LFM Pure

grandes-ecoles 2022 Q5 True/False Justification View

Let $F$ be a vector subspace of a symplectic space $(E , \omega)$, and let $F^{\omega} = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$. Is the subspace $F ^ { \omega }$ necessarily in direct sum with $F$?

grandes-ecoles 2022 Q5 True/False Justification View

Let $F$ be a vector subspace of a symplectic space $(E , \omega)$, with $\omega$-orthogonal $F^{\omega} = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$. Is the subspace $F ^ { \omega }$ necessarily in direct sum with $F$?

grandes-ecoles 2022 Q5 Direct Proof of an Inequality View

We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. Conclude the case $K = 2$ by showing the interpolation inequality $$\forall f \in \mathcal{C}^{2}([0,1]), \quad \max\left(\|f\|_{\infty}, \left\|f^{\prime}\right\|_{\infty}\right) \leqslant \left\|f^{\prime\prime}\right\|_{\infty} + C\left(\left|f\left(x_{1}\right)\right| + \left|f\left(x_{2}\right)\right|\right)$$ with $C = 1 + \frac{1}{x_{2} - x_{1}}$.

grandes-ecoles 2022 Q5 Direct Proof of an Inequality View

We fix two distinct real numbers $x_1 < x_2$ in $[0,1]$. Conclude the case $K = 2$ by showing the interpolation inequality $$\forall f \in \mathcal{C}^2([0,1]), \quad \max\left(\|f\|_\infty, \left\|f^\prime\right\|_\infty\right) \leqslant \left\|f^{\prime\prime}\right\|_\infty + C\left(\left|f\left(x_1\right)\right| + \left|f\left(x_2\right)\right|\right)$$ with $C = 1 + \frac{1}{x_2 - x_1}$.

grandes-ecoles 2022 Q5 Deduction or Consequence from Prior Results View

Deduce that, for all $P \in \mathbb { R } _ { n - 1 } [ X ]$, $$P = \sum _ { i = 1 } ^ { n } P \left( a _ { i } \right) L _ { i }.$$

grandes-ecoles 2022 Q5 Proof of Set Membership, Containment, or Structural Property View

For all $x \in \mathbb { R } _ { + } ^ { * }$ and all $t \in \mathbb { R } _ { + } ^ { * }$, we denote $k _ { x } ( t ) = \mathrm { e } ^ { \min ( x , t ) } - 1$ where $\min ( x , t )$ denotes the smaller of the real numbers $x$ and $t$. Draw a graph of the function $k _ { x }$. Show that $k _ { x }$ belongs to $E$, where $E$ is the set of continuous functions $f$ from $\mathbb { R } _ { + } ^ { * }$ to $\mathbb { R }$ such that the integral $\int _ { 0 } ^ { + \infty } f ^ { 2 } ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$ converges.

grandes-ecoles 2022 Q5.1 Proof of Set Membership, Containment, or Structural Property View

Recall that $\Gamma$ denotes the subgroup of $G_0$ formed by elements $g$ such that $g(V_\mathbb{Z})=V_\mathbb{Z}$.
Show that for all $v,w\in\mathcal{H}$ and all $R\geq 0$, the set $$\{g\in\Gamma \text{ such that } d(gv,w)\leq R\}$$ is finite.

grandes-ecoles 2022 Q5.3 Proof of Set Membership, Containment, or Structural Property View

We consider the three vectors $$w_1 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad w_2 = \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ We denote by $T$ the set of vectors $v\in\mathcal{H}$ such that $B(v,w_i)\geq 0$ for all $i\in\{1,2,3\}$.
Show that $T$ is compact and contains $v_0 = \begin{pmatrix}0\\0\\1\end{pmatrix}$.

grandes-ecoles 2022 Q5.4 Existence Proof View

We consider the three vectors $$w_1 = \begin{pmatrix}0\\1\\0\end{pmatrix}, \quad w_2 = \begin{pmatrix}1\\-1\\0\end{pmatrix}, \quad w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ Let $S_{1,2}$ be the subgroup of $\Gamma$ generated by $s_{w_1}$ and $s_{w_2}$. Let $v\in\mathcal{H}$.
Show that there exists $g\in S_{1,2}$ such that $$B(gv,w_1)\geq 0 \quad \text{and} \quad B(gv,w_2)\geq 0.$$

grandes-ecoles 2022 Q5.5 Direct Proof of an Inequality View

We consider the vector $$w_3 = \begin{pmatrix}-1\\0\\-1\end{pmatrix}.$$ Show that if $B(v,w_3)<0$, then $d(v_0, s_{w_3}(v)) < d(v_0,v)$.

grandes-ecoles 2022 Q5.6 Existence Proof View

Show that for all $v\in\mathcal{H}$, there exists $g\in\Gamma$ such that $gv\in T$.

grandes-ecoles 2022 Q6 Computation of a Limit, Value, or Explicit Formula View

Write the matrix $H$ of the inner product $\phi(P,Q) = \int_0^1 P(t)Q(t)\,\mathrm{d}t$ in the canonical basis of $\mathbb{R}_{n-1}[X]$, that is, the matrix with general term $h_{i,j} = \phi\left(X^i, X^j\right)$ where the indices $i$ and $j$ vary between 0 and $n-1$.