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 denote $$\Sigma_{N}(\bar{g}, g) = \left\{p \in \Sigma_{N} \mid \sum_{i=1}^{N} p_{i} g_{k}(i) = \bar{g}_{k}, 1 \leqslant k \leqslant d\right\}.$$
Show that if $\theta$ is a critical point of $L$ (that is, a point where the gradient of $L$ vanishes) then $M^{T} p(\theta) = M^{T} q$ and $p(\theta) \in \Sigma_{N}(\bar{g}, g)$.

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 $$g_0(x) = \begin{cases} \mathbb{P}(X > x) & \text{if } x \geqslant 0 \\ 0 & \text{if } x < 0 \end{cases}$$ and $Lg_0$ denotes the unique bounded solution of (E) with support in $\mathbb{R}^+$ for $g = g_0$. Show that $Lg_0(x) = 1$ for $x \geqslant 0$ and $Lg_0(x) = 0$ for $x < 0$.

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 $t \geqslant 0$.

The purpose of this question is to show that for any strictly positive integers $n$ and $k$, there exists a constant $\alpha(k,n) \in ]0,1[$ such that, if $t_1, \ldots, t_n$ are strictly positive real numbers satisfying $1 > \sum_{i=1}^n t_i > 1 - \alpha(k,n)$, then there exist non-negative integers $p_1, \ldots, p_n \geqslant 0$ and $q$ such that $$\sum_{i=1}^n p_i = q > 0, \quad \text{and for all } i = 1, \ldots, n, \quad (kq+1)t_i > kp_i.$$ We proceed by induction on $n$.
17a. Handle the case $n = 1$ by showing that the constant $\alpha(k,1) = \frac{1}{k+1}$ works.
We assume the statement is true up to rank $n-1 \geqslant 1$. In particular, $\alpha(k,n-1) > 0$ is defined for all $k \geqslant 1$. We set for $k \geqslant 1$ $$\alpha(k,n) = \frac{1}{4kN^{n-1}} \quad \text{where} \quad N = 1 + \max\left(\frac{4k}{\alpha(k,n-1)}, 2kn(n+1)\right).$$ We are given $t_1 \geqslant t_2 \geqslant \cdots \geqslant t_n > 0$, and we assume that $\sum_{i=1}^n t_i = 1 - \alpha$ with $0 < \alpha < \alpha(k,n)$.
17b. If $t_n < \alpha(k,n-1) - \alpha$, establish the statement at rank $n$.
17c. If $t_n \geqslant \alpha(k,n-1) - \alpha$, apply the result of question 16 to the $\frac{t_i}{1-\alpha}$, $i = 1, \ldots, n$. With its notation, show that $$\alpha(k,n) < \min\left(\frac{1}{n+1}, \frac{1}{2}\alpha(k,n-1)\right) \quad \text{and} \quad 1 - qk\frac{\alpha}{1-\alpha} \geqslant \frac{1}{2}.$$ Conclude by distinguishing the cases $i \geqslant 2$ and $i = 1$.

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 $L(\theta) = \ln(Z(\theta)) - q^{T} M\theta$.
Show that $L$ is of class $\mathscr{C}^{2}$ and that for all integers $1 \leqslant l, k \leqslant d$ we have $$\frac{\partial^{2} L}{\partial \theta_{l} \partial \theta_{k}}(\theta) = \sum_{i=1}^{N} p_{i}(\theta)(M_{il} - m_{l}(\theta))(M_{ik} - m_{k}(\theta))$$ where $m(\theta) = M^{T} p(\theta)$.

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.

Let $\mathcal{S}$ be an integer simplex of $\mathbb{R}^n$ with vertices $0, s_1, \ldots, s_n$ having exactly $k$ interior integer points and let $x = \sum_{i=1}^n t_i s_i$ be an interior integer point of $\mathcal{S}$.
18a. Show that $\sum_{i=1}^n t_i \leqslant 1 - \alpha(k,n)$. (One may reason by contradiction and construct then $k+1$ distinct integer points interior to $\mathcal{S}$.)
18b. Show that $\frac{\alpha(k,n)}{1-\alpha(k,n)} x \in (\mathcal{S} - x)$.
18c. Deduce that $a(\mathcal{S} - x) \geqslant \frac{\alpha(k,n)}{1-\alpha(k,n)}$.

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 $\ker \widetilde{M} = \{0\}$ where $\widetilde{M} = (M \mid \mathbf{1})$.
We are interested in this question in the number of points at which the function $L$ attains its minimum.
(a) Show that if $\theta$ and $\theta'$ are two distinct points of $\mathbb{R}^{N}$ such that $L$ has a critical point at $\theta$, then the derivative of $t \rightarrow L(t\theta + (1-t)\theta')$ is strictly increasing on $[0,1]$ and vanishes at $t = 1$.
(b) Deduce that there is at most one critical point for $L$ and conclude on the number of points at which $L$ attains its minimum.

We now assume that $S _ { + } = S _ { - }$ and we assume that $\left( x _ { + } \mid q \right) = 0$. We denote by $z = \binom { x _ { + } } { 0 }$, $R ^ { + } = \left( \begin{array} { c c } S _ { + } & 0 \\ 0 & + 1 \end{array} \right) , R ^ { - } = \left( \begin{array} { c c } S _ { + } & 0 \\ 0 & - 1 \end{array} \right)$.
(a) Show that $O z = R^{+} z = R^{-} z$.
(b) We now write $$O = \left( \begin{array} { c c } \alpha ^ { \prime } & { } ^ { t } q ^ { \prime } \\ r ^ { \prime } & P ^ { \prime } \end{array} \right)$$ where $P ^ { \prime } \in M _ { n - 1 } ( \mathbb { R } )$. Construct then $z ^ { \prime } = \binom { \eta ^ { \prime } } { x ^ { \prime } } \in \mathbb { R } ^ { n }$ with $x ^ { \prime } \in \mathbb { R } ^ { n - 1 }$ strictly positive and $\eta ^ { \prime } \geq 0$ such that there exists a sign diagonal matrix $R ^ { \prime }$ satisfying $O z ^ { \prime } = R ^ { \prime } z ^ { \prime }$.
(c) In the case where $\eta ^ { \prime } = 0$, and using question 1(c), show that there exists a sign diagonal matrix $S$ such that $O \left( z + z ^ { \prime } \right) = S \left( z + z ^ { \prime } \right)$ and conclude.

Conclude the proof of Theorem 2, which states: For every strictly positive integer $k$, there exists a strictly positive constant $C(n,k)$ such that for every integer simplex $\mathcal{S}$ in $\mathbb{R}^n$ having exactly $k$ interior integer points, $\operatorname{Vol}(\mathcal{S}) \leqslant C(n,k)$.

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.

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.

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 $E_{n}$.

Let $m$ be a measure. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function that admits a variance relative to $m$. Show that $fm$ is integrable. As a consequence, the real $$\operatorname { Var } _ { m } ( f ) = \int f ( x ) ^ { 2 } m ( x ) d x - \left( \int f ( x ) m ( x ) d x \right) ^ { 2 }$$ is well defined. Show that $\operatorname { Var } _ { m } ( f ) \geqslant 0$.

Let $m$ be a measure. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function that admits an entropy relative to $m$. We consider the function $h : [ 0 , + \infty [ \rightarrow \mathbb { R }$ defined by $h ( 0 ) = 0$ and for $x > 0$, $h ( x ) = x \ln ( x )$.
2a. Show that $f ^ { 2 } m$ is integrable. As a consequence, the real $$\operatorname { Ent } _ { m } ( f ) = \int h \left( f ( x ) ^ { 2 } \right) m ( x ) d x - h \left( \int f ( x ) ^ { 2 } m ( x ) d x \right)$$ is well defined.
2b. Let $a > 0$. Show that $$\forall x \geqslant 0 , \quad h ( x ) \geqslant ( x - a ) h ^ { \prime } ( a ) + h ( a ) ,$$ with strict inequality if $x \neq a$.
2c. Show that $\operatorname { Ent } _ { m } ( f ) \geqslant 0$. You may use the previous question with $a = \int f ( x ) ^ { 2 } m ( x ) d x$.
2d. We assume here that for all $x \in \mathbb { R } , m ( x ) > 0$. Characterize the functions $f$ such that $\operatorname { Ent } _ { m } ( f ) = 0$.

We consider a function $f \in \mathscr { C } _ { b } ^ { 0 }$. We define for $( t , x ) \in \mathbb { R } ^ { 2 }$ $$\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$$ where $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$. Show that the function $\Phi _ { f } : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R }$ is well defined and continuous.

Let $f : \mathbb { R } \rightarrow \mathbb { R } _ { + }$ be a positive function in $\mathscr { C } _ { b } ^ { 0 }$. We define for $t \in \mathbb { R }$ $$J ( t ) = \int h \left( \Phi _ { f } ( t , x ) \right) \mu ( x ) d x$$ where $\Phi _ { f } ( t , x ) = \int f ( x \cos t + y \sin t ) \mu ( y ) d y$, $\mu ( x ) = \frac { 1 } { \sqrt { \pi } } e ^ { - x ^ { 2 } }$, and $h(x) = x\ln(x)$ for $x > 0$, $h(0) = 0$. Show that $J : \mathbb { R } \rightarrow \mathbb { R }$ is continuous, and calculate $J ( 0 )$ and $J \left( \frac { \pi } { 2 } \right)$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1): $T^{2m} \neq 0_{\mathcal{L}(E)}$ and $T^{2m+1} = 0_{\mathcal{L}(E)}$, and (H4): $T \circ M + M \circ T = 0_{\mathcal{L}(E)}$.
Let $k \in \mathbb{N}$.
(a) Show that $M \circ T^k = (-1)^k T^k \circ M$.
(b) Deduce that $\operatorname{Im}(T^k)$ and $\operatorname{ker}(T^k)$ are stable under $M$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$.
Show that one of the two following assertions is true: (i) $\operatorname{ker}(T) \subset F^+$, (ii) $\operatorname{ker}(T) \subset F^-$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$, where $S(v,w) = (v \mid T(w)) + (T(v) \mid w)$.
We assume here that $\operatorname{ker}(T) \subset F^+$.
(a) Show that $\forall z \in F^-, T^{2m}(z) = 0_E$.
(b) Show that $\operatorname{Im}(T)^\perp \subset F^+$ and that $\operatorname{Im}(T^2)^\perp \cap \operatorname{Im}(T) \subset F^-$.
(c) Let $z \in \operatorname{Im}(T)^\perp$ with $z \neq 0_E$. Show that $T(z) \in G^\perp$ and that $T(z) \neq 0_E$.
(d) Let $z \in \operatorname{Im}(T^2)^\perp \cap \operatorname{Im}(T)$ with $z \neq 0_E$. Show that $T(z) \in G^\perp$ and that $T(z) \neq 0_E$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$.
We now say that a pair $(w_1, w_2) \in E \times E$ is a characterizing pair of $G$ if $w_1$ and $w_2$ satisfy the three properties:
(A) $w_1 \in F^+$, $T(w_1) \in G^\perp$ and $T(w_1) \neq 0_E$,
(B) $w_2 \in F^-$, $T(w_2) \in G^\perp$ and $T(w_2) \neq 0_E$,
(C) $w_i \in \operatorname{Im}(T^2)^\perp$ for $i = 1$ and $i = 2$.
Deduce from the previous questions the existence of a characterizing pair of $G$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$.
Deduce that $\operatorname{dim}(G) \leq 2m-2$.

Let $m \geq 2$ be a natural integer and $E$ an $\mathbb{R}$-vector space of dimension $2m+1$ equipped with a scalar product $(.|.)$. Let $T, M$ be two endomorphisms of $E$ satisfying (H1)–(H4). We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$. Let $G$ be the set of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$.
We assume that $G$ satisfies hypothesis (H5): $\operatorname{dim}(G) = 2m-2$.
Show that if $(w_1, w_2)$ is a characterizing pair of $G$ then $(T(w_1), T(w_2))$ constitutes a basis of $G^\perp$.

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 $\left( \mathcal { F } _ { 2 } \right)$?