grandes-ecoles

Papers (176)
2025
centrale-maths1__official 40 centrale-maths2__official 36 mines-ponts-maths1__mp 17 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 28 mines-ponts-maths2__pc 23 mines-ponts-maths2__psi 25 polytechnique-maths-a__mp 35 polytechnique-maths__fui 9 polytechnique-maths__pc 27 x-ens-maths-a__fui 10 x-ens-maths-a__mp 18 x-ens-maths-b__mp 6 x-ens-maths-c__mp 6 x-ens-maths-d__mp 31 x-ens-maths__pc 27 x-ens-maths__psi 30
2024
centrale-maths1__official 21 centrale-maths2__official 28 geipi-polytech__maths 9 mines-ponts-maths1__mp 23 mines-ponts-maths1__psi 9 mines-ponts-maths2__mp 14 mines-ponts-maths2__pc 19 mines-ponts-maths2__psi 20 polytechnique-maths-a__mp 42 polytechnique-maths-b__mp 27 x-ens-maths-a__mp 43 x-ens-maths-b__mp 29 x-ens-maths-c__mp 22 x-ens-maths-d__mp 41 x-ens-maths__pc 20 x-ens-maths__psi 23
2023
centrale-maths1__official 37 centrale-maths2__official 32 e3a-polytech-maths__mp 4 mines-ponts-maths1__mp 14 mines-ponts-maths1__pc 21 mines-ponts-maths1__psi 21 mines-ponts-maths2__mp 21 mines-ponts-maths2__pc 13 mines-ponts-maths2__psi 22 polytechnique-maths__fui 3 x-ens-maths-a__mp 24 x-ens-maths-b__mp 10 x-ens-maths-c__mp 10 x-ens-maths-d__mp 10 x-ens-maths__pc 22
2022
centrale-maths1__mp 22 centrale-maths1__pc 33 centrale-maths1__psi 42 centrale-maths2__mp 26 centrale-maths2__pc 37 centrale-maths2__psi 40 mines-ponts-maths1__mp 26 mines-ponts-maths1__pc 20 mines-ponts-maths1__psi 23 mines-ponts-maths2__mp 22 mines-ponts-maths2__pc 9 mines-ponts-maths2__psi 18 x-ens-maths-a__mp 8 x-ens-maths-b__mp 19 x-ens-maths-c__mp 17 x-ens-maths-d__mp 47 x-ens-maths1__mp 13 x-ens-maths2__mp 26 x-ens-maths__pc 7 x-ens-maths__pc_cpge 14 x-ens-maths__psi 22 x-ens-maths__psi_cpge 26
2021
centrale-maths1__mp 34 centrale-maths1__pc 36 centrale-maths1__psi 28 centrale-maths2__mp 21 centrale-maths2__pc 38 centrale-maths2__psi 28 x-ens-maths2__mp 35 x-ens-maths__pc 29
2020
centrale-maths1__mp 42 centrale-maths1__pc 36 centrale-maths1__psi 38 centrale-maths2__mp 2 centrale-maths2__pc 35 centrale-maths2__psi 39 mines-ponts-maths1__mp_cpge 22 mines-ponts-maths2__mp_cpge 19 x-ens-maths-a__mp_cpge 10 x-ens-maths-b__mp_cpge 19 x-ens-maths-c__mp 10 x-ens-maths-d__mp 13 x-ens-maths1__mp 13 x-ens-maths2__mp 20 x-ens-maths__pc 6
2019
centrale-maths1__mp 37 centrale-maths1__pc 40 centrale-maths1__psi 38 centrale-maths2__mp 37 centrale-maths2__pc 39 centrale-maths2__psi 46 x-ens-maths1__mp 24 x-ens-maths__pc 18 x-ens-maths__psi 9
2018
centrale-maths1__mp 21 centrale-maths1__pc 31 centrale-maths1__psi 39 centrale-maths2__mp 23 centrale-maths2__pc 35 centrale-maths2__psi 30 x-ens-maths1__mp 18 x-ens-maths2__mp 13 x-ens-maths__pc 17 x-ens-maths__psi 20
2017
centrale-maths1__mp 45 centrale-maths1__pc 22 centrale-maths1__psi 17 centrale-maths2__mp 30 centrale-maths2__pc 28 centrale-maths2__psi 44 x-ens-maths1__mp 24 x-ens-maths2__mp 7 x-ens-maths__pc 17 x-ens-maths__psi 19
2016
centrale-maths1__mp 41 centrale-maths1__pc 31 centrale-maths1__psi 33 centrale-maths2__mp 25 centrale-maths2__pc 42 centrale-maths2__psi 17 x-ens-maths1__mp 10 x-ens-maths2__mp 32 x-ens-maths__pc 1 x-ens-maths__psi 20
2015
centrale-maths1__mp 18 centrale-maths1__pc 11 centrale-maths1__psi 42 centrale-maths2__mp 44 centrale-maths2__pc 1 centrale-maths2__psi 14 x-ens-maths1__mp 16 x-ens-maths2__mp 19 x-ens-maths__pc 30 x-ens-maths__psi 20
2014
centrale-maths1__mp 28 centrale-maths1__pc 26 centrale-maths1__psi 36 centrale-maths2__mp 24 centrale-maths2__pc 23 centrale-maths2__psi 29 x-ens-maths2__mp 13
2013
centrale-maths1__mp 3 centrale-maths1__pc 45 centrale-maths1__psi 20 centrale-maths2__mp 32 centrale-maths2__pc 50 centrale-maths2__psi 32 x-ens-maths1__mp 14 x-ens-maths2__mp 10 x-ens-maths__pc 22 x-ens-maths__psi 9
2012
centrale-maths1__pc 23 centrale-maths1__psi 20 centrale-maths2__mp 27 centrale-maths2__psi 20
2011
centrale-maths1__mp 27 centrale-maths1__pc 15 centrale-maths1__psi 21 centrale-maths2__mp 29 centrale-maths2__pc 8 centrale-maths2__psi 28
2010
centrale-maths1__mp 7 centrale-maths1__pc 23 centrale-maths1__psi 9 centrale-maths2__mp 10 centrale-maths2__pc 36 centrale-maths2__psi 27
2023 mines-ponts-maths2__pc

13 maths questions

Q9 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View
Let $E$ be a Euclidean space of dimension $N$. We denote by $(|)$ the inner product and $\|\cdot\|$ the associated Euclidean norm. Let $u$ be a self-adjoint endomorphism of $E$. We define $q_u : E \rightarrow \mathbf{R}$ by $q_u : x \mapsto (u(x) \mid x)$ and we assume that for all $x \in E$, $q_u(x) \geq 0$. State the spectral theorem for the endomorphism $u$. What can be said about the eigenvalues of $u$?
Q10 Invariant lines and eigenvalues and vectors Eigenvalue interlacing and spectral inequalities View
Let $E$ be a Euclidean space of dimension $N$. Let $u$ be a self-adjoint endomorphism of $E$ such that for all $x \in E$, $q_u(x) = (u(x) \mid x) \geq 0$. We assume that 0 is a simple eigenvalue of $u$ and we denote by $\lambda_2$ the smallest nonzero eigenvalue of $u$. We denote by $p : E \rightarrow E$ the orthogonal projection onto the vector line $\ker(u)$. Show that for all $x \in E$, $q_u(x - p(x)) \geq \lambda_2 \|x - p(x)\|^2$.
Q11 Matrices Eigenvalue and Characteristic Polynomial Analysis View
We consider a Markov kernel $K$. We assume that 1 is a simple eigenvalue of $K$. We assume that there exists a probability $\pi \in \mathscr{M}_{1,N}(\mathbf{R})$ such that:
(a) For all $j \in \llbracket 1;N \rrbracket$, $\pi[j] \neq 0$.
(b) $\forall (i,j) \in \llbracket 1;N \rrbracket^2$, $\pi[i] K[i,j] = K[j,i] \pi[j]$; we say that $K$ is $\pi$-reversible. Show that $\pi K = \pi$.
Q12 Matrices Matrix Algebra and Product Properties View
For $X, Y \in \mathscr{M}_{N,1}(\mathbf{R})^2$, we define $$\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$$ where $\pi \in \mathscr{M}_{1,N}(\mathbf{R})$ is a probability with $\pi[j] \neq 0$ for all $j$. Show that $(X, Y) \mapsto \langle X, Y \rangle$ is an inner product on $\mathscr{M}_{N,1}(\mathbf{R})$.
Q13 Matrices Linear Transformation and Endomorphism Properties View
We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, where $\pi$ is a $\pi$-reversible probability for the Markov kernel $K$. We consider the endomorphism of $E$ defined by $u : X \mapsto (I_N - K)X$. Show that $\ker(u) = \operatorname{Vect}(U)$ and that $u$ is a self-adjoint endomorphism of $E$.
Q14 Matrices Eigenvalue and Characteristic Polynomial Analysis View
We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, and the endomorphism $u : X \mapsto (I_N - K)X$ with $q_u(X) = (u(X) \mid X)$. Show that for all $X \in E$, $$q_u(X) = \frac{1}{2} \sum_{i=1}^{N} \sum_{j=1}^{N} (X[i] - X[j])^2 K[i,j] \pi[i]$$ What can be said about the eigenvalues of $u$?
Q15 Matrices Qualitative and asymptotic analysis of solutions View
We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, and the matrix $H_t$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, H_t[i,j] = e^{-t} \sum_{n=0}^{+\infty} \frac{t^n K^n[i,j]}{n!}$. Let $X \in E$. We denote by $\psi_X$ the function defined from $\mathbf{R}$ to $E$ by $\psi_X : t \mapsto H_t X$ and $\varphi_X$ the function defined from $\mathbf{R}$ to $\mathbf{R}$ by $\varphi_X : t \mapsto \|H_t X\|^2$. Justify that $\psi_X$ is differentiable and that for all $t$ in $\mathbf{R}$, $$\psi_X'(t) = -(I_N - K) H_t X$$
Q16 Matrices Deduction or Consequence from Prior Results View
We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, the endomorphism $u : X \mapsto (I_N - K)X$, and for $X \in E$, the functions $\psi_X : t \mapsto H_t X$ and $\varphi_X : t \mapsto \|H_t X\|^2$. Deduce that $\varphi_X$ is differentiable and express $\varphi_X'(t)$ in terms of $q_u$.
Q17 Matrices Proof of Stability or Invariance View
We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, the endomorphism $u : X \mapsto (I_N - K)X$ with $\ker(u) = \operatorname{Vect}(U)$, and the matrix $H_t$. We denote by $p : E \rightarrow E$ the orthogonal projection onto $\ker(u)$. Let $t \in \mathbf{R}_+$. Show that $p(H_t X) = p(X)$.
Q18 Matrices Derivative of Composite Function in Applied/Modeling Context View
We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, the endomorphism $u : X \mapsto (I_N - K)X$, the orthogonal projection $p : E \rightarrow E$ onto $\ker(u)$, and for $X \in E$, the function $\varphi_X : t \mapsto \|H_t X\|^2$. We set $Y = X - p(X)$. We denote by $\lambda$ the smallest nonzero eigenvalue of $u$. Show that for all real $t \in \mathbf{R}_+$, $\varphi_Y'(t) \leq -2\lambda \varphi_Y(t)$. Deduce that $\forall t \in \mathbf{R}_+, \|H_t X - p(X)\|^2 \leq e^{-2\lambda t} \|X - p(X)\|^2$.
Q19 Matrices Derivative of Composite Function in Applied/Modeling Context View
We consider the Euclidean space $E = \mathscr{M}_{N,1}(\mathbf{R})$ equipped with the inner product $\langle X, Y \rangle = \sum_{i=1}^{N} X[i] Y[i] \pi[i]$, the endomorphism $u : X \mapsto (I_N - K)X$, the orthogonal projection $p : E \rightarrow E$ onto $\ker(u) = \operatorname{Vect}(U)$, and $\lambda$ the smallest nonzero eigenvalue of $u$. We have established that $\forall t \in \mathbf{R}_+, \|H_t X - p(X)\|^2 \leq e^{-2\lambda t} \|X - p(X)\|^2$. Let $i \in \llbracket 1;N \rrbracket$ and $t \in \mathbf{R}_+$. Show that $\|H_t E_i - \pi[i] U\| \leq e^{-\lambda t} \sqrt{\pi[i]}$.
Q20 Matrices Derivative of Composite Function in Applied/Modeling Context View
We consider the matrix $H_t$ defined by $\forall (i,j) \in \llbracket 1;N \rrbracket^2, H_t[i,j] = e^{-t} \sum_{n=0}^{+\infty} \frac{t^n K^n[i,j]}{n!}$, and $\pi$ the stationary probability. Show that for all $(i,j) \in \llbracket 1;N \rrbracket^2$ and all $t \in \mathbf{R}_+$, $$H_t[i,j] - \pi[j] = \sum_{k=1}^{N} \left(H_{t/2}[i,k] - \pi[k]\right)\left(H_{t/2}[k,j] - \pi[j]\right)$$ One may use question 5.
Q21 Matrices Convergence of Expectations or Moments View
We consider the matrix $H_t$, the stationary probability $\pi$, and $\lambda$ the smallest nonzero eigenvalue of $u : X \mapsto (I_N - K)X$. We have established that $\|H_t E_i - \pi[i] U\| \leq e^{-\lambda t} \sqrt{\pi[i]}$ and that $$H_t[i,j] - \pi[j] = \sum_{k=1}^{N} \left(H_{t/2}[i,k] - \pi[k]\right)\left(H_{t/2}[k,j] - \pi[j]\right)$$ Deduce that for all $(i,j) \in \llbracket 1;N \rrbracket^2$ and all $t \in \mathbf{R}_+$, $$\left|H_t[i,j] - \pi[j]\right| \leq e^{-\lambda t} \sqrt{\frac{\pi[j]}{\pi[i]}}$$ Determine $\lim_{t \rightarrow +\infty} H_t[i,j]$.