Lx=d: SundayResearch

This is just a side talk and you can skip this. It is a detail about relationship between Fredholm equation of the second kind and the max determinant problem. The basic idea is as following. The discrete form of Fredholm equation of the second kind is a matrix form (a linear system). To get the limit of the discrete form, the dimension of matrix n goes to infinity. The solution of linear system is obtained by a variant of Cramer's rule. This needs the determinant. The absolute value of the determinant relates with the system's convergence. Therefore, the max determinant problem was interesting. Fredholm equation of the second kind is as following. Assume this equation as a discrete problem, the range a,b is n-subdivided, then Set λ (b-a)/n = h, the coefficient of this equation becomes following. When we let the subdivision number n to infinity, the dimension of matrix becomes infinite. When we solve this linear system by Cramer's rule, we need determinant. Please...

I would like to talk about why mathematicians are interested in max determinant problem. This is just my personal theory and I could not find an article that say this directly. So, I warn you that I might be completely wrong. The max determinant problem is mentioned in a context of partial differential equation.  Is a partial differential equation interesting? I safely say, yes. This includes heat and wave problem. We can design buildings, computers, cars, ships, airplanes, ... and so on. There are so many applications of this in our world. Hadamard is one of the mathematicians who contributed the max determinant problem. His one of the interests was partial differential equation. A basic partial differential equation, for instance, a wave equation is like this. This can be re-written as (By the way, if we wrote it as above, an operator d^2/d x^2 looks like to have an eigenvalue λ. Like Mu = -λu. This is a clue of relationship between integral equation and linear algebra.) ...

Abstract Gilbert Strang asked us what is the maximal determinant if the matrix has only specific numbers in his book, Introduction to Linear algebra. I enjoyed this problem for almost three weeks also as a programming problem. So I would like to introduce this problem in this article. Introduction My primary school has words, ``Be one day as one step of your life (一日生きることが一歩生きることであれ．)'' by Yukawa Hideki. These days I finally start to understand these words. I can only do something if I could do every day. Even for five minutes, if I do something every day, I found quite difference. Recently, I joined an activity. It took some significant time from my Sunday research time, though I would like to continue both my activity and my Sunday research. At the end of March, I learn max determinant problem that exists. I didn't have any dedicated time for this problem. But, I use my commune time and elevator waiting time, I solved this problem. (Our company's elevator giv...

It is too short notice, but in case you have a chance and interest... ----- Time    Wednesday, March 16 · 12:00pm - 6:00pm Location    Pappelallee 86 10437 Berlin, bei enishi http://www.facebook.com/event.php?eid=195135683854242&index=1 Liebe Freunde in Berlin, eine Spendenaktion für Japan wird organisiert. Bitte weiter sagen!! OPERATION ONIGIRI Was können wir aus Berlin für Japan tun? Wir sammeln in Berlin die Spende für die Unterstützung der Erdbebenopfer in Japan. Als Dankeschön geben wir den Spendern Reisbällchen (ONIGIRI). Wer praktisch helfen möchte oder wer irgendwas zur Hilfe überlegt, bitte helft Ihr uns beim Kochen (Mit Reiskocher ist besonders hilfsreich). Wer spenden möchte, bitte kommt Ihr vor Ort. Auf jeden Euro/Cent freuen wir uns, es ist eine große Hilfe für Japan. Wer leider keine Zeit dafür hat, reicht es aus, nur Ideen und Tipps uns zu geben, und den Freunden/Bekannte weiterzusagen. Wo genau? Ist noch nic...

One day, human being predicts earthquake and so on, then they use the energy to light schools of poet, music, and mathematics. That day, I would say the human becomes a bit clever. --- Hitoshi Unless they use it to destroy the world. -- Leo On a conversation at Potsdamerplatz, 2011-03-13, Berlin.

gluLookAt implementation in python Here is my gluLookAt matrix computation code by python (numpy) and its test code. This code cares rather comprehensiveness than efficiency, therefore, there are matrix multiplication and matrix transpose. You can remove this if you want to have more efficient code. The test compares the result of this implementation and the OpenGL's computation result with less than 1.0e-6 error. As far as I tested, the result is in the error threshold in Ubuntu 10.04, python 2.6, glx version 1.4 ----- # Copyright (C) 2010-2011 H. Yamauchi # under New (3-clause) BSD license # # get lookat matrix (gluLookAt compatible matrix) python, numpy # # \param[in]  _eye    eye point # \param[in]  _lookat lookat point # \param[in]  _up     up vector # \return 4x4 gluLookAt matrix def getLookAtMatrix(_eye, _lookat, _up):   ez = _eye - _lookat   ez = ez / numpy.linalg.norm(ez)   ex = numpy.cross(_up,...

gluLookAt matrix (why Gimbal lock happens?)  The camera posture is represented by a rotation matrix, and camera position is represented by a translation matrix. Because in OpenGL, the default camera position and posture are defined and we move this camera around in the program. The camera is at origin, look into the Z axis minus direction, and up direction is Y axis plus direction. A rotation matrix and a translation matrix look like the following. where, x : camera basis X axis, ``right'' in the figure, ``side'' in the Mesa program in the last blog entry y : camera basis Y axis, ``up'' in the figure, ``up'' in the Mesa program in the last blog entry z : camera basis z axis, ``Z'' in the figure, ``-forward'' in the Mesa program in the last blog entry e : camera position, ``eye'' in the figure, ``eyex, eyey, eyez'' in the Mesa program in the last blog entry In the Mesa program, eye , lookat , up are given to t...