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Posts Tagged ‘linear algebra’

IMC 2016 – Day 1 – Problem 2

July 27, 2016 Leave a comment

Problem 2. Let {k} and {n} be positive integers. A sequence {(A_1,...,A_k)} of {n\times n} matrices is preferred by Ivan the Confessor if {A_i^2 \neq 0} for {1\leq i \leq k}, but {A_iA_j = 0} for {1\leq i,j \leq k} with {i \neq j}. Show that if {k \leq n} in al preferred sequences and give an example of a preferred sequence with {k=n} for each {n}.

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SEEMOUS 2016 – Problems

March 5, 2016 3 comments

Problem 1. Let {f} be a continuous and decreasing real valued function defined on {[0,\pi/2]}. Prove that

\displaystyle \int_{\pi/2-1}^{\pi/2} f(x)dx \leq \int_0^{\pi/2} f(x)\cos x dx \leq \int_0^1 f(x) dx.

When do we have equality?

Problem 2. a) Prove that for every matrix {X \in \mathcal{M}_2(\Bbb{C})} there exists a matrix {Y \in \mathcal{M}_2(\Bbb{C})} such that {Y^3 = X^2}.

b) Prove that there exists a matrix {A \in \mathcal{M}_3(\Bbb{C})} such that {Z^3 \neq A^2} for all {Z \in \mathcal{M}_3(\Bbb{C})}.

Problem 3. Let {A_1,A_2,...,A_k} be idempotent matrices ({A_i^2 = A_i}) in {\mathcal{M}_n(\Bbb{R})}. Prove that

\displaystyle \sum_{i=1}^k N(A_i) \geq \text{rank} \left(I-\prod_{i=1}^k A_i\right),

where {N(A_i) = n-\text{rank}(A_i)} and {\mathcal{M}_n(\Bbb{R})} is the set of {n \times n} matrices with real entries.

Problem 4. Let {n \geq 1} be an integer and set

\displaystyle I_n = \int_0^\infty \frac{\arctan x}{(1+x^2)^n}dx.

Prove that

a) {\displaystyle \sum_{i=1}^\infty \frac{I_n}{n} =\frac{\pi^2}{6}.}

b) {\displaystyle \int_0^\infty \arctan x \cdot \ln \left( 1+\frac{1}{x^2}\right) dx = \frac{\pi^2}{6}}.

Some hints follow.

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Vojtech Jarnik Competition 2015 – Problems Category 2

March 27, 2015 Leave a comment

Problem 1. Let {A} and {B} be two {3 \times 3} matrices with real entries. Prove that

\displaystyle A - (A^{-1}+(B^{-1}-A)^{-1})^{-1} = ABA,

provided all the inverses appearing on the left-hand side of the equality exist.

Problem 2. Determine all pairs {(n,m)} of positive integers satisfying the equation

\displaystyle 5^n = 6m^2+1.

Problem 3. Determine the set of real values {x} for which the following series converges, and find its sum:

\displaystyle \sum_{n=1}^\infty \left( \sum_{k_i \geq 0, k_1+2k_2+...+nk_n = n} \frac{(k_1+...+k_n)!}{k_1!...k_n!} x^{k_1+...+k_n}\right).

Problem 4. Find all continuously differentiable functions {f : \Bbb{R} \rightarrow \Bbb{R}}, such that for every {a \geq 0} the following relation holds:

\displaystyle \int_{D(a)} xf\left( \frac{ay}{\sqrt{x^2+y^2}}\right) dxdydz = \frac{\pi a^3}{8}(f(a)+\sin a -1),

where {D(a) = \left\{ (x,y,z) : x^2+y^2+z^2 \leq a^2,\ |y| \leq \frac{x}{\sqrt{3}}\right\}}

Existence of Sylow subgroups

June 19, 2014 Leave a comment

Let {G} be a finite group such that {|G|=mp^a} where {p} is a prime number, {a \geq 1} and {\gcd(m,p)=1}. Then there exists a subgroup {H \leq G} such that {|H|=p^a}. (such a subgroup is called a Sylow subgroup).

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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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Sierpinski’s Theorem for Additive Functions

January 26, 2014 1 comment

We say that {f: \Bbb{R} \rightarrow \Bbb{R}} is an additive function if

\displaystyle f(x+y)=f(x)+f(y),\ \forall x,y \in \Bbb{R}.

1. Prove that there exist additive functions which are discontinuous with or without the Darboux Property.

2. Prove that for every additive function {f} there exist two functions {f_1,f_2:\Bbb{R} \rightarrow \Bbb{R}} which are additive, have the Darboux Property, and {f=f_1+f_2}.

The second part is similar to Sierpinski’s Theorem which states that every real function can be written as the sum of two real functions with Darboux property.

(A function {g:I \rightarrow \Bbb{R}} has the Darboux property if for every {[a,b]\subset I}, {g([a,b])} is an interval.)

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