We define $$\varphi _ { p + 1 } \left( v , v ^ { \prime } \right) = { } ^ { t } V \Delta _ { p + 1 } V ^ { \prime } = v _ { 1 } v _ { 1 } ^ { \prime } - \sum _ { i = 2 } ^ { p + 1 } v _ { i } v _ { i } ^ { \prime }$$ and $$q _ { p + 1 } ( v ) = \varphi _ { p + 1 } ( v , v )$$ If $L = \left( l _ { i , j } \right) _ { i , j } \in O ( 1 , p ) , v = ( 1,0 , \ldots , 0 )$ and $v ^ { \prime } = ( 0,1,0 , \ldots , 0 )$, give the equations on the $l _ { i , j }$ corresponding to $$\varphi _ { p + 1 } \left( f ( v ) , f \left( v ^ { \prime } \right) \right) = \varphi _ { p + 1 } \left( v , v ^ { \prime } \right) , \quad q _ { p + 1 } ( f ( v ) ) = q _ { p + 1 } ( v ) \quad \text { and } \quad q _ { p + 1 } \left( f \left( v ^ { \prime } \right) \right) = q _ { p + 1 } \left( v ^ { \prime } \right)$$ What do we obtain similarly with ${ } ^ { t } L$ ?

Let $a , b , c$ and $d$ be four real numbers. We consider the matrix of $\mathcal { M } _ { 2 } ( \mathbb { R } )$ $$L = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)$$ Write the equations on $a , b , c , d$ expressing the membership of $L$ in $O ( 1,1 )$.

We denote, for every real $\gamma$, $L ( \gamma ) = \left( \begin{array} { c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma \end{array} \right)$.
Show, for all real numbers $\gamma$ and $\gamma ^ { \prime }$, the equality: $$L ( \gamma ) L \left( \gamma ^ { \prime } \right) = L \left( \gamma + \gamma ^ { \prime } \right)$$ Deduce that $O ^ { + } ( 1,1 ) \cap \tilde { O } ( 1,1 )$ is a commutative subgroup of the group $O ^ { + } ( 1,1 )$.

Is the group $O ^ { + } ( 1,1 ) \cap \tilde { O } ( 1,1 )$ compact?

Show that the group $O ^ { + } ( 1,1 )$ is commutative.

Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ( 1,3 )$. Show the inequality $\ell _ { 1,1 } ^ { 2 } \geqslant 1$.

Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 }$ and $L ^ { \prime } = \left( \ell _ { i , j } ^ { \prime } \right) _ { 1 \leqslant i , j \leqslant 4 }$ be two elements of $\tilde { O } ( 1,3 )$. We set $L ^ { \prime \prime } = L L ^ { \prime } = \left( \ell _ { i , j } ^ { \prime \prime } \right) _ { 1 \leqslant i , j \leqslant 4 }$.
Prove the following inequalities: $$0 \leqslant \sqrt { \sum _ { k = 2 } ^ { 4 } \ell _ { 1 , k } ^ { 2 } } \sqrt { \sum _ { k = 2 } ^ { 4 } \ell _ { k , 1 } ^ { \prime 2 } } + \sum _ { k = 2 } ^ { 4 } \ell _ { 1 , k } \ell _ { k , 1 } ^ { \prime } < \ell _ { 1,1 } ^ { \prime \prime }$$ Deduce that the set $\tilde { O } ( 1,3 )$ is a subgroup of the Lorentz group $O ( 1,3 )$.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ 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 $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$.
Show that, if the vector $a$ is zero, then the matrix $L$ belongs to the group $G$.

In the usual Euclidean space $\mathbb { R } ^ { 3 }$, show that, for all vectors $u$ and $v$ of $\mathbb { R } ^ { 3 }$ of the same norm, there exists a rotation $r$ such that $r ( u ) = v$.

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.

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 $L_1 \in G$ and $R_2 \in SO(3)$ 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 $L_2 = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R _ { 2 } \end{array} \right)$.
Show that the real numbers $\beta _ { 2 } , \beta _ { 3 } , \delta _ { 2 }$ and $\delta _ { 3 }$ are zero.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Deduce that every matrix $L$ of $O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ can be written in the form of a product of the type $$L = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) \left( \begin{array} { c c c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma & 0 & 0 \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R ^ { \prime } \end{array} \right)$$ where $R$ and $R ^ { \prime }$ are two elements of $S O ( 3 )$ and $\gamma$ is a real number.

Is the decomposition $$L = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) \left( \begin{array} { c c c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma & 0 & 0 \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R ^ { \prime } \end{array} \right)$$ obtained unique?

Prove that $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ is indeed a quadratic form on $\mathbb { K } ^ { n }$, where $\left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$ denotes the quadratic form $q$ defined on $\mathbb { K } ^ { n }$ by the formula $$q \left( x _ { 1 } , \ldots , x _ { n } \right) = a _ { 1 } x _ { 1 } ^ { 2 } + \cdots + a _ { n } x _ { n } ^ { 2 }$$

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$: $$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Show that $T \in \mathcal{L}(E)$.

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$: $$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$ 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 $\exists M \geq 0, \forall f \in E, \|T(f)\|_E \leq M\|f\|_E$.

Let $E$ be the set of continuous functions from $[0,1]$ to $\mathbb{R}$ equipped with the norm $\|.\|_{\infty}$: $$\|f\|_{\infty} = \max_{x \in [0,1]} |f(x)|$$ We denote by $T$ the application defined on $E$ such that: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Determine $\operatorname{Ker}(T)$ and $\operatorname{Im}(T)$.

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: $$\forall f \in E, \quad \forall x \in [0,1], \quad T(f)(x) = x f\left(\frac{x}{2}\right)$$ Show that $T \in \mathcal{L}(E)$ with this norm.

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 consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix}.$$
(a) Show that we define a group law $*$ on $\mathbf{L}$ by setting for $M, N \in \mathbf{L}$: $$M * N = M + N + \frac{1}{2}[M, N]$$ Explicitly determine the inverse of $M_{p,q,r}$.
(b) Determine the matrices $M_{p,q,r} \in \mathbf{L}$ that commute with all elements of $\mathbf{L}$ for the law $*$. Is $(\mathbf{L}, *)$ commutative?

We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix},$$ and the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Show that for all matrices $M, N \in \mathbf{L}$, we have: $$(\exp M) \times (\exp N) = \exp(M * N)$$

We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix},$$ and the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Let $M$ and $N$ be two elements of $\mathbf{L}$. Show that $$\exp([M,N]) = \exp(M)\exp(N)\exp(-M)\exp(-N)$$