grandes-ecoles

Papers (191)
2025
centrale-maths1__official 40 centrale-maths2__official 42 mines-ponts-maths1__mp 20 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 24 mines-ponts-maths2__psi 26 polytechnique-maths-a__mp 27 polytechnique-maths__fui 16 polytechnique-maths__pc 27 x-ens-maths-a__mp 18 x-ens-maths-c__mp 9 x-ens-maths-d__mp 38 x-ens-maths__pc 27 x-ens-maths__psi 38
2024
centrale-maths1__official 28 centrale-maths2__official 29 geipi-polytech__maths 9 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 19 mines-ponts-maths2__mp 23 mines-ponts-maths2__pc 21 mines-ponts-maths2__psi 21 polytechnique-maths-a__mp 44 polytechnique-maths-b__mp 37 x-ens-maths-a__mp 43 x-ens-maths-b__mp 35 x-ens-maths-c__mp 22 x-ens-maths-d__mp 45 x-ens-maths__pc 24 x-ens-maths__psi 26
2023
centrale-maths1__official 44 centrale-maths2__official 33 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 15 mines-ponts-maths1__pc 23 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 18 mines-ponts-maths2__psi 22 polytechnique-maths__fui 23 x-ens-maths-a__mp 25 x-ens-maths-b__mp 24 x-ens-maths-c__mp 20 x-ens-maths-d__mp 20 x-ens-maths__pc 18 x-ens-maths__psi 15
2022
centrale-maths1__mp 48 centrale-maths1__official 48 centrale-maths1__pc 37 centrale-maths1__psi 43 centrale-maths2__mp 32 centrale-maths2__official 32 centrale-maths2__pc 39 centrale-maths2__psi 45 mines-ponts-maths1__mp 25 mines-ponts-maths1__pc 24 mines-ponts-maths1__psi 24 mines-ponts-maths2__mp 24 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 x-ens-maths-a__mp 13 x-ens-maths-b__mp 40 x-ens-maths-c__mp 27 x-ens-maths-d__mp 46 x-ens-maths1__mp 13 x-ens-maths2__mp 40 x-ens-maths__pc 15 x-ens-maths__pc_cpge 15 x-ens-maths__psi 22 x-ens-maths__psi_cpge 23
2021
centrale-maths1__mp 40 centrale-maths1__official 40 centrale-maths1__pc 36 centrale-maths1__psi 29 centrale-maths2__mp 30 centrale-maths2__official 29 centrale-maths2__pc 38 centrale-maths2__psi 37 x-ens-maths2__mp 39 x-ens-maths__pc 44
2020
centrale-maths1__mp 42 centrale-maths1__official 42 centrale-maths1__pc 36 centrale-maths1__psi 40 centrale-maths2__mp 38 centrale-maths2__official 38 centrale-maths2__pc 40 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 24 mines-ponts-maths2__mp_cpge 21 x-ens-maths-a__mp_cpge 18 x-ens-maths-b__mp_cpge 20 x-ens-maths-d__mp 14 x-ens-maths1__mp 18 x-ens-maths2__mp 20 x-ens-maths__pc 18
2019
centrale-maths1__mp 37 centrale-maths1__official 37 centrale-maths1__pc 40 centrale-maths1__psi 39 centrale-maths2__mp 37 centrale-maths2__official 37 centrale-maths2__pc 39 centrale-maths2__psi 49 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 26
2018
centrale-maths1__mp 47 centrale-maths1__official 47 centrale-maths1__pc 41 centrale-maths1__psi 44 centrale-maths2__mp 44 centrale-maths2__official 44 centrale-maths2__pc 35 centrale-maths2__psi 38 x-ens-maths1__mp 19 x-ens-maths2__mp 17 x-ens-maths__pc 22 x-ens-maths__psi 24
2017
centrale-maths1__mp 45 centrale-maths1__official 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__official 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 26 x-ens-maths2__mp 16 x-ens-maths__pc 18 x-ens-maths__psi 26
2016
centrale-maths1__mp 42 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 47 centrale-maths2__psi 27 x-ens-maths1__mp 18 x-ens-maths2__mp 46 x-ens-maths__pc 15 x-ens-maths__psi 20
2015
centrale-maths1__mp 42 centrale-maths1__pc 18 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 18 centrale-maths2__psi 33 x-ens-maths1__mp 16 x-ens-maths2__mp 31 x-ens-maths__pc 30 x-ens-maths__psi 22
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 27 centrale-maths2__mp 24 centrale-maths2__pc 26 centrale-maths2__psi 27 x-ens-maths1__mp 9 x-ens-maths2__mp 16 x-ens-maths__pc 4 x-ens-maths__psi 24
2013
centrale-maths1__mp 22 centrale-maths1__pc 45 centrale-maths1__psi 29 centrale-maths2__mp 31 centrale-maths2__pc 52 centrale-maths2__psi 32 x-ens-maths1__mp 24 x-ens-maths2__mp 35 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__mp 36 centrale-maths1__pc 28 centrale-maths1__psi 33 centrale-maths2__mp 27 centrale-maths2__psi 18
2011
centrale-maths1__mp 27 centrale-maths1__pc 17 centrale-maths1__psi 24 centrale-maths2__mp 29 centrale-maths2__pc 17 centrale-maths2__psi 10
2010
centrale-maths1__mp 19 centrale-maths1__pc 30 centrale-maths1__psi 13 centrale-maths2__mp 32 centrale-maths2__pc 37 centrale-maths2__psi 27
2025 centrale-maths2__official

42 maths questions

Q1 Matrices Diagonalizability and Similarity View
Explain why the matrix $J_n$ is diagonalizable.
Q2 Matrices Eigenvalue and Characteristic Polynomial Analysis View
We denote by $\lambda_{\text{max}}$ the largest of the eigenvalues of $J_n$ and $\lambda_{\text{min}}$ the smallest. Show that $$\forall x \in \Lambda_n, \quad n\lambda_{\min} \leqslant \sum_{1 \leqslant i,j \leqslant n} J_n(i,j) x_i x_j \leqslant n\lambda_{\max}$$
Q4 Matrices Matrix Norm, Convergence, and Inequality View
Give a bound for $H_n$ in the case where $J_n$ is moreover an orthogonal matrix distinct from $\pm I_n$.
Q5 Matrices Eigenvalue and Characteristic Polynomial Analysis View
We denote by $J_n^{(\mathrm{C})}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ whose coefficients all equal $\frac{1}{n}$, except for its diagonal coefficients, which are zero. Each particle thus interacts in the same way with all other particles.
Determine the spectrum of $U_n = nJ_n^{(\mathrm{C})} + I_n$, then that of $J_n^{(\mathrm{C})}$.
Q6 Addition & Double Angle Formulae Trigonometric Identity Proof or Derivation View
We denote by $J_n^{(\mathrm{s})}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad J_n^{(\mathrm{S})}(i,j) = \frac{2}{\sqrt{2n+1}} \sin\left(\frac{2\pi ij}{2n+1}\right).$$
Show that, for all $p \in \mathbb{N}^*$ and $x \in \mathbb{R} \backslash \pi\mathbb{Z}$, $$\sum_{k=1}^{p} \cos(2kx) = \frac{1}{2}\left(\frac{\sin((2p+1)x)}{\sin(x)} - 1\right)$$
Q7 Matrices Structured Matrix Characterization View
We denote by $J_n^{(\mathrm{s})}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad J_n^{(\mathrm{S})}(i,j) = \frac{2}{\sqrt{2n+1}} \sin\left(\frac{2\pi ij}{2n+1}\right).$$
Deduce that $J_n^{(\mathrm{s})}$ is a symmetric orthogonal matrix.
Q8 Matrices Structured Matrix Characterization View
For all $k \in \llbracket 1,n \rrbracket$, we denote by $C_{n,k}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad C_{n,k}(i,j) = \begin{cases} 1 & \text{if } (i \in \llbracket 1,k \rrbracket \text{ and } j = i+n-k) \text{ or } (i \in \llbracket k+1,n \rrbracket \text{ and } j = i-k) \\ 0 & \text{otherwise} \end{cases}$$ We note that $C_{n,n} = I_n$. We set $J_n^{(1)} = C_{n,1} + C_{n,n-1}$.
Verify that, in the case where $n = 9$, $J_n^{(1)}$ is the matrix such that, for all $(i,j) \in \llbracket 1,9 \rrbracket^2$, the coefficient with index $(i,j)$ equals 1 if the vertices $i$ and $j$ of the graph are connected by an edge and equals 0 otherwise. This means that each particle interacts only with its two nearest neighbors.
Q10 Matrices Matrix Power Computation and Application View
For all $k \in \llbracket 1,n \rrbracket$, we denote by $C_{n,k}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad C_{n,k}(i,j) = \begin{cases} 1 & \text{if } (i \in \llbracket 1,k \rrbracket \text{ and } j = i+n-k) \text{ or } (i \in \llbracket k+1,n \rrbracket \text{ and } j = i-k) \\ 0 & \text{otherwise} \end{cases}$$ We note that $C_{n,n} = I_n$. We set $J_n^{(1)} = C_{n,1} + C_{n,n-1}$.
Show that, for all $k \in \llbracket 1,n \rrbracket$, $C_{n,1}^k = C_{n,k}$.
Q11 Invariant lines and eigenvalues and vectors Annihilating or minimal polynomial and spectral deductions View
For all $k \in \llbracket 1,n \rrbracket$, we denote by $C_{n,k}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad C_{n,k}(i,j) = \begin{cases} 1 & \text{if } (i \in \llbracket 1,k \rrbracket \text{ and } j = i+n-k) \text{ or } (i \in \llbracket k+1,n \rrbracket \text{ and } j = i-k) \\ 0 & \text{otherwise} \end{cases}$$ We note that $C_{n,n} = I_n$. We set $J_n^{(1)} = C_{n,1} + C_{n,n-1}$.
Deduce an annihilating polynomial of $C_{n,1}$, then its spectrum.
Q12 Invariant lines and eigenvalues and vectors Spectral properties of structured or special matrices View
For all $k \in \llbracket 1,n \rrbracket$, we denote by $C_{n,k}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad C_{n,k}(i,j) = \begin{cases} 1 & \text{if } (i \in \llbracket 1,k \rrbracket \text{ and } j = i+n-k) \text{ or } (i \in \llbracket k+1,n \rrbracket \text{ and } j = i-k) \\ 0 & \text{otherwise} \end{cases}$$ We note that $C_{n,n} = I_n$. We set $J_n^{(1)} = C_{n,1} + C_{n,n-1}$.
Deduce that $J_n^{(1)}$ admits the following eigenvalues, enumerated with their multiplicity: $$\lambda_k = 2\cos\left(\frac{2\pi k}{n}\right), \quad k \in \llbracket 0, n-1 \rrbracket.$$
Q13 Matrices Matrix Algebra and Product Properties View
Let $(u,v,r,s) \in (\mathbb{N}^*)^4$. Let $A = (a_{ij})_{\substack{1 \leqslant i \leqslant u \\ 1 \leqslant j \leqslant v}} \in \mathcal{M}_{u,v}(\mathbb{R})$ and $B \in \mathcal{M}_{r,s}(\mathbb{R})$. We define the Kronecker product of $A$ by $B$, and we denote $A \otimes B$, the matrix of $\mathcal{M}_{ur,vs}(\mathbb{R})$ which is defined by $uv$ blocks of size $r \times s$ in such a way that, for all $(i,j) \in \llbracket 1,u \rrbracket \times \llbracket 1,v \rrbracket$, the block with index $(i,j)$ is $a_{i,j}B$.
Besides $u,v,r$ and $s$, we are also given two non-zero natural integers, $w$ and $t$.
Show that $\otimes$ is a bilinear map from $\mathcal{M}_{u,v}(\mathbb{R}) \times \mathcal{M}_{r,s}(\mathbb{R})$ to $\mathcal{M}_{ur,vs}(\mathbb{R})$.
Q14 Matrices Matrix Algebra and Product Properties View
Let $(u,v,r,s) \in (\mathbb{N}^*)^4$. Let $A = (a_{ij})_{\substack{1 \leqslant i \leqslant u \\ 1 \leqslant j \leqslant v}} \in \mathcal{M}_{u,v}(\mathbb{R})$ and $B \in \mathcal{M}_{r,s}(\mathbb{R})$. We define the Kronecker product of $A$ by $B$, and we denote $A \otimes B$, the matrix of $\mathcal{M}_{ur,vs}(\mathbb{R})$ which is defined by $uv$ blocks of size $r \times s$ in such a way that, for all $(i,j) \in \llbracket 1,u \rrbracket \times \llbracket 1,v \rrbracket$, the block with index $(i,j)$ is $a_{i,j}B$.
Besides $u,v,r$ and $s$, we are also given two non-zero natural integers, $w$ and $t$.
Show that, for all matrices $A \in \mathcal{M}_{u,v}(\mathbb{R})$, $A' \in \mathcal{M}_{v,w}(\mathbb{R})$, $B \in \mathcal{M}_{r,s}(\mathbb{R})$ and $B' \in \mathcal{M}_{s,t}(\mathbb{R})$, $$(A \otimes B)(A' \otimes B') = (AA') \otimes (BB').$$
Q15 Matrices Diagonalizability and Similarity View
Let $(u,v,r,s) \in (\mathbb{N}^*)^4$. Let $A = (a_{ij})_{\substack{1 \leqslant i \leqslant u \\ 1 \leqslant j \leqslant v}} \in \mathcal{M}_{u,v}(\mathbb{R})$ and $B \in \mathcal{M}_{r,s}(\mathbb{R})$. We define the Kronecker product of $A$ by $B$, and we denote $A \otimes B$, the matrix of $\mathcal{M}_{ur,vs}(\mathbb{R})$ which is defined by $uv$ blocks of size $r \times s$ in such a way that, for all $(i,j) \in \llbracket 1,u \rrbracket \times \llbracket 1,v \rrbracket$, the block with index $(i,j)$ is $a_{i,j}B$.
Show that if $A \in \mathcal{M}_u(\mathbb{R})$ and $B \in \mathcal{M}_r(\mathbb{R})$ are diagonalizable, then $A \otimes B$ is diagonalizable and $$\operatorname{Sp}(A \otimes B) = \{\lambda\mu \mid \lambda \in \operatorname{Sp}(A), \mu \in \operatorname{Sp}(B)\}.$$ One may start by determining the inverse of the Kronecker product of two invertible matrices.
Q16 Matrices Matrix Entry and Coefficient Identities View
We set $N = n^2$ and $$J_N^{(2)} = I_n \otimes J_n^{(1)} + J_n^{(1)} \otimes I_n \in \mathcal{M}_N(\mathbb{R})$$
Verify that, in the case where $n = 3$, $J_N^{(2)}$ is the matrix such that, for all $(i,j) \in \llbracket 1,9 \rrbracket^2$, the coefficient with index $(i,j)$ equals 1 if the vertices $i$ and $j$ of the graph are connected by an edge (the edges count whether they are dashed or solid), and equals 0 otherwise.
Q17 Matrices Eigenvalue and Characteristic Polynomial Analysis View
We set $N = n^2$ and $$J_N^{(2)} = I_n \otimes J_n^{(1)} + J_n^{(1)} \otimes I_n \in \mathcal{M}_N(\mathbb{R})$$
Show that the eigenvalues of $J_N^{(2)}$ are the $\lambda_j + \lambda_k$, for $(j,k) \in \llbracket 1,n \rrbracket^2$.
Q18 Differentiating Transcendental Functions Differentiation under the integral sign with transcendental kernels View
Justify that the function $\psi_n$ is differentiable on $\mathbb{R}_+$ and that $\psi_n' = m_n$.
We admit that, when $\psi$ is differentiable on $\mathbb{R}_+^*$, then $(\lim \psi_n)' = \lim \psi_n'$, that is $\psi' = m$, on $\mathbb{R}_+^*$.
Q19 Discrete Probability Distributions Deriving or Identifying a Probability Distribution from a Random Process View
In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV. We adopt the following convention: for all $x = (x_1, \ldots, x_n) \in \Lambda_n$, we denote $x_{n+1} = x_1$ and $x_0 = x_n$.
Verify that $$Z_n(h) = \sum_{x \in \Lambda_n} \prod_{i=1}^n \mathrm{e}^{\beta x_i x_{i+1} + h x_i}$$
Q20 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View
Show that, for all natural integers $p$ and $q$ greater than or equal to 2, for any matrix $M = (M(i,j))_{1 \leqslant i,j \leqslant q} \in \mathcal{M}_q(\mathbb{R})$ and for all $(i,j) \in \llbracket 1,q \rrbracket^2$, the coefficient with index $(i,j)$ of the matrix $M^p$ is $$\sum_{(k_2,\ldots,k_p) \in \llbracket 1,q \rrbracket^{p-1}} M(i,k_2)\left(\prod_{r=2}^{p-1} M(k_r, k_{r+1})\right) M(k_p, j),$$ the product being equal to 1 in the case where $p = 2$.
Q21 Matrices Matrix Power Computation and Application View
In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV. We adopt the following convention: for all $x = (x_1, \ldots, x_n) \in \Lambda_n$, we denote $x_{n+1} = x_1$ and $x_0 = x_n$.
We set $A = \begin{pmatrix} \mathrm{e}^{\beta - h} & \mathrm{e}^{-\beta - h} \\ \mathrm{e}^{-\beta + h} & \mathrm{e}^{\beta + h} \end{pmatrix}$.
Show that $Z_n(h) = \operatorname{tr}(A^n)$, where $\operatorname{tr}$ denotes the trace of a square matrix.
Q22 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View
In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV.
We set $A = \begin{pmatrix} \mathrm{e}^{\beta - h} & \mathrm{e}^{-\beta - h} \\ \mathrm{e}^{-\beta + h} & \mathrm{e}^{\beta + h} \end{pmatrix}$.
Determine the eigenvalues of the matrix $A$.
Q23 Invariant lines and eigenvalues and vectors Properties of eigenvalues under matrix operations View
In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV.
We set $A = \begin{pmatrix} \mathrm{e}^{\beta - h} & \mathrm{e}^{-\beta - h} \\ \mathrm{e}^{-\beta + h} & \mathrm{e}^{\beta + h} \end{pmatrix}$.
Show then that $$\psi_n(h) \underset{n \rightarrow +\infty}{\longrightarrow} \ln\left(\mathrm{e}^{\beta} \operatorname{ch}(h) + \sqrt{\mathrm{e}^{2\beta} \operatorname{ch}^2(h) - 2\operatorname{sh}(2\beta)}\right).$$
Q24 Differentiating Transcendental Functions Compute derivative of transcendental function View
In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV.
Deduce an expression for the function $m$ and conclude that $m^+ = 0$.
Recall that $m = \psi'$ when $\psi$ is differentiable on $\mathbb{R}_+^*$.
Q25 Matrices Matrix Entry and Coefficient Identities View
In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
For all $x \in \Lambda_n$, we set $s_n(x) = \sum_{k=1}^n x_i$. Verify that $$Z_n(h) = \mathrm{e}^{-\frac{\beta}{2}} \sum_{x \in \Lambda_n} \exp\left(\frac{\beta}{2n}\left(s_n(x)\right)^2 + h s_n(x)\right).$$
Q26 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View
Show that $\int_{-\infty}^{+\infty} \mathrm{e}^{-\frac{x^2}{2}} \mathrm{~d}x$ converges. We admit that its value is $\sqrt{2\pi}$.
Q27 Integration by Substitution Substitution to Prove an Integral Identity or Equality View
Show that $$\forall u \in \mathbb{R}, \quad \forall a \in \mathbb{R}_+^*, \quad \mathrm{e}^{\frac{au^2}{2}} = \int_{-\infty}^{+\infty} \mathrm{e}^{ut - \frac{t^2}{2a}} \frac{\mathrm{~d}t}{\sqrt{2\pi a}}$$
Q28 Integration by Substitution Substitution within a Multi-Part Proof or Derivation View
In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
Deduce that $$Z_n(h) = \sqrt{\frac{n}{2\mathrm{e}^{\beta}\pi\beta}} \int_{-\infty}^{+\infty} \left(\sum_{x \in \Lambda_n} \mathrm{e}^{(t+h)s_n(x)}\right) \mathrm{e}^{-\frac{nt^2}{2\beta}} \mathrm{~d}t$$
Q29 Sequences and Series Functional Equations and Identities via Series View
Show that, for all $t \in \mathbb{R}$, $$\sum_{x \in \Lambda_n} \prod_{i=1}^n \mathrm{e}^{(t+h)x_i} = (2\operatorname{ch}(t+h))^n$$
Q30 Integration by Substitution Substitution within a Multi-Part Proof or Derivation View
In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
Show that $$Z_n(h) = \sqrt{\frac{n}{2\mathrm{e}^{\beta}\pi\beta}} \int_{-\infty}^{+\infty} \mathrm{e}^{-nG_h(x)} \mathrm{d}x$$
Q31 Stationary points and optimisation Find critical points and classify extrema of a given function View
In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
By distinguishing cases according to whether $\beta \leqslant 1$ or $\beta > 1$, show that $G_h$ admits a minimum on $\mathbb{R}_+$ attained at a unique point $u_h$. Show moreover that
  • [(a)] if $\beta \leqslant 1$ and $h = 0$, then $u_h = 0$;
  • [(b)] if $\beta > 1$, then $u_h > 0$;
  • [(c)] for any $\beta \in \mathbb{R}_+^*$,
    • [(i)] $G_h'(u_h) = 0$;
    • [(ii)] $h = \beta G_0'(u_h)$;
    • [(iii)] $G_h''(u_h) > 0$ when $h > 0$.
Q32 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
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 $h > 0$. Justify that $G_h$ attains its minimum on $\mathbb{R}$ at a unique point which is $u_h$.
Q33 Integration by Substitution Substitution within a Multi-Part Proof or Derivation View
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 denote $\widehat{G}_h : x \longmapsto G_h(x + u_h) - \min G_h$. Show that $$\psi_n(h) = -G_h(u_h) - \frac{1}{2n}\ln\left(2\mathrm{e}^{\beta}\pi\beta\right) + \frac{1}{n}\ln\left(\int_{-\infty}^{+\infty} \mathrm{e}^{-n\widehat{G}_h\left(\frac{t}{\sqrt{n}}\right)} \mathrm{d}t\right)$$
Q34 Proof Proof That a Map Has a Specific Property View
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 $f_h : x \longmapsto \frac{\widehat{G}_h(x)}{x^2}$. Show that $f_h$ can be extended to a continuous function on all of $\mathbb{R}$ by setting $f_h(0) = \frac{\gamma_h}{2}$.
Q35 Proof Bounding or Estimation Proof View
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).
Q36 Proof Computation of a Limit, Value, or Explicit Formula View
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 $f_h : x \longmapsto \frac{\widehat{G}_h(x)}{x^2}$.
Show then that $$\int_{-\infty}^{+\infty} \mathrm{e}^{-n\widehat{G}_h\left(\frac{t}{\sqrt{n}}\right)} \mathrm{d}t \xrightarrow[n \rightarrow +\infty]{} \sqrt{\frac{2\pi}{\gamma_h}}$$ then conclude that $\psi(h) = -G_h(u_h)$.
Q37 Proof Proof That a Map Has a Specific Property View
In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
Show that $G_0'$ establishes a continuous bijection from $[u_0; +\infty[$ to $\mathbb{R}_+$. Deduce that the function $u : h \longmapsto u_h$ is continuous on $\mathbb{R}_+$ and differentiable on $\mathbb{R}_+^*$.
Q38 Proof Deduction or Consequence from Prior Results View
In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
We set $G_h : x \longmapsto \frac{(x-h)^2}{2\beta} - \ln(2\operatorname{ch}(x))$.
Deduce that $\psi$ is differentiable on $\mathbb{R}_+^*$ and that $m(h) = \frac{u(h) - h}{\beta}$ for all $h \in \mathbb{R}_+^*$.
Q39 Proof Characterization or Determination of a Set or Class View
In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
Show then that $m^+ = 0$ if $\beta \leqslant 1$, and $m^+ > 0$ if $\beta > 1$.
Q40 Proof Existence Proof View
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 $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
Justify that, for any continuous and bounded function $f$ on $\mathbb{R}$, the quantity $$E_{n,f} = \int_{-\infty}^{+\infty} \mathbb{E}\left(f\left(\frac{t}{n^{1/4}} + n^{1/4} M_n\right)\right) \exp\left(-\frac{t^2}{2}\right) \frac{\mathrm{d}t}{\sqrt{2\pi}}$$ is well defined.
Q41 Proof Bounding or Estimation Proof View
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 $$\mathbb{E}\left(f\left(n^{1/4} M_n\right)\right) \xrightarrow[n \rightarrow +\infty]{} \int_{-\infty}^{+\infty} f(u) \varphi_\infty(u) \mathrm{d}u$$
Q42 Proof Proof That a Map Has a Specific Property View
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 $\mathbb{R}$.
Q43 Proof Deduction or Consequence from Prior Results View
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 $n \geqslant n_0$, $$\mathbb{P}\left(n^{1/4} M_n \leqslant x\right) \leqslant \frac{\varepsilon}{2} + \int_{-\infty}^{x + \frac{1}{k}} \varphi_\infty(u) \mathrm{d}u$$
Q44 Proof Computation of a Limit, Value, or Explicit Formula View
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 $$\varphi_\infty : x \longmapsto \frac{1}{Z_\infty} \exp\left(-\frac{x^4}{12}\right).$$
Show finally that $$\mathbb{P}\left(n^{1/4} M_n \leqslant x\right) \underset{n \rightarrow +\infty}{\longrightarrow} \int_{-\infty}^{x} \varphi_\infty(u) \mathrm{d}u$$