The function $q$ associates to any real $x$ the real number $q ( x ) = x - \lfloor x \rfloor - \frac { 1 } { 2 }$.
Show, for all real $t > 0$, the identity
$$\int _ { 1 } ^ { + \infty } \frac { t q ( u ) } { e ^ { t u } - 1 } \mathrm {~d} u = - \frac { 1 } { 2 } \ln \left( 1 - e ^ { - t } \right) - \ln P \left( e ^ { - t } \right) - \int _ { 1 } ^ { + \infty } \ln \left( 1 - e ^ { - t u } \right) \mathrm { d } u$$

Show, for all real $t > 0$, the identity $$\int_{1}^{+\infty} \frac{tq(u)}{e^{tu}-1} \mathrm{~d}u = -\frac{1}{2}\ln(1-e^{-t}) - \ln P(e^{-t}) - \int_{1}^{+\infty} \ln(1-e^{-tu}) \mathrm{d}u$$

$\mathbf{15}$ ▷ Let $M \in \mathcal{M}_n(\mathbf{R})$, which can also be considered as a complex matrix, and let the application $\delta_M : \mathbf{R} \rightarrow \mathbf{R},\ t \mapsto \delta_M(t) = \operatorname{det}\left(I_n + tM\right)$. Using a Taylor expansion to order 1, show that $\delta_M$ is differentiable at 0 and compute $\delta_M'(0)$.

Let $A$ and $B$ be two polynomials in $\mathbf{R}[X]$ whose coefficients are all strictly positive. Prove that the coefficients of the product $AB$ are also strictly positive.

Let $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathscr{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$. We equip $\mathscr{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III.
Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a path-connected subset of $\mathscr{A}_p(E, \mathbb{R})$ if and only if $p \leqslant d-1$. (Hint: One may use question 3d.)

We consider a power series $f = \lambda z + F, F \in O_2, \lambda = (f)_1 \neq 0$, with $h$ the unique series in $O_1$ such that $h \circ f = I$ and $g$ the unique series in $O_1$ such that $f \circ g = I$. Show that $g = h$.

Let $p \in \llbracket 1, d \rrbracket$. We denote by $\widetilde{\operatorname{Gr}}(p, E)$ the set of oriented subspaces of dimension $p$ of $E$. For all oriented vector subspace $(V, C)$ of dimension $p$ of $E$, $\Psi(V, C)$ denotes the unique element of $\mathcal{A}_p(E, \mathbb{R})$ such that for all $e \in C$ we have $\Omega_p(e) = \operatorname{vol}_p(e) \Psi(V, C)$.
We equip $\mathcal{A}_p(E, \mathbb{R})$ with the inner product introduced in Part III.
Show that $\Psi(\widetilde{\operatorname{Gr}}(p, E))$ is a path-connected subset of $\mathcal{A}_p(E, \mathbb{R})$ if and only if $p \leqslant d-1$. (Hint: One may use question 3d.)

Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Show that $K = \operatorname{co}(\operatorname{Ext}(K))$.

Let $K$ be a non-empty, convex, closed and bounded subset of $\mathbb{R}^d$. Show that $K = \operatorname{co}(\operatorname{Ext}(K))$.

Let $u \in \mathcal { L } \left( \mathbb { R } ^ { n } \right)$ be an endomorphism of $\mathbb { R } ^ { n }$. We denote by $M$ the matrix of $u$ in the canonical basis of $\mathbb { R } ^ { n }$. The standard symplectic form is $b_s(x,y) = \langle x, j(y) \rangle$ where $j$ is canonically associated with $J = \left( \begin{array}{cc} 0 & -I_m \\ I_m & 0 \end{array} \right)$. Show that $u$ is a symplectic endomorphism of the standard symplectic space $\left( \mathbb { R } ^ { n } , b _ { s } \right)$ if and only if $M ^ { \top } J M = J$.

Let $u \in \mathcal { L } \left( \mathbb { R } ^ { n } \right)$ be an endomorphism of $\mathbb { R } ^ { n }$. We denote by $M$ the matrix of $u$ in the canonical basis of $\mathbb { R } ^ { n }$. Show that $u$ is a symplectic endomorphism of the standard symplectic space $\left( \mathbb { R } ^ { n } , b _ { s } \right)$ if and only if $M ^ { \top } J M = J$.

Let $d \in \llbracket 1 , n \rrbracket , \left( U _ { 1 } , \ldots , U _ { d } \right)$ be a linearly independent family in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$ and $H = \operatorname { Vect } \left( U _ { 1 } , \ldots , U _ { d } \right)$.
Prove that there exist integers $i _ { 1 } , \ldots , i _ { d }$ satisfying $1 \leqslant i _ { 1 } < \cdots < i _ { d } \leqslant n$ such that the application $$\left\lvert \, \begin{array} { c c c } H & \rightarrow & \mathcal { M } _ { d , 1 } ( \mathbb { R } ) \\ \left( \begin{array} { c } x _ { 1 } \\ \vdots \\ x _ { n } \end{array} \right) & \mapsto & \left( \begin{array} { c } x _ { i _ { 1 } } \\ \vdots \\ x _ { i _ { d } } \end{array} \right) \end{array} \right.$$ is bijective.
One may consider the rank of the matrix in $\mathcal { M } _ { n , d } ( \mathbb { R } )$ whose columns are $U _ { 1 } , \ldots , U _ { d }$.

Show that the function series $\sum f_n(x)$ converges normally on any segment contained in $]0, +\infty[$ and that the function $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$ is of class $\mathcal{C}^2$ on $]0, +\infty[$.

For all $n \in \mathbb{N}^\star$, let $f_n \in \mathcal{C}^2(]0,+\infty[)$ satisfy $f_n(1) = 0$, $f_n(2) = 0$ and $f_n^{\prime\prime}(x) = (-1)^n 2^{-nx^2}$ for all $x > 0$. Show that the function series $\sum f_n(x)$ converges normally on any segment contained in $]0,+\infty[$ and that the function $F : x \mapsto \sum_{n=1}^{+\infty} f_n(x)$ is of class $\mathcal{C}^2$ on $]0,+\infty[$.

Let $n \in \mathbb{N}^*$, $W$ be a monic polynomial of degree $n$, $Q = \frac { 1 } { 2 ^ { n - 1 } } T _ { n } - W$, and for all $k \in \llbracket 0 , n \rrbracket$, $z _ { k } = \cos \left( \frac { k \pi } { n } \right)$. We assume that $\sup _ { x \in [ - 1,1 ] } | W ( x ) | = \frac { 1 } { 2 ^ { n - 1 } }$. Deduce that $Q = 0$, then that $W = \frac { 1 } { 2 ^ { n - 1 } } T _ { n }$.
One may consider the sum of the inequalities from the previous question and exploit question 6 applied to suitable data.

To each function $f \in E$, we associate the function $U ( f )$ defined for all $x > 0$ by $$U ( f ) ( x ) = \left\langle k _ { x } \mid f \right\rangle = \int _ { 0 } ^ { + \infty } \left( \mathrm { e } ^ { \min ( x , t ) } - 1 \right) f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$$ Using the Cauchy-Schwarz inequality, show that for all functions $f \in E$, $$\lim _ { \substack { x \rightarrow 0 \\ x > 0 } } U ( f ) ( x ) = 0$$

Let $E$ be a non-empty subset of $\mathbb{R}^d$. The polar cone of $E$ is defined by $$E^+ := \left\{p \in \mathbb{R}^d : p \cdot x \geqslant 0, \forall x \in E\right\}$$ and its bi-polar cone by $$E^{++} = (E^+)^+ := \left\{\xi \in \mathbb{R}^d : \xi \cdot p \geqslant 0, \forall p \in E^+\right\}.$$ Show that $E^+$ and $E^{++}$ are closed convex cones and that $E \subset E^{++}$.

Let $E$ be a non-empty subset of $\mathbb{R}^d$. The polar cone of $E$ is defined by $$E^+ := \left\{p \in \mathbb{R}^d : p \cdot x \geq 0, \forall x \in E\right\}$$ and its bi-polar cone by $$E^{++} = (E^+)^+ := \left\{\xi \in \mathbb{R}^d : \xi \cdot p \geq 0, \forall p \in E^+\right\}.$$ Show that $E^+$ and $E^{++}$ are closed convex cones and that $E \subset E^{++}$.

Let $\mathcal { W }$ be a vector subspace of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$ of dimension $d$. Prove that $$\operatorname { card } \left( \mathcal { W } \cap \mathcal { V } _ { n , 1 } \right) \leqslant 2 ^ { d }$$

Let $r$ be a real-valued function of class $\mathcal { C } ^ { n }$ on $I = [a,b]$ and vanishing at $n + 1$ distinct points of $I$. Show that there exists $c \in I$ such that $r ^ { ( n ) } ( c ) = 0$.

Let $E$ be a non-empty subset of $\mathbb{R}^d$. Using the definitions of $E^+$ and $E^{++}$ from question 16, show that $E = E^{++}$ if and only if $E$ is a closed convex cone.

Let $E$ be a non-empty subset of $\mathbb{R}^d$. With $E^+$ and $E^{++}$ as defined in question 16, show that $E = E^{++}$ if and only if $E$ is a closed convex cone.

Let $n$ be a nonzero natural integer, $I = [a,b]$ with $a < b$, and $a_1 < \cdots < a_n$ distinct real numbers in $I$. Let $W = \prod_{i=1}^n (X - a_i)$ and let $f$ be a real-valued function of class $\mathcal { C } ^ { n }$ on $I$. Let $P = \Pi ( f )$ be the interpolation polynomial of $f$ associated with the real numbers $a _ { 1 } , \ldots , a _ { n }$, defined by $$\Pi ( f ) = \sum _ { i = 1 } ^ { n } f \left( a _ { i } \right) L _ { i }.$$ For all $x \in I$, show that there exists $c \in I$ such that $$f ( x ) - P ( x ) = \frac { f ^ { ( n ) } ( c ) } { n ! } W ( x ).$$ For $x$ distinct from the $a _ { i }$, one may consider the function $r$ defined on $I$ by $$r ( t ) = f ( t ) - P ( t ) - K W ( t )$$ where the real number $K$ is chosen so that $r ( x ) = 0$.

Let $\mathcal{C}^{1}$ be the space of functions of class $C^{1}$ from $[-\pi, \pi]$ to $\mathbf{C}$. For $f \in \mathcal{C}^{1}$, we set $$\|f\|_{\infty} = \max\{|f(t)|; t \in [-\pi, \pi]\} \quad \text{and} \quad V(f) = \int_{-\pi}^{\pi} |f^{\prime}|.$$
Let $F \in \mathcal{R}_{n}$, $P$ and $Q$ be two elements of $\mathbf{C}_{n}[X]$ satisfying $F = \frac{P}{Q}$ and $\forall z \in \mathbb{U},\ Q(z) \neq 0$. For $t \in [-\pi, \pi]$, we set $f(t) = F(e^{it}) = g(t) + ih(t)$. For $u \in [-\pi, \pi]$, we define $f_{u}(t) = g(t)\cos(u) + h(t)\sin(u)$.
We admit the equality $$\int_{-\pi}^{\pi} \left(\int_{-\pi}^{\pi} \left|f_{u}^{\prime}(t)\right| \mathrm{d}u\right) \mathrm{d}t = \int_{-\pi}^{\pi} \left(\int_{-\pi}^{\pi} \left|f_{u}^{\prime}(t)\right| \mathrm{d}t\right) \mathrm{d}u$$ We also admit that, for $u \in [-\pi, \pi]$ such that $f_{u}$ is not constant, the set of points in $]-\pi, \pi[$ where the function $f_{u}^{\prime}$ vanishes is finite.
Deduce the inequality $$(3) \quad V(f) \leq 2\pi n \|f\|_{\infty}.$$

Let $\mu_1$ and $\mu_2$ be two probabilities on $\mathbb{N}^*$. We assume that $\forall r \in \mathbb{N}^*, \mu_1(\mathbb{N}^* r) = \mu_2(\mathbb{N}^* r)$. We want to show that $\mu_1 = \mu_2$.
We recall that we denote by $(p_i)_{i \in \mathbb{N}^*}$ the sequence of prime numbers, ordered in increasing order. Show that for all $r \in \mathbb{N}^*$ and all integer $n \geqslant 1$: $$\bigcup_{i=1}^{n+1} \mathbb{N}^* r p_i = \left(\bigcup_{i=1}^{n} \mathbb{N}^* r p_i\right) \cup \left(\mathbb{N}^* r p_{n+1} \backslash \bigcup_{i=1}^{n} \mathbb{N}^* r p_{n+1} p_i\right).$$