In this part, $\varphi : [ a , b ] \rightarrow \mathbb { R }$ and $f : [ a , b ] \rightarrow \mathbb { R }$ are two functions of class $\mathcal { C } ^ { \infty }$. We are interested in integrals of the form $$I ( \lambda ) = \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } f ( x ) d x$$ where $\lambda$ is a strictly positive real parameter.
(a) We assume that $\left| \varphi ^ { \prime } ( x ) \right| \geq 1$ for all $x \in [ a , b ]$ and that $\varphi ^ { \prime }$ is monotone on $[ a , b ]$. Show that there exists a constant $c _ { 1 } > 0$, independent of $\lambda , \varphi$ and of $a , b$, such that $$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x \right| \leq c _ { 1 } \lambda ^ { - 1 }$$ Hint. One can write $$\int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x = \int _ { a } ^ { b } i \lambda \varphi ^ { \prime } ( x ) e ^ { i \lambda \varphi ( x ) } \frac { 1 } { i \lambda \varphi ^ { \prime } ( x ) } d x$$ and integrate by parts.
(b) Let $\delta > 0$. We assume that $\left| \varphi ^ { \prime } ( x ) \right| \geq \delta$ for all $x \in [ a , b ]$ and that $\varphi ^ { \prime }$ is monotone on $[ a , b ]$. Show that $$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x \right| \leq c _ { 1 } ( \lambda \delta ) ^ { - 1 }$$

We assume $\mathbb{K} = \mathbb{R}$. We denote by $O ( n )$ the usual orthogonal group of $\mathbb { R } ^ { n }$ (which identifies with $O _ { n , 0 }$). We denote by $K _ { r , s } : = O _ { r , s } \cap O ( n )$.
Prove that $K _ { r , s }$ is compact and in bijection with $O ( r ) \times O ( s )$.

In this part, $\varphi : [ a , b ] \rightarrow \mathbb { R }$ and $f : [ a , b ] \rightarrow \mathbb { R }$ are two functions of class $\mathcal { C } ^ { \infty }$. We are interested in integrals of the form $$I ( \lambda ) = \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } f ( x ) d x$$ where $\lambda$ is a strictly positive real parameter.
Case where the phase can be stationary. Throughout this question, we assume that $\left| \varphi ^ { \prime \prime } ( x ) \right| \geq 1$ for all $x \in [ a , b ]$.
(a) Show that $\varphi ^ { \prime }$ is strictly monotone on $[ a , b ]$ and that there exists a unique point $c \in [ a , b ]$ such that $\left| \varphi ^ { \prime } ( c ) \right| = \inf _ { x \in [ a , b ] } \left| \varphi ^ { \prime } ( x ) \right|$.
(b) If $x \in [ a , b ]$, show that $\left| \varphi ^ { \prime } ( x ) \right| \geq | x - c |$.
(c) Show that for all $\delta > 0$, $$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x \right| \leq 2 c _ { 1 } ( \lambda \delta ) ^ { - 1 } + 2 \delta$$ (d) Deduce that there exists a constant $c _ { 2 }$, independent of $\lambda , \varphi , a$ and $b$ such that $$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } d x \right| \leq c _ { 2 } \lambda ^ { - 1 / 2 }$$ (e) Show that $$\left| \int _ { a } ^ { b } e ^ { i \lambda \varphi ( x ) } f ( x ) d x \right| \leq c _ { 2 } \lambda ^ { - 1 / 2 } \left( | f ( b ) | + \int _ { a } ^ { b } \left| f ^ { \prime } ( x ) \right| d x \right)$$

We assume $\mathbb{K} = \mathbb{R}$. Prove that $S O ( 2 ) = \{ M \in O ( 2 ) \mid \operatorname { det } ( M ) = 1 \}$ is path-connected.

We assume $\mathbb{K} = \mathbb{R}$. Let $H : = \left\{ ( x , y , z ) \in \mathbb { R } ^ { 3 } \mid z ^ { 2 } = x ^ { 2 } + y ^ { 2 } + 1 \right\}$ be a hyperboloid of two sheets.
(a) Prove that if $f \in O \left( Q _ { 2,1 } \right)$, then $f ( H ) = H$.
(b) We denote by $S O _ { 2,1 } : = \left\{ M \in O _ { 2,1 } \mid \operatorname { det } ( M ) = 1 \right\}$. Prove that $S O _ { 2,1 }$ is a closed subgroup of $O _ { 2,1 }$.

We assume $\mathbb{K} = \mathbb{R}$. For $f \in O \left( Q _ { 2,1 } \right)$, we denote by $( x _ { f } , y _ { f } , z _ { f } )$ the vector $f ( 0,0,1 )$. We also denote by $S O _ { 2,1 } ^ { + } : = \left\{ M = j ( f ) \in S O _ { 2,1 } \mid z _ { f } > 0 \right\}$.
(a) Prove that, for all $t \in \mathbb { R }$, the linear map $r _ { t }$ whose matrix (in the canonical basis of $\mathbb { R } ^ { 3 }$ ) equals $$\left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \operatorname { ch } ( t ) & \operatorname { sh } ( t ) \\ 0 & \operatorname { sh } ( t ) & \operatorname { ch } ( t ) \end{array} \right]$$ is in $S O _ { 2,1 } ^ { + }$ (one may subsequently call such a linear map a hyperbolic rotation).
(b) Let $M = j ( f )$. Suppose that $M \in S O _ { 2,1 } ^ { + }$. Show that there exists a rotation (in the usual sense) $\rho$ with axis $( 0,0,1 )$ and $t \in \mathbb { R }$ such that $r _ { t } \circ \rho \circ f \in S O _ { 2,1 } ^ { + }$ and satisfies $r _ { t } \circ \rho \circ f ( 0,0,1 ) = ( 0,0,1 )$.
(c) Prove that $S O _ { 2,1 } ^ { + }$ is path-connected.

The Carnot sphere is the set: $$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$
The functions $f$ and $g$ on $[0, 2\pi]$ are defined by: $$f(s) = \frac{2(1-\cos s)}{s^2} \quad \text{and} \quad g(s) = \frac{s - \sin s}{2s^2}$$ (extended by continuity at $0$).
Show that if $(p,q,r) \in B(1)$ with $r \geq 0$ then $r = g \circ f^{-1}(p^2 + q^2)$.
State and establish a converse.
One may give the shape of the function $s \mapsto g \circ f^{-1}(s^2)$ for $s \in [0,1]$ and in particular the tangent lines at $s=0$ and $s=1$.

We assume $\mathbb{K} = \mathbb{R}$. Deduce from question 14 that $O _ { 2,1 }$ is the union of four closed subsets pairwise disjoint and path-connected.

The Carnot sphere is the set: $$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$
Show the existence of a constant $c_1 > 0$ such that for all $(p,q,r) \in B(1)$, we have $$c_1^{-1} \leq p^2 + q^2 + |r| \leq c_1$$

We assume $\mathbb{K} = \mathbb{R}$. Prove that there exists a surjective group homomorphism $\psi : O _ { 2,1 } \rightarrow \mathbb { Z } / 2 \mathbb { Z } \times \mathbb { Z } / 2 \mathbb { Z }$ whose kernel is $S O _ { 2,1 } ^ { + }$.

The Carnot sphere is the set: $$B(1) = \left\{(p,q,r) \in \mathbf{R}^3 \mid \exists (\theta,\varphi) \in [-\pi,\pi] \times [-2\pi,2\pi], \quad \gamma_{\theta,\varphi}(1) = \exp\left(M_{p,q,r}\right)\right\}.$$
(a) Show that for all $(p,q,r) \in \mathbf{R}^3 \setminus \{(0,0,0)\}$, there exists a unique $\lambda > 0$ such that: $$(\lambda p, \lambda q, \lambda^2 r) \in B(1).$$
(b) Deduce that for every point $A \in \mathbf{H}$, there exists a positive real $T(A)$ and parameters $(\theta, \varphi)$ (also depending on $A$) such that $A$ is the endpoint of the Carnot path controlled by $(u_{\theta,\varphi}, v_{\theta,\varphi}) \in E(T(A))$.
(c) Show the existence of a constant $c_2 > 0$ such that for all $(p,q,r) \in \mathbf{R}^3$, $$c_2^{-1}\sqrt{p^2 + q^2 + |r|} \leq T\left(\exp\left(M_{p,q,r}\right)\right) \leq c_2\sqrt{p^2 + q^2 + |r|}$$

We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. If $V$ and $V ^ { \prime }$ are two $\mathbb { K }$-vector spaces of finite dimension, $q \in \mathcal { Q } ( V )$ and $q ^ { \prime } \in \mathcal { Q } \left( V ^ { \prime } \right)$ are two non-degenerate quadratic forms, the orthogonal sum $q \perp q ^ { \prime }$ of $q$ and $q ^ { \prime }$ is the quadratic form on $V \times V ^ { \prime }$ defined by $$q \perp q ^ { \prime } \left( x , x ^ { \prime } \right) = q ( x ) + q ^ { \prime } \left( x ^ { \prime } \right)$$ for all $x \in V$ and all $x ^ { \prime } \in V ^ { \prime }$.
Let $V , V ^ { \prime }$ and $V ^ { \prime \prime }$ be three $\mathbb { K }$-vector spaces of finite dimension and $\left( q , q ^ { \prime } , q ^ { \prime \prime } \right) \in \mathcal { Q } ( V ) \times \mathcal { Q } \left( V ^ { \prime } \right) \times \mathcal { Q } \left( V ^ { \prime \prime } \right)$.
(a) Show that $q \perp q ^ { \prime } \in \mathcal { Q } \left( V \times V ^ { \prime } \right)$ and then that $\left( q \perp q ^ { \prime } \right) \perp q ^ { \prime \prime } \cong q \perp \left( q ^ { \prime } \perp q ^ { \prime \prime } \right)$.
(b) Show that if $q ^ { \prime } \cong q ^ { \prime \prime }$ then $q \perp q ^ { \prime } \cong q \perp q ^ { \prime \prime }$.
(c) Prove that if $V = V ^ { \prime } \oplus V ^ { \prime \prime }$ and $\tilde { q } ( x , y ) = 0$ for all $x \in V ^ { \prime }$ and all $y \in V ^ { \prime \prime }$, then $q \cong q ^ { \prime } \perp q ^ { \prime \prime }$ where $q ^ { \prime }$ is the restriction of $q$ to $V ^ { \prime }$ and $q ^ { \prime \prime }$ is the restriction of $q$ to $V ^ { \prime \prime }$.

We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. Let $V$ be a $\mathbb { K }$-vector space of finite dimension, $q \in \mathcal { Q } ( V )$ and $v , w \in V$ be two distinct vectors of $V$ such that $q ( v ) = q ( w ) \neq 0$.
We want to show in this question that there then exists an isometry $h \in O ( q )$ such that $h ( v ) = w$.
(a) Let $x \in V$ such that $q ( x ) \neq 0$. We denote by $s _ { x }$ the endomorphism of $V$ defined by $y \mapsto s _ { x } ( y ) = y - 2 \frac { \widetilde { q } ( x , y ) } { q ( x ) } x$. Show that $s _ { x }$ and $- s _ { x }$ belong to $O ( q )$.
(b) Suppose here that $q ( w - v ) \neq 0$. Show that the map $s _ { w - v }$ is an isometry such that $s _ { w - v } ( v ) = w$.
(c) Suppose here that $q ( w - v ) = 0$. Show that there exists an isometry $g \in O ( q )$ such that $g ( v ) = w$ and conclude.

We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. Let $\left( V _ { i } \right) _ { 1 \leq i \leq 3 }$ be three $\mathbb { K }$-vector spaces of finite dimension and $q _ { i } \in \mathcal { Q } \left( V _ { i } \right)$ for $1 \leq i \leq 3$ satisfying $q _ { 1 } \perp q _ { 3 } \cong q _ { 2 } \perp q _ { 3 }$. Show that $q _ { 1 } \cong q _ { 2 }$.
Hint: one may reason by induction and use questions 17 and 18.

We return to the case where $\mathbb { K }$ is an arbitrary field of characteristic zero. Let $V$ be a $\mathbb { K }$-vector space of finite dimension and $q \in \mathcal { Q } ( V )$. Show that there exists a unique non-negative integer $m$ and an anisotropic quadratic form $q _ { \text {an} }$, unique up to isometry, such that $q \cong q _ { an } \perp m \cdot h$ where $m \cdot h = h \perp \cdots \perp h$ is the orthogonal sum of $m$ copies of $h$ and $h$ is the quadratic form defined by $h \left( x _ { 1 } , x _ { 2 } \right) = x _ { 1 } x _ { 2 }$ (introduced in question 6b).
Hint: one may use question 6b and the previous question.

In this problem, $\mathbb{K}$ denotes the field $\mathbb{R}$ or the field $\mathbb{C}$ and $E$ is a non-zero $\mathbb{K}$-vector space. If $f$ is an endomorphism of $E$, for every subspace $F$ of $E$ stable by $f$ we denote by $f_F$ the endomorphism of $F$ induced by $f$.
Show that a line $F$ generated by a vector $u$ is stable by $f$ if and only if $u$ is an eigenvector of $f$.

In this problem, $\mathbb{K}$ denotes the field $\mathbb{R}$ or the field $\mathbb{C}$ and $E$ is a non-zero $\mathbb{K}$-vector space. $f$ is an endomorphism of a $\mathbb{K}$-vector space $E$.
I.B.1) Show that there exist at least two subspaces of $E$ stable by $f$ and give an example of an endomorphism of $\mathbb{R}^2$ which admits only two stable subspaces.
I.B.2) Show that if $E$ is of finite dimension $n \geqslant 2$ and if $f$ is non-zero and non-injective, then there exist at least three subspaces of $E$ stable by $f$ and at least four when $n$ is odd.
Give an example of an endomorphism of $\mathbb{R}^2$ which admits only three stable subspaces.

In this problem, $\mathbb{K}$ denotes the field $\mathbb{R}$ or the field $\mathbb{C}$ and $E$ is a non-zero $\mathbb{K}$-vector space. $f$ is an endomorphism of a $\mathbb{K}$-vector space $E$.
I.C.1) Show that every subspace generated by a family of eigenvectors of $f$ is stable by $f$. Specify the endomorphism induced by $f$ on every eigenspace of $f$.
I.C.2) Show that if $f$ admits an eigenspace of dimension at least equal to 2 then there exist infinitely many lines of $E$ stable by $f$.
I.C.3) What can be said about $f$ if all subspaces of $E$ are stable by $f$?

In this problem, $\mathbb{K}$ denotes the field $\mathbb{R}$ or the field $\mathbb{C}$ and $E$ is a non-zero $\mathbb{K}$-vector space. $f$ is an endomorphism of a $\mathbb{K}$-vector space $E$. In this subsection, $E$ is a space of finite dimension.
I.D.1) Show that if $f$ is diagonalisable then every subspace of $E$ admits a complement in $E$ stable by $f$.
One may start from a basis of $F$ and a basis of $E$ consisting of eigenvectors of $f$.
I.D.2) Show that if $\mathbb{K} = \mathbb{C}$ and if every subspace of $E$ stable by $f$ admits a complement in $E$ stable by $f$, then $f$ is diagonalisable.
What about the case if $\mathbb{K} = \mathbb{R}$?

In this part, $n$ and $p$ are two natural integers at least equal to 2, $f$ is a diagonalisable endomorphism of a $\mathbb{K}$-vector space $E$ of dimension $n$, which admits $p$ distinct eigenvalues $\{\lambda_1, \ldots, \lambda_p\}$ and, for all $i$ in $\llbracket 1, p \rrbracket$, we denote by $E_i$ the eigenspace of $f$ associated with the eigenvalue $\lambda_i$.
The goal here is to show that a subspace $F$ of $E$ is stable by $f$ if and only if $F = \bigoplus_{i=1}^{p} (F \cap E_i)$.
II.A.1) Show that every subspace $F$ of $E$ such that $F = \bigoplus_{i=1}^{p} (F \cap E_i)$ is stable by $f$.
II.A.2) Let $F$ be a subspace of $E$ stable by $f$ and $x$ a non-zero vector of $F$.
Justify the existence and uniqueness of $(x_i)_{1 \leqslant i \leqslant p}$ in $E_1 \times \cdots \times E_p$ such that $x = \sum_{i=1}^{p} x_i$.
II.A.3) If we denote $H_x = \{i \in \llbracket 1, p \rrbracket \mid x_i \neq 0\}$, $H_x$ is non-empty and, up to reordering the eigenvalues (and the eigenspaces), we can assume that $H_x = \llbracket 1, r \rrbracket$ with $1 \leqslant r \leqslant p$. Thus we have $x = \sum_{i=1}^{r} x_i$ with $x_i \in E_i \setminus \{0\}$ for all $i$ in $\llbracket 1, r \rrbracket$.
We denote $V_x = \operatorname{Vect}(x_1, \ldots, x_r)$.
Show that $\mathcal{B}_x = (x_1, \ldots, x_r)$ is a basis of $V_x$.
II.A.4) Show that for all $j$ in $\llbracket 1, r \rrbracket$, $f^{j-1}(x)$ belongs to $V_x$ and give the matrix of the family $(f^{j-1}(x))_{1 \leqslant j \leqslant r}$ in the basis $\mathcal{B}_x$.
II.A.5) Show that $(f^{j-1}(x))_{1 \leqslant j \leqslant r}$ is a basis of $V_x$.
II.A.6) Deduce that for all $i$ in $\llbracket 1, p \rrbracket$, $x_i$ belongs to $F$ and conclude.

In this part, $n$ and $p$ are two natural integers at least equal to 2, $f$ is a diagonalisable endomorphism of a $\mathbb{K}$-vector space $E$ of dimension $n$, which admits $p$ distinct eigenvalues $\{\lambda_1, \ldots, \lambda_p\}$ and, for all $i$ in $\llbracket 1, p \rrbracket$, we denote by $E_i$ the eigenspace of $f$ associated with the eigenvalue $\lambda_i$. In this subsection, we place ourselves in the case where $p = n$.
II.B.1) Specify the dimension of $E_i$ for all $i$ in $\llbracket 1, p \rrbracket$.
II.B.2) How many lines of $E$ are stable by $f$?
II.B.3) If $n \geqslant 3$ and $k \in \llbracket 2, n-1 \rrbracket$, how many subspaces of $E$ of dimension $k$ and stable by $f$ are there?
II.B.4) How many subspaces of $E$ are stable by $f$ in this case? Give them all.

We consider the differentiation endomorphism $D$ on $\mathbb{K}[X]$ defined by $D(P) = P'$ for all $P$ in $\mathbb{K}[X]$.
III.A.1) Verify that for all $n$ in $\mathbb{N}$, $\mathbb{K}_n[X]$ is stable by $D$ and give the matrix $A_n$ of the endomorphism induced by $D$ on $\mathbb{K}_n[X]$ in the canonical basis of $\mathbb{K}_n[X]$.
III.A.2) Let $F$ be a subspace of $\mathbb{K}[X]$, of finite non-zero dimension, stable by $D$.
a) Justify the existence of a natural integer $n$ and a polynomial $R$ of degree $n$ such that $R \in F$ and $F \subset \mathbb{K}_n[X]$.
b) Show that the family $(D^i(R))_{0 \leqslant i \leqslant n}$ is a free family of $F$.
c) Deduce that $F = \mathbb{K}_n[X]$.
III.A.3) Give all subspaces of $\mathbb{K}[X]$ stable by $D$.

Draw a graph of $\Delta\left(0, \vec{e}_1\right)$ and $\Delta\left(2, \frac{\vec{e}_1 + \vec{e}_2}{\sqrt{2}}\right)$.

We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$.
Draw $\Psi(M(A, \vec{b}))$ in the case $A = R_{\pi/6}$ and $\vec{b} = \binom{1}{2}$.

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 $\Psi\left(M\left(I_2, \overrightarrow{0}\right)\right)$.