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## Agregation 2014 – Mathematiques Generales – Parts 4-6

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. 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.

Categories: Algebra, Linear Algebra

## 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 ${\Delta_{I,J}(A)}$.

## Agreg 2012 Analysis Part 4

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 ${\|f(x)-f_n(x) \|\leq \frac{1}{n}}$.

Categories: Analysis, Fixed Point

## Agreg 2012 Analysis Part 2

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.

Categories: Functional Analysis

## Agreg 2012 Analysis Part 1

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 ${v}$.

## Agregation 2013 – Analysis – Part 3

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 ${\{n \in \Bbb{N} | \lambda_n\leq s\}}$.

## Agregation 2013 – Analysis – Part 2

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.)