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
2011 centrale-maths2__pc

17 maths questions

Q1 Matrices Linear System and Inverse Existence View
Prove that the real symmetric matrix $A$ is invertible. (One may consider the kernel of the map $x \mapsto A x$).
Q2 Matrices Linear System and Inverse Existence View
Prove that $\forall x , y \in \mathbb { R } ^ { n } , \left\langle A ^ { - 1 } x ; y \right\rangle = \left\langle x ; A ^ { - 1 } y \right\rangle$. Deduce that the matrix $A ^ { - 1 }$ is symmetric.
Q3 Matrices Projection and Orthogonality View
For $x , y \in \mathbb { R } ^ { n }$, we set: $( x ; y ) _ { A } = \langle A x ; y \rangle$. We denote by $E$ the endomorphism of the vector space $\mathbb { R } ^ { n }$ defined by $\forall x \in \mathbb { R } ^ { n } , E ( x ) = A ^ { - 1 } K x$.
Prove that $( ; ) _ { A }$ defines an inner product on $\mathbb { R } ^ { n }$. Then show that $$\forall x , y \in \mathbb { R } ^ { n } , ( E ( x ) ; y ) _ { A } = ( x ; E ( y ) ) _ { A } .$$
Q4 Invariant lines and eigenvalues and vectors Compute eigenvectors or eigenspaces View
Show that there exists a basis $\left( e _ { i } \right) _ { 1 \leq i \leq n }$ of $\mathbb { R } ^ { n }$ and $n$ strictly positive real numbers $\lambda _ { i } \in \mathbb { R } ^ { + * } ( 1 \leq i \leq n )$ such that $$\forall i \in \{ 1 , \ldots , n \} , A ^ { - 1 } K e _ { i } = \lambda _ { i } e _ { i }$$
Q5 Second order differential equations Structure of the solution space View
We consider the differential equation: $$\forall t \in \left[ 0 , + \infty \left[ , A x ^ { \prime \prime } ( t ) = - K x ( t ) \right. \right. \tag{1}$$ with unknown function $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$.
Show that $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$ is a solution of the differential equation (1) if and only if there exist $2 n$ real numbers $\left( a _ { i } \right) _ { 1 \leq i \leq n } , \left( b _ { i } \right) _ { 1 \leq i \leq n }$ such that: $$\forall t \in \left[ 0 , + \infty \left[ , x ( t ) = \sum _ { i = 1 } ^ { n } \left( a _ { i } \cos \left( t \sqrt { \lambda _ { i } } \right) + b _ { i } \sin \left( t \sqrt { \lambda _ { i } } \right) \right) e _ { i } \right. \right.$$ Deduce that the set of solutions of (1) is a finite-dimensional vector space and specify its dimension.
Q6 Chain Rule Higher-Order Derivatives of Products/Compositions View
Let $x , y \in C ^ { 1 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$. Prove that $$\forall t \in \left[ 0 , + \infty \left[ , \frac { d } { d t } ( \langle A x ; y \rangle ) ( t ) = \left\langle A x ^ { \prime } ( t ) ; y ( t ) \right\rangle + \left\langle A x ( t ) ; y ^ { \prime } ( t ) \right\rangle \right. \right.$$
Q7 Work done and energy Potential energy function and energy diagram interpretation View
Let $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$ be a solution of the differential equation (1). For each real $t \geq 0$ we set, $T \left( x ^ { \prime } \right) ( t ) = \frac { 1 } { 2 } \left\langle A x ^ { \prime } ( t ) ; x ^ { \prime } ( t ) \right\rangle$ and $U ( x ) ( t ) = \frac { 1 } { 2 } \langle K x ( t ) ; x ( t ) \rangle$. Show then that the quantity $T \left( x ^ { \prime } \right) ( t ) + U ( x ) ( t )$ does not depend on $t \in [ 0 , + \infty [$.
Q8 Reduction Formulae Bound or Estimate a Parametric Integral View
Let $k \in \mathbb { N } ^ { * }$. We denote by $c _ { k }$ the positive real number such that: $c _ { k } \int _ { 0 } ^ { 2 \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } d t = 1$, and for every real $t$ we set $R _ { k } ( t ) = c _ { k } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k }$.
Calculate $\int _ { 0 } ^ { \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } \sin t \, d t$. Deduce that $c _ { k } \leq \frac { k + 1 } { 4 }$.
Q9 Sequences and series, recurrence and convergence Coefficient and growth rate estimation View
Let $k \in \mathbb { N } ^ { * }$. We denote by $c _ { k }$ the positive real number such that: $c _ { k } \int _ { 0 } ^ { 2 \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } d t = 1$, and for every real $t$ we set $R _ { k } ( t ) = c _ { k } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k }$.
Let $\epsilon \in ] 0 , \pi [$. We set: $d _ { k } ( \epsilon ) = \sup _ { t \in [ \epsilon , 2 \pi - \epsilon ] } R _ { k } ( t )$. Prove then that $$\lim _ { k \rightarrow + \infty } d _ { k } ( \epsilon ) = 0$$
Q10 Sequences and series, recurrence and convergence Sequence of functions convergence View
Let $k \in \mathbb { N } ^ { * }$. We denote by $c _ { k }$ the positive real number such that: $c _ { k } \int _ { 0 } ^ { 2 \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } d t = 1$, and for every real $t$ we set $R _ { k } ( t ) = c _ { k } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k }$.
Let $\epsilon \in ] 0 , \pi [$, and $k \in \mathbb { N }$. Prove that for every $h \in C _ { 2 \pi } ( \mathbb { R } ; \mathbb { C } )$ that is of class $C ^ { 1 }$ on $\mathbb { R }$ and every real $u$, we have: $$\int _ { 0 } ^ { 2 \pi } R _ { k } ( u - t ) h ( t ) d t = \int _ { 0 } ^ { 2 \pi } R _ { k } \left( t _ { 1 } \right) h \left( u - t _ { 1 } \right) d t _ { 1 }$$ and $$\left| \int _ { 0 } ^ { 2 \pi } R _ { k } ( u - t ) h ( t ) d t - h ( u ) \right| \leq 2 \left\| h ^ { \prime } \right\| \epsilon + 4 \pi \| h \| d _ { k } ( \epsilon )$$ (We recall that $\int _ { 0 } ^ { 2 \pi } R _ { k } \left( t _ { 1 } \right) d t _ { 1 } = 1$ and that $\| h \|$ is defined at the beginning of the problem statement. To establish the inequality, one may use that $h \left( u - t _ { 1 } \right) = h \left( u - t _ { 1 } + 2 \pi \right)$ when $t _ { 1 } \in [ 2 \pi - \epsilon , 2 \pi ]$).
Q11 Second order differential equations Solving homogeneous second-order linear ODE View
In what follows we restrict to the case $n = 2$ from Part I.
Let $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { 2 } \right) \right. \right.$ be a solution of equation (1). Show that there exist four real numbers $c _ { 1 } , c _ { 2 } , \varphi _ { 1 } , \varphi _ { 2 }$ such that: $$\forall t \in \left[ 0 , + \infty \left[ , x ( t ) = \sum _ { i = 1 } ^ { 2 } c _ { i } \cos \left( t \sqrt { \lambda _ { i } } + \varphi _ { i } \right) e _ { i } \right. \right.$$ (We recall that the two vectors $e _ { 1 } , e _ { 2 }$ are introduced in Question 4).
Q12 Proof Direct Proof of a Stated Identity or Equality View
We denote by $C _ { 2 \pi , 2 \pi } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$ the set of continuous functions $f : \mathbb { R } ^ { 2 } \rightarrow \mathbb { C }$ such that: $$\forall \left( \theta _ { 1 } , \theta _ { 2 } \right) \in \mathbb { R } ^ { 2 } , f \left( \theta _ { 1 } + 2 \pi , \theta _ { 2 } \right) = f \left( \theta _ { 1 } , \theta _ { 2 } \right) = f \left( \theta _ { 1 } , \theta _ { 2 } + 2 \pi \right)$$
Let $f \in C _ { 2 \pi , 2 \pi } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. Prove that $$\sup _ { \left( \theta _ { 1 } , \theta _ { 2 } \right) \in \mathbb { R } ^ { 2 } } \left| f \left( \theta _ { 1 } , \theta _ { 2 } \right) \right| = \sup _ { \left( \theta _ { 1 } , \theta _ { 2 } \right) \in [ 0,2 \pi ] ^ { 2 } } \left| f \left( \theta _ { 1 } , \theta _ { 2 } \right) \right|$$ Deduce that $\left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto \left| f \left( \theta _ { 1 } , \theta _ { 2 } \right) \right|$ is bounded on $\mathbb { R } ^ { 2 }$ and attains its supremum.
Q13 Proof Computation of a Limit, Value, or Explicit Formula View
We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We denote by $C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$ the set of functions $f \in C _ { 2 \pi , 2 \pi } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$ such that the two partial derivatives $\frac { \partial f } { \partial \theta _ { 1 } } , \frac { \partial f } { \partial \theta _ { 2 } }$ exist at every point of $\mathbb { R } ^ { 2 }$ and both define continuous functions on $\mathbb { R } ^ { 2 }$.
We set $\forall t \in \left[ 0 , + \infty \left[ , \theta ( t ) = \left( t \sqrt { \lambda _ { 1 } } + \varphi _ { 1 } , t \sqrt { \lambda _ { 2 } } + \varphi _ { 2 } \right) \right. \right.$.
The Ergodic Theorem states: Let $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. Then, $$\lim _ { T \rightarrow + \infty } \frac { 1 } { T } \int _ { 0 } ^ { T } f \circ \theta ( t ) d t = ( 2 \pi ) ^ { - 2 } \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } \tag{4}$$
Let $j , l \in \mathbb { Z }$. Prove the Ergodic Theorem in the special case of the function $\left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto f \left( \theta _ { 1 } , \theta _ { 2 } \right) = e ^ { i \theta _ { 1 } j } e ^ { i \theta _ { 2 } l }$. (In the case where $( j , l ) \neq ( 0,0 )$ one may verify that $j \sqrt { \lambda _ { 1 } } + l \sqrt { \lambda _ { 2 } }$ is non-zero and then one may calculate each side of (4) separately in this special case).
Q14 Proof Deduction or Consequence from Prior Results View
We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set: $$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$
Let $k \in \mathbb { N } ^ { * }$. Prove that there exist $( 2 k + 1 ) ^ { 2 }$ complex numbers $\left( a _ { j , l } \right) _ { - k \leq j , l \leq k }$ such that for every $( u , v ) \in \mathbb { R } ^ { 2 } : f _ { k } ( u , v ) = \sum _ { - k \leq j , l \leq k } a _ { j , l } e ^ { i u j } e ^ { i v l }$. Justify that the function $f _ { k }$ satisfies the Ergodic Theorem.
Q15 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set: $$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$
Let $\epsilon \in ] 0 , \pi [$ and $k \in \mathbb { N } ^ { * }$. By writing $f _ { k } ( u , v ) - f ( u , v )$ as a sum of two terms and applying Question 10, prove that for every $( u , v ) \in \mathbb { R } ^ { 2 }$: $$\left| f _ { k } ( u , v ) - f ( u , v ) \right| \leq 2 \epsilon \left( \left\| \frac { \partial f } { \partial \theta _ { 1 } } \right\| + \left\| \frac { \partial f } { \partial \theta _ { 2 } } \right\| \right) + 8 \pi \| f \| d _ { k } ( \epsilon )$$
Q16 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We fix an arbitrary element $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. For each $k \in \mathbb { N } ^ { * }$ we set: $$\forall ( u , v ) \in \mathbb { R } ^ { 2 } , f _ { k } ( u , v ) = \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } R _ { k } \left( u - \theta _ { 1 } \right) R _ { k } \left( v - \theta _ { 2 } \right) f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 }$$
Prove the Ergodic Theorem for the function $f$. (One may set $M = 2 \left( \left\| \frac { \partial f } { \partial \theta _ { 1 } } \right\| + \left\| \frac { \partial f } { \partial \theta _ { 2 } } \right\| \right) + 8 \pi \| f \|$. For given $\epsilon > 0$, one may choose $k \in \mathbb { N } ^ { * }$ such that $d _ { k } ( \epsilon ) < \epsilon$. Next, one may apply Question 14 to $f _ { k }$ and consider $T _ { 0 } > 0$ such that for every $T \geq T _ { 0 }$: $$\left| \frac { 1 } { T } \int _ { 0 } ^ { T } f _ { k } \circ \theta ( t ) d t - ( 2 \pi ) ^ { - 2 } \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } f _ { k } \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } \right| < \epsilon .)$$
Q17 Sequences and Series Uniform or Pointwise Convergence of Function Series/Sequences View
Let $a , b \in ] 0,2 \pi [$ such that $a < b$. We denote by $\phi _ { a , b } : \mathbb { R } \rightarrow \mathbb { R }$ the continuous $2 \pi$-periodic function defined as follows. The function $\phi _ { a , b }$ is zero on $[ 0 , a ]$ and $[ b , 2 \pi ]$. For every $t \in [ a , b ] , \phi _ { a , b } ( t ) = \sin ^ { 2 } \left( \frac { \pi } { b - a } ( t - a ) \right)$.
Recall that every non-empty open set of $] - 1,1 [ ^ { 2 }$ contains a rectangle of the form $] \cos b , \cos a [ \times ] \cos d , \cos c [$ where $0 < a < b < \pi$ and $0 < c < d < \pi$.
Consider the solution $x ( t ) = \sum _ { i = 1 } ^ { 2 } \cos \left( t \sqrt { \lambda _ { i } } + \varphi _ { i } \right) e _ { i }$ of (1) obtained by taking $c _ { 1 } = c _ { 2 } = 1$ in (2). Let $\Omega$ be a non-empty open set of $\left\{ u e _ { 1 } + v e _ { 2 } \mid u , v \in ] - 1,1 [ \right\}$. Prove that there exists $t \in [ 0 , + \infty [$ such that $x ( t ) \in \Omega$. (One may reason by contradiction and justify the existence of a function of the type $\left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto \phi _ { a , b } \left( \theta _ { 1 } \right) \phi _ { c , d } \left( \theta _ { 2 } \right) = \Phi \left( \theta _ { 1 } , \theta _ { 2 } \right)$ such that $\Phi ( \theta ( t ) )$ is zero for all $t \in [ 0 , + \infty [)$.