## 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 we denote the set of pairs such that . We call the set of inversions of a permutation the set

and we denote the cardinal of .

**1.** For which permutations is the number maximum?

For we denote the transposition which changes and .

**4.2** (a) Let . Prove that

and that is obtained from by adding or removing an element of .

(b) Find explicitly in function of the element of which makes it differ from .

Let . We call word a finite sequence of elements of . We say that is the length of and that the elements are the letters of . The case of a void word is authorized.

A writing of a permutation is a word such that . We make the convention that the permutation which corresponds to the void word is the identity.

## Agregation 2014 – Mathematiques Generales – Parts 1-3

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

**Introduction and notations**

For we denote . For an integer we denote the group of permutations of .

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

For two integers and we denote the family of -element subsets of .

Let be two positive integers and a matrix with elements in a field . (all fields are assumed commutative in the sequel) A minor of is the determinant of a square matrix extracted from . We can define for and the minor

where (respectively ) are the elements of (respectively ) arranged in increasing order. We denote this minor .

## 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 a finite dimensional normed vector space. Consider a convex, closed, bounded non-void set. If is a continuous application, then has a fixed point in .

1. In endowed with we consider the following application:

Prove that is continuous with values in the unit sphere of , but does not admit any fixed points.

2. Consider a normed vector space, a closed, bounded non-void subset of and a compact application (not necessarily linear). (a compact application maps bounded sets into relatively compact sets)

i) Let . We can cover (which is compact) by a finite number of open balls of radius : with for every . For we define

Prove that is continuous and that there exists such that for we have .

ii) We introduce the application defined by

Prove that for every we have .

## 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 be a complex Banach space which is not reduced to . (it is known that ) For we define as the set of those such that is bijective, and denote .

We define the spectrum by . In particular, if is an eigenvalue for we have and therefore . (but note that may contain elements which are not eigenvalues)

1. Suppose that . Prove that and .

2. Prove that if then and

3. Prove that is an open set in and for every the application is analytic in a neighborhood of any point .

4. Deduce that for every , is a non-void and compact.

## Agreg 2012 Analysis Part 1

**Part 1. Finite dimension**

The goal is to prove the following theorem:

**Theorem 1.** Let be a square matrix with non-negative coefficients. Suppose that for every with non-negative coordinates, the vector has strictly positive components. Then

- (i) the spectral radius is a simple eigenvalue for ;
- (ii) there exists an eigenvector of associated to with strictly positive coordinates.
- (iii) any other eigenvalue of verifies ;
- (iv) there exists an eigenvector of associated to with strictly positive components.

1. Consider such that . Prove that for distinct we have . Deduce that there exists such that .

2. Prove that the coefficients of are strictly positive.

3. For we denote . Prove that if and only if there exists such that .

4. Denote . Consider and denote . Prove that

5. Denote . Prove that is an interval which does not reduces to , it is bounded and closed.

6. Denote . Prove that if verifies then we have . Deduce that is an eigenvalue of and that for this eigenvalue there exists an eigenvector with coordinates strictly positive.

7. Consider . Prove that and implies . Deduce that and every other eigenvalue of verifies .

8. Prove that every eigenvector of which has positive coordinates is proportional to .

## Agregation 2013 – Analysis – Part 3

**Part III: Muntz spaces and the Clarkson-Edros Theorem**

Recall that for every we define and that is a strictly increasing sequence of positive integers.

1. Suppose that and . Let . Define and by recurrence for define by

(a) Calculate and prove that

(b) Prove that for every is a linear combination of .

(c) Prove that for every we have .

(d) Deduce that .

(e) Conclude that .

From here on suppose that is arbitrary and the series converges. For denote . For denote the cardinal of the set .

## Agregation 2013 – Analysis – Part 2

**Part II: A Blaschke product**

1. Let be a sequence of complex numbers.

(a) Prove that for every we have

(b) Prove that for every we have

2. Let be a sequence of holomorphic functions on an open set such that the series of general terms converges normally on every compact subset of . Prove that the sequence of functions defined for all by

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

(a series of functions converges normally on a set if for every we have that on and the series is convergent.)