Show the existence of constants $C_1 > C_2 > 0$ and $R_0 > 0$ such that, for all $R\geq R_0$, $$C_2 e^R \leq |\Gamma(R)| \leq C_1 e^R.$$

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 $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
Show that there exists $\varepsilon _ { 0 } > 0$ such that, for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$.

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 $n \geqslant 0$, $f^{(n)}(x_{0}) = 0$.

We fix $x \in \mathcal{C}$ such that $\|Ax\| = 1$. Let $B \in \mathrm{SO}(\mathbb{R}^2)$ be a matrix such that $x = B\binom{1}{0}$. a) Show that for all $r \in ]0,1[$ there exists $x_r \in \mathcal{C}$ such that $$\left\| AB\begin{pmatrix} r & 0 \\ 0 & \frac{1}{r} \end{pmatrix} x_r \right\| > 1$$ b) Show that if $x_r = \binom{y_r}{z_r}$, then $z_r^2 > \dfrac{r^2}{1+r^2}$.

Using the above, show that there exists a basis $(e_1, e_2)$ of $\mathbb{R}^2$ such that $\|Ax\| = \|x\|_2$ for $x \in \{e_1, e_2\}$.

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 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 $\phi$.

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 $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t) \end{array}$$ Let $M_1$ and $M_2$ be two elements of $(\mathbb{C}[A])^*$. Show that there exists $a \in \mathbb{R}$ such that $$\forall t \in [0,1], \quad M(t) = Z_a(t) M_1 + \left(1 - Z_a(t)\right) M_2 \in (\mathbb{C}[A])^*.$$

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t). \end{array}$$ Let $M_1$ and $M_2$ be two elements of $(\mathbb{C}[A])^*$. Show that there exists $a \in \mathbb{R}$ such that $$\forall t \in [0,1], \quad M(t) = Z_a(t) M_1 + \left(1 - Z_a(t)\right) M_2 \in (\mathbb{C}[A])^*.$$

We have $I _ { n } = \frac { p _ { n } + \zeta ( 2 ) q _ { n } } { d _ { n } ^ { 2 } }$ where $p _ { n }$ and $q _ { n }$ are non-zero integers for all $n \in \mathbb { N } ^ { * }$.
Show that there exists $N \in \mathbb { N } ^ { * }$ such that for all $n \geqslant N$,
$$0 < \left| p _ { n } + \zeta ( 2 ) q _ { n } \right| \leqslant \zeta ( 2 ) \left( \frac { 5 } { 6 } \right) ^ { n }$$
One may use, without proving it, the inequality $9 \frac { 5 \sqrt { 5 } - 11 } { 2 } \leqslant \frac { 5 } { 6 }$.

In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We denote $Z_\infty = \int_{-\infty}^{+\infty} \exp\left(-\frac{x^4}{12}\right) \mathrm{d}x$ and we consider the function $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
Justify that, for any continuous and bounded function $f$ on $\mathbb{R}$, the quantity $$E_{n,f} = \int_{-\infty}^{+\infty} \mathbb{E}\left(f\left(\frac{t}{n^{1/4}} + n^{1/4} M_n\right)\right) \exp\left(-\frac{t^2}{2}\right) \frac{\mathrm{d}t}{\sqrt{2\pi}}$$ is well defined.

Let $\mathcal{U}_n$ be the vector subspace of functions $\mathbb{R}^n \rightarrow \mathbb{R}$ generated by indicator functions of polytopes of $\mathbb{R}^n$. For a polytope $P$ of $\mathbb{R}^n$, the relative interior $P^\circ$ is defined as $P^\circ = \{x \in P : \ell_i(x) = a_i \Leftrightarrow i \in S_F\}$ where $S_F = \{i \in I, \ell_i(x) = a_i\ \forall x \in F\}$.
Show that for every polytope $P$ of $\mathbb{R}^n$ and for all $x \in P \backslash P^\circ$, there exists a face $F \subset P$ such that $F \neq P$ and $x \in F$.

A triangulation of a polytope $P$ is a complex formed of simplices whose realization equals $P$. Show that every polytope admits a triangulation.

Let $n \geq 1$ be an integer. We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$ and $\mathcal{M}_n(\mathbb{Z})$ the set of $n \times n$ matrices with integer coefficients.
Suppose that $M \in B_n \backslash \mathcal{M}_n(\mathbb{Z})$. Show that there exists a sequence $(r_1, s_1), (r_2, s_2), \ldots, (r_k, s_k)$ of pairs of indices with $k \geq 2$ such that $$0 < M_{r_i, s_i} < 1, \quad 0 < M_{r_i, s_{i+1}} < 1 \quad \text{and} \quad (r_k, s_k) = (r_1, s_1)$$ then that we can assume that all the pairs $(r_1, s_1), (r_1, s_2), (r_2, s_2), \ldots, (r_{k-1}, s_{k-1}), (r_{k-1}, s_k)$ are distinct.

Let $n \geq 1$ be an integer. Let $\mathbb{C}[[\mathbb{Z}^n]]$, $\mathcal{R}$, $\mathbb{C}(\mathbb{Z}^n)$, and $\mathrm{I} : \mathcal{R} \rightarrow \mathbb{C}(\mathbb{Z}^n)$ be as defined previously.
Let $u : \mathbb{Z}^n \rightarrow \mathbb{R}$ be an injective group homomorphism. Show that there exists a unique map $s_u : \mathbb{C}(\mathbb{Z}^n) \rightarrow \mathcal{R}$ satisfying the following three conditions:

• [(a)] $s_u(Pf) = P\,s_u(f)$ for all $f \in \mathbb{C}(\mathbb{Z}^n)$ and $P \in \mathbb{C}[\mathbb{Z}^n]$.
• [(b)] $\mathrm{I}(s_u(f)) = f$ for all $f \in \mathbb{C}(\mathbb{Z}^n)$.
• [(c)] $s_u\left(\frac{1}{1-g}\right) = \sum_{n \in \mathbb{N}} g^n$ if $g$ is a finite linear combination of elements of the form $x^\gamma$ with $\gamma \in \mathbb{Z}^n$ satisfying $u(\gamma) > 0$.

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$, and we are given $f \in \mathcal{C}^1(\mathbb{R}^d)$. Show that $f$ admits a minimizer on $C$, which we denote $x_*$ in the following questions.

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$, and we are given $f \in \mathcal{C}^1(\mathbb{R}^d)$. Let $x_*$ be a minimizer of $f$ on $C$. Suppose in this question that $\|x_*\| = 1$. The objective is to show that $$\exists \lambda \geq 0,\, \nabla f(x_*) = -\lambda x_*.$$ a) Let $x, y \in \mathbb{R}^d$ such that $x \neq y$ and $\|x\| = \|y\| = 1$. Show that $\langle x, v \rangle > 0$ and $\langle y, v \rangle < 0$, where $v := x - y$. b) Suppose by contradiction that (7) is not satisfied. Show that there exists $v \in \mathbb{R}^d$ such that $\langle v, \nabla f(x_*) \rangle > 0$ and $\langle v, x_* \rangle > 0$. Deduce a contradiction and conclude. Hint: consider the quantities $f(x_* - tv)$ and $\|x_* - tv\|^2$, in the limit $t \rightarrow 0^+$.

Prove that $\varphi$ admits an extension $\psi$ to $V$ compatible with $u$.

Prove that there exists an injective linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. According to Part III, this linear application $\varphi$ admits an extension $\psi : V \rightarrow \mathcal{D}$ compatible with $u$.

Let $a \geq b \geq c > 0$ be real numbers such that for all $n \in \mathbb { N }$, there exist triangles of side lengths $a ^ { n } , b ^ { n } , c ^ { n }$. Prove that the triangles are isosceles.

A subset $S$ of the plane is called convex if given any two points $x$ and $y$ in $S$, the line segment joining $x$ and $y$ is contained in $S$. A quadrilateral is called convex if the region enclosed by the edges of the quadrilateral is a convex set. Show that given a convex quadrilateral $Q$ of area 1, there is a rectangle $R$ of area 2 such that $Q$ can be drawn inside $R$.