = f^{-1}(W) \cap B(a,r)$.
Justify that $V$ and $W$ are open sets of $E$.