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
2022 mines-ponts-maths2__mp

24 maths questions

Q1 Matrices Matrix Algebra and Product Properties View
We are given two matrices $A$ and $B$ in $\mathcal{M}_n(\mathbf{K})$. We assume that $A$ and $B$ commute.
$\mathbf{1}$ ▷ Show that the matrices $A$ and $e^{B}$ commute.
Q3 Matrices Matrix Algebra and Product Properties View
$\mathbf{3}$ ▷ Conversely, suppose the relation $\forall t \in \mathbf{R} \quad e^{t(A+B)} = e^{tA} e^{tB}$ is satisfied. By differentiating this relation twice with respect to the real variable $t$, show that the matrices $A$ and $B$ commute.
Q4 Matrices Matrix Norm, Convergence, and Inequality View
$\mathbf{4}$ ▷ For any matrix $A \in \mathcal{M}_n(\mathbf{K})$, prove the relation $\left\| e^{A} \right\| \leq e^{\|A\|}$.
Q6 Proof Bounding or Estimation Proof View
In this part, we denote by $A$ and $B$ two arbitrary matrices in $\mathcal{M}_n(\mathbf{K})$. For every nonzero natural integer $k$, we set $$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$
$\mathbf{6}$ ▷ Prove the inequalities $$\forall k \in \mathbf{N}^* \quad \left\| X_k \right\| \leq \exp\left(\frac{\|A\| + \|B\|}{k}\right) \text{ and } \left\| Y_k \right\| \leq \exp\left(\frac{\|A\| + \|B\|}{k}\right).$$
Q7 Proof Bounding or Estimation Proof View
In this part, we denote by $A$ and $B$ two arbitrary matrices in $\mathcal{M}_n(\mathbf{K})$. For every nonzero natural integer $k$, we set $$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$
We introduce the function $$\begin{aligned} h : \mathbf{R} & \longrightarrow \mathcal{M}_n(\mathbf{K}) \\ t & \longmapsto h(t) = e^{tA} e^{tB} - e^{t(A+B)} \end{aligned}$$
$\mathbf{7}$ ▷ Show that $$X_k - Y_k = O\left(\frac{1}{k^2}\right) \text{ as } k \rightarrow +\infty.$$
Q8 Matrices Matrix Norm, Convergence, and Inequality View
In this part, we denote by $A$ and $B$ two arbitrary matrices in $\mathcal{M}_n(\mathbf{K})$. For every nonzero natural integer $k$, we set $$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$
The objective is to prove the relation $$\lim_{k \rightarrow +\infty} \left(e^{\frac{A}{k}} e^{\frac{B}{k}}\right)^k = e^{A+B}.$$
$\mathbf{8}$ ▷ Verify the relation $$X_k^k - Y_k^k = \sum_{i=0}^{k-1} X_k^i \left(X_k - Y_k\right) Y_k^{k-i-1}$$ Deduce from this the relation $\lim_{k \rightarrow +\infty} \left(e^{\frac{A}{k}} e^{\frac{B}{k}}\right)^k = e^{A+B}$.
Q9 Matrices Matrix Group and Subgroup Structure View
In this part, $\mathbf{K} = \mathbf{R}$. For every natural integer $n$, $n \geq 2$, we introduce the set, called the special linear group: $$\mathrm{SL}_n(\mathbf{R}) = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \operatorname{det}(M) = 1 \right\}.$$ If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$
$\mathbf{9}$ ▷ Determine $\mathcal{A}_G$ when $G = \mathrm{SL}_n(\mathbf{R})$.
Q10 Matrices Matrix Group and Subgroup Structure View
In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$
$\mathbf{10}$ ▷ If $G = \mathrm{O}_n(\mathbf{R})$, show that $\mathcal{A}_G = \mathcal{A}_n(\mathbf{R})$, the set of antisymmetric matrices.
Q11 Matrices Matrix Group and Subgroup Structure View
In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{11}$ ▷ Using part 2, show that $\mathcal{A}_G$ is a vector subspace of $\mathcal{M}_n(\mathbf{R})$.
Q12 Matrices Matrix Group and Subgroup Structure View
In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{12}$ ▷ Let $A \in \mathcal{A}_G$ and $B \in \mathcal{A}_G$. Show that the application $$\begin{aligned} u : \mathbf{R} & \longrightarrow \mathcal{M}_n(\mathbf{R}) \\ t & \longmapsto u(t) = e^{tA} \cdot B \cdot e^{-tA} \end{aligned}$$ takes values in $\mathcal{A}_G$.
Q13 Groups Algebra and Subalgebra Proofs View
In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ If $A$ and $B$ are two matrices in $\mathcal{M}_n(\mathbf{K})$, their Lie bracket is defined by $[A, B] = AB - BA$. In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{13}$ ▷ Deduce from question 12) that $\mathcal{A}_G$ is stable under the Lie bracket, i.e. $$\forall A \in \mathcal{A}_G, \forall B \in \mathcal{A}_G, [A, B] \in \mathcal{A}_G.$$
Q14 Groups Symplectic and Orthogonal Group Properties View
In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ We recall that, if $M$ is a matrix in $\mathcal{M}_n(\mathbf{R})$, we say that $M$ is tangent to $G$ at $I_n$ if there exist $\varepsilon > 0$ and an application $\left.\gamma : \right]-\varepsilon, \varepsilon[ \rightarrow G$, differentiable, such that $\gamma(0) = I_n$ and $\gamma'(0) = M$. The set of matrices tangent to $G$ at $I_n$ is called the tangent space to $G$ at $I_n$, and is denoted $\mathcal{T}_{I_n}(G)$. In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{14}$ ▷ Prove the inclusion $\mathcal{A}_G \subset \mathcal{T}_{I_n}(G)$.
Q15 Proof Computation of a Limit, Value, or Explicit Formula View
$\mathbf{15}$ ▷ Let $M \in \mathcal{M}_n(\mathbf{R})$, which can also be considered as a complex matrix, and let the application $\delta_M : \mathbf{R} \rightarrow \mathbf{R},\ t \mapsto \delta_M(t) = \operatorname{det}\left(I_n + tM\right)$. Using a Taylor expansion to order 1, show that $\delta_M$ is differentiable at 0 and compute $\delta_M'(0)$.
Q16 Matrices Determinant and Rank Computation View
$\mathbf{16}$ ▷ Show that the differential at the point $I_n$ of the application $\det: \mathcal{M}_n(\mathbf{R}) \rightarrow \mathbf{R}$ is the linear form ``trace''.
Q17 Groups Symplectic and Orthogonal Group Properties View
In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ The tangent space to $G$ at $I_n$ is denoted $\mathcal{T}_{I_n}(G)$.
$\mathbf{17}$ ▷ Show that, in the particular cases $G = \mathrm{SL}_n(\mathbf{R})$ and $G = \mathrm{O}_n(\mathbf{R})$, we have $\mathcal{T}_{I_n}(G) = \mathcal{A}_G$.
Q18 Systems of differential equations View
We consider two distinct complex numbers $\alpha$ and $\beta$. We assume that a matrix $A \in \mathcal{M}_3(\mathbf{C})$ has $\alpha$ as a simple eigenvalue and $\beta$ as a double eigenvalue.
$\mathbf{18}$ ▷ Show that $A$ is similar to a matrix of the form $$T = \left( \begin{array}{ccc} \alpha & 0 & 0 \\ 0 & \beta & a \\ 0 & 0 & \beta \end{array} \right)$$ where $a$ is a certain complex number. Compute $T^n$ for $n$ a natural integer, then $e^{tT}$ for $t$ real. Deduce from this a necessary and sufficient condition on $\alpha$ and $\beta$ for $\lim_{t \rightarrow +\infty} e^{tA} = 0_3$.
Q19 Second order differential equations Qualitative and asymptotic analysis of solutions View
In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$ a square matrix with complex coefficients, and we denote by $u$ the endomorphism of $\mathbf{C}^n$ canonically associated with this matrix. We set $\alpha = \max_{\lambda \in \operatorname{Sp}(A)} \operatorname{Re}(\lambda)$. For all real $t$ and for $(i,j) \in \llbracket 1,n \rrbracket^2$, we denote by $v_{i,j}(t)$ the coefficient with indices $(i,j)$ of the matrix $e^{tA}$.
$\mathbf{19}$ ▷ Show that, if $\lim_{t \rightarrow +\infty} f_A(t) = 0_n$, then $\alpha < 0$.
Q20 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View
In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. For every eigenvalue $\lambda$ of the matrix $A$, we denote by $m_\lambda$ its multiplicity, and we introduce the vector subspace $$F_\lambda = \operatorname{Ker}\left((A - \lambda I_n)^{m_\lambda}\right) = \operatorname{Ker}\left((u - \lambda \operatorname{Id}_E)^{m_\lambda}\right).$$
$\mathbf{20}$ ▷ Show that $\mathbf{C}^n = \bigoplus_{\lambda \in \operatorname{Sp}(A)} F_\lambda$.
Q21 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning View
In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. For every eigenvalue $\lambda$ of the matrix $A$, we denote by $m_\lambda$ its multiplicity, and we introduce the vector subspace $$F_\lambda = \operatorname{Ker}\left((A - \lambda I_n)^{m_\lambda}\right).$$
$\mathbf{21}$ ▷ Deduce from question 20) the existence of three matrices $P, D$ and $N$ in $\mathcal{M}_n(\mathbf{C})$ such that $A = P(D + N)P^{-1}$, where $D$ is diagonal, $N$ is nilpotent, $DN = ND$, and $P$ is invertible.
Q22 Second order differential equations Qualitative and asymptotic analysis of solutions View
In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. We set $\alpha = \max_{\lambda \in \operatorname{Sp}(A)} \operatorname{Re}(\lambda)$. For all real $t$ and for $(i,j) \in \llbracket 1,n \rrbracket^2$, we denote by $v_{i,j}(t)$ the coefficient with indices $(i,j)$ of the matrix $e^{tA}$.
$\mathbf{22}$ ▷ Deduce from question 21) that there exists a natural integer $p$ such that, for all $(i,j) \in \llbracket 1,n \rrbracket^2$, we have $$v_{i,j}(t) = O\left(t^p e^{\alpha t}\right) \text{ as } t \rightarrow +\infty.$$
Q23 Second order differential equations Qualitative and asymptotic analysis of solutions View
In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. We set $\alpha = \max_{\lambda \in \operatorname{Sp}(A)} \operatorname{Re}(\lambda)$.
$\mathbf{23}$ ▷ Study the converse of question 19): that is, show that if $\alpha < 0$ then $\lim_{t \rightarrow +\infty} f_A(t) = 0_n$.
Q24 Invariant lines and eigenvalues and vectors Properties of eigenvalues under matrix operations View
In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$.
$\mathbf{24}$ ▷ We assume, in this question only, that all eigenvalues of the matrix $A$ have real parts that are positive or zero. Show that, if $X \in \mathbf{C}^n$, we have $$\lim_{t \rightarrow +\infty} e^{tA} X = 0 \Longleftrightarrow X = 0.$$
Q25 Invariant lines and eigenvalues and vectors Invariant subspaces and stable subspace analysis View
In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. We introduce the following polynomials: $$\begin{aligned} & P_s(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) < 0}} (X - \lambda)^{m_\lambda}, \\ & P_i(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) > 0}} (X - \lambda)^{m_\lambda}, \\ & P_n(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) = 0}} (X - \lambda)^{m_\lambda}, \end{aligned}$$ and the subspaces $E_s = \operatorname{Ker}(P_s(A))$, $E_i = \operatorname{Ker}(P_i(A))$ and $E_n = \operatorname{Ker}(P_n(A))$ of $E = \mathbf{C}^n$.
$\mathbf{25}$ ▷ After justifying that $E = E_s \oplus E_i \oplus E_n$, show that $$E_s = \left\{ X \in E \mid \lim_{t \rightarrow +\infty} e^{tA} X = 0 \right\}.$$
Q26 Second order differential equations Qualitative and asymptotic analysis of solutions View
In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$. We introduce the following polynomials: $$\begin{aligned} & P_s(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) < 0}} (X - \lambda)^{m_\lambda}, \\ & P_i(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) > 0}} (X - \lambda)^{m_\lambda}, \\ & P_n(X) = \prod_{\substack{\lambda \in \operatorname{Sp}(A) \\ \operatorname{Re}(\lambda) = 0}} (X - \lambda)^{m_\lambda}, \end{aligned}$$ and the subspaces $E_s = \operatorname{Ker}(P_s(A))$, $E_i = \operatorname{Ker}(P_i(A))$ and $E_n = \operatorname{Ker}(P_n(A))$ of $E = \mathbf{C}^n$. We have $E = E_s \oplus E_i \oplus E_n$.
$\mathbf{26}$ ▷ Show that $$E_n = \left\{ X \in E \mid \exists C \in \mathbf{R}_+^* \quad \exists p \in \mathbf{N} \quad \forall t \in \mathbf{R} \quad \left\| e^{tA} X \right\|_E \leq C(1 + |t|)^p \right\}.$$