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Agregation 2014 – Mathematiques Generales – Parts 4-6

March 21, 2014 Leave a comment

This is the second part of the Mathematiques Generales French Agregation written exam 2014. For the complete notation list and the first three parts look at this post.

Part 4 – Reduced form of permutations

For {n \geq 2} we denote {\Gamma} the set of pairs {(i,j)} such that {1\leq i<j \leq n}. We call the set of inversions of a permutation {\sigma \in S_n} the set

\displaystyle I(\sigma) = \{(i,j) \in \Gamma : \sigma(i)>\sigma(j)\}

and we denote {N(\sigma)} the cardinal of {I(\sigma)}.

1. For which permutations {\sigma \in S_n} is the number {N(\sigma)} maximum?

For {k \in [1..n-1]} we denote {\tau_k \in S_k} the transposition which changes {k} and {k+1}.

4.2 (a) Let {(k,\sigma) \in [1..n-1]\times S_n}. Prove that

\displaystyle N(\tau_k \circ \sigma) = \begin{cases} N(\sigma)+1 & \text{ if }\sigma^{-1}(k) < \sigma^{-1}(k+1)\\ N(\sigma)-1 & \text{ if }\sigma^{-1}(k) > \sigma^{-1}(k+1), \end{cases}

and that {I(\tau_k \circ \sigma)} is obtained from {I(\sigma)} by adding or removing an element of {\Gamma}.

(b) Find explicitly {\sigma^{-1} \circ \tau_k \circ \sigma} in function of the element of {I(\tau_k \circ \sigma)} which makes it differ from {I(\sigma)}.

Let {T= \{ \tau_1,...,\tau_{n-1} \}}. We call word a finite sequence {m=(t_1,...,t_l)} of elements of {T}. We say that {l} is the length of {m} and that the elements {t_1,..,t_l} are the letters of {m}. The case of a void word {(l=0)} is authorized.

A writing of a permutation {\sigma \in S_n} is a word {m=(t_1,...,t_l)} such that {\sigma =(t_1,...,t_l)}. We make the convention that the permutation which corresponds to the void word is the identity.

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Agregation 2014 – Mathematiques Generales – Parts 1-3

March 20, 2014 1 comment

This post contains the first three parts of the the Mathematiques Generales part French Agregation contest 2014.

Introduction and notations

For {m \leq n} we denote { [m..n] =\{m,m+1,..,n\} }. For an integer {n \geq 1} we denote {S_n} the group of permutations of {[1..n]}.

We say that a square matrix is inferior (superior) unitriangular if it is inferior (superior) triangular and all its diagonal elements are equal to {1}.

For two integers { n \geq 1} and {k \geq 0} we denote {\mathcal{P}_k(n)} the family of {k}-element subsets of {[1..n]}.

Let {m,n} be two positive integers and {A} a {m\times n} matrix with elements in a field {\Bbb{K}}. (all fields are assumed commutative in the sequel) A minor of {A} is the determinant of a square matrix extracted from {A}. We can define for {k \in [1..\min(m,n)]} and {(I,J) \in \mathcal{P}_k(m) \times \mathcal{P}_k(n)} the minor

\displaystyle \left| \begin{matrix}a_{i_1,j_1} & ... & a_{i_1,j_k} \\ \vdots & \ddots & \vdots \\ a_{i_k,j_1} & ... & a_{i_k,j_k} \end{matrix} \right|

where {i_1,..,i_k} (respectively {j_1,..,j_k}) are the elements of {I} (respectively {J}) arranged in increasing order. We denote this minor {\Delta_{I,J}(A)}.

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Agreg 2012 Analysis Part 4

October 20, 2013 Leave a comment

A Fixed Point Theorem

This parts wishes to extend the next result (which can be used without proof) to infinite dimension.

Theorem. (Browuer) Consider {F} a finite dimensional normed vector space. Consider {C\subset F} a convex, closed, bounded non-void set. If {f:C\rightarrow C} is a continuous application, then {f} has a fixed point in {C}.

1. In {\ell^2(\Bbb{N})} endowed with {\|\cdot \|_2} we consider the following application:

\displaystyle f: B(0,1) \subset \ell^2(\Bbb{N}) \rightarrow \ell^2(\Bbb{N})

\displaystyle f(x) =(\sqrt{1-\|x\|_2},x_0,x_1,...).

Prove that {f} is continuous with values in the unit sphere of {\ell^2(\Bbb{N})}, but {f} does not admit any fixed points.

2. Consider {E} a normed vector space, {B} a closed, bounded non-void subset of {E} and {f:B \rightarrow E} a compact application (not necessarily linear). (a compact application maps bounded sets into relatively compact sets)

i) Let {n \in \Bbb{N}\setminus \{0\}}. We can cover {\overline{f(B)}} (which is compact) by a finite number {N_n} of open balls of radius {\frac{1}{n}}: {\overline{f(B)} \subset \displaystyle \bigcup_{i=1}^{N_n} \mathring{B}(y_i,\frac{1}{n})} with {y_i \in \overline{f(B)}} for every {i}. For {y \in E} we define

\displaystyle \psi(y) =\begin{cases} \frac{1}{n}-\|y-y_i\| & \text{ if } y \in B(y_i,\frac{1}{n}) \\ 0 & \text{ otherwise} \end{cases}

Prove that {\Psi : y \in \overline{f(B)} \mapsto \sum_{i=1}^{N_n} \psi_i(y)} is continuous and that there exists {\delta>0} such that for {y \in \overline{f(B)}} we have {\Psi(y)\geq \delta}.

ii) We introduce the application {f_n : B \rightarrow E} defined by

\displaystyle f_n(x)= \left( \sum_{i=1}^{N_n} \psi_i(f(x))\right)^{-1} \sum_{i=1}^{N_n} \psi_i(f(x))y_i.

Prove that for every {x \in B} we have {\|f(x)-f_n(x) \|\leq \frac{1}{n}}.

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Agreg 2012 Analysis Part 2

October 16, 2013 Leave a comment

Part 2. Some elements of Spectral Analysis

In this part we prove that the spectrum of a bounded linear operator is non-empty, and we look at the characteristics of the spectrum of a compact operator.

Let {E} be a complex Banach space which is not reduced to {\{0\}}. (it is known that {E'\neq \{0\}}) For {T \in \mathcal{L}(E)} we define {res(T)} as the set of those {\lambda \in \Bbb{C}} such that {\lambda I-T} is bijective, and denote {R_\lambda(T)=(\lambda I-T)^{-1} \in \mathcal{L}(E)}.

We define the spectrum by {\sigma(T)=\Bbb{C} \setminus res(T)}. In particular, if {\lambda} is an eigenvalue for {T} we have {\ker(\lambda I-T)\neq \{0\}} and therefore {\lambda \in \sigma(T)}. (but note that {\sigma(T)} may contain elements which are not eigenvalues)

1. Suppose that {\|T\|<1}. Prove that {1 \in res(T)} and {(I-T)^{-1}=\sum_{k=0}^\infty T^k}.

2. Prove that if {|\lambda |> \|T\|} then {\lambda \in res(T)} and

\displaystyle \lim_{|\lambda| \rightarrow \infty} \|R_\lambda(T)\|=0.

3. Prove that {res(T)} is an open set in {\Bbb{C}} and for every {x \in E,\ell\in E'} the application {\phi : \lambda \mapsto \ell(R_\lambda(T)x)} is analytic in a neighborhood of any point {\lambda_0 \in res(T)}.

4. Deduce that for every {T \in \mathcal{L}(E)}, {\sigma(T)} is a non-void and compact.

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Agreg 2012 Analysis Part 1

October 14, 2013 Leave a comment

Part 1. Finite dimension

The goal is to prove the following theorem:

Theorem 1. Let {A \in M_n(\Bbb{R})} be a square matrix with non-negative coefficients. Suppose that for every {x \in \Bbb{R}^n\setminus \{0\}} with non-negative coordinates, the vector {Ax} has strictly positive components. Then

  • (i) the spectral radius {\rho = \sup \{ |\lambda | : \lambda \in \Bbb{C} \text{ is an eigenvalue for }A\}} is a simple eigenvalue for {A};
  • (ii) there exists an eigenvector {v} of {A} associated to {\rho} with strictly positive coordinates.
  • (iii) any other eigenvalue of {A} verifies {|\lambda|<\rho};
  • (iv) there exists an eigenvector of {A^T} associated to {\rho} with strictly positive components.

1. Consider {(w_1,..,w_n) \in \Bbb{C}^n} such that {|w_1+..+w_n|=|w_1|+...+|w_n|}. Prove that for distinct {j,l \in \{1,..,n\}} we have {\text{re}(\overline{w_j}w_l)=|w_j||w_l|}. Deduce that there exists {\theta \in [0,2\pi)} such that {w_j=e^{i\theta}|w_j|,\ j=1..n}.

2. Prove that the coefficients of {A} are strictly positive.

3. For {z \in \Bbb{C}^n} we denote {|z|=(|z_1|,..,|z_n|)}. Prove that {A|z|=|Az|} if and only if there exists {\theta \in [0,2\pi)} such that {z_j=e^{i\theta}|z_j|,\ j=1..n}.

4. Denote {\mathcal{C}= \{x \in \Bbb{R}^n : x_i \geq 0, i=1..n\}}. Consider {x \in \mathcal{C}} and denote {e=(1,1,..,1) \in \Bbb{R}^n}. Prove that

\displaystyle 0 \leq (Ax|e)\leq (x|e)\max_j \sum_{k=1}^n a_{kj}.

5. Denote {\mathcal{E}= \{ t \geq 0 : \text{ there exists }x \in \mathcal{C} \setminus \{0\} \text{ such that } Ax-tx \in \mathcal{C}\}}. Prove that {\mathcal{E}} is an interval which does not reduces to {\{0\}}, it is bounded and closed.

6. Denote {\rho=\max \mathcal{E}>0}. Prove that if {x \in \mathcal{C}\setminus \{0\}} verifies {Ax-\rho x \in \mathcal{C}} then we have {Ax=\rho x}. Deduce that {\rho} is an eigenvalue of {A} and that for this eigenvalue there exists an eigenvector {v} with coordinates strictly positive.

7. Consider {z \in \Bbb{C}^n}. Prove that {Az=\rho z} and {(z|v)=0} implies {z=0}. Deduce that {\ker(A-\rho I)=\text{span}\{v\}} and every other eigenvalue of {A} verifies {|\lambda | <\rho}.

8. Prove that every eigenvector of {A} which has positive coordinates is proportional to {v}.

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Agregation 2013 – Analysis – Part 3

June 19, 2013 Leave a comment

Part III: Muntz spaces and the Clarkson-Edros Theorem

Recall that for every {\lambda \in \Bbb{N}} we define {nu_\lambda(t)=t^\lambda,\ t \in [0,1]} and that {\Lambda=(\lambda_n)_{n \in \Bbb{N}}} is a strictly increasing sequence of positive integers.

1. Suppose that {\lambda_0=0} and {\displaystyle \sum_{n \geq 1}\frac{1}{\lambda_n} =\infty}. Let {k \in \Bbb{N}\setminus \Lambda}. Define {Q_0=\nu_k} and by recurrence for {n \in \Bbb{N}} define {Q_{n+1}} by

\displaystyle Q_{n+1}(x)=(\lambda_{n+1}x^{\lambda_{n+1}}\int_x^1 Q_n(t)t^{-1-\lambda_{n+1}}dt.

(a) Calculate {Q_1} and prove that {\|Q_1\|_\infty \leq \displaystyle \left|1-\frac{k}{\lambda_1}\right|.}

(b) Prove that for every {n \geq 1} {Q_n-\nu_k} is a linear combination of {\nu_{\lambda_1},\nu_{\lambda_2},..,\nu_{\lambda_n}}.

(c) Prove that for every {n \geq 1} we have {\displaystyle\|Q_n\|_\infty \leq \prod_{i=1}^n \left|1-\frac{k}{\lambda_j}\right|}.

(d) Deduce that {\nu_k \in \overline{M_\Lambda}}.

(e) Conclude that {C([0,1])=\overline{M_\Lambda}}.

From here on suppose that {\lambda_0} is arbitrary and the series {\displaystyle \sum_{n \geq 1}\frac{1}{\lambda_n}} converges. For {p \in \Bbb{N}} denote {\rho_p(\Lambda)=\sum_{\lambda_n>p} \frac{1}{\lambda_n}}. For {s \in \Bbb{N}} denote {N_s(\Lambda)} the cardinal of the set {\{n \in \Bbb{N} | \lambda_n\leq s\}}.

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Agregation 2013 – Analysis – Part 2

June 16, 2013 3 comments

Part II: A Blaschke product

1. Let {(z_n)_{n \in \Bbb{N}}} be a sequence of complex numbers.

(a) Prove that for every {N \in \Bbb{N}} we have

\displaystyle \left| \left[ \prod_{j=0}^N (1+z_j) -1 \right] \right|\leq \left[\prod_{j=0}^N (1+|z_j|) \right] -1.

(b) Prove that for every {N \in \Bbb{N}} we have

\displaystyle \prod_{j=0}^N (1+|z_j|) \leq \exp \left(\sum_{j=0}^n |z_j|\right).

2. Let {(g_j)} be a sequence of holomorphic functions on an open set {U \subset \Bbb{C}} such that the series of general terms {g_j} converges normally on every compact subset of {U}. Prove that the sequence of functions {(G_N)} defined for all {z \in U} by

\displaystyle G_N(z)= \prod_{j=0}^N (1+g_j(z))

converges uniformly on every compact subset of {U} to a function which is holomorphic on {U}.

(a series of functions {\sum_{j \geq 0}g_j} converges normally on a set {K} if for every {j} we have that {|g_j|\leq u_j} on {K} and the series {\sum_{j \geq 0}u_j|} is convergent.)

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