Laplacian matrix and its eigenvalues

Japanese version

Today, I read a paper, ``A signal processing approach to fair surface design'' (G. Taubin), but I did not understand it. It said that the eigenvalues of a Laplacian matrix like Equation 1 is Equation 2.
Eq 1Eq 2

Today, my friend CR taught me why this is. Actually, it is not so difficult. Here, we can back to the basics of eigenvalues, a characteristics function:
Also we know a Laplacian matrix is a second order differentiation. Therefore, it is
One of the solution of this differential equation is exponential function (since the second derivative is back to the original one except a constant factor). If you remember the Euler's equation, you can get Equation 2.

Additionally, Equation 1 is symmetric. Therefore, its eigenvalues are real and eigenvectors are orthogonal. Moreover, this is a diffusion equation and you can see the relationship with Fourier

Now, I can see why the paper's title is ``signal processing approach.'' But, one who is not familiar with mathematics like me, it is hard to understand the logic as ``The Equation 1's eigenvalues can be written as Equation 2. Therefore, we could decompose the signal...''

Of course, this is easy for the professionals and it is too verbose to write this like this article. However, at least I need this kind of explanations. Thanks CR.


Foldable Computer

Japanese version

If I remember correctly, when I was a high school student, there was a exhibition near a station. The theme was Children's dream and it was provided by a Kindergarten. Each child draws a picture with a title "I wish it was ..." Some already drew a picture to please their parents. Some others were totally free from anything. One of my friend G is a designer and once told me that "A picture by a child is very strong and I sometimes feel I can never overcome their drawing." He also said, "They never consider an object must have some three dimensional constraint. They draw as they feel when they saw a thing, which is totally irrelevant what they saw. Their drawings have overwhelming power." When a child feel "The mouth is big!", then she/he draws a mouth that is larger than the face without any doubt. He even feels some kind of fear in their picture with incredible freedom.

I remember one of them, "Flexible sized television." A boy wanted to have a television, which expands when he pulled it. All his family pulled their television to make it larger in the picture.

For more than 10 years, quite a few times I was disappointed to see a new television. It is just larger and still I can not change the screen size. Still we can not make a resizable TV, a notebook, or iPod. I may buy one if it is resizable.

I remember I told about this story with a friend he passed away last week. We told a lot of things including this TV. I told him other rather boring stories, like "Current computer graphics is not good at to make the scene dirty. A city has no garbage, all the pitching street has no defect, all the windows are perfectly clean, and all the cars are new." I remember he told me "Indeed, the real world is filled by dirtiness. This poster has a fold." when we walked along a super market just has opened. He always listened my such boring stories.

One day, the resizable TV would be invented. Today, I heard about the foldable display. Oh my friend, I miss you that we could not talk with how the world is changing. I never knew we spend such a precious time.

Foldable Displays http://jp.youtube.com/watch?v=nhSR_6-Y5Kg
UIST 2008 (http://www.acm.org/uist/uist2008/)



Yesterday I lost a friend.

I don't know why.

I feel something wrong in this world, but I wish he never left.

I liked him.

The world became a bit less attractive.

All my wish was just him alive.



I miss him.

May rest in peace.

Sorry for our loss.



Japanese version

Last week, I visited to my old friends in Edinburgh and Durham. Walking in the old city with my friends, that was fun holidays.

The history of the train system started in the United Kingdom. Therefore, I thought the train system is also advanced, however, it seems not so. For example, you can see the hand-written cargo numbers. Our train was missing one coach J, and the announce was "Today, we are missing one coach J. If you have a reservation in the coach J, you can find your seat in coach B. If you have a reservation in the coach B and can not find your seat, you can find it in coach F. Sorry for the inconvenience." We thought, wow, in England, the system is so difficult. How the people can manager that? But soon we found out many people were walking around and asked "Do you know where the coach x's seat?" We told a woman about the announce and she said, "What a beautiful organization!" I see why the English comedy is so good.