Deduce that, for all $t \in \mathbb{R}$, the application $(x,y) \mapsto \dfrac{1 - \left((x + \cos t)^2 + (y + \sin t)^2\right)}{x^2 + y^2}$ is harmonic on $\mathbb{R}^2 \backslash \{(0,0)\}$.

Prove that, for any continuous application $f : C(0,1) \rightarrow \mathbb{R}$, the set $\mathcal{D}_f$ admits exactly one element.

Let $m$ be an integer greater than or equal to 2. We consider a polynomial $P \in \mathcal{P}_m$ and we denote by $P_C$ the restriction of $P$ to the circle $C(0,1)$.
Show that the application $$\phi_{m-2} : \left|\begin{array}{rll} \mathcal{P}_{m-2} & \rightarrow & \mathcal{P} \\ Q & \mapsto & \Delta \tilde{Q} \end{array}\right. \quad \text{where} \quad \tilde{Q}(x,y) = (1 - x^2 - y^2) Q(x,y)$$ is linear and injective and that $\operatorname{Im} \phi_{m-2} \subset \mathcal{P}_{m-2}$.

Let $m$ be an integer greater than or equal to 2. We consider a polynomial $P \in \mathcal{P}_m$.
Deduce that there exists a polynomial $T \in \mathcal{P}_{m-2}$ such that $P + (1 - x^2 - y^2) T$ is a harmonic polynomial.

Let $m$ be an integer greater than or equal to 2. We consider a polynomial $P \in \mathcal{P}_m$ and we denote by $P_C$ the restriction of $P$ to the circle $C(0,1)$.
Show that the unique element of the set $\mathcal{D}_{P_C}$ is the restriction to $\bar{D}(0,1)$ of a polynomial of degree less than or equal to $m$.

Let $m$ be an integer greater than or equal to 2. We consider a polynomial $P \in \mathcal{P}_m$ and we denote by $P_C$ the restriction of $P$ to the circle $C(0,1)$.
Explicitly determine the set $\mathcal{D}_{P_C}$ when the polynomial $P$ is defined by $P(x,y) = x^3$.

Let $P \in \mathcal{P}$. Show that $P$ decomposes uniquely in the form: $$P(x,y) = H(x,y) + (1 - x^2 - y^2) Q(x,y)$$ where $H$ is a harmonic polynomial and $Q \in \mathcal{P}$.

Let $\alpha \in \mathbb{R}$. We say that $f$ is a solution of equation $\left(E_{\alpha}\right)$ if $f \in \mathcal{C}^{2}(\mathbb{R}, \mathbb{C})$ and $$\left(E_{\alpha}\right) \quad \forall x \in \mathbb{R}, f^{\prime\prime}(x) + \frac{ix}{2} f^{\prime}(x) + \frac{f(x)}{2}\left(\alpha|f(x)|^{2} + 1\right) = 0$$ In questions 1 to 7 of Part I, we assume that $\alpha > 0$. Moreover, we assume that there exists a solution $f$ of $(E_{\alpha})$ which satisfies $f(0) = 1$ and $f^{\prime}(0) = 0$.
We set $F_{1} = \Re(f)$ and $f_{2} = \Im(f)$. Express $f_{1}^{\prime\prime}$ and $f_{2}^{\prime\prime}$ in terms of $f_{1}^{\prime}, f_{2}^{\prime}, f_{1}$ and $f_{2}$.

Let $\alpha \in \mathbb{R}$. We say that $f$ is a solution of equation $\left(E_{\alpha}\right)$ if $f \in \mathcal{C}^{2}(\mathbb{R}, \mathbb{C})$ and $$\left(E_{\alpha}\right) \quad \forall x \in \mathbb{R}, f^{\prime\prime}(x) + \frac{ix}{2} f^{\prime}(x) + \frac{f(x)}{2}\left(\alpha|f(x)|^{2} + 1\right) = 0$$ In questions 1 to 7 of Part I, we assume that $\alpha > 0$. Moreover, we assume that there exists a solution $f$ of $(E_{\alpha})$ which satisfies $f(0) = 1$ and $f^{\prime}(0) = 0$.
Show that $$\forall x \in \mathbb{R}, \left|f^{\prime}(x)\right|^{2} + \frac{1}{4\alpha}\left(\alpha|f(x)|^{2} + 1\right)^{2} = \frac{1}{4\alpha}(\alpha + 1)^{2}$$

We consider the space $\mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { d } \right)$ of functions $f : \mathbb { R } ^ { d } \rightarrow \mathbb { C }$ continuous and 1-periodic in each of their variables, equipped with the uniform norm $\| f \| _ { \infty } = \sup _ { \theta \in \mathbb { R } ^ { d } } | f ( \theta ) |$. A trigonometric polynomial (in $d$ variables) is any function of the form $\theta \mapsto \sum _ { k \in K } c _ { k } e ^ { 2 i \pi k \cdot \theta }$ where $K$ is a finite subset of $\mathbb { Z } ^ { d }$. We work in dimension $d = 2$. The subspace $\mathscr { C } _ { \text {sep} } \left( \mathbb { R } ^ { 2 } \right)$ of $\mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$ consists of functions of the form $\theta = \left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto \sum _ { i = 1 } ^ { n } f _ { i } \left( \theta _ { 1 } \right) g _ { i } \left( \theta _ { 2 } \right)$, where $n \in \mathbb { N } ^ { * }$ and $f _ { 1 } , \ldots , f _ { n } , g _ { 1 } , \ldots , g _ { n } \in \mathscr { C } _ { \text {per} } ( \mathbb { R } )$. We admit that trigonometric polynomials in one variable are dense in $\mathscr{C}_{\text{per}}(\mathbb{R})$.
Show that the set of trigonometric polynomials in two variables is dense in $\mathscr { C } _ { \text {sep} } \left( \mathbb { R } ^ { 2 } \right)$.

For $j \geqslant 2$ an integer, the function $\psi _ { j } : \mathbb { R } \rightarrow \mathbb { R }$ is 1-periodic and defined on $] - 1 / 2,1 / 2 ]$ by $$\forall t \in ] - 1 / 2,1 / 2 ] , \quad \psi _ { j } ( t ) = \max ( 0,1 - j | t | ) .$$ For integers $0 \leqslant k < j$, the functions $\psi _ { j , k } : \mathbb { R } \rightarrow \mathbb { R }$ are defined by $$\forall t \in \mathbb { R } , \quad \psi _ { j , k } ( t ) = \psi _ { j } \left( t - \frac { k } { j } \right) .$$
Show that $\psi _ { j , k } \in \mathscr { C } _ { \text {per} } ( \mathbb { R } )$.

For $j \geqslant 2$ an integer, the function $\psi _ { j } : \mathbb { R } \rightarrow \mathbb { R }$ is 1-periodic and defined on $] - 1 / 2,1 / 2 ]$ by $\psi _ { j } ( t ) = \max ( 0,1 - j | t | )$. For integers $0 \leqslant k < j$, $\psi _ { j , k } ( t ) = \psi _ { j } \left( t - \frac { k } { j } \right)$. We are given $f \in \mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$ and $j \geqslant 2$ an integer, and we set $$S _ { j } ( f ) \left( \theta _ { 1 } , \theta _ { 2 } \right) = \sum _ { k _ { 1 } = 0 } ^ { j - 1 } \sum _ { k _ { 2 } = 0 } ^ { j - 1 } f \left( \frac { k _ { 1 } } { j } , \frac { k _ { 2 } } { j } \right) \psi _ { j , k _ { 1 } } \left( \theta _ { 1 } \right) \psi _ { j , k _ { 2 } } \left( \theta _ { 2 } \right) .$$
Show that $S _ { j } ( f ) \in \mathscr { C } _ { \text {sep} } \left( \mathbb { R } ^ { 2 } \right)$ and coincides with $f$ at the points $\left( \frac { \ell _ { 1 } } { j } , \frac { \ell _ { 2 } } { j } \right)$ for $\left( \ell _ { 1 } , \ell _ { 2 } \right) \in \mathbb { Z } ^ { 2 }$.

For $j \geqslant 2$ an integer, the function $\psi _ { j } : \mathbb { R } \rightarrow \mathbb { R }$ is 1-periodic and defined on $] - 1 / 2,1 / 2 ]$ by $\psi _ { j } ( t ) = \max ( 0,1 - j | t | )$. For integers $0 \leqslant k < j$, $\psi _ { j , k } ( t ) = \psi _ { j } \left( t - \frac { k } { j } \right)$. We are given $f \in \mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$ and $j \geqslant 2$ an integer, and $$S _ { j } ( f ) \left( \theta _ { 1 } , \theta _ { 2 } \right) = \sum _ { k _ { 1 } = 0 } ^ { j - 1 } \sum _ { k _ { 2 } = 0 } ^ { j - 1 } f \left( \frac { k _ { 1 } } { j } , \frac { k _ { 2 } } { j } \right) \psi _ { j , k _ { 1 } } \left( \theta _ { 1 } \right) \psi _ { j , k _ { 2 } } \left( \theta _ { 2 } \right) .$$
Let $j \geqslant 2 , k _ { 1 }$ and $k _ { 2 }$ be two integers such that $0 \leqslant k _ { 1 } , k _ { 2 } < j$, and $$\theta \in \left[ \frac { k _ { 1 } } { j } , \frac { k _ { 1 } + 1 } { j } \left[ \times \left[ \frac { k _ { 2 } } { j } , \frac { k _ { 2 } + 1 } { j } [ . \right. \right. \right.$$
Express $S _ { j } ( f ) ( \theta )$ as a barycenter of the points $f \left( \frac { \ell _ { 1 } } { j } , \frac { \ell _ { 2 } } { j } \right)$ where $\ell _ { 1 } \in \left\{ k _ { 1 } , k _ { 1 } + 1 \right\}$ and $\ell _ { 2 } \in \left\{ k _ { 2 } , k _ { 2 } + 1 \right\}$. Deduce that $\left\| S _ { j } ( f ) - f \right\| _ { \infty } \rightarrow 0$ when $j \rightarrow + \infty$.

Using the results of questions 8, 9, 10a and 10b, conclude that the set of trigonometric polynomials in two variables is dense in $\mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$.

We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ and $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ two functions continuous and 1-periodic in each of their arguments, and $\omega \in \mathbb { R } ^ { 2 } , x \in \mathbb { R }$ two parameters. We consider the following problem $$\left\{ \begin{array} { l } F ^ { \prime } ( t ) = f ( \alpha ( t ) ) \\ \alpha ^ { \prime } ( t ) = \omega + x g ( \alpha ( t ) ) \end{array} \right.$$ with the initial conditions $F ( 0 ) = 0$ and $\alpha ( 0 ) = ( 0,0 )$, where $\alpha : \mathbb { R } \rightarrow \mathbb { R } ^ { 2 }$ and $F : \mathbb { R } \rightarrow \mathbb { C }$ are the unknown functions. We assume that $f$ has zero average, that is $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } = 0$.
We assume $x = 0$. Determine the unique solution $( F , \alpha )$ of system (3) with initial conditions $F ( 0 ) = 0$ and $\alpha ( 0 ) = ( 0,0 )$.

We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ and $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ two functions continuous and 1-periodic in each of their arguments, and $\omega \in \mathbb { R } ^ { 2 }$. We consider the system $$\left\{ \begin{array} { l } F ^ { \prime } ( t ) = f ( \alpha ( t ) ) \\ \alpha ^ { \prime } ( t ) = \omega \end{array} \right.$$ with $F(0)=0$, $\alpha(0)=(0,0)$. We assume that $f$ has zero average. The vector $\omega = \left( \omega _ { 1 } , \omega _ { 2 } \right)$ is said to be resonant if there exists $\left( k _ { 1 } , k _ { 2 } \right) \in \mathbb { Z } ^ { 2 } \backslash \{ ( 0,0 ) \}$ such that $k _ { 1 } \omega _ { 1 } + k _ { 2 } \omega _ { 2 } = 0$.
Show that, if $\omega$ is resonant, there exists a function $f$ for which $F ( t ) = t$.

We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ continuous and 1-periodic in each of its arguments, with zero average $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } = 0$. We consider the system with $x=0$: $$F'(t) = f(\alpha(t)), \quad \alpha'(t) = \omega, \quad F(0)=0, \quad \alpha(0)=(0,0).$$ Suppose that $\omega$ is not resonant.
Show that, if $f$ is a trigonometric polynomial, then $F$ is bounded on $\mathbb { R }$.

We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ continuous and 1-periodic in each of its arguments, with zero average $\int _ { 0 } ^ { 1 } \int _ { 0 } ^ { 1 } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } = 0$. We consider the system with $x=0$: $$F'(t) = f(\alpha(t)), \quad \alpha'(t) = \omega, \quad F(0)=0, \quad \alpha(0)=(0,0).$$ Suppose that $\omega$ is not resonant.
Show that more generally, if $f \in \mathscr { C } _ { \text {per} } \left( \mathbb { R } ^ { 2 } \right)$, then $F ( t ) = o ( t )$ when $t \rightarrow + \infty$.

We are given $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ and $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ two functions continuous and 1-periodic in each of their arguments, and $\omega \in \mathbb { R } ^ { 2 }$. We assume $x \neq 0$ (but close to 0). We consider a new unknown function $\tilde { \alpha } : \mathbb { R } \rightarrow \mathbb { R } ^ { 2 }$, of the form $$\tilde { \alpha } ( t ) = \alpha ( t ) + x h ( \alpha ( t ) ) ,$$ where $h : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ is an auxiliary function, 1-periodic in each of its arguments, of class $\mathscr { C } ^ { 1 }$ and of zero average, and which moreover, for some $\nu \in \mathbb { R } ^ { 2 }$, satisfies the equation $$\forall \theta \in \mathbb { R } ^ { 2 } , \quad d h ( \theta ) \cdot \omega + g ( \theta ) = \nu . \tag{4}$$
Determine $\nu$ as a function of $g$. In the case where the two components $g _ { 1 }$ and $g _ { 2 }$ of $g$ are trigonometric polynomials, deduce the existence of a solution $h$ of equation (4), which you will make explicit.

We are given $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ continuous and 1-periodic in each of its arguments, of class $\mathscr{C}^1$. We assume $x \neq 0$ (but close to 0), and that a solution $h$ of equation (4) exists (1-periodic, $\mathscr{C}^1$, zero average). We set $\tilde{\alpha}(t) = \alpha(t) + x h(\alpha(t))$ where $\alpha$ satisfies $\alpha'(t) = \omega + x g(\alpha(t))$.
Show that there exists a function $\varepsilon : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ such that $$\tilde { \alpha } ^ { \prime } ( t ) = \omega + x \nu + x \varepsilon ( x , t )$$ and $\sup _ { t \in \mathbb { R } } \| \varepsilon ( x , t ) \| \rightarrow 0$ when $x \rightarrow 0$.

We are given $g : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ continuous and 1-periodic in each of its arguments, of class $\mathscr{C}^1$. We assume $x \neq 0$ (but close to 0), and that a solution $h$ of equation (4) exists (1-periodic, $\mathscr{C}^1$, zero average). We set $\tilde{\alpha}(t) = \alpha(t) + x h(\alpha(t))$ where $\alpha$ satisfies $\alpha'(t) = \omega + x g(\alpha(t))$, $\alpha(0)=(0,0)$. From question 15b, $\tilde{\alpha}'(t) = \omega + x\nu + x\varepsilon(x,t)$.
Let $T > 0$ be fixed. Show that there exists a function $\eta : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }$ such that $$\alpha ( t ) = ( \omega + x \nu ) t + x ( h ( 0,0 ) - h ( \omega t ) ) + x \eta ( x , t )$$ and $\sup _ { t \in [ 0 , T ] } \| \eta ( x , t ) \| \rightarrow 0$ when $x \rightarrow 0$.

Let $H \in C^{\infty}(\mathbb{R}, \mathbb{R}^{3})$ such that $\|H^{\prime}(x)\| = 1$ and $\|H^{\prime\prime}(x)\| \neq 0$ for all $x \in \mathbb{R}$. We denote $T_{H}(x) = H^{\prime}(x)$, $n_{H}(x) = T_{H}^{\prime}(x)/\|T_{H}^{\prime}(x)\|$ and $b_{H}(x) = T_{H}(x) \wedge n_{H}(x)$. We admit that there exist $k_{H}(x)$ and $\tau_{H}(x)$ such that $$\left\{\begin{array}{l} T_{H}^{\prime}(x) = k_{H}(x)\, n_{H}(x) \\ n_{H}^{\prime}(x) = -k_{H}(x)\, T_{H}(x) + \tau_{H}(x)\, b_{H}(x) \\ b_{H}^{\prime}(x) = -\tau_{H}(x)\, n_{H}(x) \end{array}\right.$$
We assume that $m = 0$ and there exists $\lambda > 0$ such that $G(0) = (0,0,2\lambda)$, $G^{\prime}(0) = (1,0,0)$, and $G \in C^{\infty}(\mathbb{R}, \mathbb{R}^{3})$ satisfies $\forall x \in \mathbb{R},\, G^{\prime\prime}(x) = \frac{1}{2}G(x) \wedge G^{\prime}(x)$, with $T(x) = G^{\prime}(x)$, $n(x) = T^{\prime}(x)/\lambda$, $b(x) = T(x) \wedge n(x)$.
Express $k_{G}$ and $\tau_{G}$, then show that there exists $\alpha \in \mathbb{R}$ such that the function $$\Psi(t, x) = \frac{1}{\sqrt{t}}\, k_{G}\!\left(\frac{x}{\sqrt{t}}\right) \exp\!\left(i\int_{0}^{x} \frac{1}{\sqrt{t}}\, \tau_{G}\!\left(\frac{y}{\sqrt{t}}\right) dy\right)$$ is a solution of equation $(F_{\alpha})$, that is $$\forall (t, x) \in \mathbb{R}_{+}^{*} \times \mathbb{R}, \quad i\frac{\partial \Psi}{\partial t}(t, x) + \frac{\partial^{2} \Psi}{\partial x^{2}}(t, x) + \frac{1}{2}\Psi(t, x)\left(\alpha|\Psi(t, x)|^{2} + \frac{1}{t}\right) = 0$$

Let $p \in ]0,1[$. We start from the zero matrix of $\mathcal{M}_n(\mathbb{R})$, denoted $M_0$. For all $k \in \mathbb{N}$, we construct the matrix $M_{k+1}$ from the matrix $M_k$ by scanning through the matrix in one pass and changing each zero coefficient to 1 with probability $p$, independently.
Throughout this question we use the Python language. M denotes a square matrix of order $n$. Its rows and columns are numbered from 0 to $n-1$. The expression $\mathrm{M[i,j]}$ allows access to the element at the intersection of row $i$ and column $j$ and len(M) gives the order of the matrix M.
a) Write a function Somme(M) that returns the sum of the coefficients of the matrix M.
b) Write a function Bernoulli(p) that returns 1 with probability $p$ and 0 with probability $1-p$. You may use the expression random() which returns a real number in the interval $[0,1[$ according to the uniform distribution.
c) Using the function Bernoulli(p), write a function Modifie(M,p) that randomly modifies the matrix M according to the principle described above.
d) Write a function Simulation(n,p) that returns the smallest integer $k$ such that $M_k$ is completely filled starting from a random filling of the zero matrix of order $n$ (which can be obtained by zeros$((n,n))$). It is not required to store the $M_k$.

We define the matrix $Z_n = (z_{i,j}) \in \mathcal{M}_n(\mathbb{R})$ by $z_{i,j} = \begin{cases} 1 & \text{if } 1 \leqslant i < n \text{ and } j = i+1 \\ 1 & \text{if } (i,j) \in \{(n-1,1),(n,2)\} \\ 0 & \text{in all other cases} \end{cases}$
Show that the matrix $Z_n$ is irreducible.

For a nonzero polynomial $P \in \mathbb{R}_n[X]$, express $\operatorname{deg}(\tau(P))$ and $\operatorname{cd}(\tau(P))$ in terms of $\operatorname{deg}(P)$ and $\operatorname{cd}(P)$.