Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_2$: $$\|f\|_2 = \sqrt{\int_0^1 |f(x)|^2 \, dx}$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Calculate the minimal possible value for the constant $M$ in the relation $\exists M \geq 0, \forall f \in E, \|T(f)\|_2 \leq M\|f\|_2$. For this, you may consider the family $(f_n)_{n \geq 2}$ of elements of $E$ such that: (i) $f_n$ is piecewise affine, (ii) $f_n(0) = f_n\left(\frac{1}{2} - \frac{1}{n}\right) = f_n\left(\frac{1}{2} + \frac{1}{n^2}\right) = f_n(1) = 0$ and $f_n\left(\frac{1}{2}\right) = 1$.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Justify that $G$ is a subgroup of $O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ isomorphic to $S O ( 3 )$.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$. We assume that the vector $a$ is non-zero.
Deduce from question III.E.1 that there exists an element $L _ { 1 }$ of $G$ such that: $$L _ { 1 } L = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \ell _ { 1,2 } & \ell _ { 1,3 } & \ell _ { 1,4 } \\ \alpha & \lambda _ { 1 } & \lambda _ { 2 } & \lambda _ { 3 } \\ 0 & \mu _ { 1 } & \mu _ { 2 } & \mu _ { 3 } \\ 0 & \nu _ { 1 } & \nu _ { 2 } & \nu _ { 3 } \end{array} \right)$$ where $\alpha$ is a strictly positive real number that we will specify, $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } , \mu _ { 1 } , \mu _ { 2 } , \mu _ { 3 } , \nu _ { 1 } , \nu _ { 2 }$ and $\nu _ { 3 }$ are real numbers that we will not seek to determine.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$. We assume that the vector $a$ is non-zero. We fix coefficients $\alpha , \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } , \mu _ { 1 } , \mu _ { 2 } , \mu _ { 3 } , \nu _ { 1 } , \nu _ { 2 }$ and $\nu _ { 3 }$ such that $$L _ { 1 } L = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \ell _ { 1,2 } & \ell _ { 1,3 } & \ell _ { 1,4 } \\ \alpha & \lambda _ { 1 } & \lambda _ { 2 } & \lambda _ { 3 } \\ 0 & \mu _ { 1 } & \mu _ { 2 } & \mu _ { 3 } \\ 0 & \nu _ { 1 } & \nu _ { 2 } & \nu _ { 3 } \end{array} \right)$$ Let $v _ { 2 } = \left( \begin{array} { l } \mu _ { 1 } \\ \mu _ { 2 } \\ \mu _ { 3 } \end{array} \right)$ and $v _ { 3 } = \left( \begin{array} { l } \nu _ { 1 } \\ \nu _ { 2 } \\ \nu _ { 3 } \end{array} \right)$. Show that $v _ { 2 }$ and $v _ { 3 }$ are two unit vectors orthogonal to each other in $\mathbb { R } ^ { 3 }$ equipped with its usual Euclidean structure.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$. We assume that the vector $a$ is non-zero. We fix coefficients $\alpha , \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } , \mu _ { 1 } , \mu _ { 2 } , \mu _ { 3 } , \nu _ { 1 } , \nu _ { 2 }$ and $\nu _ { 3 }$ such that $$L _ { 1 } L = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \ell _ { 1,2 } & \ell _ { 1,3 } & \ell _ { 1,4 } \\ \alpha & \lambda _ { 1 } & \lambda _ { 2 } & \lambda _ { 3 } \\ 0 & \mu _ { 1 } & \mu _ { 2 } & \mu _ { 3 } \\ 0 & \nu _ { 1 } & \nu _ { 2 } & \nu _ { 3 } \end{array} \right)$$ Let $R _ { 2 } \in S O ( 3 )$. We set $L _ { 2 } = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R _ { 2 } \end{array} \right) \in G$. Show that we can choose $R _ { 2 }$ such that $$L _ { 1 } L L _ { 2 } = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \beta _ { 1 } & \beta _ { 2 } & \beta _ { 3 } \\ \alpha & \delta _ { 1 } & \delta _ { 2 } & \delta _ { 3 } \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$$ where $\beta _ { 1 } , \beta _ { 2 } , \beta _ { 3 } , \delta _ { 1 } , \delta _ { 2 }$ and $\delta _ { 3 }$ are real numbers that we will not seek to determine.

Let $(A, \vec{b})$ and $(A^{\prime}, \vec{b}^{\prime})$ be in $\mathrm{SO}(2) \times \mathbb{R}^2$. Show that $M(A, \vec{b}) M\left(A^{\prime}, \vec{b}^{\prime}\right) = M\left(A A^{\prime}, A \vec{b}^{\prime} + \vec{b}\right)$.

We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$.
Determine $\Psi\left(M\left(I_2, \overrightarrow{0}\right)\right)$.

We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$.
Verify that $\Psi\left(M\left(R_\theta, q\vec{u}_\theta\right)\right) = \Delta\left(q, \vec{u}_\theta\right)$; deduce that $\Psi$ is surjective.

We now assume that $f$ satisfies the hypotheses allowing us to define its Radon transform.
Demonstrate that if two lines $\Delta\left(q_1, \vec{u}_{\theta_1}\right)$ and $\Delta\left(q_2, \vec{u}_{\theta_2}\right)$ coincide, then $\hat{f}\left(q_1, \theta_1\right) = \hat{f}\left(q_2, \theta_2\right)$.

Let $\lambda \in \mathbb{R}^*$ and $(x_0, y_0) \in \mathbb{R}^2$ be fixed. We define: $$\Omega_{x_0, y_0, \lambda} = \left\{ \lambda(x,y) + (x_0, y_0) \mid (x,y) \in \Omega \right\}$$ By which geometric transformation(s) is the set $\Omega_{x_0, y_0, \lambda}$ the image of $\Omega$? Justify that $\Omega_{x_0, y_0, \lambda}$ is an open set of $\mathbb{R}^2$.

For all permutations $\rho, \rho' \in \mathfrak{S}_n$, show that $P_{\rho\rho'} = P_\rho P_{\rho'}$. Deduce that, for all permutations $\sigma, \tau \in \mathfrak{S}_n$, if $\sigma$ and $\tau$ are conjugate then $P_\sigma$ and $P_\tau$ are similar.

Show that $\alpha$ is a group morphism and describe its kernel, where $$\begin{aligned} \alpha : S \times S & \longrightarrow \mathrm{GL}(\mathbb{H}) \\ (u, v) & \longmapsto (Z \mapsto uZv^{-1}) \end{aligned}$$

Show that $\alpha$ is continuous and that the image of $\alpha$ is contained in $\mathrm{SO}(\mathbb{H})$. One may begin by showing that $\alpha(u,v) \in \mathrm{O}(\mathbb{H})$ for $(u,v) \in S \times S$.

Let $\theta \in \mathbb{R}$ and $v \in \mathbb{H}^{\mathrm{im}} \cap S$, and let $u = (\cos\theta)E + (\sin\theta)v$. a) Show that $u \in S$ and that $u^{-1} = (\cos\theta)E - (\sin\theta)v$. b) Let $w \in \mathbb{H}^{\mathrm{im}} \cap S$ be a vector orthogonal to $v$. Describe the matrix of $C_u$ in the direct orthonormal basis $(v, w, vw)$ of $\mathbb{H}^{\mathrm{im}}$.

Show that the map $u \mapsto C_u$ induces a surjective group morphism $S \rightarrow \mathrm{SO}(\mathbb{H}^{\mathrm{im}})$ and describe its kernel.

Show that $(f(I), f(J), f(K))$ is a direct orthonormal basis of $\mathbb{H}^{\mathrm{im}}$ for all $f \in \mathrm{Aut}(\mathbb{H})$.

Let $f : A \rightarrow B$ be a morphism of commutative rings. Let $F$ be an element of $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right]$. Show that we have $f \left( F \left( a _ { 1 } , \ldots , a _ { n } \right) \right) = F \left( f \left( a _ { 1 } \right) , \ldots , f \left( a _ { n } \right) \right)$ for all $a _ { 1 } , \ldots , a _ { n } \in A$.

Let $B$ be a commutative ring. Let $n$ be a strictly positive integer and $b _ { 1 } , \ldots , b _ { n }$ be elements of $B$. a) Show that there exists a unique ring morphism $f$ from $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right]$ to $B$ such that $f \left( X _ { i } \right) = b _ { i }$ for all $i \in \{ 1 , \ldots , n \}$. b) Deduce that $B$ has property (TF) if and only if there exist an integer $n \geq 1$ and a surjective ring morphism $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right] \rightarrow B$. c) Show that an abelian group $M$ has property (F) if and only if there exist an integer $r \geq 1$ and a surjective group morphism $\mathbf { Z } ^ { r } \rightarrow M$. d) Let $A$ and $B$ be commutative rings such that there exists a surjective ring morphism from $A$ to $B$. Show that if $A$ has property (TF), then so does $B$. State and prove an analogous statement for property (F).

Let $M$ be an additive subgroup of $\mathbf { Z } ^ { n }$ with $n \in \mathbf { N }$ (we agree that $\mathbf { Z } ^ { 0 }$ is the trivial group). We propose to prove by induction on $n$ the following result: (*) There exists $r \in \mathbf { N }$ such that the abelian group $M$ is isomorphic to $\mathbf { Z } ^ { r }$. a) Verify the cases $n = 0$ and $n = 1$. We now assume the result is true for $n - 1$. Let $p : \mathbf { Z } ^ { n } \rightarrow \mathbf { Z }$ be the projection onto the first coordinate, we denote by $N$ the kernel of $p$ and $N _ { 1 } = M \cap N$, then we set $p ( M ) = a \mathbf { Z }$ with $a \in \mathbf { Z }$. We choose $e _ { 1 } \in M$ such that $p \left( e _ { 1 } \right) = a$. Show that if $a \neq 0$, then the application $$N _ { 1 } \times \mathbf { Z } \rightarrow M , ( x , m ) \mapsto x + m e _ { 1 }$$ is a group isomorphism. b) Deduce (*). c) Show that the integer $r$ such that $M$ is isomorphic to $\mathbf { Z } ^ { r }$ is unique (one may consider the rank of a family of vectors of $\mathbf { Z } ^ { r }$ in the $\mathbf { Q }$-vector space $\mathbf { Q } ^ { r }$).

Throughout this part, we denote by $r$ and $s$ strictly positive integers. Let $A$ be a commutative ring not reduced to $\{ 0 \}$. We say that two matrices $M$ and $N$ of $M _ { s , r } ( A )$ are $A$-equivalent if $M \sim N$ (where $M \sim N$ if and only if there exist $U \in G L _ { s } ( A )$ and $V \in G L _ { r } ( A )$ such that $N = U M V$). If $M \in M _ { s , r } ( \mathbf { Z } )$ and $k$ is an integer at most equal to $\min ( r , s )$, we denote by $m _ { k } ( M )$ the gcd of the minors of size $k$ of $M$. Let $M$ and $N$ be two $\mathbf { Z }$-equivalent matrices of $M _ { s , r } ( \mathbf { Z } )$. Show that for all $k \leq \min ( r , s )$, we have $m _ { k } ( M ) = m _ { k } ( N )$ (one may begin by showing that $m _ { k } ( M )$ divides $m _ { k } ( N )$).