User beni bogosel - MathOverflow

Books about Capacity theory

2012-10-28T15:45:01Z

While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for every compact by:

$$ \operatorname{cap}(K)=\inf \lbrace \|u\|_{H^1(\Bbb{R}^N)} : v\in C_0^\infty(\Bbb{R}^N), v \geq 1 \text{ on }K\rbrace $$

The definition can be extended to open sets and then to every set of $\Bbb{R}^N$, relative capacity with respect to an open set can be defined by restricting the integral and the smooth function space to an open set D, etc.

The capacity has some strange properties which are unnatural at a first sight, like the fact that the capacity of $\partial K$ is the same as the capacity of $K$ for a compact $K$.

I want to understand better what capacity really means, and for that I tried to find all sort of books about potential theory (even the ones referred in the mentioned book), and all seem to have the same way of dealing with the subject: the setting is very general and abstract and the definition presented above just as a particular case.

Do you know any book, article or course notes which deal with this specific capacity in detail explaining:

the definition and the intuition behind the capacity;

examples of capacity computation for simple sets (using capacitary potentials);

the connection between the capacity and the Sobolev spaces ?

In the mentioned book the study of capacity is made in section 3.3. It contains all the definitions and all the needed properties of the capacity, but I still feel that I need a better understanding. That's why I asked this question.

Weak divergence implies weak differentiability of components?

2012-10-23T17:01:15Z

Suppose $\Omega$ is an open set in $\Bbb{R}^N$ and $\sigma : \Omega \to \Bbb{R}^N$ is a field with all components belonging to $L^2(\Omega)$.

We say that $\sigma$ has weak divergence if there exists a function $w \in L^2(\Omega)$ such that for all $\varphi \in C_c^\infty (\Omega)$ we have

$$ \int_\Omega \sigma \cdot \nabla \varphi=-\int_\Omega w \varphi. $$

My question is:

Can we establish a result of the form: if $\sigma$ has weak divergence then each component of $\sigma$ is weakly differentiable?

The idea is that I've seen this technique in proving that if a function $u$ is $H^1(\Omega)$ and it satisfies some convenient weak condition then $\nabla u$ has a weak divergence and therefore $u \in H^2(\Omega)$. The book where I've seen this technique is aimed for engineers, and therefore it is not very rigorous. That is why I've asked this question.

Applications for p-Sylow subgroups theorem

2011-04-04T19:36:29Z

I have searched for such a question and didn't find it. I recently had a presentation in which I introduced $p$-Sylow subgroups and proved Sylow's theorems. I will have another one soon, concerning applications of Sylow's theorem.

My question is:

Are there any spectacular applications of Sylow's theorem in group theory and other fields of mathematics (which are of course related to groups)?

Answer by Beni Bogosel for minimum distance between two arbitrary circles in space

2012-04-12T21:14:07Z

I didn't manage to solve the problem (edit: in the meantime an answer was posted which says a precise formula using radicals cannot be found), but I can post a proof that the line joining the points where the minimal/maximal distance is achieved is perpendicular to the tangent line at the circles in those contact points. (inspired by the comment of Gerhard Paseman)

To do this, choose $\vec{a}$ and $\vec{d}$ the position vectors of the centers and $\vec{b},\vec{c}$, respectively $\vec{e},\vec{f}$ be two pairs of orthogonal unit vectors which span the planes of the first and respectively the second circle. Denote by $r,s$ the radii of the two circles. Consider the circles parametrized as (in fact, the argument works for any parametrization) $$ p(\theta)=\vec{a}+r\cos\theta\ \vec{b}+r\sin\theta\ \vec{c}, \ \theta \in [0,2\pi] $$ $$ q(\tau)=\vec{d}+s\cos\tau\ \vec{e}+s\sin\tau\ \vec{f}, \tau \in [0,2\pi]$$ and denote $F(\theta,\tau)=|p(\theta)-q(\tau)|^2$. Then the pair of points which realize the minimal/maximal distance must satisfy $$ \frac{\partial F}{\partial \theta}=\frac{\partial F}{\partial \tau}=0. $$

We have $$ \frac{\partial F}{\partial \theta}=2\sum_{i=1}^3 [p_i(\theta)-q_i(\tau)]p_i'(\theta)=2 (p(\theta)-q(\tau))\cdot p'(\theta) $$ $$ \frac{\partial F}{\partial \tau}=-2\sum_{i=1}^3 [ p_i(\theta)-q_i(\tau) ] q_i'(\tau)=-2 (p(\theta)-q(\tau))\cdot q'(\tau) $$ where "$\cdot$" is the usual dot product. Therefore when $\theta,\tau$ correspond to the minimum/maximum value, the partial derivatives vanish and $p'(\theta)\perp (p(\theta)-q(\tau))$ and $q(\tau)'\perp (p(\theta)-q(\tau))$ where $p'(\theta),q'(\tau)$ are the tangent vectors in the contact points and $p(\theta)-q(\tau)$ is the vector connecting the points where minimal/maximal distance is achieved.

Articles with examples of Darboux functions without fixed points

2012-04-10T09:16:44Z

A function $f: I \to J$ ($I,J$ intervals) has the Darboux property or the Intermediate value property if for every $a < b \in I$ and for every $\lambda$ between $f(a)$ and $f(b)$ there exists $c \in [a,b]$ such that $f(c)=\lambda$. Equivalently, the image of any interval under $f$ is an interval.

I know that there are functions $f: [0,1] \to [0,1]$ which have the Darboux property, but have no fixed points.

What are some articles which can be taken as references for this non-existence theorem? The only one I found was this, but I guess that there are older articles which deal with this subject. I searched Google and Mathscinet, but didn't find any except the one above (maybe I don't know how to search...).

Inequality involving perimeter and area

2012-02-15T10:36:21Z

I am studying an article: The parametric problem of capillarity: the case of two and three fluids, by U. Massari. In one of his proofs, he uses an inequality I can't manage to prove. It is like this:

Let $\Omega \subset \Bbb{R}^n$ be an open, bounded set, with Lipschitz boundary with constant $L$. The following inequality holds: $$(1) \ \ \int_{\partial \Omega} \chi_E d\mathcal{H}^{n-1} \leq \sqrt{1+L^2}\int_{\Omega_\varepsilon}| \nabla \chi_E|+c \int_{\Omega_\varepsilon} \chi_E dV $$ where $E$ is a measurable set of finite perimeter (i.e. $\chi_E \in BV(\Omega)$), and the integral on $\partial \Omega$ is in fact the integral of the trace of $\chi_E$ on the boundary of $\Omega$.

where $\Omega_\varepsilon = \lbrace x \in \Omega : d(x,\partial \Omega) <\varepsilon \rbrace$

The previous inequality is stated without proof or reference in the article, but there is another similar inequality, with a reference to a proof:

Let $\Omega \subset \Bbb{R}^n$ be an open, bounded set, with the property that there exists $\rho>0$ such that for every $x \in \Omega$ there is a ball $B_\rho$ of radius $\rho$ (not necessarily centered in $x$) with $x \in B_\rho \subset \Omega$. The following inequality holds: $$ (2) \ \ \int_{\partial \Omega} \chi_E d\mathcal{H}^{n-1} \leq \int_{\Omega_\varepsilon}| \nabla \chi_E|+c \int_{\Omega_\varepsilon} \chi_E dV $$ where $E$ is a measurable set of finite perimeter (i.e. $\chi_E \in BV(\Omega)$), and the integral on $\partial \Omega$ is in fact the integral of the trace of $\chi_E$ on the boundary of $\Omega$. The constant $c$ depends on $\varepsilon, \rho, \Omega$ and $n$.

This inequality is proved in I. Tamanini: Il problema della capillarita su domini non regolari. The hypothesys with the interior spheres of radius $\rho$ is used heavily in the proof. First $\Omega$ is written as a countable union of balls of radius $\rho$.

The question is: do you know an article which proves the inequality $(1)$? If not, it is possible to deduce $(1)$ using $(2)$? Thank you.

Answer by Beni Bogosel for Maximum area of intersection between annulus and circle?

2012-04-10T15:01:22Z

There is a formula for the area of the intersection of two circles of given radii in terms of the distance between the centers. The formula can be found here: http://mathworld.wolfram.com/Circle-CircleIntersection.html

If the radius of $C_3$ is smaller than $R_2-R_1$ or greater than $R_2$ then the answer is obvious. In the other cases you can also calculate the explicit area formula of the intersection. If we denote $\mathcal{A}(C_i,C_j)$ the area of the intersection of the circles $C_i,C_j$. Then the area of the intersection of $C_3$ with the annulus is $\mathcal{A}(C_2,C_3)-\mathcal{A}(C_1,C_3)$, and using the formulas presented in the link you can write the exact formula in terms of $d$, and then optimize with respect to $d$ the formula you get.

Answer by Beni Bogosel for Inequality involving perimeter and area

2012-04-09T10:16:57Z

I have found an article which deals with this kind of inequalities. It is available in the following link: Funzioni BV e Tracce

Darboux function on $[0,1]$ with interesting property

2011-06-10T10:20:21Z

I have proved a few years ago the following proposition:

There exists $f: [0,1] \to [0,1]$ with Darboux property such that there exist $A,B \subset[0,1]$ with $A\cap B=\emptyset,\ A \cup B=[0,1]$ with $f(A)\subset B$ and $f(B)\subset A$. (of course $A,B\neq \emptyset$)

A function $f : I\subset \Bbb{R} \to \Bbb{R}$ ($I$ is an interval) has the Darboux property if $f([a,b])$ is an interval forall $[a,b]\subset I$.

The proof resembles the proof of Sierpinski's Therem, that any function $f : \Bbb{R} \to \Bbb{R}$ can be written as the sum of two functions each of them having the Darboux property.

My question is:

have I proved something new, or it is a known fact that such a function exists?
if the proposition is original can it be useful, I mean, can I submit this as an article?

[edit:] I know I should have done my job and send this to a some magazines to see if it is worth publishing. One of my teachers said that I send the article to JMAA, and of course it got rejected, because it's not that good. I tried at another magazine, but didn't even get an answer if it is rejected or not. I thought then that the proposition is not worthy of an article and proposed it as a problem to AMM. They said its too hard to be published as a problem. As I am a beginner and don't have any paper published until now. I don't know where should I try to send it.

Could you please name some magazines where I could try and send the paper and recieve an answer to wether the proposition can be published or not?

Volume correction of a sequence

2012-04-08T12:46:03Z

I study the article of Baldo: Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids. The main point of the article is to prove a $\Gamma$-convergence theorem result. In this cases, as can be seen in the definition of $\Gamma$ convergence the first part is easy, but the last part is almost always constructive, and things get messy.

In the article, in the proof of the second part of the definition of $\Gamma$-convergence a sequence $(u_\varepsilon) \subset L^1(\Omega \subset \Bbb{R}^N;\Bbb{R}^n)$ ($N$ is not necessarily equal to $n$) which converges to $u$ in $L^1(\Omega;\Bbb{R}^n)$ is constructed, and we know that $\int_\Omega |u_\varepsilon -u| \leq K\varepsilon $. We denote $\eta_\varepsilon = \int_\Omega u_\varepsilon(x)dx -\int_\Omega u(x)dx \in \Bbb{R}^n$. Then we have $|\eta_\varepsilon|\leq K\varepsilon$.

The problem is that the sequence $u_\varepsilon$ must be corrected, hopefully on a small set and preserving continuity, such that $\int_\Omega u_\varepsilon=\int_\Omega u$. To do this, in the article (pages 79-80) the author chooses a ball $B_\varepsilon=B(x_0,\varepsilon^{1/N})$ such that $B_\varepsilon \subset \lbrace u_\varepsilon =\alpha \rbrace \subset\Omega$, and defines

$$ v_\varepsilon(x) =\begin{cases} u_\varepsilon(x) & x \in \Omega \setminus B_\varepsilon \newline \alpha+ h_\varepsilon(1- \varepsilon^{-1/N}|x-x_0|) & x \in B_\varepsilon \end{cases}$$ where $h_\varepsilon=-N\omega_{N-1}^{-1}\eta_\varepsilon \varepsilon^{(1-N)/N} $ and $\omega_{N-1}$ is the volume of the $N-1$ dimensional unit ball.

In the article it says that from here it follows that $\int_\Omega v_\varepsilon =\int_\Omega u$, but doing some calculations leads to $$ \int_\Omega v_\varepsilon(x)dx=\int_\Omega u_\varepsilon(x)dx+h_\varepsilon\int_{B_\varepsilon}(1-\varepsilon^{-1/N}|x-x_0|)dx= \int_\Omega u_\varepsilon(x)dx +h_\varepsilon C \varepsilon $$ where $C$ is a constant, and we would need that the last term be equal to $\eta_\varepsilon$, but the calculations show that the last integral is of order $\varepsilon$, and cannot cancel the power of $\varepsilon$ in the expression of $h_\varepsilon$. Moreover, the expression of $h_\varepsilon$ looks exactly like the integral was calculated in $N-1$ dimensions, not $N$. Anyway, this could be corrected if the expression of $h_\varepsilon$ wouldn't be used in the sequel of the article in an essential way (page 80 bottom).

It is used in proving that

$$ \limsup_{\varepsilon \to 0} \int_{B_\varepsilon}\left[ \varepsilon |\nabla v_\varepsilon|^2+\frac{1}{\varepsilon}W(v_\varepsilon)dx \right]= $$ $$ \limsup_{\varepsilon \to 0} \left[ \varepsilon |h_\varepsilon|^2\varepsilon^{-2/N}|B_\varepsilon|+\frac{1}{\varepsilon}\int_{B_\varepsilon}W(\alpha+ h_\varepsilon(1- \varepsilon^{-1/N}|x-x_0|) )dx \right]= 0$$ where $W$ is continuous on $\Bbb{R}^n$ with $W(\alpha)=0$.

The gradient term easily converges to zero for the correct expression of $h_\varepsilon$ (and a pretty wide range of powers of $\varepsilon$ in $h_\varepsilon$), but for the term with $W$ to converge to zero we need $h_\varepsilon \to 0$ or the volume of $B_\varepsilon$ to have order greater than $\varepsilon$ and if my calculations are correct, this cannot happen at the same time.

Since the article is widely cited, and the same formula for $h_\varepsilon$ is used in the article of Luciano Modica: Gradient theory of phase transitions with boundary contact energy, which is again very cited, I guess that the calculations can be corrected, or I am not getting this right.

Is there any mistake in my calculations? If I am correct, then is there another way to correct the integrals of $u_\varepsilon$ such that the limsup remains zero?

I know the question seems long, but I included some of the details for it to be a bit self contained. Anyway, the main issue is the choice of the correction of the integral term.

Does regularity of the boundary imply interior sphere condition

2012-03-19T10:04:50Z

In the article of Massari presented here there is a trace inequality which is said to be true for domains which satisfy the interior sphere condition:

There exists $\rho>0$ such that for every $x \in \Omega$ there is a ball $B_\rho$ of radius $\rho$ such that $x \in B_\rho \subset \Omega$. This rhoughly means that the curvature of the domain is bounded from above.

In some other article of Anzellotti and Giaquinta they prove a similar trace inequality for bounded domains with $C^1$ boundary. My question is:

If a bounded open set $\Omega$ has $C^1$ boundary, is it true that it satisfies the interior sphere condition mentioned above?

[edit] If the answer is negative for $C^1$ boundary, is it possible that for a $C^k$ with $k \geq 2$ or $C^\infty$ boundary the result becomes true?

Inequality involving BV norm and a regularizing kernel

2012-03-02T11:51:32Z

In the same article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# (related to this question http://mathoverflow.net/questions/89801/a-limit-involving-a-regularizing-kernel) I encountered an inequality, which I didn't manage to prove.

It is like this:

$$ \int_{\Bbb{R}^d}\left[ |u^0(x)|-\int_{\Bbb{R}}\left(\chi(\xi,u^0(x))\star \varphi_\varepsilon \right)^2d \xi\right]dx \leq C \|u^0\|_{BV}\cdot\varepsilon $$

where $$ \chi(\xi,u)=\begin{cases} 1 & {0\leq \xi\leq u} \newline -1 & u \leq \xi \leq 0 \newline 0 & \text{otherwise} \end{cases}$$

and $\varphi_\varepsilon$ is a regularization kernel in $x$ and $u_0$ is regular enough for all objects to be well defined.

In the article, the inequality is stated as obvious, and no indication, reference or attempt to prove it is made. It is possible to prove that the LHS tends to zero as $\varepsilon \to 0$. Still, I cannot get the majorization by $\varepsilon$ in the RHS, which clearly depends only on $\varphi_{\varepsilon}$.

Do you have some ideas in proving this inequality? Thank you.

Singular conformally-Euclidean metrics

2012-02-27T10:00:24Z

Suppose $W : \Bbb{R}^n \to \Bbb{R}_+$ is a continuous, positive function, with exactly $n$ zeros $\alpha_1,...,\alpha_n$. Define the following 'distance':

$$ d(\alpha_i,\alpha_j)=\inf{\int_0^1 \sqrt{W(\gamma(t))}| \gamma'(t)|dt : \gamma \in C^1([0,1];\Bbb{R}^n), \gamma(0)=\alpha_i,\ \gamma(1)=\alpha_j}$$

Suppose I have a set of real, positive numbers $\sigma_{ij}>0,\ i \neq j$ with the property that $\sigma_{ij}=\sigma_{ji}$ and $\sigma_{ij} \leq \sigma_{ik}+\sigma_{kj},i,j,k=1,...,n$.

My question is:

Can we find $\alpha_i, i=1..n$ and $W$ with the desired properties, such that $d(\alpha_i,\alpha_j)=\sigma_{ij}$?

I feel that the fact that we can choose $W$ and the zeros of $W$, $\alpha_1,...,\alpha_n$ gives enough freedom for us to solve this system. Thank you.

I should say why I need to know if this result is true. I am studying an article of Baldo: Minimal interface criterion for phase transitions in mixtures of Cahn-Hilliard fluids, where he proves that the following functional $$ \mathcal{F}(E_1,...,E_n) = \sum_{1\leq i < j\leq n} d(\alpha_i,\alpha_j) \mathcal{H}^{N-1}(\partial^*E_i \cap \partial^*E_j \cap \Omega) $$ is a $\Gamma$-limit of certain functionals, and therefore it is lower semicontinuous, where $d(\alpha_i,\alpha_j)$ is defined as above. I was wondering if it is possible to prove that for any $\sigma_{ij}$ which satisfy the triangle inequality (which is a necessary physical condition), the lower semicontinuity, and therefore the existence of a minimum for the given energy, $$ \mathcal{F}(E_1,...,E_n) = \sum_{1\leq i < j\leq n} \sigma_{ij} \mathcal{H}^{N-1}(\partial^*E_i \cap \partial^*E_j \cap \Omega) $$ still holds.

Answer by Beni Bogosel for Are weak and strong convergence of sequences not equivalent?

2012-03-01T15:23:01Z

A remark in H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Chapter 3:

In any infinite dimensional space the weak topology is strictly coarser than the strong topology.

In the same place there are two examples:

The unit sphere $S={x \in E : \|x \|=1}$, with $E$ infinite dimensional, is never closed in the weak topology $\sigma(E,E^*)$
The unit ball $U={x \in E : \|x\|<1}$, with $E$ infinite dimensional is never open in the weak topology $\sigma(E,E^*)$

The proofs of these two facts can be found in the book.

You can find some useful facts and examples in the following documents: http://www.uio.no/studier/emner/matnat/math/MAT4380/v06/Weakconvergence.pdf

http://people.sissa.it/~bianchin/Courses/Functionanal/lecture06.weaktopologies.pdf

A property of sets of finite perimeter

2012-02-15T21:00:05Z

I have a question regarding sets of finite perimeter. I feel that it should be true, but I didn't manage to prove or find a reference about it. Suppose $D$ is an open, bounded subset of $\Bbb{R}^n$, and define the perimeter of a measurable set $A \subset D$ as

$$P_D(A)=\sup \left \lbrace\ \int_D \chi_A {\rm div} \varphi \ dx \ : \ \varphi \in C_c^1(U;\Bbb{R}^N),\ |\varphi|\leq 1\ \right\rbrace $$

If $|A|=V$ with $V\in (0,|\Omega|)$ and $P_D(A)<\infty$ is it true that we can modify $A$ up to a set of measure zero in order to find a small ball included in $A$?

If yes, is there any reference, or easy proof for this?

Thank you.

A limit involving a regularizing kernel

2012-02-28T21:37:09Z

I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/#

Somewhere in the middle of it, I'm stuck at proving a certain limit equality. Maybe it's obvious and I can't get it.

$$ \int_{(\Bbb{R})} \left(\chi(\xi,u)\star \varphi_\varepsilon \right)^2d \xi \to |u| \text{ in } {L}^1_{loc} $$

where $\varphi_\varepsilon(t,x)$ is a regularizing kernel, $u$ satisfies $$\partial_t u +\text{div}A(u)=0 \text{ and }\text{ in }\mathcal{D}^\prime((o,\infty)\times \Bbb{R}^d) $$ and

$$ \chi(\xi,u)=\begin{cases} 1 & {0\leq \xi\leq u} \newline -1 & u \leq \xi \leq 0 \newline 0 & \text{otherwise} \end{cases}$$

Thank you.

[edit] Sorry. I forgot to mention that $u \in L^1_{loc}$.

Sperner Lemma Applications

2012-02-27T22:22:50Z

I was always fascinated with this result. Sperner's lemma is a combinatorial result which can prove some pretty strong facts, as Brouwer fixed point theorem. I know at least another application of this lemma, namely, Monsky's theorem, which states that it is impossible to dissect a square into an odd number of triangles having equal areas.

Browsing through a few questions this evening I found two references to Sperner's lemma with respect to totally different applications. I searched the site, and didn't found a question which asks about other applications of Sperner's lemma, so I thought I'll ask the question myself.

What other applications of Sperner's lemma are there?

(I made the question community wiki.)

Answer by Beni Bogosel for A property of sets of finite perimeter

2012-02-16T09:41:30Z

I'm sorry that I answer my own question, but I found out the answer this morning from my teacher. There are examples of sets of finite perimeter with positive measure, which do not contain any open ball.

For example, take $D=B(0,1)$, the unit ball in $\Bbb{R}^2$ and denote $S=D \cap \Bbb{Q}^2=(x_n)_{n \geq 0}$. Then, we can find a sequence of positive real numbers $(r_n)$ such that

$\sum_{n=1}^\infty 2\pi r_n < \infty$
$\sum_{n=1}^\infty \pi r_n^2 < \pi$

Take $B=\bigcup_{n=1}^\infty B(x_n,r_n) $. Then $C=D\setminus B$ has the desired property. Indeed, $|C|=|D|-|B|>0$, and $Per_D(C)=Per_D(B)<\infty$.

If $C$ would contain an open ball then that ball would intersect $S \subset B$, which is not possible.

How to solve geometry problems using involutions

2011-07-13T13:50:47Z

Some geometry problems ( like this and this ) have short solutions if we use involutions. What references are there for solving geometry problems using involutions? I am particularly interested in theoretical aspects of involutions, and maybe a reference with solved problems like these two presented in the links.

There are quite a few problems on AoPS that are solved using involutions. I searched there for help, but it seems that the user that posted most of the solutions (grobber) is not active anymore. There was a suggestion to make a teaching article for such problems but there was no one to finish the task. This question was on StackExchange for almost a month now without any answers or comments, that's why I've posted it here.

$C_0$-semigroups applications

2011-06-01T12:13:34Z

My graduation paper was about stability theorems for $C_0$-semigroups (see the Wikipedia article for the definitions: http://en.wikipedia.org/wiki/C0-semigroup). I am about to go in another direction for my master degree paper, namely shape optimisation. But before I leave behind $C_0$-semigroups, I would like to know if there is some aplicability of the stability theorems I know in this field. The only applications I found for my paper were about the Hille-Yosida theorem and some of its applications to existence and uniqueness of solutions of partial differential equations.

I will not put any names to my theorems, since maybe they are not known to the world as my teachers name them. Here are some of them:

The $C_0$-semigroup ${T(t)}_{t \geq 0}$ is exponentially stable if and only if there exists $p \geq 1$ such that $\int_0^\infty \|T(t)\|^pdt <\infty$.

The $C_0$-semigroup ${T(t)}_{t \geq 0}$ is exponentially stable if and only if it satisfies the following condition: For any $f \in \mathcal{C}$ it follows that $x_f \in \mathcal{C}$ where $x_f: \Bbb{R}_+ \to X,\ x_f(t)=\int_0^t T(t-s)f(s)ds$, and $\mathcal{C} = { f : \Bbb{R}_+ \to X,\ f \text{ continuous and bounded } }$.

The last theorem can be formulated and proved in some cases for $(L^p,L^q)$ spaces with $(p,q) \neq (1,\infty)$. A more general concept, dichotomy can be formulated (the space splits into two spaces, on one of them there is stability, and on the other one there is instability.

All these sound very nice, and have quite beautiful proofs, but are they applicable to some branches of applied math, such as ordinary or partial differential equations, or they are just pure math, and thats it?

Car movement - differential geometry interpretation.

2011-05-31T17:58:43Z

I've posted this on Math Stack Exchange and I didn't get any answer in a couple of days, so I'll try and post it here too.

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 $[X,Y]=XY-YX$ (I can't remember the English word now). All these transformations seem very logical. My question is:

How can we justify the mathematical interpretation made above, especially the part with the rotations and translations?

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.

Answer by Beni Bogosel for Dissecting a square

2011-05-26T13:19:00Z

I think there are very few such solutions. The pieces must be identical, and they must touch the center. Consider the segment joining the center with one of the vertices. Then all small figures (in which you split the square) must contain a segment of this length, and there are only four such segments. Any such segment belongs to at most two small figures, and we find that there are at most $8$ small figures. From here on it is easy to see that the possible splits are:

the square itself
the square cut by a diagonal
the square cut by two diagonals
the square cut by parallel lines through the center
the square cut by parallel lines through the center and by its diagonals
the square cut by a line through the center
the square cut by two orthogonal lines through its center.
the square cut by any smooth curve symmetric by its center.
the square cut by any smooth curve symmetric by its center, and the rotate of this curve by $\pi/2$.

There are indeed many solutions. Sorry for my initial remark. I think that essentially the square can be dissected in 2,4 or 8 parts. The 8 parts is unique. The 2 parts cutting must be symmetric by its center, and the 4 parts cutting must be made such that is invariant by a $\pi/2$ rotation.

Morera type theorems

2011-05-25T10:15:09Z

In Stein and Shakarchi, Complex Analysis, Princeton lectures in Analysis, Chapter 2, Problem 2 an interesting question is posed. The problem section in each chapter contains more complicated problems, with a research taste.

Morera's theorem simply states that if a function $f$ is continuous on $\Bbb{C}$ and $\int_D f(z)dz=0$ for any triangle(rectangle) $D$, then $f$ is holomorphic in $\Bbb{C}$. (the theorem is still valid if we replace $\Bbb{C}$ by a disk).

The problem presented above, states that

Morera's theorem is still valid if we replace the contours of integration from triangles/rectangles to circles, and more generally, to any contour which is a translate and dilate of a toy contour $\Gamma$.

Is there a simple proof for this problem, or maybe a reference to an article in which I can find the proofs?

[I will post the hint given after the problem in the book but is quite long and I don't have the necessary time right now.]

Haar measure of a subgroup

2011-05-23T09:00:54Z

What is the connection between the normalized Haar measure of a compact group and the normalized Haar measure of one of its compact subgroups?

I am trying to solve the following problem:

Given $G$ a compact group with normalized measure $\mu$ and ${H_n}$ an increasing sequence of compact subgroups of $G$ with normalized Haar measures $\mu_k$ such that $\bigcup H_n$ is dense in $G$. Prove that $\mu_k$ converges in the weak star topology to $\mu$.

[edit] The problem is indeed an exercise, as you can see from my comments, but I don't know why this is so relevant. I asked a question which could enlighten me in order to solve the given problem, and I think that the given question about Haar measures is not so trivial, since no one gave an answer until now.

Sum of two closed operators closable

2011-04-09T19:43:09Z

I found this question on another forum, and after processing it a bit, I didn't find a good answer. The question is:

Is the sum of two closed operators closable? If not, give an example of two closed operators such that their sum is not closable.

Baire Category Theorem Application

2011-02-22T20:59:34Z

In Antoine Henrot Michel Pierre - Variation et optimisation de formes, Une analyse geometrique, a book I'm studying I found an interesting problem. The problem is listed below. The first 3 points of the problem are pretty easy, and I solved them. The 4-th seems a little harder. The only indication I get is to use point 3) and the Baire theorem for $(\Sigma,\delta)$.

Denote by $\Sigma$ the quotient space of the family of Lebesgue measurable sets of $\Bbb{R}^N$ by the equivalence relation $E_1 \sim E_2 \Leftrightarrow \chi_{E_1}=\chi_{E_2} a.e.$. Denote by $|X|$ the Lebesgue measure of the measurable set $X$.

1) Prove that $\delta(E_1,E_2)=\arctan( |E_1 \Delta E_2|)$ is a distance on $\Sigma$.

2) Prove that given $(E_n)_{n \geq 1}, E$ measurable sets in $\Bbb{R}^N$ the following three properties are equivalent.

$\delta(E_n,E) \to 0$;

$\chi_{E_n}-\chi_E \xrightarrow{\sigma(L^1,L^\infty)} 0$;

$\chi_{E_n}-\chi_E \xrightarrow{L^1} 0$.

3) Prove that $(\Sigma,\delta)$ is a complete metric space.

4) Given the sequence $ (f_n)$ of integrable real valued functions on $\Bbb{R}^N$, such that for any measurable set $E$ of $\Bbb{R}^N$ there exists $\displaystyle \lim_{n \to \infty}\int_E f_n$, prove that if $|E| \to 0$ then $\displaystyle \sup_n\int_E |f_n| \to 0$. (Hint: Use the Baire category theorem for $(\Sigma,\delta)$)

The question is: How can I apply Baire theorem to solve the 4-th point in the problem?

Reduce number of dimensions in an inequality

2011-03-05T13:40:39Z

Suppose I want to prove an inequality involving only norms and inner products between $k$ vectors in $\Bbb{R}^n$ where $n >k$. Is it possible to reduce it to vectors in $\Bbb{R}^k$?

My argument is the following:

change the basis in $\Bbb{R}^n$ to another orthonormal basis for which the first $k$ vectors span the subspace generated by the initial $k$ vectors in my inequality.
orthonormal transformations preserve the inner product and the norm.
in the new basis, my $k$ vectors have non-zero terms only on the first $k$ positions.

Does this allow me to say that is enough to prove my inequality over vectors in $\Bbb{R}^k$?

Important lines in triangle - reverse problem

2011-02-28T18:12:31Z

It is known that if three numbers $x,y,z$ are the lengths of the edges of some triangle, then there exists a triangle with medians of length $x,y,z$. Also, if $x,y,z>0$ (no condition imposed) there exists a triangle with angle bisectors of length $x,y,z$ (the proof of this is very beautiful and uses Brouwer fixed point theorem).

I was wondering if there are some other results like this:

if $x,y,z>0$ satisfy the family of conditions $ { C_1,C_2,...,C_n }$(possibly void) then there exists a triangle for which the lengths of some important lines (for eg. symmedians) are $x,y,z$.

Do you know any such results?

Pseudo-alternate series

2011-02-20T05:46:44Z

Suppose $(a_n)$ is a non-increasing sequence of positive real numbers and $\varepsilon_i = {\pm 1},\ \forall i \in \mathbb{N}$ such that $\sum\limits_{i=1}^\infty \varepsilon_i a_i$ is convergent. Is it true that $\lim\limits_{n\to \infty}(\varepsilon_1+\varepsilon_2+...+\varepsilon_n) a_n=0$?

Tangent surfaces curvature inequality

2011-02-20T11:49:49Z

I found this lemma in a few surface geometry proofs:

If we have two surfaces, $S$ and $S'$, which are tangent in the point $p$ then if: (i) $S'$ has positive curvature in $p$; (ii) $S$ is, locally around $p$, situated on the same side of $S'$, then the curvature of $S$ in $p$ is greater or equal to the curvature of $S'$ in $p$.

I am interested in a book/reference where I can find a proof for this lemma. Thank you.

Comment by Beni Bogosel

2012-10-30T10:49:43Z

I think this is more suitable for math.stackexchange.com

Comment by Beni Bogosel

2012-10-23T17:35:22Z

That is what I was thinking, but I wanted another confirmation.

Comment by Beni Bogosel

2012-05-21T20:25:17Z

Since you say that there are some large pdf's, you can try to shrink them. You can shrink pdf's without losing quality if you are using linux, at least if the documents were processed with latex. The code for it is "gs -sDEVICE=pdfwrite -dCompatibilityLevel=1.4 -dNOPAUSE -dQUIET -dBATCH -sOutputFile=target_name.pdf input_name.pdf", and the command must be run in terminal in the folder you have the pdf file. I used it for some time and I managed to get a files which which are even 80% smaller.

Comment by Beni Bogosel

2012-04-10T20:31:31Z

The reference on Darboux functions is awesome. Thank you.

Comment by Beni Bogosel

2012-04-08T19:34:17Z

Thank you. I really appreciate your answer.

Comment by Beni Bogosel

2012-04-08T16:30:26Z

You should post this question on math.stackexchange.com, if it hadn't already been posted. The idea is that for any matrices $A,B$ you have $tr(AB)=tr(BA)$, so $tr(AB-BA)$ is always zero.

Comment by Beni Bogosel

2012-04-08T13:05:05Z

I think this was already asked on math.stackexchange.com

Comment by Beni Bogosel

2012-03-20T13:34:51Z

Thank you for your examples.

Comment by Beni Bogosel

2012-03-11T13:48:52Z

I wrote a detailed version of your answer here: mathproblems123.wordpress.com/2009/10/01/…

Comment by Beni Bogosel

2012-03-02T10:39:20Z

Thank you very much for your guidelines and for your answer. I edited my question using your suggestions, and I will say why I needed to know if this result holds in a next edit of my question.

Comment by Beni Bogosel

2012-03-01T17:23:10Z

@Gerald Edgar: Yes. I realize that now. Thank you for your comment.

Comment by Beni Bogosel

2012-02-29T16:38:47Z

@Willie Wong: Thank you for your comment. I'm starting to understand now.

Comment by Beni Bogosel

2012-02-27T11:03:55Z

Thank you for your answer. Can you please give me a reference for the first part, where you say that there is an embedding of a metric space with $n$ points in $\Bbb{R}^n$. I tried the same approach, but the answers i got here: math.stackexchange.com/questions/113727/… said that it is not possible to embed any metric space of $n$ points in $\Bbb{R}^n$.

Comment by Beni Bogosel

2012-02-15T22:05:31Z

Thank you very much. I edited the question.

Comment by Beni Bogosel

2012-02-15T21:02:17Z

The usual brackets {} do not show in math mode. I don't know why is that. That's why I used [].