Solution Ecole Polytechnique -Ecole Normales Superieures Concours 2013

We shall walk through the problem at little steps and day-by-day, not so much interest in the final solution per se , but with a higher aim to revise the modern algebra lessons along the way.

The French professors who designed this problem had done beautifully using all the concepts learned in the 2-year Classe Préparatoire (or Prépa, equivalent to Bachelor degree in Math & Science) – it is like an orchestra composer who pieces together all instruments to play a beautiful symphony – the catch is that the student must have a good grasp of all algebra topics.

SOLUTION

I – Operators on the functions with finite support

1. a. Prove that V is a vector subspace of \mathbb{C} ^{\mathbb{Z}}.

Proof:
Recall the definitions:

\mathbb{C} ^{\mathbb{Z}} = v.s.{f:\mathbb{Z} \mapsto \mathbb{C}}

Support = supp(ƒ) = {k \in \mathbb{Z} \mid f(k) \neq 0}

V = {f | supp(f) is a finite set}.

To prove V a vector subspace of \mathbb{C} ^{\mathbb{Z}},
1) V must be non-empty?

V contains null function, so not empty subset of \mathbb{C} ^{\mathbb{Z}}

2) closed under vector addition?

supp (f+g) \subset supp(f) \cup supp (g)

3) closed under scalar multiplication?

supp(\alpha f) = supp(f) for \alpha \neq 0

Given ƒ ∈ \mathbb{C} ^{\mathbb{Z}} , E(ƒ) ∈ \mathbb{C} ^{\mathbb{Z}}
by E(ƒ)(κ)= ƒ(κ+1), κ ∈ \mathbb{Z}

1. b. Show that E∈ L( \mathbb{C} ^{\mathbb{Z}} ) and V is stable by E.

E by definition is an operator of shift, hence a linear transformation, thus
E∈ L( \mathbb{C} ^{\mathbb{Z}} ).

——
[Solutions for XLC paper]:

http://www.math93.com/images/pdf/concours_cpge/X_ENS/Polytechnique_MP_A_XLC_corr_1.pdf

Translated Ecole Polytechnique & Ecole Normales Superieures Concours 2013

Math Paper A (XLC)
Duration: 4 hours

Use of calculator disallowed.

We propose to study the algebras of the remarkable endomorphisms of vector spaces of infinite dimension.

Preamble

An i^{th} root of unity is called primitive if it generates the group of i^{th} roots of unity.

In this problem, all vector spaces are over the base field of complex numbers field \mathbb{C} .

If ε is a vector space, the algebra of the endomorphisms of ε is denoted by L(ε), and the group of the automorphisms of ε is denoted by GL(ε).

Id_{\varepsilon} denotes the identity mapping of ε.

If u ∈ L(ε), \mathbb{C}[u] denotes the sub-algebra \{P(u) \mid P \in \mathbb{C}[X] \} of L(ε) of the Polynomials in u.

\mathbb{C} ^{\mathbb{Z}} denotes the vector space of the functions of \mathbb{Z} to \mathbb{C} .

If ƒ is the function of \mathbb{Z} to \mathbb{C}, supp(ƒ) denotes the set of κ ∈ \mathbb{Z} such that ƒ(κ) ≠0
We call this set the support of ƒ.

Throughout the problem, V denotes the set of functions of \mathbb{Z} to \mathbb{C} of which the support is a finite set.

I – Operators on the functions with finite support

1. a. Prove that V is a vector subspace of \mathbb{C} ^{\mathbb{Z}}.
Given ƒ ∈ \mathbb{C} ^{\mathbb{Z}} ,
we define E(ƒ) ∈ \mathbb{C} ^{\mathbb{Z}}
by
E(ƒ)(κ)= ƒ(κ+1), κ ∈ \mathbb{Z}

1. b. Show that E∈ L( \mathbb{C} ^{\mathbb{Z}} ) and V is stable by E.

In the following, E denotes uniquely the endomophism of V induced.

2. Show that E ∈ GL(V).

3. For i \in  \mathbb{Z} , we define v_i  \in  \mathbb{C} ^{\mathbb{Z} } by:

v_i(k) = \begin{cases} 1 , & \text{if } k = i\\ 0, & \text{if } k \neq i \end{cases}

3.a. Prove that the family \{v_i\}_{ i \in \mathbb{Z}} is the base of V.

3.b. Calculate E(v_i ).
Let \lambda, \mu \in \mathbb{C} ^{\mathbb{Z}} ,
we define the respective linear mappings F, H \in L(V)
by: H(v_i) = \lambda(i)v_i
and F(v_i) = \mu(i)v_{i+1},   i \in \mathbb{Z}

4. Prove that
H  \circ E = E \circ H + 2E
if and only if for all i \in \mathbb{Z}, \lambda(i) = \lambda(0)-2i

In the remaining of Section I (but not in the following Sections), we asume the conditions in question 4 are verified.

5. Prove that
E \circ F = F \circ E + H
if and only if \forall i \in \mathbb{Z},  \mu(i) = \mu(0) +I(\lambda(0) -1) -i^2

6.a. Prove that for f \in V the vector space generated by H^n(f), n\in \mathbb{N} has finite dimension.

6.b. Deduce that a vector subspace non-reduced to {0} of V, stable by H, contains at least one of the v_i.

In the remaining of Section I (but not in the following Sections), we asume the conditions in question 5 are verified and
\lambda(0)=0, \mu(0)=1

7.a. Prove that F \in GL(V)

7.b. Prove that E and F are not of finite order in the group GL(V).

7.c. Calculate the kernel of H and prove that H^r \neq Id_{v} \text{ for } r \geq 1

8. \mathbb{C}[X] denotes the polynomials with complex number coefficients in one indeterminate X.

8.a. Prove that \mathbb{C}[E] is isomorphic (as Algebra) to \mathbb{C}[X].

8.b. Prove that \mathbb{C}[F] is isomorphic (as Algebra) to \mathbb{C}[X].

8.c. Prove that \mathbb{C}[H] is isomorphic (as Algebra) to \mathbb{C}[X].

II – Interlude

In all the rest of the problem, we fix an odd interger \ell \geq 3 and q a \ell^{th} primitive root of unity.

9. Prove that q^2 is a \ell^{th} primitive root of unity.

Let W_{\ell}  = \displaystyle \bigoplus \limits_ {0 \leq i  < \ell } \mathbb{C} v_i \text{ and } a \in \mathbb{C}^{*}

10. Consider the element G_a \text { of  } L(W_{\ell}) \text { of which the matrix with base }  \{v_i\}_{0 \leq i  < \ell } is :

\begin{pmatrix} 0 & 0 & 0 & \ldots & 0 & a\\ 1 & 0 & 0 & \ldots & 0 & 0\\ 0 & 1 & 0 & \ldots & 0 & 0\\ \vdots & 0 &  \ddots & \ddots &  \vdots & \vdots\\  0 & \vdots &  \ddots & \ddots &  0& 0\\ 0 & 0 &\ldots & 0 & 1 & 0 \end{pmatrix}

10.a. Calculate {G_a}^{\ell}.
Prove that G_a \text{  is diagonalisable.}

10.b. Let b \ell^{th} root of a.
Calacute the eigenvectors of G_a and the associated eigenvalues in function of b, q and v_i .

Let’s define a linear mapping P_a : V \to V by
P_a(v_i) =a^{p}v_r \text{ where } i \in \mathbb{Z} , and
define r and p respectively the residue and the quotient of the euclidian division of i by ℓ; ie:
i = p\ell + r
0 \leq r < \ell , p \in \mathbb{Z}

11. Prove that P_a is a projector of image W_\ell.

III – Quantum Operators

12. Prove that
H  \circ E = q^{2}E \circ H if and only if
\forall  i \in \mathbb{Z}, \lambda(i) = \lambda(0) q^ {-2i}

In the following problem, we asume the conditions in question 12 are verified and
\lambda (0) \neq 0

13. Prove that H \in GL(V).

14. Prove that
E \circ F = F \circ  E  + H - H^{-1} if and only if
\forall  i \in \mathbb{Z},  \mu(i) = \mu(i-1) + \lambda(0)q^ {-2i} -\lambda(0)^{-1}q^{2i}

In the following problem, we asume the conditions in question 14 are verified.

15.a. Prove that \lambda \text{ and } \mu are periodic over \mathbb{Z}, of periods dividing \ell.

15.b. Prove that the period of \lambda \text { is equal to } \ell.

15.c. Prove that the period of \mu is also equal to \ell.

16. Let C = (q -q^{-1)}E \circ F+q^{-1}H +qH^{-1} with H^{-1} being inverse of H.

16.a. Prove that
C = (q -q^{-1)}F \circ E + qH +q^{-1}H^{-1}.

16.b. For i \in \mathbb{Z} , prove that v_i is an eigenvector of C.

16.c. Deduce that C is a homothety of v of which we calculate the ratio of R(\lambda(0), \mu(0),q)   in function of \lambda(0), \mu(0), q.

16.d. Let’s fix q \text{ and } \lambda(0). Prove that the mapping
\mu(0) \mapsto R(\lambda(0), \mu(0),q) is a bijection of \mathbb{C} onto \mathbb{C}.

16.e. Let’s fix q \text{ and } \mu(0) . Prove that the mapping
\lambda(0) \mapsto  R(\lambda(0), \mu(0),q) is a surjection of \mathbb{C}^{*} onto \mathbb{C} but not a bijection.

IV – Modular Quantum Operators

Let \ell, W_\ell, a, P_a like in the Section II. We say an element \phi of L(V) is compatible with P_a if
P_a \circ \phi \circ P_a = P_a \circ \phi

17.a. Prove that if \phi \in L(V) is commutative with P_a ,then \phi is compatible with P_a.

17.b. Prove that H \text{ and } H^{-1} are compatible with P_a.

Let U_q the set of endomorphisms \phi \in L(V) which are compatible with P_a.

18. Prove that U_q is a sub-algebra of L(V).

19. Prove that E \in U_q \text { and } F \in U_q.

20.a. Show that there exists an unique morphism of algebras \Psi_{a}: U_q \to L(W_{\ell}) such that:
\forall \phi \in U_q , \Psi_{a}(\phi) \circ P_a = P_a \circ \phi

20.b. Prove that \phi \in U_q is contained in the kernel of \Psi_{a} if and only if the image of \phi is a vector subspace of v generated by the vectors v_i - a^{p}v_{r} \text{ , } i \in \mathbb{Z} where i =p\ell + r is the euclidian division of i \text { by }  \ell.

21. Let’s study in this question \Psi_a(E).

21.a. Determine \Psi_a(E)(v_0).

21.b. Deduce \Psi_a(E^\ell).

21.c. Calculate the dimension of the vector subspace of \mathbb{C} [\Psi_a(E)]

21.d. Calculate the eigenvectors of \Psi_a(E)

22. Let W a non zero sub-space of W_\ell stable by \Psi_a(H).

22.a. Show that W contains at least one of the vectors v_i.

22.b. What do you say if W is in addition stable by \Psi_a(E) ?

23. Give a necessary and sufficient condition on R(\lambda(0), \mu(0),q) in order for the operator \Psi_a(F) to be nilpotent.

—End—

[Source]:
http://www.ilemaths.net/maths_p-concours-polytechnique-mp-2013-01.php

Ecole Polytechnique Concours 2013

French Math Exams paper is called “Composition”, it is unlike English Math paper solving different independent questions. In fact “Composition” is made up of many inter-dependent smaller questions, they together step-by-step lead to proving some Math theorems or corollary.

Before every Math composition, the French Math professor would tell the students the test scope covers all Math they learn thus far from primary school till today. Quite similar to sitting for any English language Composition, the scope of  vocaburary and grammar covers everything we learn since day 1 in primary school. Math is, after all, a “language” of science and logic.

Look at this year Ecole Polytchnique (and Ecole Normales Supérieures) Math Composition below:

http://www.ilemaths.net/maths_p-concours-polytechnique-mp-2013-01.php

[Note: My next few blogs will contain the English translation, and hopefully the solution contributed from the comments by blog readers. ]

It is notoriously famous for being very tough. It needs 2 years of preparation  after Baccalaureat (A-level) in the Classes Préparatoires (Maths Supérieures, Maths Spéciales), located not in universities but in few prestigious ancient Lycées (High Schools) selected by Napoléon eg. Lycée Louis Le Grand (Paris), Lycée Henri IV, Lycée Pierre de Fermat (Toulous), Lycée Du Park (Lyon)… taking in only the brightest Baccalaureat high-school students in Math (only 7.5% of each year High-school cohorts from Baccalaureat).

200 years ago the 19th century Math genius (father of Group Theory and Modern Algebra) Evariste Galois failed this Ecole Polytechnique “Composition” Exams twice because he was too good for the Examiners to understand him. The inventor of Topology Henri Poincaré topped in this Exams, while Charles Hermite (Galois’s 15 years junior from the same professor Richard of Lycée Louis Le Grand) was in the last position, almost failed!

Note: Ecole Normales Supérieures and Ecole Polytechnique combine their Concours entrance exams together in recent years.

French Taupe: 3/2 & 5/2

French elite Grandes Écoles (Engineering College), established since Napoleon with the first Military College (1794) École Polytechnique (nickname X because the College logo shows two crossed swords like X), entry only through very competitive ‘Concours’ Entrance Exams – to gauge its difficulty, Évariste Galois failed in two consecutive years.

Before taking Concours, there are two years of Prépas, or Classe Préparatoire (Preparatory class) housed in a Lycée (High school) to prepare the top Math / Science post-Baccalaureat students. These two undergraduate years are so torturous that French call these students Taupes (Moles) – they don’t see sunlight because most of the time they are studying 24×7, minus sleeping and meal time.

Most students take 2 years to prepare (Year 1: Mathématiques Supérieures, Year 2: Mathématiques Spéciales) for the Concours in order to enter X. These students are nicknamed 3/2 (Trois-Demi), so called playfully by the integration of X:

\displaystyle\int_{1}^{2} xdx= \frac{1}{2}x^{2}\Bigr|_{1}^{2}=\frac{3}{2}

If by the end of second year some students fail the Concours, they can repeat the second year, then these repeat students are called 5/2 (Cinq-Demi) – integrating X from Year 2 to Year 3:

\displaystyle\int_{2}^{3} x dx=\frac{1}{2}x^{2}\Bigr\vert_{2}^{3}=\frac{5}{2}

Évariste Galois was 5/2 yet he still failed X, not because of his intelligence but the incompetent X Examiner at whom the angry Galois threw the chalk duster. (Well done !)

Another famous 5/2 is René Thom (Fields medal 1958) who discovered ‘Chaos Theory’.

There are few rare cases of 7/2 (Sept-Demi):
\displaystyle\int_{3}^{4} x dx=\frac{1}{2}x^{2}\Bigr\vert_{3}^{4}=\frac{7}{2}
for those who insist on attempting 3 times to enter X or other elite Grandes Écoles. Equally good – if not better – is École Normale Supérieure (ENS) where Galois finally entered after having failed X twice. The tragic Galois was expelled by ENS for his involvement in the Revolution.

Note: Only 200 years later that ENS officially apologized in recent year, during the Évariste Galois Anniversary ceremony, for wrongfully expelled the greatest Math genius of France and mankind.

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One of the top Classe Préparatoire “Lycée Pierre de Fermat” named after the 17th century great Mathematician of the “Last Theorem of Fermat”, in his hometown Toulouse, Southern France.

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