Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$, where $S(v,w) = (v \mid T(w)) + (T(v) \mid w)$.
We assume here that $\operatorname{ker}(T) \subset F^+$.
(a) Show that $\forall z \in F^-, T^{2m}(z) = 0_E$.
(b) Show that $\operatorname{Im}(T)^\perp \subset F^+$ and that $\operatorname{Im}(T^2)^\perp \cap \operatorname{Im}(T) \subset F^-$.
(c) Let $z \in \operatorname{Im}(T)^\perp$ with $z \neq 0_E$. Show that $T(z) \in G^\perp$ and that $T(z) \neq 0_E$.
(d) Let $z \in \operatorname{Im}(T^2)^\perp \cap \operatorname{Im}(T)$ with $z \neq 0_E$. Show that $T(z) \in G^\perp$ and that $T(z) \neq 0_E$.

Deduce that if $u$, $v$ and $v'$ in $E$ satisfy $v \neq v'$ and $\|u - v\| = \|u - v'\|$ then $\left\|u - \frac{v + v'}{2}\right\| < \|u - v\|$.

Deduce that if $C$ is a non-empty closed convex set of $E$ and $u$ is a vector of $E$, then there exists a unique $v$ in $C$ such that
$$\forall w \in F, \quad \|u - v\| \leqslant \|u - w\|$$
We say that $v$ is the projection of $u$ onto $C$ and we denote $d(u, C) = \|u - v\|$.

We consider the space $E = \mathcal{M}_{k,d}(\mathbb{R})$ equipped with the inner product defined by
$$\forall (A, B) \in E^{2}, \quad \langle A \mid B \rangle = \operatorname{tr}\left(A^{\top} \cdot B\right)$$
We denote by $\|\cdot\|_{F}$ the associated Euclidean norm. We fix a vector $(u_{1}, \ldots, u_{d})$ in $\mathbb{R}^{d}$ with $\|u\| = 1$, and define
$$g : \left\lvert \, \begin{aligned} & \mathcal{M}_{k,d}(\mathbb{R}) \rightarrow \mathbb{R} \\ & M \mapsto \|M \cdot u\| \end{aligned} \right.$$
Let $C = \left\{M \in \mathcal{M}_{k,d}(\mathbb{R}) \mid g(M) \leqslant r\right\}$. Let $r$ and $t$ be two real numbers, with $t > 0$. Show that for every matrix $M$ in $\mathcal{M}_{k,d}(\mathbb{R})$
$$d(M, C) < t \quad \Longrightarrow \quad g(M) < r + t$$

Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$). Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ and harmonic on $U$. For all $\varepsilon > 0$ we set $g_\varepsilon(x) = f(x) + \varepsilon \|x\|^2$. Deduce that $\forall x \in U, f(x) \leqslant \sup_{y \in \partial U} f(y)$.

Let $n$ be a non-zero natural number and $\Phi_n : \{0,1\}^n \rightarrow \llbracket 0, 2^n - 1 \rrbracket$, $(x_j)_{j \in \llbracket 1,n \rrbracket} \mapsto \sum_{j=1}^{n} x_j 2^{n-j}$.
Using the results of Q8--Q10, deduce that $\Phi_n$ is bijective.

Using the results of Q33--Q36, conclude that $[0,1[$ is not countable.

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 $\lambda^*\left(x_i\right)$.

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 $x \in \operatorname{Im} u^{p-1}$.

Deduce that, for all $n \in \mathbb{N}$ and $P \in \mathbb{C}_n[X]$, the function from $\mathbb{R}$ to $\mathbb{C}$, $\theta \mapsto P(\cos\theta)$ is in $\mathcal{S}_n$.
Recall that $\mathcal{S}_n$ is the $\mathbb{C}$-vector space of functions $f : \mathbb{R} \rightarrow \mathbb{C}$ satisfying $$\exists (a_0, \ldots, a_n) \in \mathbb{C}^{n+1}, \quad \exists (b_1, \ldots, b_n) \in \mathbb{C}^n, \quad \forall t \in \mathbb{R}, \quad f(t) = a_0 + \sum_{k=1}^{n}\left(a_k \cos(kt) + b_k \sin(kt)\right)$$

Let $\left(f_n\right)_{n \geqslant 1}$ be a sequence of functions from $\mathbb{N}^*$ to $\mathbb{R}$ such that, for all $x \in \mathbb{N}^*$, the sequence $\left(f_n(x)\right)_{n \geqslant 1}$ converges to a real number $f(x)$ as $n$ tends to $+\infty$. We assume that there exists a function $h : \mathbb{N}^* \rightarrow [0, +\infty[$ such that $h(X)$ has finite expectation and such that $\left|f_n(m)\right| \leqslant h(m)$ for all $m$ and $n$ in $\mathbb{N}^*$. Justify that $E(f(X))$ has finite expectation and show that $$\lim_{n \rightarrow +\infty} E\left(f_n(X)\right) = E(f(X)).$$

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: $$\frac{\left\|Q_1\right\|_K \left\|R_1\right\|_K}{\left\|Q_1 R_1\right\|_K} = C_{n,m}^K.$$

Let $z \in \stackrel{\circ}{\mathbb{D}}$. Using the result that for all $z \in \stackrel{\circ}{\mathbb{D}}$, $\ln|1-z| = -\operatorname{Re}\left(\sum_{n=1}^{\infty}\frac{z^n}{n}\right)$, deduce that the Mahler measure of the polynomial $X - z$ is 1 and, in the case where $z \neq 0$, that of the polynomial $X - z^{-1}$ is $|z|^{-1}$.

Let $\lambda$ be the leading coefficient of $Q$ and let $\alpha_1, \ldots, \alpha_n$ be the roots of $Q$ counted with multiplicity. Deduce from the previous questions that: $$M(Q) = |\lambda| \prod_{i=1}^n \max\left\{1, \left|\alpha_i\right|\right\}$$

We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. Deduce from question 4.32 that there exists a polynomial $Q_3$ whose roots are all in $[-1, +\infty[$ and such that the pair $(Q_3, R_2)$ forms a good extremal pair.

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. Deduce from question 4.41 that: $$\left(1 - X^2\right)P'' - XP' + (n+m)^2 P = 0$$

We choose $I = [-1,1]$ and fix any very good extremal pair $(Q, R)$. We set $P = QR$. Deduce from question 4.43 that: $$C_{n,m} = 2^{n+m-1} \cdot \left[\prod_{k=1}^n \left(1 + \cos\left(\frac{2k-1}{2(n+m)}\pi\right)\right)\right] \cdot \left[\prod_{k=1}^m \left(1 + \cos\left(\frac{2k-1}{2(n+m)}\pi\right)\right)\right].$$

Show that, if $\omega$ is a symplectic form on $E$, then for every vector $x$ in $E$, $\omega ( x , x ) = 0$.

Deduce that, for all $P \in \mathbb { R } _ { n - 1 } [ X ]$, $$P = \sum _ { i = 1 } ^ { n } P \left( a _ { i } \right) L _ { i }.$$

Let $n \in \mathbb{N}^*$, $W$ be a monic polynomial of degree $n$, $Q = \frac { 1 } { 2 ^ { n - 1 } } T _ { n } - W$, and for all $k \in \llbracket 0 , n \rrbracket$, $z _ { k } = \cos \left( \frac { k \pi } { n } \right)$. We assume that $\sup _ { x \in [ - 1,1 ] } | W ( x ) | = \frac { 1 } { 2 ^ { n - 1 } }$. Deduce that $Q = 0$, then that $W = \frac { 1 } { 2 ^ { n - 1 } } T _ { n }$.
One may consider the sum of the inequalities from the previous question and exploit question 6 applied to suitable data.

We assume that $f$ is a function from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ of class $\mathcal { C } ^ { 1 }$ satisfying $$\left\{ \begin{array} { l } \lim _ { x \rightarrow 0 } f ( x ) = 0 \\ \exists C > 0 ; \forall x > 0 , \quad \left| f ^ { \prime } ( x ) \right| \leqslant C \frac { \mathrm { e } ^ { x / 2 } } { \sqrt { x } } \end{array} \right.$$ For $x \in \mathbb { R } _ { + } ^ { * }$, we set $\Phi ( x ) = \frac { 4 \sqrt { x } \mathrm { e } ^ { x / 2 } } { 1 + x } - \int _ { 0 } ^ { x } \frac { \mathrm { e } ^ { t / 2 } } { \sqrt { t } } \mathrm {~d} t$. Show that $\Phi$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R } _ { + } ^ { * }$, that $\lim _ { x \rightarrow 0 } \Phi ( x ) = 0$ and that, for all $x > 0 , \Phi ^ { \prime } ( x ) \geqslant 0$. Deduce that $\Phi ( x ) \geqslant 0$ for all $x > 0$.

We say that the element $M$ of $\mathcal{M}_{n}(\mathbf{C})$ satisfies $\mathcal{P}$ if, for every $(i,j)$ in $\{1,\ldots,n\}^{2}$, there exists an element $P_{M,i,j}$ of $\mathbf{C}_{n-1}[X]$ such that $$\forall z \in \mathbf{C} \backslash \sigma(M), \quad \left(R_{z}(M)\right)_{i,j} = \frac{P_{M,i,j}(z)}{\chi_{M}(z)}$$
We admit that every matrix in $\mathcal{M}_{n}(\mathbf{C})$ satisfies $\mathcal{P}$. Deduce that, if $M \in \mathcal{M}_{n}(\mathbf{C})$ and $(X,Y) \in \mathcal{M}_{n,1}(\mathbf{C})^{2}$, there exists an element $P_{M,X,Y}$ of $\mathbf{C}_{n-1}[X]$ such that $$\forall z \in \mathbf{C} \backslash \sigma(M), \quad X^{T}R_{z}(M)Y = \frac{P_{M,X,Y}(z)}{\chi_{M}(z)}.$$

Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set $$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
Let $F \in \mathcal{R}_{n}$, $P$ and $Q$ be two elements of $\mathbf{C}_{n}[X]$ satisfying $F = \frac{P}{Q}$ and $\forall z \in \mathbb{U},\ Q(z) \neq 0$. For $t \in [-\pi, \pi]$, we set $f(t) = F(e^{it}) = g(t) + ih(t)$. For $u \in [-\pi, \pi]$, we define $f_{u}(t) = g(t)\cos(u) + h(t)\sin(u)$.
We admit the equality $$\int_{-\pi}^{\pi} \left(\int_{-\pi}^{\pi} \left|f_{u}^{\prime}(t)\right| \mathrm{d}u\right) \mathrm{d}t = \int_{-\pi}^{\pi} \left(\int_{-\pi}^{\pi} \left|f_{u}^{\prime}(t)\right| \mathrm{d}t\right) \mathrm{d}u$$ We also admit that, for $u \in [-\pi, \pi]$ such that $f_{u}$ is not constant, the set of points in $]-\pi, \pi[$ where the function $f_{u}^{\prime}$ vanishes is finite.
Deduce the inequality $$(3) \quad V(f) \leq 2\pi n \|f\|_{\infty}.$$

Let $E$ be a countably infinite subset of $\mathbb{R}$. Let $(\Omega, \mathscr{A}, P)$ be a probability space. We denote by $\mathbb{E}(X)$ the expectation of a real random variable $X$. Let $\mathscr{P}(E)$ be the set of subsets of $E$ and $\mathscr{B}(\mathscr{P}(E), \mathbb{R})$ the $\mathbb{R}$-vector space of bounded functions from $\mathscr{P}(E)$ to $\mathbb{R}$ with norm $\|f\| = \sup\{|f(A)|, \quad A \in \mathscr{P}(E)\}$.
Show that for all random variables $X$ and $Y$ on $(\Omega, \mathscr{A}, P)$ and for every subset $A$ of $E$: $$|\mu_X(A) - \mu_Y(A)| \leqslant \mathbb{E}\left(|\mathbb{1}_{\{X \in A\}} - \mathbb{1}_{\{Y \in A\}}|\right)$$ and deduce that $\|\mu_X - \mu_Y\| \leqslant P(X \neq Y)$, where $\{X \neq Y\} = \{\omega \in \Omega \text{ such that } X(\omega) \neq Y(\omega)\}$.