Let $N$ be the set of non-negative integers. Suppose $f : N \rightarrow N$ is a function such that $f ( f ( f ( n ) ) ) < f ( n + 1 )$ for every $n \in N$. Prove that $f ( n ) = n$ for all $n$ using the following steps or otherwise. a) If $f ( n ) = 0$, then $n = 0$. b) If $f ( x ) < n$, then $x < n$. (Start by considering $n = 1$.) c) $f ( n ) < f ( n + 1 )$ and $n < f ( n + 1 )$ for all $n$. d) $f ( n ) = n$ for all $n$.

A compactification of a topological space $X$ is a compact topological space $Y$ which contains a dense subspace homeomorphic to $X$. Let $X = (0,1]$, in the subspace topology of $\mathbb{R}$ and $f : X \longrightarrow \mathbb{R},\ x \mapsto \sin\frac{1}{x}$. Show the following:
(A) $Y := [0,1]$ is a compactification of $X$, but $f$ does not extend to a continuous function $Y \longrightarrow \mathbb{R}$, i.e., there does not exist a continuous function $g : Y \longrightarrow \mathbb{R}$ such that $\left.g\right|_X = f$.
(B) $X$ is homeomorphic to the set $X_1 := \left\{\left.\left(t, \sin\frac{1}{t}\right)\right\rvert\, t \in X\right\} \subseteq \mathbb{R}^2$.
(C) The closure $Y_1$ of $X_1$ in $\mathbb{R}^2$ is a compactification of $X$.
(D) $f$ extends to a continuous function $Y_1 \longrightarrow \mathbb{R}$.

Let $f : [0,1] \longrightarrow \mathbb{R}$ be a continuous function. Define $g(0) = f(0)$ and $g(x) = \max\{f(y) \mid 0 \leq y \leq x\}$ for $0 < x \leq 1$. Show that $g$ is well-defined and that $g$ is a monotone continuous function.

Let $(X, d)$ be a compact metric space. For $x \in X$ and $\epsilon > 0$, define $B_{\epsilon}(x) := \{y \in X \mid d(x, y) < \epsilon\}$. For $C \subseteq X$ and $\epsilon > 0$, define $B_{\epsilon}(C) := \cup_{x \in C} B_{\epsilon}(x)$. Let $\mathcal{K}$ be the set of non-empty compact subsets of $X$. For $C, C^{\prime} \in \mathcal{K}$, define $\delta\left(C, C^{\prime}\right) = \inf\{\epsilon \mid C \subseteq B_{\epsilon}\left(C^{\prime}\right)$ and $C^{\prime} \subseteq B_{\epsilon}(C)\}$. Show that $(\mathcal{K}, \delta)$ is a compact metric space.

Let $F \subseteq \mathbb{R}^{3}$ be a non-empty finite set, and $X = \mathbb{R}^{3} \backslash F$, taken with the subspace topology of $\mathbb{R}^{3}$. Show that $X$ is homeomorphic to a complete metric space. (Hint: Look for a suitable continuous function from $X$ to $\mathbb{R}$.)

Throughout this question every mentioned function is required to be a differentiable function from $\mathbb { R }$ to $\mathbb { R }$. The symbol $\circ$ denotes composition of functions.
(a) Suppose $f \circ f = f$. Then for each $x$, one must have $f ^ { \prime } ( x ) = $ \_\_\_\_ or $f ^ { \prime } ( f ( x ) ) = $ \_\_\_\_. Complete the sentence and justify.
(b) For a non-constant $f$ satisfying $f \circ f = f$, it is known and you may assume that the range of $f$ must have one of the following forms: $\mathbb { R } , ( - \infty , b ] , [ a , \infty )$ or $[ a , b ]$. Show that in fact the range must be all of $\mathbb { R }$ and deduce that there is a unique such function $f$. (Possible hints: For each $y$ in the range of $f$, what can you say about $f ( y )$? If the range has a maximum element $b$ what can you say about the derivative of $f$?)
(c) Suppose that $g \circ g \circ g = g$ and that $g \circ g$ is a non-constant function. Show that $g$ must be onto, $g$ must be strictly increasing or strictly decreasing and that there is a unique such increasing $g$.

Let $p \in \mathbb{N}^*$. We denote by $c = (c_1, \ldots, c_{2p})$ the canonical basis of $\mathbb{K}^{2p}$. $$\text{For all } x = \sum_{i=1}^{2p} x_i c_i \in \mathbb{K}^{2p}, \text{ we set } q_p(x) = 2\sum_{i=1}^{p} x_i x_{i+p}.$$
a) Show that $q_p$ is a quadratic form on $\mathbb{K}^{2p}$ and compute $\operatorname{mat}(q_p, c)$.
b) We call an Artin space (or artinian space) of dimension $2p$ any pair $(F,q)$, where $F$ is a $\mathbb{K}$-vector space of dimension $2p$, and where $q$ is a quadratic form on $F$ such that $(F,q)$ and $(\mathbb{K}^{2p}, q_p)$ are isometric. Show that in this case, $q$ is non-degenerate. When $p=1$, we say that $(F,q)$ is an artinian plane.
c) We assume that $\mathbb{K} = \mathbb{C}$ and for all $$x = \sum_{k=1}^{2p} x_k c_k \in \mathbb{C}^{2p}, \text{ we set } q(x) = \sum_{k=1}^{2p} x_k^2.$$ Show that $(\mathbb{C}^{2p}, q)$ is an Artin space.
d) We assume that $\mathbb{K} = \mathbb{R}$ and for all $$x = \sum_{i=1}^{2p} x_i c_i \in \mathbb{R}^{2p}, \text{ we set } q'(x) = \sum_{i=1}^{p} x_i^2 - \sum_{i=p+1}^{2p} x_i^2.$$ Show that $(\mathbb{R}^{2p}, q')$ is an Artin space.
e) If $(F,q)$ is an Artin space of dimension $2p$, show that there exists a vector subspace $G$ of $F$ of dimension $p$ such that the restriction of $q$ to $G$ is identically zero.

We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself, that is, the set of automorphisms $f$ of $E$ satisfying: for all $x \in E$, $q(f(x)) = q(x)$.
Let $f$ be an endomorphism of $E$.
a) Show that $f \in O(E,q)$ if and only if, for all $(x,y) \in E^2$: $\varphi(f(x),f(y)) = \varphi(x,y)$. Show that if $F$ is a vector subspace of $E$ and if $f \in O(E,q)$, then $f(F^\perp) = (f(F))^\perp$.
b) Let $e$ be a basis of $E$. Compute the matrix of the bilinear form: $$(x,y) \mapsto \varphi(f(x),f(y)) \text{ in terms of } \operatorname{mat}(f,e) \text{ and } \operatorname{mat}(\varphi,e).$$
c) Let us set $M = \operatorname{mat}(f,e)$ and $\Omega = \operatorname{mat}(\varphi,e)$. Show that $f \in O(E,q)$ if and only if $\Omega = {}^t M \Omega M$.
d) Show that if $f \in O(E,q)$, then $\operatorname{det}(f) \in \{1,-1\}$. We will denote: $$O^+(E,q) = \{f \in O(E,q) \mid \operatorname{det}(f) = 1\} \text{ and } O^-(E,q) = \{f \in O(E,q) \mid \operatorname{det}(f) = -1\}.$$

We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $F$ and $G$ be two vector subspaces of $E$ such that $E = F \oplus G$. We denote by $s$ the symmetry with respect to $F$ parallel to $G$.
a) Show that $s \in O(E,q)$ if and only if $F$ and $G$ are orthogonal (for $\varphi$).
b) Deduce that the symmetries in $O(E,q)$ are the symmetries with respect to $F$ parallel to $F^\perp$, where $F$ is a non-singular subspace of $E$.
c) When $H$ is a non-singular hyperplane, we will call reflection along $H$ the symmetry with respect to $H$ parallel to $H^\perp$. Show that every reflection of $E$ is an element of $O^-(E,q)$.
d) Let $(x,y) \in E^2$ such that $q(x) = q(y)$ and $q(x-y) \neq 0$. We denote by $s$ the reflection along $H = \{x-y\}^\perp$. Show that $s(x) = y$.

We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Suppose that $E$ is an artinian space of dimension $2p$ and that $F$ is a subspace of $E$ of dimension $p$ such that $q_{/F} = 0$.
If $f \in O(E,q)$ with $f(F) = F$, show that $f \in O^+(E,q)$.

We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $F$ be a subspace of $E$ such that $\bar{F} = E$ (where $\bar{F}$ is a non-singular completion of $F$). Show that if $f \in O(E,q)$ with $f_{/F} = \operatorname{Id}_F$ (where $\operatorname{Id}_F$ is the identity application from $F$ to $F$), then $f \in O^+(E,q)$.

We denote by $O(E,q)$ the set of isometries of $(E,q)$ into itself.
Let $f \in O(E,q)$. We assume that for all $x \in E$ such that $q(x) \neq 0$, we have $f(x) - x \neq 0$ and $q(f(x)-x) = 0$.
We propose to demonstrate that $f \in O^+(E,q)$ and that $E$ is an Artin space.
a) Show that $\operatorname{dim}(E) \geq 3$.
b) We denote by $V = \operatorname{Ker}(f - \operatorname{Id}_E)$. Show that $q_{/V} = 0$.
c) Let $x \in E$ such that $q(x) = 0$. We denote $H = \{x\}^\perp$. Show that $q_{/H}$ is not identically zero. Deduce that there exists $y \in E$ such that $q(x+y) = q(x-y) = q(y) \neq 0$.
d) We denote by $U = \operatorname{Im}(f - \operatorname{Id}_E)$. Show that $q_{/U} = 0$.
e) Show that $U^\perp = V = U$.
f) Deduce that $E$ is an Artin space and that $f \in O^+(E,q)$.

Let $N : \mathbb{R}^{n} \rightarrow \mathbb{R}_{+}$ be a norm on the vector space $\mathbb{R}^{n}$. Prove that the application defined by $$\forall x \in \mathbb{R}^{n}, \quad f(x) = \exp\left(-N(x)^{2}\right),$$ is continuous and log-concave on $\mathbb{R}^{n}$. (One may observe that the function $u \mapsto u^{2}$ is convex on $\mathbb{R}_{+}$).

In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$.
Show that the application which associates to $f$ the function $Lf$ is injective.

We denote $\mathcal{I} = \{(j, k) \in \mathbf{N}^{2} \mid j \in \mathbf{N} \text{ and } 0 \leq k < 2^{j}\}$; for $j \in \mathbf{N}$, $\mathcal{T}_{j} = \{k \in \mathbf{N} \mid 0 \leq k < 2^{j}\}$. For all $(j, k) \in \mathcal{I}$, $\theta_{j,k} : [0,1] \rightarrow [0,1]$ is defined by $$\theta_{j,k}(x) = \left\{ \begin{array}{l} 1 - |2^{j+1} x - 2k - 1| \quad \text{if } x \in [k 2^{-j}, (k+1) 2^{-j}] \\ 0 \text{ otherwise} \end{array} \right.$$ Show that for all $(j, k) \in \mathcal{I}$, the function $\theta_{j,k}$ is continuous, affine on each interval of the form $[\ell 2^{-n}, (\ell+1) 2^{-n}]$ where $n > j$ and $\ell \in \mathcal{T}_{n}$.

Let $n \in \mathbf{N}$. Show that $S_{n}$ is a projector on $\mathcal{C}_{0}$, whose subordinate norm (to $\|\cdot\|_{\infty}$) equals 1.

We denote $\alpha$ a real number such that $\alpha > -1/2$, $E$ the $\mathbb{R}$-vector space of functions of class $\mathcal{C}^\infty$ on $[-1,1]$ with real values, and $$S_\alpha(f,g) = \int_{-1}^{1} f(t)g(t)\left(1-t^2\right)^{\alpha - \frac{1}{2}} \mathrm{~d}t$$ Verify that $S_\alpha$ is an inner product on $E$.

Let $f : C(0,1) \rightarrow \mathbb{R}$ be a continuous application. We define: $$u(x,y) = \begin{cases} \mathrm{N}_f(x,y) & \text{if } (x,y) \in D(0,1) \\ f(x,y) & \text{if } (x,y) \in C(0,1) \end{cases}$$ on $\bar{D}(0,1)$.
Prove that $u$ is an application continuous at every point of $C(0,1)$. What can be concluded about the application $u$?

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 $\operatorname{Im} \phi_{m-2} \subset \mathcal{P}_{m-2}$.

We denote by $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ the set of $n \times n$ symmetric matrices whose eigenvalues are all simple. Show that the first component $s_{1}^{\downarrow}$ of $s^{\downarrow}$ is of class $\mathscr{C}^{1}$ on $\mathcal{S}_{2}^{\dagger}(\mathbb{R})$, but not on $\mathcal{S}_{2}(\mathbb{R})$. (One may use question 1d.)

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 $\left( \ell _ { 1 } , \ell _ { 2 } \right) \in \mathbb { Z } ^ { 2 }$.

Two simplexes $\mathcal{S}$ and $\mathcal{S}'$ in $\mathbb{R}^n$ are called equivalent if there exist an enumeration of the vertices $s_0, s_1, \ldots, s_n$ of $\mathcal{S}$, and $s_0', s_1', \ldots, s_n'$ of $\mathcal{S}'$, and a matrix $A$ in $\mathrm{GL}_n(\mathbb{Z})$ such that $A(s_i - s_0) = s_i' - s_0'$ for all $i = 1, \ldots, n$.
Show that the volume, the number of integer points and the number of interior integer points are the same for two equivalent integer simplexes.

Let $f \in \mathbb{R}^{N}$ and $J_{f,*} = \ln\left(\sum_{i=1}^{N} e^{f_{i}}\right)$. We consider $F : ]0, +\infty[ \rightarrow \mathbb{R}$ the function defined by $F(\beta) = \frac{1}{\beta} \ln\left(\sum_{i=1}^{N} e^{\beta f_{i}}\right)$.
Show that $F$ is differentiable and calculate its derivative $F'$. Show further that for all $\beta \in ]0, +\infty[$, there exists $p(\beta) \in \Sigma_{N}(\beta f)$ such that $F'(\beta) = -\frac{1}{\beta^{2}} H_{N}(p(\beta))$.

Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$. For all $\theta \in \mathbb{R}^{d}$, let $f(\theta) = M\theta \in \mathbb{R}^{N}$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$ and $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N}$$ where $f(\theta) = (f_{1}(\theta), \ldots, f_{N}(\theta))$. The function $L : \mathbb{R}^{d} \rightarrow \mathbb{R}$ is defined by $$L(\theta) = \ln(Z(\theta)) - q^{T} M\theta.$$
Show that $L$ is of class $\mathscr{C}^{1}$ and calculate its gradient.

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Let $P \in \mathbb{R}_n[X]$ be integer-valued on the integers. Show that $\delta(P)$ is also integer-valued on the integers.