22. Show that if $Q \in \mathbb { R } [ X ]$ is a polynomial such that $\operatorname { deg } ( Q ) < \operatorname { deg } ( P )$ and, for every integer $1 \leqslant i \leqslant m$ and every integer $0 \leqslant k < k _ { i } , Q ^ { ( k ) } \left( t _ { i } \right) = 0$, then $Q = 0$. 2b. Show that there exists a unique polynomial $H ( f , P ) \in \mathbb { R } [ X ]$ such that $\operatorname { deg } ( H ( f , P ) ) < \operatorname { deg } ( P )$ and such that, for every integer $1 \leqslant i \leqslant m$ and every integer $0 \leqslant k < k _ { i }$,
$$H ( f , P ) ^ { ( k ) } \left( t _ { i } \right) = f ^ { ( k ) } \left( t _ { i } \right) .$$
For $t \in [ a , b ] \backslash \left\{ t _ { 1 } , \ldots , t _ { m } \right\}$. We set
$$Q ( f , P ) ( t ) = \frac { f ( t ) - H ( f , P ) ( t ) } { \left( t - t _ { 1 } \right) ^ { k _ { 1 } } \cdots \left( t - t _ { m } \right) ^ { k _ { m } } } .$$
3a. We set $g = f - H ( f , P )$. Show that, for every integer $1 \leqslant i \leqslant m$ and every real $x \in [ a , b ]$, we have
$$f ( x ) - H ( f , P ) ( x ) = \left( x - t _ { i } \right) ^ { k _ { i } } \int _ { 0 } ^ { 1 } \frac { v ^ { k _ { i } - 1 } } { \left( k _ { i } - 1 \right) ! } g ^ { \left( k _ { i } \right) } \left( t _ { i } v + x ( 1 - v ) \right) d v$$
3b. Show that the function $Q ( f , P )$ extends uniquely to a function of class $\mathscr { C } ^ { \infty }$ from $[ a , b ]$ to $\mathbb { R }$.
4a. Let $s _ { 0 } \in [ a , b ]$ and let an integer $n \geqslant 1$. Show that
$$Q \left( f , \left( X - s _ { 0 } \right) ^ { n } \right) \left( s _ { 0 } \right) = \frac { f ^ { ( n ) } \left( s _ { 0 } \right) } { n ! }$$

• 4b. Let $P _ { 1 } , P _ { 2 } \in \mathbb { R } [ X ]$ be two monic polynomials split in $] a , b [$. Show that

$$H \left( f , P _ { 1 } P _ { 2 } \right) = H \left( f , P _ { 1 } \right) + P _ { 1 } H \left( Q \left( f , P _ { 1 } \right) , P _ { 2 } \right) \quad \text { and } \quad Q \left( f , P _ { 1 } P _ { 2 } \right) = Q \left( Q \left( f , P _ { 1 } \right) , P _ { 2 } \right)$$
We fix $t \in [ a , b ] \backslash \left\{ t _ { 1 } , \ldots , t _ { m } \right\}$. For all $s \in [ a , b ]$, we set
$$Q _ { t } ( s ) = f ( s ) - H ( f , P ) ( s ) - Q ( f , P ) ( t ) \prod _ { i = 1 } ^ { m } \left( s - t _ { i } \right) ^ { k _ { i } }$$
5a. Show that the function $Q _ { t }$ vanishes to order $\operatorname { deg } ( P ) + 1$ in the interval [ $\min \left( t , t _ { 1 } \right) , \max \left( t , t _ { m } \right)$ ]. 5b. Deduce that if $P$ is monic, there exists $\xi \in \left[ \min \left( t , t _ { 1 } \right) , \max \left( t , t _ { m } \right) \right]$ such that
$$f ( t ) - H ( f , P ) ( t ) = \frac { f ^ { ( \operatorname { deg } ( P ) ) } ( \xi ) } { \operatorname { deg } ( P ) ! } P ( t )$$
We say that a function $h$ from $[ a , b ]$ to $\mathbb { R }$ is absolutely monotone on an interval $[ a , b ]$ if it is of class $\mathscr { C } ^ { \infty }$ on $[ a , b ]$ and if, for every integer $n \geqslant 0$, the function $h ^ { ( n ) }$ takes positive values on $[ a , b ]$. In particular $h$ takes positive values.

Show that
$$\mathbf { E } \left[ \Delta \tilde { S } _ { n } \right] = - \mathbf { E } \left[ \tilde { S } _ { n } p \left( \tilde { I } _ { n } \right) \right]$$
then deduce the equation satisfied by $\mathbf { E } \left[ \Delta \tilde { I } _ { n } \right]$.
Here $\Delta \tilde{S}_n = \tilde{S}_{n+1} - \tilde{S}_n$, $\Delta \tilde{I}_n = \tilde{I}_{n+1} - \tilde{I}_n$, $\Delta \tilde{R}_n = \tilde{R}_{n+1} - \tilde{R}_n$, and $\tilde{S}_n + \tilde{I}_n + \tilde{R}_n = M$ for all $n$.

Let $n \geq 1$ be an integer. We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$ and $\mathcal{M}_n(\mathbb{Z})$ the set of $n \times n$ matrices with integer coefficients.
Suppose that $M \in B_n \backslash \mathcal{M}_n(\mathbb{Z})$. Deduce that there exists a nonzero matrix $Q$ and $\epsilon > 0$ such that $\{M + tQ, t \in [-\epsilon, \epsilon]\} \subset B_n$, and conclude that every vertex of $B_n$ is of the form $P^\sigma$.

Let $n \geq 1$ be an integer. We denote by $\mathbb{C}[[\mathbb{Z}^n]]$ the $\mathbb{C}$-vector space of functions $f : \mathbb{Z}^n \rightarrow \mathbb{C}$. We say that $f \in \mathbb{C}[[\mathbb{Z}^n]]$ is rational if there exists a nonzero $P \in \mathbb{C}[\mathbb{Z}^n]$ such that $Pf \in \mathbb{C}[\mathbb{Z}^n]$. We say that $f$ is torsion if there exists a nonzero $P \in \mathbb{C}[\mathbb{Z}^n]$ such that $Pf = 0$. We denote by $\mathcal{R}$ the $\mathbb{C}$-vector space of rational elements and $\mathcal{T}$ the $\mathbb{C}$-vector space of torsion elements of $\mathbb{C}[[\mathbb{Z}^n]]$.
In the case where $n = 1$, show that the inclusions $0 \subset \mathcal{T} \subset \mathcal{R} \subset \mathbb{C}[[\mathbb{Z}^n]]$ are strict.

Let $n \geq 1$ be an integer. We denote by $\mathbb{C}[[\mathbb{Z}^n]]$ the $\mathbb{C}$-vector space of functions $f : \mathbb{Z}^n \rightarrow \mathbb{C}$, $\mathcal{R}$ the $\mathbb{C}$-vector space of rational elements, $\mathcal{T}$ the $\mathbb{C}$-vector space of torsion elements, and $\mathbb{C}(\mathbb{Z}^n)$ the field of fractions of $\mathbb{C}[\mathbb{Z}^n]$.
We define a $\mathbb{C}$-linear map $\mathrm{I} : \mathcal{R} \rightarrow \mathbb{C}(\mathbb{Z}^n)$ as follows. If $f \in \mathcal{R}$ satisfies $Qf = P$ with $P, Q \in \mathbb{C}[\mathbb{Z}^n]$, we set $\mathrm{I}(f) = \frac{P}{Q}$. Show that $\mathrm{I}$ is well defined, and that it is a linear map with kernel $\mathcal{T}$ satisfying $\mathrm{I}(Pf) = P\,\mathrm{I}(f)$ for all $f \in \mathcal{R}$ and $P \in \mathbb{C}[\mathbb{Z}^n]$.

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$. Show that $\nabla f(x) = -Mx$, for all $x \in \mathbb{R}^d$.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$. We now assume that $h > 0$.
We set $\gamma_h = G_h''(u_h)$ and we denote $f_h : x \longmapsto \frac{\widehat{G}_h(x)}{x^2}$.
Justify that $f_h$ is bounded below by a strictly positive real number $c_h$ (which we do not seek to determine).

Let $n \geq 1$ be an integer. Let $\mathbb{C}[[\mathbb{Z}^n]]$, $\mathcal{R}$, $\mathbb{C}(\mathbb{Z}^n)$, and $\mathrm{I} : \mathcal{R} \rightarrow \mathbb{C}(\mathbb{Z}^n)$ be as defined previously.
Let $u : \mathbb{Z}^n \rightarrow \mathbb{R}$ be an injective group homomorphism. Show that there exists a unique map $s_u : \mathbb{C}(\mathbb{Z}^n) \rightarrow \mathcal{R}$ satisfying the following three conditions:

• [(a)] $s_u(Pf) = P\,s_u(f)$ for all $f \in \mathbb{C}(\mathbb{Z}^n)$ and $P \in \mathbb{C}[\mathbb{Z}^n]$.
• [(b)] $\mathrm{I}(s_u(f)) = f$ for all $f \in \mathbb{C}(\mathbb{Z}^n)$.
• [(c)] $s_u\left(\frac{1}{1-g}\right) = \sum_{n \in \mathbb{N}} g^n$ if $g$ is a finite linear combination of elements of the form $x^\gamma$ with $\gamma \in \mathbb{Z}^n$ satisfying $u(\gamma) > 0$.

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$. Describe the set of minimizers of $f$ on $C$.

Let $n \geq 1$ be an integer. For $A \subset \mathbb{R}^n$, let $E_A = \sum_{\gamma \in A \cap \mathbb{Z}^n} x^\gamma \in \mathbb{C}[[\mathbb{Z}^n]]$.
Show that if $A$ and $B$ are two subsets of $\mathbb{R}^n$ and $\gamma \in \mathbb{Z}^n$ we have $$E_{A \cup B} + E_{A \cap B} = E_A + E_B \quad \text{and} \quad E_{\gamma + A} = x^\gamma E_A.$$

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$, so that $\nabla f(x) = -Mx$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $$x_{n+1} := P_C(x_n - \tau \nabla f(x_n)), \quad \text{with} \quad P_C(x) := \begin{cases} x & \text{if } \|x\| \leq 1, \\ x/\|x\| & \text{otherwise.} \end{cases}$$ Suppose in this question that $\|x_0\| \geq 1$. a) Show that $$\forall n \in \mathbb{N} \setminus \{0\},\, x_n = \frac{(\mathrm{I}_d + \tau M)^n x_0}{\|(\mathrm{I}_d + \tau M)^n x_0\|}$$ b) Calculate $\lim_{n \rightarrow \infty} x_n$. Hint. Decompose $x_0 = \sum_{1 \leq i \leq d} \alpha_i e_i$ in an orthonormal basis of eigenvectors $(e_1, \cdots, e_d)$, associated with the eigenvalues $\lambda_1, \cdots, \lambda_d$ of $M$. Introduce the set of indices $I := \{i \in \llbracket 1, d \rrbracket \mid \alpha_i \neq 0\}$, the eigenvalue $\lambda := \max_{i \in I} \lambda_i$, and the vector $x_0' := \sum_{i \in I'} \alpha_i e_i$ where $I' := \{i \in I \mid \lambda_i = \lambda\}$.

Let $n \geq 1$ be an integer. For $A \subset \mathbb{R}^n$, let $E_A = \sum_{\gamma \in A \cap \mathbb{Z}^n} x^\gamma \in \mathbb{C}[[\mathbb{Z}^n]]$.
Let $\gamma_1, \ldots, \gamma_k$ be a family of vectors in $\mathbb{Z}^n \subset \mathbb{R}^n$ and $$C(\gamma_1, \ldots, \gamma_k) = \left\{\sum_{i=1}^k t_i \gamma_i : (t_1, \ldots, t_k) \in [0, +\infty[^k\right\}.$$ Show that if $\gamma_1, \ldots, \gamma_k$ is a free family, $E_{v + C(\gamma_1, \ldots, \gamma_k)}$ is rational for all $v \in \mathbb{R}^n$.

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$, so that $\nabla f(x) = -Mx$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $$x_{n+1} := P_C(x_n - \tau \nabla f(x_n)), \quad \text{with} \quad P_C(x) := \begin{cases} x & \text{if } \|x\| \leq 1, \\ x/\|x\| & \text{otherwise.} \end{cases}$$ How does the sequence behave when $\|x_0\| < 1$?

Let $n \geq 1$ be an integer. For $A \subset \mathbb{R}^n$, let $E_A = \sum_{\gamma \in A \cap \mathbb{Z}^n} x^\gamma \in \mathbb{C}[[\mathbb{Z}^n]]$. Let $$C(\gamma_1, \ldots, \gamma_k) = \left\{\sum_{i=1}^k t_i \gamma_i : (t_1, \ldots, t_k) \in [0, +\infty[^k\right\}.$$
Generalize the previous question in the case where $\gamma_1, \ldots, \gamma_k \in \mathbb{Z}^n$ is a family of vectors not necessarily free but for which there exists a linear form $\ell : \mathbb{R}^n \rightarrow \mathbb{R}$ such that $\ell(\gamma_i) > 0$ for $i = 1, \ldots, k$.
Hint. One may triangulate the polytope $P = \{x \in C(\gamma_1, \ldots, \gamma_k) : \ell(x) = 1\}$.

We fix an integer $d \in \mathbb{N}^*$, and we equip $\mathbb{R}^d$ with the usual inner product denoted $\langle \cdot, \cdot \rangle$ and the associated Euclidean norm $\|\cdot\|$. We denote $C := \{x \in \mathbb{R}^d \mid \|x\| \leq 1\}$ the closed unit ball of $\mathbb{R}^d$. Let $M$ be a nonzero real symmetric matrix of size $d \times d$ such that $\forall x \in \mathbb{R}^d,\, \langle x, Mx \rangle \geq 0$. We define $f(x) := -\frac{1}{2}\langle x, Mx \rangle$, so that $\nabla f(x) = -Mx$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $$x_{n+1} := P_C(x_n - \tau \nabla f(x_n)), \quad \text{with} \quad P_C(x) := \begin{cases} x & \text{if } \|x\| \leq 1, \\ x/\|x\| & \text{otherwise.} \end{cases}$$ Show that there exists a hyperplane $H \subset \mathbb{R}^d$ such that, for all $x_0 \in \mathbb{R}^d \setminus H$, we have $\lim_{n \rightarrow \infty} f(x_n) = \min\{f(x) \mid x \in C\}$.

In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We denote $Z_\infty = \int_{-\infty}^{+\infty} \exp\left(-\frac{x^4}{12}\right) \mathrm{d}x$ and we consider the function $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
An analogous proof to that of the previous subsection allows us to show that, for any continuous and bounded function $f$ on $\mathbb{R}$, $$E_{n,f} \xrightarrow[n \rightarrow +\infty]{} \int_{-\infty}^{+\infty} f(u) \varphi_\infty(u) \mathrm{d}u$$
Let $K \in \mathbb{R}_+^*$, and let $f$ be a $K$-Lipschitz function and bounded on $\mathbb{R}$. Show that $$\left|E_{n,f} - \mathbb{E}\left(f\left(n^{1/4} M_n\right)\right)\right| \leqslant \frac{2K}{n^{1/4}\sqrt{2\pi}}$$ and deduce that $$\mathbb{E}\left(f\left(n^{1/4} M_n\right)\right) \xrightarrow[n \rightarrow +\infty]{} \int_{-\infty}^{+\infty} f(u) \varphi_\infty(u) \mathrm{d}u$$

In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We are given $x \in \mathbb{R}$ and $\varepsilon > 0$. Let $k$ be a non-zero natural integer such that $k \geqslant \frac{2}{\varepsilon Z_\infty}$. We define the function $$f_k : u \in \mathbb{R} \longmapsto \begin{cases} 1 & \text{if } u \leqslant x \\ 1 - k(u-x) & \text{if } x < u \leqslant x + \frac{1}{k} \\ 0 & \text{otherwise} \end{cases}$$
Show that $f_k$ is $k$-Lipschitz on $\mathbb{R}$.

In this subsection, we still assume that $J_n = J_n^{(\mathrm{C})}$. Moreover, we assume that $\beta = 1$ and $h = 0$.
We denote $Z_\infty = \int_{-\infty}^{+\infty} \exp\left(-\frac{x^4}{12}\right) \mathrm{d}x$ and we consider the function $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
We are given $x \in \mathbb{R}$ and $\varepsilon > 0$. Let $k$ be a non-zero natural integer such that $k \geqslant \frac{2}{\varepsilon Z_\infty}$. We define the function $$f_k : u \in \mathbb{R} \longmapsto \begin{cases} 1 & \text{if } u \leqslant x \\ 1 - k(u-x) & \text{if } x < u \leqslant x + \frac{1}{k} \\ 0 & \text{otherwise} \end{cases}$$
Deduce that there exists $n_0 \in \mathbb{N}$ such that, for all $n \geqslant n_0$, $$\mathbb{P}\left(n^{1/4} M_n \leqslant x\right) \leqslant \frac{\varepsilon}{2} + \int_{-\infty}^{x + \frac{1}{k}} \varphi_\infty(u) \mathrm{d}u$$

1) As / talk 2) However / talking 3) While / to talk 4) Although / talking

18. The meaning of the verse: ``Like a fish that falls into a well, and like autumn that wanders through the garden, seeing'' and ``like a sea that wanders empty and like a barren shore'' --- which of the following verses is most appropriate?

1. [(1)] That the sky is like a deceiving friend that from the arrow it is not shown to us
2. [(2)] That God placed such a nature in the world that people are estranged from it and unhappy
3. [(3)] The camel's ascent was not a flight from the ground the camel's pride, beware of what it is
4. [(4)] From this floor the sky is high and without Kayin's fortress is tall and without

19. Which verse has a different meaning from the other verses?

1. [(1)] Go to the Ka'ba, build a heart in which God himself made that the Ka'ba of the heart is the hidden Ka'ba
2. [(2)] I came from within the Ka'ba, every direction that is calm came to you from the side of others
3. [(3)] The Ka'ba is far from the sanctuary and the shrine is one the sanctuary is close to the heart
4. [(4)] O Lord, this Ka'ba's intended spectacle is what that the marigold of my path is Nasrin

20. The verse: ``The veil is lifted from inside and outside'' --- with which verse does it have the closest meaning?

1. [(1)] Someone whose face the heart saw is the people of reason someone whose stature found life is the people of integrity
2. [(2)] For the wandering mind, the world revolves around that the mirror is attached to you and the mirror is around the world
3. [(3)] Unseen, the witness of my intention is apparent this eye of mine is hidden, watching
4. [(4)] Your imagination, like a candle, came to the sixty inside the house, the candle of life is full

21. Which verse has a different meaning from the other verses?

1. [(1)] If I did not go, at dusk, dawn will go that I look from the front, my view went
2. [(2)] But I am the same first love and more that the mind of everyone becomes the fortune of your will
3. [(3)] That neither love nor loyalty between me and you to the right of love and loyalty between me and you
4. [(4)] Sa'di, to the fortune-tellers, a love is written on the heart outside, we can turn ``Ella'' to the fortune-tellers

22. The meaning of the text: ``It is proper that you help each other to be saved from where we are in it. They commanded him and tamed him.'' --- with all the verses as evidence, the verse \ldots\ldots\ldots\ldots is appropriate.

1. [(1)] The lion's skin was brought to the coat the merchants agreed, peace was offered
2. [(2)] By chance, the greeting of peace was offered to you, the pasture, how the wind of no trace
3. [(3)] A thousand taunts of the enemy to us, no laughter two friends if one does the work together
4. [(4)] Yes, to the agreement of the world, one can take goodness to the agreement of the world's virtue was taken

23. Which verse has a different meaning from the other verses?

1. [(1)] The heart is poorly cooked, less fortune from the company of men the temptation that catches in you is my elixir
2. [(2)] My wish is that it should be on the lip, take this covenant that the drunkards are all still asleep in the dream
3. [(3)] Saqi, in the limit of what is described, the pie that the drunkards are all still asleep in the dream
4. [(4)] My dear, this mind of joy is not my dear, the mind of joy is not listening