Let $M \in \mathscr{M}_{N,d}(\mathbb{R})$ defined by $M_{i,j} = g_{j}(i)$, $q \in \Sigma_{N}$, and for all $\theta \in \mathbb{R}^{d}$, $f(\theta) = M\theta$, $Z(\theta) = \sum_{i=1}^{N} e^{f_{i}(\theta)}$, $$p(\theta) = \left(\frac{e^{f_{1}(\theta)}}{Z(\theta)}, \ldots, \frac{e^{f_{N}(\theta)}}{Z(\theta)}\right) \in \Sigma_{N},$$ and $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$. We assume that $\ker \widetilde{M} = \{0\}$ and that the function $L$ has a global minimum attained at $\theta_{*}$. We denote $$\Sigma_{N}(\bar{g}, g) = \left\{p \in \Sigma_{N} \mid \sum_{i=1}^{N} p_{i} g_{k}(i) = \bar{g}_{k}, 1 \leqslant k \leqslant d\right\}.$$
(a) Show that $H_{N}(p(\theta_{*})) \geqslant H_{N}(q)$ and then that $H_{N}(p(\theta_{*}))$ is the maximum value of $H_{N}$ on $\Sigma_{N}(\bar{g}, g)$.
(b) Show that $p(\theta_{*})$ is the unique point of $\Sigma_{N}(\bar{g}, g)$ at which $H_{N}$ attains its maximum.

Deduce from question 6 that $x > 0$ and $x + O x \geq 0$ as well as $x - O x \geq 0$. Conclude.

Deduce from Broyden's theorem that there exist a strictly positive vector $x$ and a sign diagonal matrix $S$ such that $O x = S x$ and deduce that $u = x + S x$ is the positive vector of Tucker's theorem.

We prove Broyden's theorem by induction on the dimension. We assume $| \alpha | < 1$. Using the induction hypothesis for $Q _ { + }$ (resp. for $Q _ { - }$), we denote by $x _ { + } > 0$ (resp. $x _ { - } > 0$) a vector of $\mathbb { R } ^ { n - 1 }$ and $S _ { + }$ (resp. $S _ { - }$) the sign diagonal matrix, such that $Q _ { + } x _ { + } = S _ { + } x _ { + }$, resp. $Q _ { - } x _ { - } = S _ { - } x _ { - }$. We set

• $\eta _ { + } = - \frac { \left( x _ { + } \mid q \right) } { \alpha + 1 } , \quad \eta _ { - } = - \frac { \left( x _ { - } \mid q \right) } { \alpha - 1 }$
• $z _ { + } = \binom { x _ { + } } { \eta _ { + } } , \quad z _ { - } = \binom { x _ { - } } { \eta _ { - } }$
• $S ^ { + } = \left( \begin{array} { c c } S _ { + } & 0 \\ 0 & + 1 \end{array} \right) , \quad S ^ { - } = \left( \begin{array} { c c } S _ { - } & 0 \\ 0 & - 1 \end{array} \right)$

Show using question 1(a) that in the case where $S _ { + } \neq S _ { - }$ then one of the pairs $(z _ { + } , S ^ { + })$ or $(z _ { - } , S ^ { - })$ satisfies Broyden's theorem.

For $A \in M _ { n , m } ( \mathbb { R } )$ and $b \in \mathbb { R } ^ { n }$ as in Farkas' lemma, we set $$B = \left( \begin{array} { c c c c } 0 & 0 & A & - b \\ 0 & 0 & - A & b \\ - { } ^ { t } A & { } ^ { t } A & 0 & 0 \\ { } ^ { t } b & - { } ^ { t } b & 0 & 0 \end{array} \right)$$ Let, by Tucker's theorem, $y = { } ^ { t } \left( z _ { 1 } , z _ { 2 } , x , t \right) \geq 0$ such that $B y \geq 0$ and $y + B y > 0$.
Show that if $t > 0$ then for $z = z _ { 1 } - z _ { 2 }$, we have $- { } ^ { t } A z \geq 0$ and $( b \mid z ) > 0$.

For $A \in M _ { n , m } ( \mathbb { R } )$ and $b \in \mathbb { R } ^ { n }$ as in Farkas' lemma, we set $$B = \left( \begin{array} { c c c c } 0 & 0 & A & - b \\ 0 & 0 & - A & b \\ - { } ^ { t } A & { } ^ { t } A & 0 & 0 \\ { } ^ { t } b & - { } ^ { t } b & 0 & 0 \end{array} \right)$$ Let, by Tucker's theorem, $y = { } ^ { t } \left( z _ { 1 } , z _ { 2 } , x , t \right) \geq 0$ such that $B y \geq 0$ and $y + B y > 0$.
If $t > 0$ show that $A x = t b$ and conclude.

Deduce the inversion formula: for every integer $k \in \mathbb{N}$, $$u_k = \sum_{j=0}^{k} (-1)^{k-j} \binom{k}{j} v_j$$

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Deduce a polynomial $P \in \mathbb{R}_5[X]$ such that $$\delta^2(P) = X^3 + 2X^2 + 5X + 7$$

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Deduce that $H_n(\mathbb{Z}) \subset \mathbb{Z}$, that is, $H_n$ is integer-valued on the integers.

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 $P \in \mathbb{R}[X]$ of degree $d \in \mathbb{N}$. Show that if $P$ is integer-valued on the integers then $d! P$ is a polynomial with integer coefficients. Study the converse.

We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
Show that for every integer $k$ strictly positive, $k^\alpha$ belongs to $\mathbb{N}^*$.

We consider a real $\alpha$ such that for every prime number $p$, $p^\alpha$ is a natural number. We apply relation $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ to the function $f_\alpha(x) = x^\alpha$ and to the integer $n = \lfloor \alpha \rfloor + 1$.
Conclude that $\alpha$ is a natural number.

Let $\Lambda$ be a non-empty subset of $\mathbb{R}_*^+$ closed under addition, with $r(\Lambda) > 0$ and $d = b - a$ as defined above. Deduce that $\Lambda \subset d\mathbb{Z}$, where $d\mathbb{Z} = \{kd \mid k \in \mathbb{Z}\}$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. Using the results of questions 13a and 13b, deduce that $h(-x) \rightarrow h(0)$ when $x \rightarrow +\infty$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$. Conclude that $h$ is a constant function.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. The function $f$ is the unique bounded and uniformly continuous solution of equation (E), and $f'$ is bounded and uniformly continuous. Prove that the function $x \mapsto \sup_{t \geqslant x} f'(t)$ admits a finite limit when $x \rightarrow +\infty$. We denote $$c := \lim_{x \rightarrow +\infty} \sup_{t \geqslant x} f'(t)$$

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. We admit that there exists a subsequence $\left(t_k\right)_{k \geqslant 0}$ of $\left(y_n\right)_{n \geqslant 0}$ such that the sequence of functions $\left(\xi_k\right)_{k \geqslant 0}$ defined by $$\xi_k : \mathbb{R} \rightarrow \mathbb{R}, \quad t \mapsto \xi_k(t) = f'\left(t + t_k\right)$$ converges uniformly on every segment of $\mathbb{R}$ to a function denoted $\xi$. Show that $\xi$ is constant, equal to $c$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. With $\xi$ constant equal to $c$ as shown in question 14c, conclude that $c = 0$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. We admit that $\lim_{x \rightarrow +\infty} \inf_{t \geqslant x} f'(t) = 0$. Deduce that $f'(t) \rightarrow 0$ when $t \rightarrow +\infty$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that $g$ is of class $\mathscr{C}^1$, with support in $[0, K]$ with $K > 0$. Using the result that $f'(t) \rightarrow 0$ when $t \rightarrow +\infty$, show that for all $\ell \geqslant 0$, $f(t+\ell) - f(t) \rightarrow 0$ when $t \rightarrow +\infty$.

We assume that for all $d \geqslant 0$, $\mathbb{P}(X \in d\mathbb{Z}) < 1$, and that only finitely many $p_i$ are strictly positive. We set $$g_0(x) = \begin{cases} \mathbb{P}(X > x) & \text{if } x \geqslant 0 \\ 0 & \text{if } x < 0 \end{cases}$$ For all $g \in \mathscr{F}$ (the set of positive bounded functions with support in a segment of $\mathbb{R}^+$), we denote by $Lg$ the unique solution of (E) bounded with support in $\mathbb{R}^+$. A sequence $\left(t_k\right)_{k \geqslant 0}$ satisfies property $(\mathscr{P})$ if $t_k \rightarrow +\infty$ and there exists a continuous bounded function $\mu : \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for every piecewise continuous $g \in \mathscr{F}$, $$Lg\left(t_k\right) \rightarrow \int_0^{+\infty} g(t)\mu(t)\,dt \quad \text{when} \quad k \rightarrow +\infty$$ Show, using question 14f, that for all $g \in \mathscr{F} \cap \mathscr{C}^1(\mathbb{R}, \mathbb{R}^+)$ and $\ell \geqslant 0$, $$\int_0^{+\infty} g(t)(\mu(t+\ell) - \mu(t))\,dt = 0$$

Under the same assumptions as question 15a, deduce that $\mu$ is constant.

Under the same assumptions as question 16a, and using the fact that $\mu$ is constant (question 15b) and that $\int_0^{+\infty} g_0(t)\,dt = \mathbb{E}(X)$, deduce that $\mu(t) = \dfrac{1}{\mathbb{E}(X)}$ for all $t \geqslant 0$.

We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Conclude that if $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$, then $A$ is $F$-regular for every vector subspace $F$ of dimension $n-2$ of $E_{n}$.

We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$. In light of the results of questions 24 and 25, what can we conclude regarding the relationship between propositions $\left( \mathcal { F } _ { 1 } \right)$ and $\left( \mathcal { F } _ { 2 } \right)$?