User xuxuzhu - MathOverflow

Existence and uniqueness of a matrix differential equation with L^1 coefficients

2013-05-24T03:38:21Z

I came across the following differential equation when considering some direct scattering problems: $$ N'_x(x,z)=G(x,z)N(x,z) $$

where $N(x,z)$ is a $2\times2$ complex matrix with variables $x$ and $z$, $N'_x$ is its derivative with respect to $x$, $G$ is the matrix given by $$\begin{pmatrix} 0 & e^{-izx}u(x)\\ e^{izx}\bar{u}(x)& 0 \end{pmatrix} $$ where $u(x)\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$. The boundary condition is $N(\infty, z)=1$.My questions are:Firstly, if I take $z$ to be a real number $s$, does there exist a unique continuous solution of this differential equation? Why or why not? Since general ODE theory requires a better regularity properties on the coefficients, does the theory still hold for the not-that-good coefficients? Secondly, does the solution has a well defined limit as $x\to-\infty$? Why or why not? I would appreciate it if anyone could help me with my questions!

Norm estimation of an area integral

2013-05-24T03:41:47Z

I am solving a certain kind of integral equations using iteration and Volterra series. Now I get a formal solution and in order to prove convergence I need to estimate the $L^1$ and $L^2$ norm of the following integral: $$ f(x,\zeta)=\int_{x\le y_1\leq y_2, y_2-y_1=\zeta}u(y_1)\bar{u}(y_2)dS_2 $$ where $u\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$, $dS_2$ is the surface(line) measure on the hypersurface ${(y_1, y_2), y_2-y_1=\zeta}$. Also denote $\eta(x)=\int_x^{\infty}|u(t)|dt$, and $\gamma(x)=(\int_x^{\infty}|u(t)|^2dt)^{\frac{1}{2}}$, then how can one estimate the $L^1$ and $L^2$-norm of the integral in terms of $\eta$ and $\gamma$?

Solving systems of integral equations using Volterra series

2013-05-21T01:18:49Z

I came across this problem when trying to solve the following integral equations arising in direct scattering: $$ \begin{align} n_{11}(x,z)=1+\int_{-\infty}^xe^{-izy}u(y)n_{21}(y,z)dy, \quad n_{21}(x,z)=\int_{-\infty}^xe^{izy}\bar{u}(y)n_{11}(y,z)dy \end{align} $$

I was suggested to iterate thoses two equations to obtain Volterra series representation. However I am not familiar with Volterra series, so is there anyone who can kindly provide me with some kind of recipes on how to do it? Thank you very much!

A question from Otto Forster's book on Riemann surfaces

2013-04-08T09:51:27Z

I am reading section 14, A Finiteness Theorem of Otto Forster's book Lectures on Riemann Surfaces, and come across a problem on Theorem 14.15 on page 117. In the proof Forster introduces a function

$$F=\det(f\delta_{\nu\mu}-c_{\nu\mu})_{\nu\mu} $$

which is holomorphic, where $f$ is holomorphic, but I don't know why it follows that $F\xi_\nu\mid_Y=0$. I am wondering whether we should replace $F$ with the matrix $(f\delta_{\nu\mu}-c_{\nu\mu})$, but since the proof relies heavily on this claim, I get puzzled. Is there something wrong or am I misunderstanding some stuff? How should I understand this theorem?

Is there an analog of determinant for linear operators in infinite dimensions as that of finite dimensions?

2013-04-04T01:01:27Z

I am trying to find out the essence of what a determinant is. Besides, in finite dimensions, determinant is the kind of numerical invariant that determines the invertibility of a linear operator, but what about infinite case? Is there a similar invariant? Why if not?

Finiteness theorem for first-cohomology group of sheaf of holomorphic functions on compact Riemann surfaces

2013-04-07T03:37:15Z

I have been reading Otto Forster's Lectures on Riemann Surfaces recently, and came across a question on section 15, Finiteness Theorem, which asserts that $H^1(X, \mathcal{O})$ is finite dimensional, where $X$ is a compact Riemann surface and $\mathcal{O}$ is the sheaf of holomorphic functions. Forster proves this by reducing the problem into considering restriction map of Cech cohomology groups given by a shrinking sequence of relatively compact open coverings(this is done by choosing a proper coordinate patch and using Leray's theorem and Dolbeault's lemma), and introduces $L^2$-norms on such groups. Well, first of all, what's the point in introducing the $L^2$-norm, by which I mean, is that suggesting any interesting topological properties such as compactness or anything else? What's the essential problem lying in this theorem? Secondly, is there any other way of proving this finiteness theorem? I would appreciate your help!

What is the physical interpretation of canonical relations?

2013-03-08T21:14:05Z

This question arises when I am studying semi-classical analysis. Since Weinstein's creed claims that "everything is Lagarangian", where a point in the phase space of classical mechanics is just a cotangent fiber, hence Lagarangian, what can we say about canonical relations, which is Lagarangian submanifold of twisted direct product of two symplectic manifolds? As far as I know, if we quantize canonical relations we get unbounded operators between two Hilbert spaces. Why so? I just want someone to figure this out for me, just as the way we can regard a point in phase space as Lagarangian of cotangent bundle. Thanks.

A question on Wiener Process

2012-12-13T02:46:38Z

Suppose we have a Wiener process $W$, and $U_x$ is the amount of time spent below the level of $x$ during the time interval $(0,1)$. How can I calculate the probability density function of $U_x$. Does this has anything to do with local time or arc sine law?

Stephan, could you put it in more details please? Is it something like $\frac{1}{\pi\sqrt{u(1-u)}}e^{-\frac{x^2}{2u}}$?

How to include a .mp file into a LaTex file?

2012-11-15T05:44:40Z

I used metagraf to create the figure I wanted and saved it as a .mp file and I wanted to use the command "includegraphics{**.mps}" to include the corresponding figure but it said wrong, no such file founded.Now my question is how should I carry out the procedure provided I had already drawn out the figure I wanted using metagraf and saved it as a .mp file, so that I can include the figure into my output pdf file?

How to calculate the equivariant cohomology ring of $P^2$?

2012-10-25T02:27:21Z

It is well known that Kirwan's injection theorem gives an ring injection from $H^{\ast}_T(M)$ to $H^{\ast}_T(M^T)$ which is induced by the inclusion $M^T \to M$, where $T$ is a torus acting on manifold $M$ and $M^T$ is the fixed point set of this torus action.

I came across a problem when my professor tried to use Kirwan's injection theorem to explore the ring structure of $\mathbb{CP}^2$. Here $\mathbb{S}^1\times\mathbb{S}^1$ acts on $\mathbb{CP}^2$. The professor just regards $\mathbb{CP}^2$ as a triangle with edges $\mathbb{CP}^1$, with orthogonal axis $u$ and $v$. Then he said on each vertex there is a polynomial since $H^{\ast}_T(M^T)=H^{\ast}(M^T)\otimes\mathbb{C}[u,v]$. Suppose the triangle is put with two orthogonal edges parallel to the axis $u$ and $v$. Then for the two vertex on the edge of $u$ direction, set $u=0$ to obtain the relations between coefficients. For the case $\ast=2$, each vertex has a polynomial of the form $au+bv$. So there would be 6 unknowns with 3 equations, which gives the rank of $H^2_T(M^T)$ to be 3, same for $H^2_T(M)$.

Now my questions are: Firstly, how should I understand the view of $\mathbb{CP}^2$ as a triangle sitting in the orthogonal coordinate system, and why the $u$ and $v$ here coincident with the coordinate axis? Secondly, what is the intepretation of setting $u=0$ when we are trying to find the structure of the cohomology ring? Hope someone can help me with those questions.

How do you intepret "kill a cohomology class" intuitively for attaching an n-cell?

2012-10-06T03:10:56Z

I ran into this problem when studying Morse theory. My professor referred to the torus and riemann surface of genus 2 as an example. And after some manipulating on the long exact sequence of cohomology groups he came to the conclusion that attaching an n-cell(passing a critical point of index n) would either kill a class of m-1 or give birth to a class of m. Denoting $\mathcal{M}^{\pm}=f^{-1}(-\infty, p\pm\epsilon)$, and the index of critical point p is m, here is the long exact sequence of cohomology group:

$0\rightarrow\cdots\rightarrow H^{\ast-1}(\mathcal{M}^-)\rightarrow H^{\ast}(\mathcal{M}^+,\mathcal{M}^-)\rightarrow H^{\ast}(\mathcal{M}^+)\rightarrow H^{\ast}(\mathcal{M}^-)\rightarrow H^{\ast+1}(\mathcal{M}^+,\mathcal{M}^-)\rightarrow\cdots\rightarrow0$

My first question is, is it true that the map $H^{\ast}(\mathcal{M}^+,\mathcal{M}^-)\rightarrow H^{\ast}(\mathcal{M}^+)$ is either injective or the 0 map? Why or why not? And also why $H^{\ast}(\mathcal{M}^+,\mathcal{M}^-)$ is $\mathbb{Q}$ for $\ast=m$ and 0 otherwise?

Secondly, if this is true, we will arrive at the desired results. Either

$0\rightarrow H^{m-1}(\mathcal{M}^+)\rightarrow H^{m-1}(\mathcal{M}^-)\rightarrow0\rightarrow \mathbb{Q}\rightarrow H^{m}(\mathcal{M}^+)\rightarrow H^{m}(\mathcal{M}^-)\rightarrow0$

$0\rightarrow H^{m-1}(\mathcal{M}^+)\rightarrow H^{m-1}(\mathcal{M}^-)\rightarrow\mathbb{Q}\rightarrow0$

Thus the "either kill or give birth" statement is obtained. But what I cannot understand is how to view this inuitively, say, how can you tell whether attaching a cell will kill a class or give a new class from a geometric point of view? My professor tried to explain this using torus but I still fail to see it. I just didn't know how he managed to "see" it. I would really appreciate it if someone could kindly tell me how to see it. Thanks.

Comment by xuxuzhu

2013-04-10T03:48:20Z

Thanks a lot! I think the two books you provided seem to be much more readable for me.

Comment by xuxuzhu

2013-04-08T16:15:17Z

Actually I asked the very same question again, and corrected the terrible math writing by simply adding "`" in front of every dollar sign just as suggested. I don't know why, and it seems to happen when you type subscripts。

Comment by xuxuzhu

2013-04-08T16:11:27Z

Thank you so much, Professor Ben Mckay. I will check this out. Actually, I am taking part in a reading course where Forster's book is assigned as the textbook. It's a wonderful book, despite those two problems I have asked, and maybe more. Since you are both familiar with Forster's book and with Riemann surfaces, is there any other nice books you can recommend me to take as a reference? I really appreciate your help and hope to hear from you!

Comment by xuxuzhu

2013-04-08T09:37:27Z

Thank you so much!

Comment by xuxuzhu

2013-04-06T02:21:02Z

Thank you Professor Israel. Now everything is perfectly understandable!

Comment by xuxuzhu

2013-04-06T02:19:46Z

Thank you so much Qiaochu. It just occurred to me that I've run into the stuff of viewing determinant as actions on top exterior forms when studying smooth manifolds, but your comments for infinite case is really inspiring and I really appreciate your help!

Comment by xuxuzhu

2012-12-13T05:06:26Z

Could you put it in more details please? Is it something like $\frac{1}{\pi\sqrt{u(1-u)}}e^{-\frac{x^2}{2u}}$

Comment by xuxuzhu

2012-10-27T05:37:54Z

I must thank you for your help and the article you provided. Meantime please allow me to apologize for those stupid mistakes I made when asking questions

Comment by xuxuzhu

2012-10-07T17:56:52Z

Thank you for the help. This is exactly what I can understand, but I still what some more ituitive ways to "visualize" through an examine of some easy examples such as the torus or something. Thank you all the same for such a kind help!

Comment by xuxuzhu

2012-10-07T17:54:19Z

Thank you Marco! I found your answers most inspiring and detailed. I think I have got what I want. Thank you very much!

Comment by xuxuzhu

2012-10-07T17:53:10Z

Thank you for your answering of my first question. I have already been reading your book and find it quite helpful!