Let $a \in \mathbf{Q}^{\times}$ be a nonzero rational number. Deduce from Theorem 1 that, for every nonzero polynomial $P \in \mathbf{Q}[x]$, we have $P(e^a) \neq 0$.
(Theorem 1: Let $r \geq 2$ be an integer. If $a_1, \ldots, a_r \in \mathbf{Q}$ are distinct rational numbers, then the real numbers $e^{a_1}, \ldots, e^{a_r}$ are linearly independent over $\mathbf{Q}$.)

Show that $CL(\mathbf{R})$ is a vector space. Also show that $CL(\mathbf{R})$ is closed under multiplication.

Show that $CL(\mathbf{R})$ is a vector space. Also show that $CL(\mathbf{R})$ is stable under multiplication.

Let $f(x) = \sum_{n=0}^{\infty} c_n x^n \in \mathbf{Q}\llbracket x \rrbracket$ be a power series whose coefficients $c_n$ are integers. Show that if there exists a real number $\alpha \geq 1$ such that the numerical series $\sum_{n=0}^{\infty} c_n \alpha^n$ converges, then $f$ is a polynomial.

Let $t \in \mathbf{R}_{+}$. Verify that the function $P_{t}(f)$ is well defined for $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ and verify that $P_{t}$ is linear on $C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$.
Recall that $\forall x \in \mathbf{R}, \quad P_{t}(f)(x) = \int_{-\infty}^{+\infty} f\left(\mathrm{e}^{-t} x + \sqrt{1 - \mathrm{e}^{-2t}} y\right) \varphi(y) \mathrm{d}y.$

Let $t \in \mathbf{R}_+$. Verify that the function $P_t(f)$ is well defined for $f \in C^0(\mathbf{R}) \cap CL(\mathbf{R})$ and verify that $P_t$ is linear on $C^0(\mathbf{R}) \cap CL(\mathbf{R})$, where $$\forall x \in \mathbf{R}, \quad P_t(f)(x) = \int_{-\infty}^{+\infty} f\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y.$$

Let $Q \in \mathbf{Q}[x]$ be a polynomial with rational coefficients such that 0 is not a root. Show that there exists a unique power series $f \in \mathbf{Q}\llbracket x \rrbracket$ satisfying $Q \cdot f = 1$.
Show that if $Q$ has integer coefficients and its constant term $c_0$ equals 1 or $-1$, then this unique power series $f$ has integer coefficients.

Show that for all $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ and all $x \in \mathbf{R}$,
$$\lim_{t \rightarrow +\infty} P_{t}(f)(x) = \int_{-\infty}^{+\infty} f(y) \varphi(y) \mathrm{d}y$$

Show that for all $f \in C^0(\mathbf{R}) \cap CL(\mathbf{R})$ and all $x \in \mathbf{R}$, $$\lim_{t \rightarrow +\infty} P_t(f)(x) = \int_{-\infty}^{+\infty} f(y)\varphi(y)\,\mathrm{d}y,$$ where $P_t(f)(x) = \int_{-\infty}^{+\infty} f\!\left(\mathrm{e}^{-t}x + \sqrt{1-\mathrm{e}^{-2t}}\,y\right)\varphi(y)\,\mathrm{d}y$.

Let $G$ be a graph and $G ^ { \prime }$ a copy of $G$. Justify that $\chi _ { G } = \chi _ { G ^ { \prime } }$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Let $V$ be a convex open set of $E$ and $h$ a function of class $\mathscr{C}^1$ from $V$ to $E$. Suppose that there exists a real number $C \geqslant 0$ such that for all $x \in V$, $\|dh(x)\| \leqslant C$. Show that for all $x_1$ and $x_2$ in $V$, we have $\left\|h(x_2) - h(x_1)\right\| \leqslant C \left\|x_2 - x_1\right\|$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Show that there exists a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Show that for all $x \in B(a,r)$, the linear map $df(x)$ is injective.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Let $V$ be a convex open set of $E$ and $h$ a function of class $\mathscr{C}^1$ from $V$ to $E$. Suppose that there exists a real number $C \geqslant 0$ such that for all $x \in V$, $\|dh(x)\| \leqslant C$. Show that for all $x_1$ and $x_2$ in $V$, we have $\left\|h(x_2) - h(x_1)\right\| \leqslant C \left\|x_2 - x_1\right\|$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Show that there exists a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Show that for all $x \in B(a,r)$, the linear map $df(x)$ is injective.

Let $t \in \mathbf{R}_{+}$. Show that if $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$, then $P_{t}(f) \in C^{0}(\mathbf{R})$. Also show that $P_{t}(f)$ is bounded in absolute value by a polynomial function in $|x|$ independent of $t$. Deduce that $P_{t}(f) \in L^{1}(\varphi)$.

Let $g$ be the function defined by
$$\begin{aligned} g : ] - \pi ; \pi [ & \longrightarrow \mathbf { C } \\ \theta & \longmapsto e ^ { \mathrm { i } x \theta } \int _ { 0 } ^ { + \infty } \frac { t ^ { x - 1 } } { 1 + t e ^ { \mathrm { i } \theta } } \mathrm {~d} t \end{aligned}$$
where $x$ is a fixed element of $]0;1[$. Show, using the dominated convergence theorem, that:
$$\lim _ { \theta \rightarrow \pi ^ { - } } g ( \theta ) \sin ( x \theta ) = \int _ { - \infty } ^ { + \infty } \frac { \mathrm { d } u } { 1 + u ^ { 2 } }$$

Let $t \in \mathbf{R}_+$. Show that if $f \in C^0(\mathbf{R}) \cap CL(\mathbf{R})$, then $P_t(f) \in C^0(\mathbf{R})$. Also show that $P_t(f)$ is bounded in absolute value by a polynomial function in $|x|$ independent of $t$. Deduce that $P_t(f) \in L^1(\varphi)$.

Let $G = ( S , A )$ be a graph with $| S | = n \geq 2$. We write $\chi _ { G } ( X ) = X ^ { n } + \sum _ { k = 0 } ^ { n - 1 } a _ { k } X ^ { k }$. Give the value of $a _ { n - 1 }$ and express $a _ { n - 2 }$ in terms of $| A |$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Let $y_0 \in E$ such that $\left\|y_0 - f(a)\right\| \leqslant \frac{r}{4}$.
Show that the application $$\begin{array}{ccc} \overline{B(a,r)} & \longrightarrow & \mathbb{R} \\ x & \longmapsto & \left\|y_0 - f(x)\right\|^2 \end{array}$$ admits a minimum attained at a point $x_0$ of $B(a,r)$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Let $y_0 \in E$ such that $\left\|y_0 - f(a)\right\| \leqslant \frac{r}{4}$, and let $x_0 \in B(a,r)$ be a point where the application $x \mapsto \left\|y_0 - f(x)\right\|^2$ attains its minimum on $\overline{B(a,r)}$.
Show that $f(x_0) = y_0$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Let $y_0 \in E$ such that $\left\|y_0 - f(a)\right\| \leqslant \frac{r}{4}$.
Show that the application $$\begin{array}{ccc} \overline{B(a,r)} & \longrightarrow & \mathbb{R} \\ x & \longmapsto & \left\|y_0 - f(x)\right\|^2 \end{array}$$ admits a minimum attained at a point $x_0$ of $B(a,r)$.

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Let $y_0 \in E$ such that $\left\|y_0 - f(a)\right\| \leqslant \frac{r}{4}$. Let $x_0 \in B(a,r)$ be the point at which the minimum of $x \mapsto \|y_0 - f(x)\|^2$ is attained.
Show that $f(x_0) = y_0$.

For all $g = (\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$, we denote by $\phi_{g} : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ the map defined by $\phi_{g}(x) = Rx + \tau$.

• [(a)] Verify that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$, there exists a unique $g^{\prime\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$ such that $\phi_{g^{\prime\prime}} = \phi_{g^{\prime}} \circ \phi_{g}$. We denote this element by $g^{\prime}g$ in the following.
• [(b)] Verify that for all $g_{1}, g_{2}$ and $g_{3}$ in $\operatorname{Dep}(\mathbb{R}^{d})$ we have $g_{1}(g_{2}g_{3}) = (g_{1}g_{2})g_{3}$.