Let $d$ be the hyperbolic distance on $\mathcal{H}$ and $G_0$ the subgroup of endomorphisms preserving $B$ and $\mathcal{H}$. Show that $d(gu,gv) = d(u,v)$ for all $g\in G$.

Show that for all $u,v\in\mathcal{H}$, we have $$\operatorname{ch}(d(u,v)) = -B(u,v).$$

Show that the application $\phi : (P, Q) \mapsto \phi(P, Q) = \int_0^1 P(t)Q(t)\,\mathrm{d}t$ defines an inner product on $\mathbb{R}_{n-1}[X]$.

Show by induction that
$$\forall N \in \mathbf { N } ^ { * } , \forall z \in D , \prod _ { k = 1 } ^ { N } \frac { 1 } { 1 - z ^ { k } } = \sum _ { n = 0 } ^ { + \infty } p _ { n , N } z ^ { n }$$

Let $z \in D$. Verify that $P ( z ) \neq 0$, that
$$P ( z ) = \lim _ { N \rightarrow + \infty } \prod _ { n = 1 } ^ { N } \frac { 1 } { 1 - z ^ { n } }$$
and that for all real $t > 0$,
$$\ln P \left( e ^ { - t } \right) = - \sum _ { n = 1 } ^ { + \infty } \ln \left( 1 - e ^ { - n t } \right)$$
where $P ( z ) := \exp \left[ \sum _ { n = 1 } ^ { + \infty } L \left( z ^ { n } \right) \right]$ for all $z \in D$.

In what follows, for all $z \in D$ we denote $$P(z) := \exp\left[\sum_{n=1}^{+\infty} L(z^n)\right].$$ Let $z \in D$. Verify that $P(z) \neq 0$, that $$P(t) = \lim_{N \rightarrow +\infty} \prod_{n=1}^{N} \frac{1}{1-t^n}$$ and that for all real $t > 0$, $$\ln P(e^{-t}) = -\sum_{n=1}^{+\infty} \ln(1-e^{-nt}).$$

$\mathbf{5}$ ▷ Show that $\operatorname{det}\left(e^{A}\right) = e^{\operatorname{tr}(A)}$.

Let $N$ be a matrix in $M_{n}(\mathbf{R})$ similar to an almost diagonal matrix. Prove that $N$ is semi-simple.

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) Verify that the map $(x_1, \ldots, x_p) \mapsto [x_1, \ldots, x_p]$ belongs to $\mathscr{A}_p(\mathbb{R}^p, \mathbb{R})$.
(b) Verify that if $F$ is a vector space over $\mathbb{R}$ and if $f : F \rightarrow \mathbb{R}^p$ is linear, then $g : F^p \rightarrow \mathbb{R}$ defined for $u = (u_1, \ldots, u_p) \in F^p$ by $g(u) = [f(u_1), \ldots, f(u_p)]$ is an element of $\mathscr{A}_p(F, \mathbb{R})$.

If $f \in O_n, n \geqslant 0, g \in O_1, h \in O_l, l \geqslant 1$ and $r \geqslant 1$, show that $h^r \in O_{rl}$, that $f \circ h \in O_{nl}$ and $f \circ (g + h) - f \circ g \in O_{n+l-1}$.

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) Verify that the map $(x_1, \ldots, x_p) \mapsto [x_1, \ldots, x_p]$ belongs to $\mathcal{A}_p(\mathbb{R}^p, \mathbb{R})$.
(b) Verify that if $F$ is a vector space over $\mathbb{R}$ and if $f : F \rightarrow \mathbb{R}^p$ is linear, then $g : F^p \rightarrow \mathbb{R}$ defined for $u = (u_1, \ldots, u_p) \in F^p$ by $g(u) = [f(u_1), \ldots, f(u_p)]$ is an element of $\mathcal{A}_p(F, \mathbb{R})$.

Let $C$ and $D$ be two non-empty convex subsets of $\mathbb{R}^d$ such that $C$ is closed and bounded, $D$ is closed, and $C \cap D = \emptyset$. Show that $D - C$ is a convex closed subset of $\mathbb{R}^d$ not containing $0$.

Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Let $p \in \mathbb{R}^d$, set $$K_p := \{x \in K : p \cdot x \leqslant p \cdot y, \forall y \in K\}.$$ Show that $K_p$ is non-empty, convex and closed and that $\operatorname{Ext}(K_p) \subset \operatorname{Ext}(K)$.

Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Let $p \in \mathbb{R}^d$, set $$K_p := \{x \in K : p \cdot x \leqslant p \cdot y, \forall y \in K\}.$$ Show that $K_p$ is non-empty, convex and closed and that $\operatorname{Ext}(K_p) \subset \operatorname{Ext}(K)$.

Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Show that $\operatorname{Ext}(K)$ is non-empty (one may reduce to the case where $0 \in K$ and reason on the dimension of $K$).

Write in Python a function \texttt{modifie\_matrice(p, A)} that takes as arguments a probability $p$ and a numpy array representing a matrix $A \in \mathcal { V } _ { n , n }$. This function modifies the array A according to the following procedure: for every natural integer $k$, the matrix $A _ { k + 1 }$ is constructed from the matrix $A _ { k }$ by keeping each coefficient of $A _ { k }$ equal to $-1$ and by changing to $-1$ with probability $p$ each coefficient of $A _ { k }$ equal to $1$.

Using the function \texttt{modifie\_matrice}, write in Python a function \texttt{nb\_tours(p, n)} that takes as arguments a probability $p$ and the order $n$ of the matrices $A _ { k }$ and returns the smallest integer $k$ such that $A _ { k } = - A _ { 0 }$, starting from the matrix $A _ { 0 }$ (the real matrix of order $n$ whose coefficients are all equal to 1).

Write in Python a function \texttt{moyenne\_tours(p, n, nbe)} that takes as arguments a probability $p$, the order $n$ of the matrices $A _ { k }$ and an integer \texttt{nbe} and that returns the average, over \texttt{nbe} trials performed, of the number of steps necessary to go from $A _ { 0 }$ to $- A _ { 0 }$.

To every $p \in \mathbb{R}[X]$, we associate the function $J(p) = Jp$ from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad J(p)(x) = Jp(x) = \int_x^{x+1} p(t)\,\mathrm{d}t$$
Show that $J$ is an endomorphism of $\mathbb{R}[X]$.

Show that for all real $t \in \mathbf{R}_+$, $H_t$ is a Markov kernel.

Show that $S$ is a closed and path-connected subset of $\mathbb{H}$.

Let $r \in \mathbb { R } _ { + } ^ { * }$ such that $r < \rho$. Show that $\| \cdot \| _ { r }$ is a norm on $\mathscr { D } _ { \rho } ( \mathbb { R } )$ and that $\| f g \| _ { r } \leqslant \| f \| _ { r } \cdot \| g \| _ { r }$ for all $f , g \in \mathscr { D } _ { \rho } ( \mathbb { R } )$.

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 $$x_{n+1} = x_n P.$$ Show that the series $\sum_{n \geqslant 0} \|x_{n+1} - x_n\|_1$ is convergent.

To every $p \in \mathbb{K}[X]$, we associate the function $L(p) = Lp$ from $\mathbb{K}$ to $\mathbb{K}$ defined by $$\forall x \in \mathbb{K}, \quad L(p)(x) = Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$$
Show that $L$ is an endomorphism of $\mathbb{K}[X]$. Is it invertible?

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 $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with the projections $p_i$ and inclusions $q_i$ as defined, and $\|.\|_i$ the norm on $\mathcal{L}(E_i)$ and $\|.\|_c$ for $\mathcal{L}(\mathbf{C}^n)$.
Show that, for all $i \in \llbracket 1 ; r \rrbracket$, there exists a constant $C _ { i } > 0$ such that: $$\forall u \in \mathcal { L } \left( E _ { i } \right) , \quad \left\| q _ { i } u p _ { i } \right\| _ { c } \leqslant C _ { i } \| u \| _ { i }$$