LFM Pure

grandes-ecoles 2023 Q16 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 $q_u$.

grandes-ecoles 2023 Q16 Expectation and Variance of Sums of Independent Variables View

We use the setup of the third part. For $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we denote $T(u) = (T_i(u))_{1 \leqslant i \leqslant d} \in \mathbb{R}^d$ the vector defined by $$T_i(u) = \operatorname{Var}\left(\langle L_i, u \rangle\right) \quad \text{for } i \in \{1,\ldots,d\}.$$
Show that for all $n \geqslant 0$, $$\mathbb{E}\left(\langle X_n, u \rangle^2\right) = \mathbb{E}\left(\langle X_0, M^n u \rangle^2\right) + \sum_{k=0}^{n-1} \langle x_0 M^k, T\left(M^{n-1-k} u\right) \rangle$$ (with the convention that the sum indexed by $k$ is zero if $n = 0$).

grandes-ecoles 2023 Q17 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 $p(H_t X) = p(X)$.

grandes-ecoles 2023 Q17 Existence Proof View

Using the above, show that there exists a basis $(e_1, e_2)$ of $\mathbb{R}^2$ such that $\|Ax\| = \|x\|_2$ for $x \in \{e_1, e_2\}$.

grandes-ecoles 2023 Q17 Monotonicity and Convergence of Sequences Defined via Expectations View

We use the notations of the previous parts. We assume that there exists an eigenvalue $\lambda > 0$ and an associated eigenvector column $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$: $$Mh = \lambda h,$$ and that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$M_{i,j} \geqslant c\nu_j.$$
Show that there exist $\pi \in \mathscr{P}$ and $h' \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$ and $C > 0$ and $\gamma \in [0,1[$, such that $\pi M = \lambda \pi$ and for all $n \geqslant 0$, $$\sum_{i=1}^{d} \sum_{j=1}^{d} \left| \lambda^{-n} \left(M^n\right)_{i,j} - h_i' \pi_j \right| \leqslant C\gamma^n.$$

grandes-ecoles 2023 Q18 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 $\forall t \in \mathbf{R}_+, \|H_t X - p(X)\|^2 \leq e^{-2\lambda t} \|X - p(X)\|^2$.

grandes-ecoles 2023 Q18 Direct Proof of a Stated Identity or Equality View

Let $T$ be a closed subset of $\mathcal{C}$, such that there exist $x, y \in T$ with $y \notin \{-x, x\}$. We assume that for all $a, b \in T$ with $b \notin \{-a, a\}$, we have that $\dfrac{b-a}{\|b-a\|_2}$ and $\dfrac{b+a}{\|b+a\|_2}$ belong to $T$. Show that $T = \mathcal{C}$.

grandes-ecoles 2023 Q18 Matrix Power Computation and Application View

We use the notations of the previous parts. We assume that there exists an eigenvalue $\lambda > 0$ and an associated eigenvector column $h \in \mathscr{M}_{d,1}\left(\mathbb{R}_{+}^{*}\right)$: $$Mh = \lambda h,$$ and that there exist $\nu \in \mathscr{P}$ and $c > 0$ such that for all $i,j \in \{1,\ldots,d\}$, $$M_{i,j} \geqslant c\nu_j.$$
We assume, in this question only, that $\lambda \in ]0,1[$. Show then that $\mathbb{E}\left(\|X_n\|_1\right)$ tends to $0$ as $n$ tends to infinity and $\mathbb{P}\left(\exists n \geqslant 0 : X_n = 0\right) = 1$. We say that the population becomes extinct almost surely in finite time.

grandes-ecoles 2023 Q19 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 $\|H_t E_i - \pi[i] U\| \leq e^{-\lambda t} \sqrt{\pi[i]}$.

grandes-ecoles 2023 Q19 Proof of Equivalence or Logical Relationship Between Conditions View

Prove Theorem A: Let $\|\cdot\|$ be a norm on the $\mathbb{R}$-vector space $\mathbb{R}^2$. If $$\|x+y\|^2 + \|x-y\|^2 \geq 4$$ for all $x, y \in \mathbb{R}^2$ satisfying $\|x\| = \|y\| = 1$, then $\|\cdot\|$ comes from an inner product on $\mathbb{R}^2$.

grandes-ecoles 2023 Q19 Matrix Norm, Convergence, and Inequality View

We use the notations of the previous parts. For $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we denote $T(u) = (T_i(u))_{1 \leqslant i \leqslant d} \in \mathbb{R}^d$ the vector defined by $$T_i(u) = \operatorname{Var}\left(\langle L_i, u \rangle\right) \quad \text{for } i \in \{1,\ldots,d\}.$$
(a) Show that there exists $c_0 \geqslant 0$ such that for all $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we have $\|T(u)\|_1 \leqslant c_0 \|u\|_2^2$.
(b) Deduce the existence of $c_1 \geqslant 0$ such that for all $u \in \mathscr{M}_{d,1}(\mathbb{R})$, we have $\|T(u)\|_1 \leqslant c_1 \|u\|_1^2$.

grandes-ecoles 2023 Q20 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.

grandes-ecoles 2023 Q20 Ring and Field Structure View

Let $A$ be an algebraic $\mathbb{R}$-algebra without zero divisors. a) Show that $x^2 \in \mathbb{R} + \mathbb{R}x$ for all $x \in A$. b) Show that if $x \in A \setminus \mathbb{R}$, then $\mathbb{R} + \mathbb{R}x$ is an $\mathbb{R}$-algebra isomorphic to $\mathbb{C}$.

grandes-ecoles 2023 Q21 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 $\lim_{t \rightarrow +\infty} H_t[i,j]$.

grandes-ecoles 2023 Q21 Ring and Field Structure View

We assume that $A$ is not isomorphic to one of the algebras $\mathbb{R}$ or $\mathbb{C}$. Show that there exists $i_A \in A$ such that $i_A^2 = -1$.

grandes-ecoles 2023 Q22 Diagonalizability and Similarity View

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Show that $A ^ { - 1 } M$ is similar to a real symmetric matrix.
Hint: You may use question 3.

grandes-ecoles 2023 Q22 Algebra and Subalgebra Proofs View

We fix an element $i_A$ of $A$ such that $i_A^2 = -1$. We denote $U = \mathbb{R} + \mathbb{R}i_A$ and we define the map $$T : A \rightarrow A,\quad T(x) = i_A x i_A.$$ We denote $\mathrm{id} : A \rightarrow A$ the identity map of $A$. a) Show that $T(xy) = -T(x)T(y)$ for all $x, y \in A$. b) Calculate $T^2 = T \circ T$ and deduce that $A = \ker(T - \mathrm{id}) \oplus \ker(T + \mathrm{id})$.

grandes-ecoles 2023 Q23 Diagonalizability determination or proof View

Let $T$ be a delta endomorphism of $\mathbb{K}[X]$. For $n \in \mathbb{N}$, we denote by $T_n$ the restriction of $T$ to $\mathbb{K}_n[X]$.
Show that $T_n$ is an endomorphism of $\mathbb{K}_n[X]$. Is it diagonalizable?

grandes-ecoles 2023 Q23 Algebra and Subalgebra Proofs View

Show that $\ker(T + \mathrm{id}) = U$ and deduce that $\ker(T - \mathrm{id}) \neq \{0\}$.

grandes-ecoles 2023 Q24 Properties and Manipulation of Power Series or Formal Series View

Let $T$ be a delta endomorphism of $\mathbb{K}[X]$. For $n \in \mathbb{N}$, let $T_n$ denote the restriction of $T$ to $\mathbb{K}_n[X]$.
Determine $\operatorname{Im}(T_n)$ in terms of $n \in \mathbb{N}$ and deduce that $T$ is surjective.

grandes-ecoles 2023 Q24 Algebra and Subalgebra Proofs View

We fix $\beta \in \ker(T - \mathrm{id}) \setminus \{0\}$. a) Show that the map $x \mapsto \beta x$ sends $\ker(T - \mathrm{id})$ into $\ker(T + \mathrm{id})$. Deduce that $\beta^2 \in U$ and that $\ker(T - \mathrm{id}) = \beta U$. b) Show that $\beta^2 \in ]-\infty, 0[$. c) Prove Theorem B: An algebraic $\mathbb{R}$-algebra without zero divisors is isomorphic to $\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$.

grandes-ecoles 2023 Q25 Properties and Manipulation of Power Series or Formal Series View

We wish to show that, for every delta endomorphism $Q$, there exists a unique sequence of polynomials $(q_n)_{n \in \mathbb{N}}$ of $\mathbb{K}[X]$ such that:

$q_0 = 1$;
$\forall n \in \mathbb{N}, \deg(q_n) = n$;
$\forall n \in \mathbb{N}^*, q_n(0) = 0$;
$\forall n \in \mathbb{N}^*, Q q_n = q_{n-1}$.

Let $Q$ be a delta endomorphism. Show the existence and uniqueness of the sequence $(q_n)_{n \in \mathbb{N}}$ of polynomials associated with $Q$.

grandes-ecoles 2023 Q26 Algebra and Subalgebra Proofs View

Let $A$ be a $\mathbb{R}$-algebra such that there exists a norm $\|\cdot\|$ on the $\mathbb{R}$-vector space $A$ satisfying $$\forall x, y \in A,\quad \|xy\| = \|x\| \cdot \|y\|.$$ Show that $x^2 \in \mathbb{R} + \mathbb{R}x$ for all $x \in A$. One may use the result from question 25 with $y = 1$.

grandes-ecoles 2023 Q27 Properties and Manipulation of Power Series or Formal Series View

Let $(q_n)_{n \in \mathbb{N}}$ be a sequence of polynomials of $\mathbb{K}[X]$ such that $\forall n \in \mathbb{N}, \deg(q_n) = n$ and $$\forall (x,y) \in \mathbb{K}^2, \quad q_n(x+y) = \sum_{k=0}^n q_k(x) q_{n-k}(y)$$
Show that there exists a unique delta endomorphism $Q$ for which $(q_n)_{n \in \mathbb{N}}$ is the associated sequence of polynomials.

grandes-ecoles 2023 Q29 Determinant and Rank Computation View

Let $Q$ be a delta endomorphism, let $(q_n)_{n \in \mathbb{N}}$ be the sequence of polynomials associated with $Q$, and let $n$ be a natural number. The family $(q_0, q_1, \ldots, q_n)$ is a basis of $\mathbb{K}_n[X]$.
According to question 23, $Q$ induces an endomorphism of $\mathbb{K}_n[X]$ denoted $Q_n$. Give its matrix in the previous basis. Deduce its trace, its determinant and its characteristic polynomial.

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LFM Pure

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Circles 702

Simultaneous equations 134

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Proof by induction 36

Sine and Cosine Rules 185

Radians, Arc Length and Sector Area 22

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Standard trigonometric equations 142

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