Riding on gazelle

Status: On

Mark

April 22, 2013 YiMin Comments off

considering build up a package for my recent research. Will be back to this post later.
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Considering build theory on DtN map for nonhomogeneous source term, it has been thought to have the same property as homogeneous case.
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Considering compressive detecting method for point source problem.
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Considering particle-based transport equation with inverse source problem.

Categories: Numerics

Jam

April 13, 2013 YiMin Comments off

昨天接到g家邮件说今天jam要开始，下午把paper最后一点的问题给解决之后，就看了看题目，便毫无想法吃饭去了。再坐下来看的时候，便胡乱开始写起来，纠结于该用什么语言写，后来竟用了matlab，索性全都是matlab一路写下来，觉得matlab在处理jam的题目上倍感吃力。中间shift去python和haskell，后来还是没能做出来第三题的large set，后来才知道这题是要cheat……白白浪费了90 points。

重在参与了……mark下，rank进350.

Categories: Accessary, Austin Pavilion

Spams rampage

April 10, 2013 YiMin Comments off

最近打开WP，spam疯涨的速度简直不忍直视。去年一年一共600+个spam，今年已经提前完成超过去年的任务。现在每天一般能有10个左右spam。考虑把blog改private权限一段时间，或者限制一下IP之类的。

话说买了一个域名，minfun.info，买之前不知道这info域名这么臭名昭著，毫无办法，而且买之后忘记买host，现在挂在别的server上，担心哪天会挂掉，本来想解析到SF的免费空间上，试过之后发现SF不允许outbound的通信，plugin什么的都要手动安装，好原始，算了。

以后academic的post就两边都post出来，其他的类别的就只发布在这个blog。

Categories: Accessary

A trip to San Antonio

March 24, 2013 YiMin Comments off

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Categories: Accessary

Existence of random zeros::solution

March 3, 2013 YiMin Comments off

[UPDATE] This is the last post for this problem.

A couple of weeks ago, I came up with the solution by using moving frame which was learned in Differential Equation class long time ago. Briefly speaking, it is a good piece of work, in constructing special solution, and a nice try with coordinate transformation.

Inverse Problem: on point source.

Now I simply list my trials in probing the path to the solution.

As I mentioned in last post, I would like to continue using the same technique, with a CGO-like approach, however, after a long time of trials, I gave up with this method. This method will give out a very impressive and concise representation of the problem:
$u\Delta \phi + 2\nabla u \cdot \nabla \phi +k^2 u(n-1) \phi = 0$ ,

after we multiply $\overline{u}$ to the equation, it will morph into divergence form.

$\nabla(|u|^2\nabla\phi) + k^2|u|^2\phi = 0$ .

To my knowledge of this degenerate elliptic equation, we need to apply weighted Sobolev space theory, but unfortunately the theory cannot rule out the existence of singularities, and I came up with a counterexample for that. However, I still believe this method can be promising in 2D case.
For series form, if we require the coefficients of expansion as analytic series, i.e.
$\rho = \sum\rho_j \overline{z}^j$ , where $\rho_j = \sum c_{kj} z^k$

This can give a recursion formula for the problem, but the analytic property will force the problem to be unsolvable. What a pity.

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Categories: Complex Analysis, Doodle, Partial Differential Equations Tags: point source

Existence of random zeros::Explore

March 1, 2013 YiMin Comments off

Last time I considered the problems related to ’existence of random zeros’. And for No.5 problem, we can find a solution.

The governing equation:

$\Delta u + k^2 u =0$ ,

and $u(P(x_j)) = 0$ . Here $P$ is a projection operator onto the xy plane.

Our solution is $u = \prod_{j=1}^m (x+iy - a_j -ib_j) e^{-ikz} = \Phi e^{-ikz}$ , here $P(x_j) = (a_j,b_j)$ , since $\Phi$ is analytic, then $\Delta \Phi = 0$ .

Thus I only have to look at $\mathbb{R}^3$ case, for high dimension spaces, we just project the points onto a lower dimensional one.

For No.4, we just need to consider the solution to No.3

I do not think the randomness can be achieved for No.1 and No.2, but proof needs more work to do.

Existence of random zeros::Problems (zym8903.wordpress.com)

Categories: Complex Analysis, Doodle, Partial Differential Equations Tags: Helmholtz equation

Existence of random zeros::Problems[FINISHED]

February 24, 2013 YiMin 1 comment

[UPDATE:The problem has been solved completely. I posted the rough proof at minfun.info]
Recently I was thinking about the zeros of Helmholtz equation.