Let $n \geq 1$ be an integer. We say that a matrix $M \in \mathcal{M}_n(\mathbb{R})$ is doubly stochastic if for all $i, j \in \{1, \ldots, n\}$ we have $$M_{ij} \geq 0 \quad \text{and} \quad \sum_{k=1}^n M_{ik} = \sum_{k=1}^n M_{kj} = 1.$$ We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$.
Show that $B_n$ is a polytope and determine its dimension.

We consider a convex function $f \in \mathcal{C}(\mathbb{R})$, admitting a minimizer $x_* \in \mathbb{R}$, and $\tau > 0$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := p_f(x_n)$. We have a convergent subsequence $x_{\varphi(n)} \rightarrow x_{**}$ as $n \rightarrow \infty$, where $\varphi : \mathbb{N} \rightarrow \mathbb{N}$ is strictly increasing and $x_{**} \in \mathbb{R}$. Show that $x_{\varphi(n)+1} \rightarrow x_{**}$ as $n \rightarrow \infty$, then deduce that $p_f(x_{**}) = x_{**}$.

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 $S_n$ the symmetric group of order $n$. For all $\sigma \in S_n$, we define $P^\sigma \in \mathcal{M}_n(\mathbb{R})$ as follows: for $i, j \in \{1, 2, \ldots, n\}$ we set $P^\sigma_{ij} = 1$ if $j = \sigma(i)$, $P^\sigma_{ij} = 0$ otherwise. Show that $P^\sigma$ is a vertex of $B_n$ for all $\sigma \in S_n$.

We consider a convex function $f \in \mathcal{C}(\mathbb{R})$, admitting a minimizer $x_* \in \mathbb{R}$, and $\tau > 0$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := p_f(x_n)$. We have established that $p_f(x_{**}) = x_{**}$ for some $x_{**} \in \mathbb{R}$. Conclude that $x_{**}$ is a minimizer of $f$, and that $x_n \rightarrow x_{**}$ as $n \rightarrow \infty$.

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.

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.

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 $M \in B_n \backslash \mathcal{M}_n(\mathbb{Z})$. Deduce that there exists a nonzero matrix $Q$ and $\epsilon > 0$ such that $\{M + tQ, t \in [-\epsilon, \epsilon]\} \subset B_n$, and conclude that every vertex of $B_n$ is of the form $P^\sigma$.

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 $\|x_*\| < 1$. Show that $\nabla f(x_*) = 0$.

Let $n \geq 1$ be an integer. We denote by $\mathbb{C}[[\mathbb{Z}^n]]$ the $\mathbb{C}$-vector space of functions $f : \mathbb{Z}^n \rightarrow \mathbb{C}$. We say that $f \in \mathbb{C}[[\mathbb{Z}^n]]$ is rational if there exists a nonzero $P \in \mathbb{C}[\mathbb{Z}^n]$ such that $Pf \in \mathbb{C}[\mathbb{Z}^n]$. We say that $f$ is torsion if there exists a nonzero $P \in \mathbb{C}[\mathbb{Z}^n]$ such that $Pf = 0$. We denote by $\mathcal{R}$ the $\mathbb{C}$-vector space of rational elements and $\mathcal{T}$ the $\mathbb{C}$-vector space of torsion elements of $\mathbb{C}[[\mathbb{Z}^n]]$.
In the case where $n = 1$, show that the inclusions $0 \subset \mathcal{T} \subset \mathcal{R} \subset \mathbb{C}[[\mathbb{Z}^n]]$ are strict.

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 $t \rightarrow 0^+$.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$. We now assume that $h > 0$.
We set $\gamma_h = G_h''(u_h)$ and we denote $f_h : x \longmapsto \frac{\widehat{G}_h(x)}{x^2}$. Show that $f_h$ can be extended to a continuous function on all of $\mathbb{R}$ by setting $f_h(0) = \frac{\gamma_h}{2}$.

Let $n \geq 1$ be an integer. We denote by $\mathbb{C}[[\mathbb{Z}^n]]$ the $\mathbb{C}$-vector space of functions $f : \mathbb{Z}^n \rightarrow \mathbb{C}$, $\mathcal{R}$ the $\mathbb{C}$-vector space of rational elements, $\mathcal{T}$ the $\mathbb{C}$-vector space of torsion elements, and $\mathbb{C}(\mathbb{Z}^n)$ the field of fractions of $\mathbb{C}[\mathbb{Z}^n]$.
We define a $\mathbb{C}$-linear map $\mathrm{I} : \mathcal{R} \rightarrow \mathbb{C}(\mathbb{Z}^n)$ as follows. If $f \in \mathcal{R}$ satisfies $Qf = P$ with $P, Q \in \mathbb{C}[\mathbb{Z}^n]$, we set $\mathrm{I}(f) = \frac{P}{Q}$. Show that $\mathrm{I}$ is well defined, and that it is a linear map with kernel $\mathcal{T}$ satisfying $\mathrm{I}(Pf) = P\,\mathrm{I}(f)$ for all $f \in \mathcal{R}$ and $P \in \mathbb{C}[\mathbb{Z}^n]$.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$. We now assume that $h > 0$.
We set $\gamma_h = G_h''(u_h)$ and we denote $f_h : x \longmapsto \frac{\widehat{G}_h(x)}{x^2}$.
Justify that $f_h$ is bounded below by a strictly positive real number $c_h$ (which we do not seek to determine).

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 $u(\gamma) > 0$.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$. We now assume that $h > 0$.
We set $\gamma_h = G_h''(u_h)$ and we denote $f_h : x \longmapsto \frac{\widehat{G}_h(x)}{x^2}$.
Show then that $$\int_{-\infty}^{+\infty} \mathrm{e}^{-n\widehat{G}_h\left(\frac{t}{\sqrt{n}}\right)} \mathrm{d}t \xrightarrow[n \rightarrow +\infty]{} \sqrt{\frac{2\pi}{\gamma_h}}$$ then conclude that $\psi(h) = -G_h(u_h)$.

Let $n \geq 1$ be an integer. For $A \subset \mathbb{R}^n$, let $E_A = \sum_{\gamma \in A \cap \mathbb{Z}^n} x^\gamma \in \mathbb{C}[[\mathbb{Z}^n]]$.
Show that if $A$ and $B$ are two subsets of $\mathbb{R}^n$ and $\gamma \in \mathbb{Z}^n$ we have $$E_{A \cup B} + E_{A \cap B} = E_A + E_B \quad \text{and} \quad E_{\gamma + A} = x^\gamma E_A.$$

We have
$$I _ { n } = ( - 1 ) ^ { n } \int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } \frac { x ^ { n } ( 1 - x ) ^ { n } y ^ { n } ( 1 - y ) ^ { n } } { ( 1 - x y ) ^ { n + 1 } } \mathrm {~d} x \mathrm {~d} y$$
Let $n \in \mathbb { N } ^ { * }$. Deduce that
$$\left| I _ { n } \right| \leqslant \zeta ( 2 ) \left( \frac { 5 \sqrt { 5 } - 11 } { 2 } \right) ^ { n }$$

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
Show that $G_0'$ establishes a continuous bijection from $[u_0; +\infty[$ to $\mathbb{R}_+$. Deduce that the function $u : h \longmapsto u_h$ is continuous on $\mathbb{R}_+$ and differentiable on $\mathbb{R}_+^*$.

Let $n \geq 1$ be an integer. For $A \subset \mathbb{R}^n$, let $E_A = \sum_{\gamma \in A \cap \mathbb{Z}^n} x^\gamma \in \mathbb{C}[[\mathbb{Z}^n]]$.
Let $\gamma_1, \ldots, \gamma_k$ be a family of vectors in $\mathbb{Z}^n \subset \mathbb{R}^n$ and $$C(\gamma_1, \ldots, \gamma_k) = \left\{\sum_{i=1}^k t_i \gamma_i : (t_1, \ldots, t_k) \in [0, +\infty[^k\right\}.$$ Show that if $\gamma_1, \ldots, \gamma_k$ is a free family, $E_{v + C(\gamma_1, \ldots, \gamma_k)}$ is rational for all $v \in \mathbb{R}^n$.

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 assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
Deduce that $\psi$ is differentiable on $\mathbb{R}_+^*$ and that $m(h) = \frac{u(h) - h}{\beta}$ for all $h \in \mathbb{R}_+^*$.

Let $n \geq 1$ be an integer. For $A \subset \mathbb{R}^n$, let $E_A = \sum_{\gamma \in A \cap \mathbb{Z}^n} x^\gamma \in \mathbb{C}[[\mathbb{Z}^n]]$. Let $$C(\gamma_1, \ldots, \gamma_k) = \left\{\sum_{i=1}^k t_i \gamma_i : (t_1, \ldots, t_k) \in [0, +\infty[^k\right\}.$$
Generalize the previous question in the case where $\gamma_1, \ldots, \gamma_k \in \mathbb{Z}^n$ is a family of vectors not necessarily free but for which there exists a linear form $\ell : \mathbb{R}^n \rightarrow \mathbb{R}$ such that $\ell(\gamma_i) > 0$ for $i = 1, \ldots, k$.
Hint. One may triangulate the polytope $P = \{x \in C(\gamma_1, \ldots, \gamma_k) : \ell(x) = 1\}$.

Show that $\zeta ( 2 )$ is an irrational number.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
Show then that $m^+ = 0$ if $\beta \leqslant 1$, and $m^+ > 0$ if $\beta > 1$.

We admit, only in this question, that $\zeta ( 2 ) = \frac { \pi ^ { 2 } } { 6 }$. Show that $\pi$ is an irrational number.