Let $(E , \omega)$ be a symplectic vector space of dimension $n = 2m$. Using the results of questions 34--39, prove the following theorem: Every symplectic endomorphism of $E$ can be written as the composition of at most $2n = 4m$ symplectic transvections of $E$: if $u \in \operatorname { Symp } _ { \omega } ( E )$, there exist an integer $p \leqslant 4 m$ and $\tau _ { 1 } , \tau _ { 2 } , \ldots , \tau _ { p }$ symplectic transvections of $E$ such that $u = \tau _ { p } \circ \cdots \circ \tau _ { 2 } \circ \tau _ { 1 }$.

We equip the space $\mathcal { M } _ { n } ( \mathbb { R } )$ with its topology as a normed vector space. Show that the symplectic group $\mathrm { Sp } _ { n } ( \mathbb { R } )$ is an arc-connected subset of this space.

We fix $n = 2m \geqslant 4$. The closed Euclidean ball of radius $r$ is $$B ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + \cdots + x _ { m } ^ { 2 } + y _ { 1 } ^ { 2 } + \cdots + y _ { m } ^ { 2 } \leqslant r ^ { 2 } \right\}$$ and the symplectic cylinder of radius $r$ is $$Z ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + y _ { 1 } ^ { 2 } \leqslant r ^ { 2 } \right\}.$$ Show that, for all $r > 0$, there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ such that $u \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$.

We fix $n = 2m \geqslant 4$. The closed Euclidean ball of radius $r$ is $B ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + \cdots + x _ { m } ^ { 2 } + y _ { 1 } ^ { 2 } + \cdots + y _ { m } ^ { 2 } \leqslant r ^ { 2 } \right\}$.
Let $r > 0$ such that there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$. Let $U \in \mathcal { M } _ { 2 m } ( \mathbb { R } )$ denote the matrix of $u$ in the canonical basis of $\mathbb { R } ^ { 2 m }$. Let $\lambda \in \mathbb { C }$ be a complex eigenvalue of the matrix $U$. Show that $| \lambda | \leqslant r$.
For the case $\lambda$ non-real, if $P$ and $Q$ in $\mathcal { M } _ { 2 m , 1 } ( \mathbb { R } )$ are such that $Z = P + \mathrm { i } Q$ is an eigenvector column of $U$ for the eigenvalue $\lambda$, one may show that $\| U P \| ^ { 2 } + \| U Q \| ^ { 2 } = | \lambda | ^ { 2 } \left( \| P \| ^ { 2 } + \| Q \| ^ { 2 } \right)$.

We fix $n = 2m \geqslant 4$. Let $r > 0$ such that there exists $u \in \mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$, and every complex eigenvalue $\lambda$ of the matrix $U$ of $u$ satisfies $|\lambda| \leq r$. Deduce that $1 \leqslant r$.

We fix $n = 2m \geqslant 4$ and $B ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + \cdots + x _ { m } ^ { 2 } + y _ { 1 } ^ { 2 } + \cdots + y _ { m } ^ { 2 } \leqslant r ^ { 2 } \right\}$. Under what necessary and sufficient condition on $r > 0$ does there exist $u$ belonging to $\mathrm { SL } \left( \mathbb { R } ^ { 2 m } \right)$ such that $u \left( B ^ { 2 m } ( 1 ) \right) \subset B ^ { 2 m } ( r )$?

We fix $n = 2m \geqslant 4$. The closed Euclidean ball of radius $r$ is $B ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + \cdots + x _ { m } ^ { 2 } + y _ { 1 } ^ { 2 } + \cdots + y _ { m } ^ { 2 } \leqslant r ^ { 2 } \right\}$ and the symplectic cylinder is $Z ^ { 2 m } ( r ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + y _ { 1 } ^ { 2 } \leqslant r ^ { 2 } \right\}$.
Let $r > 0$ such that there exists a symplectic endomorphism $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $\psi \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$. We denote by $M \in \operatorname { Sp } _ { 2 m } ( \mathbb { R } )$ the matrix of $\psi$ in the canonical basis $\left( e _ { 1 } , \ldots , e _ { m } , f _ { 1 } , \ldots , f _ { m } \right)$ of $\mathbb { R } ^ { 2 m }$ and by $\psi ^ { \top }$ the endomorphism canonically associated with $M ^ { \top }$. Show that $\left| b _ { s } \left( \psi ^ { \top } \left( e _ { 1 } \right) , \psi ^ { \top } \left( f _ { 1 } \right) \right) \right| = 1$ then that $\left\| \psi ^ { \top } \left( e _ { 1 } \right) \right\| \geqslant 1$ or $\left\| \psi ^ { \top } \left( f _ { 1 } \right) \right\| \geqslant 1$.

We fix $n = 2m \geqslant 4$. Let $r > 0$ such that there exists a symplectic endomorphism $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ satisfying $\psi \left( B ^ { 2 m } ( 1 ) \right) \subset Z ^ { 2 m } ( r )$, where $B^{2m}(r)$ is the closed Euclidean ball and $Z^{2m}(r)$ is the symplectic cylinder. Using the result of Q47, show that $1 \leqslant r$.

We fix $n = 2m \geqslant 4$. The closed Euclidean ball of radius $R$ is $B ^ { 2 m } ( R )$ and the symplectic cylinder of radius $R'$ is $Z ^ { 2 m } ( R' ) = \left\{ \left( x _ { 1 } , \ldots , x _ { m } , y _ { 1 } , \ldots , y _ { m } \right) \in \mathbb { R } ^ { 2 m } , \quad x _ { 1 } ^ { 2 } + y _ { 1 } ^ { 2 } \leqslant R'^{ 2 } \right\}$. Prove the linear non-squeezing theorem: For $R > 0$ and $R ^ { \prime } > 0$, there exists $\psi \in \operatorname { Symp } _ { b _ { s } } \left( \mathbb { R } ^ { 2 m } \right)$ such that $\psi \left( B ^ { 2 m } ( R ) \right) \subset Z ^ { 2 m } \left( R ^ { \prime } \right)$ if and only if $R \leqslant R ^ { \prime }$.

Let $K$ be a compact set of $\mathbb{R}$. Let $k > 0$ and $B$ the set of functions from $K$ to $\mathbb{R}^d$ that are $k$-Lipschitz. Show that $B$ is equicontinuous.

Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. Show that a subset $A \subset C(K, \mathbb{R}^d)$ is relatively compact if and only if every sequence $(f_n)_{n \in \mathbb{N}} \in A^{\mathbb{N}}$ admits a subsequence that converges uniformly to a limit $f \in C(K, \mathbb{R}^d)$.

Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. By reasoning by contradiction, show that if $A$ is relatively compact then $A$ is equicontinuous.

Let $K$ be a compact set of $\mathbb{R}$ and $A$ a subset of $C(K, \mathbb{R}^d)$. We seek to show the following theorem:
Theorem 1: The following two properties are equivalent: - (P1) $A$ is relatively compact. - (P2) $A$ is equicontinuous and for all $x \in K$, the set $A(x) = \{f(x) \mid f \in A\}$ is bounded.
Show that $(P1) \Rightarrow (P2)$.

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 assume that $A$ satisfies (P2). We consider $(f_n)_{n \in \mathbb{N}}$ a sequence of elements of $A$, and $(g_n)_{n \in \mathbb{N}}$ a subsequence converging pointwise on $K$ to $g$.
(a) Show that $g$ is continuous on $K$.
(b) Show that the sequence $(g_n)_{n \in \mathbb{N}}$ converges uniformly to $g$ on $K$. (Hint: you may reason by contradiction.)
(c) Deduce that $(P2) \Rightarrow (P1)$.

We consider $\mathcal{F} : \mathbb{R}^d \rightarrow \mathcal{P}_c(\mathbb{R}^d)$ taking values in the set $\mathcal{P}_c(\mathbb{R}^d)$ of compact subsets of $\mathbb{R}^d$, and the differential inclusion problem: $$\left\{\begin{array}{l} y'(t) \in \mathcal{F}(y(t)) \\ y(0) = y_{\text{init}} \end{array}\right.$$
Show that if for every compact $K \subset \mathbb{R}^d$, there exists $C_K > 0$ such that $\mathcal{F}$ satisfies: $$\forall x, y \in K, \forall v_x \in \mathcal{F}(x), \forall v_y \in \mathcal{F}(y), \quad \langle v_x - v_y, x - y \rangle \leqslant C_K \|x - y\|^2$$ then problem (2) admits at most one maximal solution. (Hint: You may look at $\|X(t) - Y(t)\|^2$ for $X$ and $Y$ two solutions.)

We consider the differential inclusion problem given by $d = 2$ and $\mathcal{F} : \mathbb{R}^2 \rightarrow \mathcal{P}_c(\mathbb{R}^2)$ defined for all $x = (x_1, x_2) \in \mathbb{R}^2$ by: $$\mathcal{F}(x) = \begin{cases} \{v^-\} & \text{if } x_1 < 0 \\ \{v^+\} & \text{if } x_1 > 0 \\ [v_1^+, v_1^-] \times [v_2^+, v_2^-] & \text{if } x_1 = 0 \end{cases}$$ where $v^- = (v_1^-, v_2^-) \in \mathbb{R}^2$ and $v^+ = (v_1^+, v_2^+) \in \mathbb{R}^2$ with $v_1^- \geqslant v_1^+$ and $v_2^- \geqslant v_2^+$.
We set $v^- = (1, 2)$ and $v^+ = (-1, 2)$.
(a) Show that $\mathcal{F}$ satisfies condition (3).
(b) We choose $y_{\text{init}} = (0, 0)$. Find all maximal solutions of problem (2).
(c) We choose $y_{\text{init}} = (1, 0)$. Find all maximal solutions of problem (2).

We consider the differential inclusion problem given by $d = 2$ and $\mathcal{F} : \mathbb{R}^2 \rightarrow \mathcal{P}_c(\mathbb{R}^2)$ defined for all $x = (x_1, x_2) \in \mathbb{R}^2$ by: $$\mathcal{F}(x) = \begin{cases} \{v^-\} & \text{if } x_1 < 0 \\ \{v^+\} & \text{if } x_1 > 0 \\ [v_1^+, v_1^-] \times [v_2^+, v_2^-] & \text{if } x_1 = 0 \end{cases}$$ where $v^- = (v_1^-, v_2^-) \in \mathbb{R}^2$ and $v^+ = (v_1^+, v_2^+) \in \mathbb{R}^2$ with $v_1^- \geqslant v_1^+$ and $v_2^- \geqslant v_2^+$.
We set $v^- = (0, 1)$ and $v^+ = (1, 1)$.
(a) Show that $\mathcal{F}$ does not satisfy condition (3).
(b) We choose $y_{\text{init}} = (1, 0)$. Find all maximal solutions of problem (2).
(c) We choose $y_{\text{init}} = (0, 0)$. Find all maximal solutions of problem (2).

Let $A$ be a commutative ring. Show that if $A$ has property (F), then it has property (TF).

Let $A$ be a commutative ring. Let $S _ { 1 }$ and $S _ { 2 }$ be two subsets of $A$ such that $S _ { 1 } \subset \mathcal { A } \left( S _ { 2 } \right)$. Show that $\mathcal { A } \left( S _ { 1 } \right) \subset \mathcal { A } \left( S _ { 2 } \right)$.

Show that every finite abelian group and the additive group $\mathbf { Z } ^ { r }$ for $r \in \mathbf { N } ^ { * }$ have property (F).

Show that if $n$ is a strictly positive integer, the ring $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right]$ has property (TF), but not property (F).

Show that the ring $\mathbf { Q }$ of rational numbers does not have property (TF).

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 $a _ { 1 } , \ldots , a _ { n } \in A$.