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Posts Tagged ‘differentiable’

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 ${v}$.

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.

Categories: Differential Geometry

Surface with normal lines passing through a fixed point

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.
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 $\mathbb{R}$.