Let $B$ be a commutative ring. Let $n$ be a strictly positive integer and $b _ { 1 } , \ldots , b _ { n }$ be elements of $B$. a) Show that there exists a unique ring morphism $f$ from $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right]$ to $B$ such that $f \left( X _ { i } \right) = b _ { i }$ for all $i \in \{ 1 , \ldots , n \}$. b) Deduce that $B$ has property (TF) if and only if there exist an integer $n \geq 1$ and a surjective ring morphism $\mathbf { Z } \left[ X _ { 1 } , \ldots , X _ { n } \right] \rightarrow B$. c) Show that an abelian group $M$ has property (F) if and only if there exist an integer $r \geq 1$ and a surjective group morphism $\mathbf { Z } ^ { r } \rightarrow M$. d) Let $A$ and $B$ be commutative rings such that there exists a surjective ring morphism from $A$ to $B$. Show that if $A$ has property (TF), then so does $B$. State and prove an analogous statement for property (F).

Let $r$ and $m$ be strictly positive integers with $r \leq m$. We consider a subspace $V$ of the $\mathbf { C }$-vector space $M _ { m } ( \mathbf { C } )$ such that every element of $V$ is a matrix of rank at most $r$, and we assume $V$ contains the block matrix $A = \left( \begin{array} { c c } I _ { r } & 0 \\ 0 & 0 \end{array} \right)$. a) Let $B$ be an element of $V$, which we write in the form of a block matrix: $$B = \left( \begin{array} { l l } B _ { 11 } & B _ { 12 } \\ B _ { 21 } & B _ { 22 } \end{array} \right)$$ where the four matrices $B _ { 11 } , B _ { 12 } , B _ { 21 } , B _ { 22 }$ are respectively in $M _ { r } ( \mathbf { C } )$, $M _ { r , m - r } ( \mathbf { C } ) , M _ { m - r , r } ( \mathbf { C } )$ and $M _ { m - r } ( \mathbf { C } )$. Show that $B _ { 22 } = 0$ and $B _ { 21 } B _ { 12 } = 0$ (one may consider the minors of size $r + 1$ of the matrix $t A + B$ for $t \in \mathbf { C }$). b) Let $B$ and $C$ be two matrices of $V$, which we write in block matrix form as above: $$B = \left( \begin{array} { c c } B _ { 11 } & B _ { 12 } \\ B _ { 21 } & 0 \end{array} \right) ; \quad C = \left( \begin{array} { c c } C _ { 11 } & C _ { 12 } \\ C _ { 21 } & 0 \end{array} \right)$$ Show that $B _ { 21 } C _ { 12 } + C _ { 21 } B _ { 12 } = 0$.

Let $r$ and $m$ be strictly positive integers with $r \leq m$. We consider a subspace $V$ of the $\mathbf { C }$-vector space $M _ { m } ( \mathbf { C } )$ such that every element of $V$ is a matrix of rank at most $r$, and we assume $V$ contains the block matrix $A = \left( \begin{array} { c c } I _ { r } & 0 \\ 0 & 0 \end{array} \right)$. By question V.2a), every element $B$ of $V$ has the block form $B = \left( \begin{array} { c c } B_{11} & B_{12} \\ B_{21} & 0 \end{array} \right)$. We denote by $W$ the intersection of $V$ with the subspace of $M _ { m } ( \mathbf { C } )$ consisting of block matrices of the form $$\left( \begin{array} { c c } 0 & 0 \\ B _ { 21 } & 0 \end{array} \right)$$ We define a linear application $\varphi$ from $M _ { m } ( \mathbf { C } )$ to $M _ { r , m } ( \mathbf { C } )$ by $$\varphi : \left( \begin{array} { l l } B _ { 11 } & B _ { 12 } \\ B _ { 21 } & B _ { 22 } \end{array} \right) \mapsto \left( \begin{array} { l l } B _ { 11 } & B _ { 12 } \end{array} \right)$$ (with the notations of V.2a)). a) We write any matrix $C$ of $M _ { r , m } ( \mathbf { C } )$ in the form of a block matrix $C = \left( \begin{array} { l l } C _ { 11 } & C _ { 12 } \end{array} \right)$ with $C _ { 11 } \in M _ { r } ( \mathbf { C } )$ and $C _ { 12 } \in M _ { r , m - r } ( \mathbf { C } )$. Let $\psi$ be the linear map from $W$ to $M _ { r , m } ( \mathbf { C } ) ^ { \vee }$ which sends $B = \left( \begin{array} { c c } 0 & 0 \\ B _ { 21 } & 0 \end{array} \right)$ to the linear form $C \mapsto \operatorname { Tr } \left( B _ { 21 } C _ { 12 } \right)$. Let $s = \operatorname { dim } W$. Using the map $\psi$, show that $\operatorname { dim } ( \varphi ( V ) ) \leq m r - s$. b) Deduce that $\operatorname { dim } V \leq m r$.

a) Let $r , m , n$ be strictly positive integers such that $r \leq n \leq m$. Show that if $E$ is a subspace of $M _ { m , n } ( \mathbf { C } )$ such that every element of $E$ is a matrix of rank at most $r$, then $\operatorname { dim } E \leq m r$. b) Give an example of a subspace $E$ of $M _ { m , n } ( \mathbf { C } )$ satisfying $\operatorname { dim } E = m r$ and such that every element of $E$ is a matrix of rank at most $r$.

Let $A \in \mathscr{M}_{N}(\mathbf{R})$. Show that $A$ satisfies $(M_2)$ if and only if $AU = U$. Deduce that if $A$ and $B$ are two Markov kernels then $AB$ is also a Markov kernel.

If $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, the support of $\varphi$ is defined by: $$\operatorname{Supp}(\varphi) = \overline{\{x \in \mathbb{R} : \varphi(x) \neq 0\}}$$ We say that $\varphi$ has compact support if $\operatorname{Supp}(\varphi)$ is a bounded subset of $\mathbb{R}$. We denote by $\mathcal{C}_{c}(\mathbb{R})$ the set of continuous functions with compact support on $\mathbb{R}$.
Show that $\mathcal{C}_{c}(\mathbb{R})$ is a vector subspace of the space of continuous functions on $\mathbb{R}$.

Let $C \subset E$ be a convex set. Let $f$ and $g$ be two convex functions from $C$ to $\mathbb{R}$.
(a) Show that $f + g$ is convex, and strictly convex if one of the two functions $f$ or $g$ is strictly convex.
(b) Assume $f$ is strictly convex. Verify that the minimum of $f$ is attained on $C$ at most at one point of $C$.

Show that, for all $\rho > 0$ and all $m , n \in \mathbb { N } ^ { * }$, the sets $\mathscr { D } _ { \rho } ( \mathbb { R } ) , \mathscr { D } _ { \rho } \left( \mathbb { R } _ { n } [ X ] \right)$ and $\mathscr { D } _ { \rho } \left( \mathscr { M } _ { m , n } ( \mathbb { R } ) \right)$ are closed under addition.

Show that $S _ { n } ^ { + } ( \mathbf { R } )$ and $S _ { n } ^ { + + } ( \mathbf { R } )$ are convex subsets of $M _ { n } ( \mathbf { R } )$. Are they vector subspaces of $M _ { n } ( \mathbf { R } )$ ?

We fix a Markov kernel $K$. Show that for all $n \in \mathbf{N}$, $K^n$ is a Markov kernel.

If $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, the support of $\varphi$ is defined by: $$\operatorname{Supp}(\varphi) = \overline{\{x \in \mathbb{R} : \varphi(x) \neq 0\}}$$ We say that $\varphi$ has compact support if $\operatorname{Supp}(\varphi)$ is a bounded subset of $\mathbb{R}$. We denote by $\mathcal{C}_{c}(\mathbb{R})$ the set of continuous functions with compact support on $\mathbb{R}$. If $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we set $$\|\varphi\|_{\infty} = \sup_{x \in \mathbb{R}} |\varphi(x)| \text{ and } \|\varphi\|_{1} = \int_{-\infty}^{+\infty} |\varphi(t)| dt$$
Show that $\|\cdot\|_{1} : \varphi \mapsto \|\varphi\|_{1}$ is a norm on $\mathcal{C}_{c}(\mathbb{R})$. One may admit without proof that $\|\cdot\|_{\infty}$ is also a norm.

Let $A \in \mathcal{M}_{m,n}(\mathbb{R})$ be a matrix with $m$ rows and $n$ columns. We denote by $\langle u, v \rangle_{\mathbb{R}^n}$ the inner product between two vectors $u$ and $v$ of $\mathbb{R}^n$ and $\langle \mu, \nu \rangle_{\mathbb{R}^m}$ that between two vectors $\mu$ and $\nu$ of $\mathbb{R}^m$.
(a) Show that for all $(x, \nu) \in \mathbb{R}^n \times \mathbb{R}^m$, we have $$\langle Ax, \nu \rangle_{\mathbb{R}^m} = \left\langle x, A^\top \nu \right\rangle_{\mathbb{R}^n},$$ where $A^\top$ denotes the transpose matrix of $A$.
(b) Deduce that $\ker A \subset (\operatorname{Im} A^\top)^\perp$ where $E^\perp$ denotes the orthogonal complement of $E$ for the inner product on $\mathbb{R}^n$ for any vector subspace $E$ of $\mathbb{R}^n$.
(c) Show that $\ker A = (\operatorname{Im} A^\top)^\perp$.

Show that, for all $\rho > 0$ and all $n \in \mathbb { N } ^ { * }$, the sets $\mathscr { D } _ { \rho } ( \mathbb { R } )$ and $\mathscr { D } _ { \rho } \left( \mathscr { M } _ { n } ( \mathbb { R } ) \right)$ are closed under multiplication.

Let $t \in \mathbf{R}$ and $(i,j) \in \llbracket 1;N \rrbracket^2$, justify that the series $\sum_{n \geq 0} \frac{t^n K^n[i,j]}{n!}$ converges. We denote by $H_t \in \mathscr{M}_N(\mathbf{R})$ the matrix defined by $$\forall (i,j) \in \llbracket 1;N \rrbracket^2, H_t[i,j] = e^{-t} \sum_{n=0}^{+\infty} \frac{t^n K^n[i,j]}{n!}$$

If $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, the support of $\varphi$ is defined by: $$\operatorname{Supp}(\varphi) = \overline{\{x \in \mathbb{R} : \varphi(x) \neq 0\}}$$ We say that $\varphi$ has compact support if $\operatorname{Supp}(\varphi)$ is a bounded subset of $\mathbb{R}$. We denote by $\mathcal{C}_{c}(\mathbb{R})$ the set of continuous functions with compact support on $\mathbb{R}$. If $\varphi \in \mathcal{C}_{c}(\mathbb{R})$, we set $$\|\varphi\|_{\infty} = \sup_{x \in \mathbb{R}} |\varphi(x)| \text{ and } \|\varphi\|_{1} = \int_{-\infty}^{+\infty} |\varphi(t)| dt$$
Are the norms $\|\cdot\|_{\infty}$ and $\|\cdot\|_{1}$ equivalent?

Consider an open set $U \subset \mathbb{R}^n$, $h : U \rightarrow \mathbb{R}$ a $\mathcal{C}^1$ application and $b \in \mathbb{R}^m$. Assume that there exists $x_* \in U$ a minimum of $h$ on the set $V_b = \{x \in U \mid Ax + b = 0\}$.
(a) Show that for all $u \in \mathbb{R}^n$ such that $Au = 0$ we have $\left\langle \nabla h(x_*), u \right\rangle_{\mathbb{R}^n} = 0$ where $\nabla h(x)$ denotes the gradient of $h$ at $x$.
(b) Show the existence of $\nu_* \in \mathbb{R}^m$ such that $\nabla h(x_*) - A^T \nu_* = 0$.
(c) Deduce that the application $L : U \times \mathbb{R}^m \rightarrow \mathbb{R}$ such that $L(x, \nu) = h(x) - \langle \nu, Ax + b \rangle_{\mathbb{R}^m}$ satisfies $\frac{\partial L}{\partial x_k}(x_*, \nu_*) = 0$ for all $1 \leq k \leq n$ where $\frac{\partial L}{\partial x_k}(x, \nu)$ denotes the partial derivative of $L$ with respect to the $k$-th coordinate of $x \in \mathbb{R}^n$.
(d) Conclude that if $U$ is convex, and $h$ is convex on $U$, then $L$ admits a saddle point at $(x_*, \nu_*)$, that is, we have $$L(x_*, \nu) \leq L(x_*, \nu_*) \leq L(x, \nu_*)$$ for all $(x, \nu) \in U \times \mathbb{R}^m$.

Let $r \in \mathbb { R } _ { + } ^ { * }$ such that $r \leqslant \rho$. Show that the map $\mathscr { D } _ { \rho } ( \mathbb { R } ) \rightarrow \mathscr { D } _ { r } ( \mathbb { R } )$ which associates to a function $f$ its restriction to $U _ { r }$ is injective.

Show that for all real $t \in \mathbf{R}_+$, $H_t$ is a Markov kernel.

Let $[a, b]$ be a compact interval of $\mathbb{R}$ and $f$ a function continuous on $[a, b]$ and differentiable on $]a, b[$, with real values. Suppose that $f'(x)$ has a finite limit $\ell$ as $x \rightarrow a^{+}$. Show that $f$ is right-differentiable at $a$ and specify the value of $f'(a)$.

Let $X$ be a finite set and $p = (p_x)_{x \in X}$ a probability distribution on $X$. We assume that $p$ charges all points of $X$: $p_x > 0$ for all $x \in X$. We call entropy of $p$ the quantity $$H(p) = -\sum_{x \in X} p_x \ln(p_x)$$ We consider the set $Q_X = \{\boldsymbol{q} = (q_x)_{x \in X} \in \mathbb{R}^X \mid \forall x \in X, q_x \geq 0\}$. For all $\boldsymbol{q}, \boldsymbol{q}' \in Q_X$ such that $q_x' > 0$ for all $x \in X$, we define: $$\mathrm{KL}(\boldsymbol{q}, \boldsymbol{q}') = \sum_{x \in X} \varphi(q_x / q_x') q_x'$$ with $\varphi : \mathbb{R}_+ \rightarrow \mathbb{R}$ defined by $\varphi(x) = x \log(x) - x + 1$ for $x > 0$ and extended to 0 by continuity.
(a) Specify $\varphi(0)$.
(b) Verify that $\varphi$ is continuous, strictly convex, positive and that $\varphi(x) = 0$ if and only if $x = 1$.
(c) Show that $Q_X$ is convex and that $\boldsymbol{q} \mapsto \mathrm{KL}(\boldsymbol{q}, \boldsymbol{q}')$ is strictly convex, positive and vanishes if and only if $q = q'$.

Let $r \in \mathbb { R } _ { + } ^ { * }$ such that $r < \rho$. Show that $\| \cdot \| _ { r }$ is a norm on $\mathscr { D } _ { \rho } ( \mathbb { R } )$ and that $\| f g \| _ { r } \leqslant \| f \| _ { r } \cdot \| g \| _ { r }$ for all $f , g \in \mathscr { D } _ { \rho } ( \mathbb { R } )$.

In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$. We use the decomposition $\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$ where $E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$, with the projections $p_i$ and inclusions $q_i$ as defined, and $\|.\|_i$ the norm on $\mathcal{L}(E_i)$ and $\|.\|_c$ for $\mathcal{L}(\mathbf{C}^n)$.
Show that, for all $i \in \llbracket 1 ; r \rrbracket$, there exists a constant $C _ { i } > 0$ such that: $$\forall u \in \mathcal { L } \left( E _ { i } \right) , \quad \left\| q _ { i } u p _ { i } \right\| _ { c } \leqslant C _ { i } \| u \| _ { i }$$

Show that for $(t,s) \in \mathbf{R}_+^2$, $H_{t+s} = H_t H_s$. One may use a Cauchy product.

Let $\varphi_{0}$ be the function defined on $\mathbb{R}$ by
$$\left\{ \begin{array}{l} \varphi_{0}(x) = e^{-1/x^{2}} \text{ if } x \neq 0 \\ \varphi_{0}(0) = 0 \end{array} \right.$$
a. Show that for all $n \in \mathbb{N}$ there exists a polynomial $P_{n}$ such that for $x \neq 0$ we have
$$\varphi_{0}^{(n)}(x) = P_{n}\left(\frac{1}{x}\right) e^{-1/x^{2}}$$
b. Show that $\varphi_{0}$ is of class $C^{\infty}$ on $\mathbb{R}$.

Let $X$ be a non-empty finite set and $c : X \rightarrow \{0,1\}^+$ an injective application. We say that $c$ is a binary code on $X$. We further assume that $c$ is a prefix code, that is, for all $x \neq y$ in $X$, $c(x)$ is not a prefix of $c(y)$. We define $\bar{c} : X \rightarrow \{0,1\}^*$ such that for all $x \in X$, $c(x) = c(x)_1 \cdot \bar{c}(x)$ where $c(x)_1$ is the first element of the word $c(x)$.
(a) Verify that for all $x \neq y \in X$, if $c(x)_1 = c(y)_1$ then $\bar{c}(x) \neq \bar{c}(y)$ and $\bar{c}(x)$ is not a prefix of $\bar{c}(y)$.
(b) For $a \in \{0,1\}$ we denote $X_a = \{x \in X \mid c(x)_1 = a\}$. Show that if $X_a$ contains at least two elements, then the restriction of $\bar{c}$ to $X_a$ is a prefix code on $X_a$.
(c) Deduce that $\sum_{x \in X} 2^{-|c(x)|} \leq 1$. (Hint: One may decompose the sum into a sum over $X_0$ and $X_1$ and reason by induction on $L(c) = \max\{|c(x)| \mid x \in X\}$)