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Posts Tagged ‘differentiable’

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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Car movement – differential geometry application

May 26, 2011 1 comment

The problem presented below is from my differential geometry course. The initial reference is Nelson, Tensor Analysis 1967. The car is modelled as follows:

Denote by C(x,y) the center of the back wheel line, \theta the angle of the direction of the car with the horizontal direction, \phi the angle made by the front wheels with the direction of the car and L the length of the car.

The possible movements of the car are denoted as follows:

  •  steering: S=\displaystyle\frac{\partial}{\partial \phi};
  •   drive: D=\displaystyle\cos \theta \frac{\partial}{\partial x}+\sin\theta \frac{\partial}{\partial y}+\frac{\tan \phi}{L}\frac{\partial}{\partial \theta};
  •  rotation: R=[S,D]=\displaystyle\frac{1}{L\cos^2 \phi}\frac{\partial }{\partial \theta};
  •   translation: T=[R,D]=\displaystyle\frac{\cos \theta}{L\cos^2 \phi}\frac{\partial}{\partial y}-\frac{\sin\theta}{L\cos^2\phi}\frac{\partial}{\partial x}

Where [X,Y]=XY-YX. All these transformations seem very logical. The interpretations are quite interesting:

– from the expression of D, when the car is shorter, you can change the orientation of the car very easily, but when it is longer, like a truck, you it is not that easy ( see the term with \frac{\partial}{\partial \theta})
– the rotation is faster for smaller cars, and for greater steering angle
– translation is easier for smaller cars.

This is not a problem. It’s just a nice application of differential geometry. This presentation can generate different problems. For example:

Everyone knows that it’s not easy for a beginner to do a lateral parking. Find the least number of necessary moves to do a lateral parking, using the things presented above.

For more details about this interpretation, you could see this thread on Math Overflow.

 

Surface with normal lines passing through a fixed point

December 16, 2010 Leave a comment

Prove that if all the normal lines to a regular surface S pass through a fixed point, then the surface is a portion of the sphere.
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Interesting property for a differentiable function

October 1, 2009 Leave a comment

Prove that if f : \mathbb{R} \to \mathbb{R} is a differentiable function on \mathbb{R}, then the set of continuity points of f^\prime is dense in \mathbb{R}.

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