considering build up a package for my recent research. Will be back to this post later.
Considering build theory on DtN map for nonhomogeneous source term, it has been thought to have the same property as homogeneous case.
Considering compressive detecting method for point source problem.
Considering particle-based transport equation with inverse source problem.
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。
[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.
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:
after we multiply to the equation, it will morph into divergence form.
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.
This can give a recursion formula for the problem, but the analytic property will force the problem to be unsolvable. What a pity.
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:
and . Here is a projection operator onto the xy plane.
Our solution is , here , since is analytic, then .
Thus I only have to look at 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)
[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.
- Problem 1:
Suppose we have a bunch of points in , say , . Is there a solution of Helmholtz equation
such that .
- Problem 2:
What if in ?
- Problem 3:
[3D case] What if the media is in-homogeneous, the equation turns out to be
where is supported on a compact domain.
- Problem 4:
[ case] of the above one.
- Problem 5:
If we cannot find the solution for random zeros, we define a projection operator , maps points onto a plane.
Then can we find a solution to the Helmholtz equation such that
For Problem 5, I have a solution, but cannot be applied to other ones. Will be recorded next time.