Write, in Maple or Mathematica language, a function (or procedure) rotation, with parameters $U$ and $V$, returning:

• False if $U$ and $V$ do not have the same norm;
• a matrix $R$ of $S O ( 3 )$ such that $R U = V$ if $U$ and $V$ have the same norm.

Write, in Maple or Mathematica language, a function or procedure allowing to obtain such a decomposition $$L = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) \left( \begin{array} { c c c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma & 0 & 0 \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R ^ { \prime } \end{array} \right)$$ of a matrix of $O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$. You may use the rotation function written previously.

Fourier series. Let $\phi : \mathbb { R } \rightarrow \mathbb { C }$ be a periodic function with period $2 \pi$, of class $\mathcal { C } ^ { 1 }$.
(a) Show that for all $n \in \mathbb { Z } ^ { * } , c _ { n } ( \phi ) = \frac { c _ { n } \left( \phi ^ { \prime } \right) } { \text { in } }$.
(b) Show that the series $\sum _ { n \in \mathbb { Z } } \left| c _ { n } ( \phi ) \right|$ converges. Hint. Use Parseval's formula for the function $\phi ^ { \prime }$.
(c) Show that $\| \phi \| _ { \infty } \leq \sum _ { n \in \mathbb { Z } } \left| c _ { n } ( \phi ) \right|$.

We say that $q \in \mathcal{Q}(V)$ is isotropic if there exists $x \in V - \{ 0 \}$ such that $q ( x ) = 0$. Otherwise, we say that $q$ is anisotropic.
(a) Prove that there exists $x \in V$ such that $q ( x ) \neq 0$.
(b) We denote by $h$ the quadratic form on $\mathbb { K } ^ { 2 }$ defined by $h \left( x _ { 1 } , x _ { 2 } \right) = x _ { 1 } x _ { 2 }$ (we do not ask you to verify that $h$ is a quadratic form). Show that if $V$ is of dimension two and $q$ is isotropic then $q$ is isometric to $h$.
(c) Prove that if $q \in \mathcal { Q } ( V )$ is isotropic, then $q : V \rightarrow \mathbb { K }$ is surjective.

Let $\psi : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function, periodic with period $2 \pi$. Let $f : [ a , b ] \rightarrow \mathbb { R }$ be a function of class $\mathcal { C } ^ { 1 }$ on $[ a , b ]$. For every parameter $\varepsilon > 0$, we set $$J _ { \varepsilon } = \int _ { a } ^ { b } \psi \left( \frac { x } { \varepsilon } \right) f ( x ) d x$$ First case. In this question, we further assume that $\psi$ is of class $\mathcal { C } ^ { 1 }$ on $\mathbb { R }$ and that $f$ has compact support in $] a , b [$.
(a) Show that for all $\varepsilon > 0$, $$\left| J _ { \varepsilon } - c _ { 0 } ( \psi ) \left( \int _ { a } ^ { b } f ( x ) d x \right) \right| \leq \varepsilon ( b - a ) \left\| f ^ { \prime } \right\| _ { \infty } \sum _ { n \in \mathbb { Z } ^ { * } } \frac { \left| c _ { n } ( \psi ) \right| } { | n | }$$ Hint. One can reduce to the case where $\int _ { 0 } ^ { 2 \pi } \psi ( y ) d y = 0$.
(b) Deduce the limit of $J _ { \varepsilon }$ as $\varepsilon \rightarrow 0$.

A basis $\left( e _ { 1 } , \ldots , e _ { n } \right)$ of $V$ is said to be orthogonal for $q$ if $\widetilde { q } \left( e _ { i } , e _ { j } \right) = 0$ for all $i \neq j$.
(a) Show that there exists an orthogonal basis for $q$.
Hint: one may consider $\{ x \} ^ { \perp } = \{ y \in V \mid \widetilde { q } ( x , y ) = 0 \}$ and use questions 4c and 6a.
(b) Deduce that there exist $a _ { 1 } , \ldots , a _ { n } \in \mathbb { K } - \{ 0 \}$ such that $q \cong \left\langle a _ { 1 } , \ldots , a _ { n } \right\rangle$.

Let $\psi : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function, periodic with period $2 \pi$. Let $f : [ a , b ] \rightarrow \mathbb { R }$ be a function of class $\mathcal { C } ^ { 1 }$ on $[ a , b ]$. For every parameter $\varepsilon > 0$, we set $$J _ { \varepsilon } = \int _ { a } ^ { b } \psi \left( \frac { x } { \varepsilon } \right) f ( x ) d x$$ Second case. We now assume only that $\psi \in \mathcal { C } ^ { 0 } ( \mathbb { R } )$ is periodic with period $2 \pi$, and $f \in \mathcal { C } ^ { 1 } ( [ a , b ] )$. Let $\varepsilon > 0$. We define a subdivision of the interval $[ a , b ]$ as follows. We denote $N _ { \varepsilon }$ the integer part of $\frac { b - a } { 2 \pi \varepsilon }$. We then define $$x _ { k } ^ { \varepsilon } = a + 2 k \pi \varepsilon , \text { for every integer } k \text { such that } 0 \leq k \leq N _ { \varepsilon } .$$ (a) Show that $\lim _ { \varepsilon \rightarrow 0 } x _ { N _ { \varepsilon } } ^ { \varepsilon } = b$.
(b) Deduce that $$\lim _ { \varepsilon \rightarrow 0 } \int _ { x _ { N _ { \varepsilon } } ^ { \varepsilon } } ^ { b } \psi \left( \frac { x } { \varepsilon } \right) f ( x ) d x = 0$$ (c) Show that for every integer $k$ such that $0 \leq k \leq N _ { \varepsilon } - 1$, for all $x \in \left[ x _ { k } ^ { \varepsilon } , x _ { k + 1 } ^ { \varepsilon } \right]$, $$\left| f ( x ) - f \left( x _ { k } ^ { \varepsilon } \right) \right| \leq 2 \pi \varepsilon \left\| f ^ { \prime } \right\| _ { \infty }$$ (d) Show that $$\sum _ { k = 0 } ^ { N _ { \varepsilon } - 1 } \int _ { x _ { k } ^ { \varepsilon } } ^ { x _ { k + 1 } ^ { \varepsilon } } \psi \left( \frac { x } { \varepsilon } \right) f \left( x _ { k } ^ { \varepsilon } \right) d x = \left( \int _ { 0 } ^ { 2 \pi } \psi ( y ) d y \right) \left( \varepsilon \sum _ { k = 0 } ^ { N _ { \varepsilon } - 1 } f \left( x _ { k } ^ { \varepsilon } \right) \right)$$ (e) Show that $$\left| \sum _ { k = 0 } ^ { N _ { \varepsilon } - 1 } \int _ { x _ { k } ^ { \varepsilon } } ^ { x _ { k + 1 } ^ { \varepsilon } } \psi \left( \frac { x } { \varepsilon } \right) \left( f ( x ) - f \left( x _ { k } ^ { \varepsilon } \right) \right) d x \right| \leq \varepsilon ( b - a ) \left\| f ^ { \prime } \right\| _ { \infty } \left( \int _ { 0 } ^ { 2 \pi } | \psi ( y ) | d y \right)$$ (f) Deduce that $\lim _ { \varepsilon \rightarrow 0 } J _ { \varepsilon } = \left( \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } \psi ( y ) d y \right) \left( \int _ { a } ^ { b } f ( x ) d x \right)$.

We assume in this part that $\mathbb { K } = \mathbb { R }$. Let $q \in \mathcal { Q } \left( \mathbb { R } ^ { n } \right) ( n \geq 1 )$. Prove that there exists a pair of integers $( r , s ) ( r + s = n )$ such that $q$ is isometric to $Q _ { r , s }$ defined on the canonical basis of $\mathbb { R } ^ { n }$ by $$Q _ { r , s } \left( x _ { 1 } , \ldots , x _ { n } \right) = \sum _ { i = 1 } ^ { r } x _ { i } ^ { 2 } - \sum _ { j = r + 1 } ^ { n } x _ { j } ^ { 2 }$$

Application. Let $\varepsilon > 0$. Let $\alpha \in \mathbb { R }$. Let $g : \mathbb { R } \rightarrow \mathbb { R }$ be a continuous function. We consider the following differential equation $$\left\{ \begin{array} { l } u ^ { \prime \prime } ( t ) + u ( t ) = g \left( \frac { t } { \varepsilon } \right) \\ u ( 0 ) = \alpha , u ^ { \prime } ( 0 ) = 0 \end{array} \right.$$ (a) Justify the existence and uniqueness of a solution of (1), defined for $t \in \mathbb { R }$.
(b) Calculate this solution using the method of variation of constants. We denote this solution $u _ { \varepsilon }$.
(c) We assume that $g$ is $2 \pi$-periodic. Show that for all $t \in \mathbb { R } , u _ { \varepsilon } ( t )$ has a limit as $\epsilon \rightarrow 0 ^ { + }$, limit which one will calculate.

We assume $\mathbb{K} = \mathbb{R}$. Let $j : \mathcal { L } \left( \mathbb { R } ^ { n } \right) \longrightarrow \mathcal { M } _ { n } ( \mathbb { R } )$ be the linear isomorphism that associates to every endomorphism its matrix in the canonical basis of $\mathbb { R } ^ { n }$. We denote by $O _ { r , s } : = j \left( O \left( Q _ { r , s } \right) \right)$ the subset of matrices associated to the orthogonal group $O \left( Q _ { r , s } \right)$ of $Q _ { r , s }$.
Let $f : \mathbb { R } ^ { n } \rightarrow \mathbb { R } ^ { n }$ be a linear map and $M = j ( f )$ its matrix in the canonical basis of $\mathbb { R } ^ { n }$. Prove that $M \in O _ { r , s }$ if and only if ${ } ^ { t } M I _ { r , s } M = I _ { r , s }$ where $I _ { r , s }$ is the matrix $$I _ { r , s } = \left[ \begin{array} { c c } I _ { r } & 0 _ { r , s } \\ 0 _ { s , r } & - I _ { s } \end{array} \right]$$ $I _ { p }$ denotes the identity matrix of size $p \times p$ and $0 _ { p , q }$ the zero matrix of size $p \times q$ for all integers $p$ and $q$.
What can be said about the determinant $\operatorname { det } ( M )$ of $M$ if $M \in O _ { r , s }$ ?

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 of a non-stationary phase. We assume in this question that $\varphi ^ { \prime } ( x ) \neq 0$ for all $x \in [ a , b ]$.
(a) We define $L : \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } ) \rightarrow \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$ and $M : \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } ) \rightarrow \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$ by: for all $g \in \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$, all $x \in [ a , b ]$, $$L g ( x ) = \frac { 1 } { i \lambda \varphi ^ { \prime } ( x ) } g ^ { \prime } ( x ) , \quad M g ( x ) = - \left( \frac { g } { i \varphi ^ { \prime } } \right) ^ { \prime } ( x )$$ i. Determine the functions $g \in \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$ such that $L g = g$. ii. Let $g , h \in \mathcal { C } ^ { \infty } ( [ a , b ] , \mathbb { C } )$. We assume that $h$ has compact support in $] a , b [$. Show that $$\int _ { a } ^ { b } h ( x ) L g ( x ) d x = \frac { 1 } { \lambda } \int _ { a } ^ { b } g ( x ) M h ( x ) d x$$ (b) Show that if $f$ has compact support in $] a , b [$, then for all $N \in \mathbb { N } ^ { * }$, there exists a constant $\gamma _ { N }$ independent of $\lambda$ such that $$| I ( \lambda ) | \leq \gamma _ { N } \lambda ^ { - N }$$

We assume $\mathbb{K} = \mathbb{R}$. We denote by $O _ { r , s } : = j \left( O \left( Q _ { r , s } \right) \right)$ the subset of matrices associated to the orthogonal group $O \left( Q _ { r , s } \right)$ of $Q _ { r , s }$. Prove that $O _ { r , s }$ is a closed subgroup of $\mathrm { GL } _ { n } ( \mathbb { R } )$ (we equip $\mathcal { M } _ { n } ( \mathbb { R } )$, the set of square matrices of size $n$ with coefficients in $\mathbb { R }$, with its topology as a $\mathbb { R }$-vector space of finite dimension).

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.

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.

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 $S O _ { 2,1 } ^ { + }$.

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 consider the function $\varphi$ defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad \varphi(x) = \begin{cases} 1 & \text{if } x \in \left[-\frac{1}{2}, \frac{1}{2}\right] \\ 0 & \text{otherwise} \end{cases}$$
Justify that $\varphi$ belongs to $E_{\mathrm{cpm}}$ and calculate its Fourier transform $\mathcal{F}(\varphi)$.

Show that $\Sigma_{N}$ is a closed, bounded and convex subset of $\mathbb{R}^{N}$.
Where $\Sigma_{N}$ denotes the set of vectors $p \in \mathbb{R}^{N}$ such that $\sum_{i=1}^{N} p_{i} = 1$ and $p_{i} \geqslant 0$ for all $1 \leqslant i \leqslant N$.

Show that $H_{N}$ is positive, continuous on $\Sigma_{N}$ and calculate the value of $H_{N}(p)$ when $p_{i} = 1/N$ for all $i \in \{1, \ldots, N\}$ (uniform distribution on $\{1, \ldots, N\}$).
Where $H_{N}(p) = \sum_{i=1}^{N} \varphi(p_{i})$ and $\varphi(t) = \begin{cases} 0 & \text{if } t = 0 \\ -t \ln(t) & \text{otherwise.} \end{cases}$