The independence of random variable is always one tricky question. Yesterday, my friend asks me one question: X and Y1 are independent, X and Y2 are independent, so are X and (Y1, Y2) independent?
I thought that this should be right. But in fact, after one hour I could not prove it, although I have tried different ways. So I turned the mind; MAYBE IT IS WRONG!
In the next 5 minutes, I gave a very simple counter example; Let X and Y1 iid Bernoulli, but Y2 = |X - Y1|. Therefore, we can verify that P(X = 1, Y1 = 1, Y2 = 1) = 1/4, but P(X = 1) = 1/2 and P(Y1 = 1, Y2 = 1) = 1/4.
My friend tells me the prof claims this assertion? What? So I review the question in the class. Ok, the question is about the gaussian, who can be a very special case since its correlation means the dependence.
2016年9月20日星期二
2016年9月9日星期五
La rentrée @ IHES
En 2016, pour se manifester que l'Université Paris Saclay a été déjà comme unité, la rentrée a lieu chez IHES, où tous les élèves de chaque labo y participent. Plusieurs lectures contenant deux cours plus concentrés sont organisées. En plus, les nouveaux post-docs partagent leur enseignements pendant le thèse avec nous.
Je suis deux cours plus "appliqués" - l'arbre aléatoire et le problème de Kakeya. Bon, à vrai dire, j'ai eu l'ambition de reprendre quelques cours pures, mais à la fin, je fais le correct choix. Parceque je comprends bien ces deux cours, la grande ligne et ses techniques. Ils m'inspire aussi de trouver mon propre idée de démonstration, comme le cyclic lemma et le résultat positve dans la localisation dans cube. C'est la maths. On a besoin de non seulement apprécier la beauté, mais attaquer les questions.
Les post-docs nous partagent leur parcours aussi. Un doctorat n'est pas facile mais il désirit les efforts. Sauf que l'on face le choix entre les maths et la famille, ou peut-être de l'argent. La compétition existe toujours, en tous cas, le métier comme un mathématicien attire les jeunes quand même.
Bon, la semaine prochaine sera 3A. C'est parti.
很难想象自己竟然也能很顺利地用法语写日记了。这是一个好兆头。
3天IHES之旅感触良多，虽然这里是圣殿，但是一样，年轻的数学家们还是要面对竞争、清贫的生活、事业和家庭等等等等。
然而我觉得我们是X的学生啊，如果很在乎钱，为什么还来做数学呢？或者如果需要钱，很困难吗？
这是所谓两年GAP的代价，也算是福利吧，是时候启程了。
2016年9月8日星期四
Dynamic system: ODE
Today, I finished the last exam in my 2A the dynamic system. I have to say that it is not easy to prepare for an additional exam, but this time, finally, I got the idea of the ODE and here I would like to say something about it.
The history of ODE is so long that we cannot recall the beginning of this subject and, in fact, the development of this subject is so closed to the analysis. Except the techniques of resolving the specific equation, the existence, unity and continuity are three most important ideas in the development of ODE. That is Cauchy problem, namely we know the initial condition and how it evolves, we should know the state of every time. This idea is so simple and it influences also the philosophy school - determinism. We believe it not only because we can solve some equations, but also because that we know the law - there is no reason why we cannot fix the state in a given point.
The proof of this theorem - I used to find very hard, but just the general version of Picard theorem. In the course of Polytechnique, we use one course to prove all the version of Picard theorem, the simple version, lip-version and C^k version. By this we derive the implicit function theorem and local inversion theorem, useful weapons in the later proof of stability. The Picard theorem in Banach space gives the famous Cauchy-Lip theorem, that is in Lip condition, the solution exists locally.
Although this is just the base, it is enough. The longtime evolution is just this basic point adding some extension theorem. In fact, the terminal of the evolution is necessarily blow-up and before this case, all the trace cannot intersect. By these two points, we can nearly decide all the evolution of ODE.
When talking about the asymptotic behaviors, the perturbation theory, the resolvant theory, and the idea from the geometry are so useful. In the case of periodic theory, generally we know that the period is stable even though there are errors and sometimes we can make the modification to make the exact periodic theorem. The study of the flot or resolvant, which generalize the evolution to study the diffimorphism between space has a high viewpoint mathematical. And also, we use the first return technique of Poincare. By this, we treat the pendulum problem totally, like in this exam. The period is determined mostly by the force but not its own period.
I believe that there are many other problems to answer, like the chaos theory and ergodic theory. I will continue some discovery in 3A.
The history of ODE is so long that we cannot recall the beginning of this subject and, in fact, the development of this subject is so closed to the analysis. Except the techniques of resolving the specific equation, the existence, unity and continuity are three most important ideas in the development of ODE. That is Cauchy problem, namely we know the initial condition and how it evolves, we should know the state of every time. This idea is so simple and it influences also the philosophy school - determinism. We believe it not only because we can solve some equations, but also because that we know the law - there is no reason why we cannot fix the state in a given point.
The proof of this theorem - I used to find very hard, but just the general version of Picard theorem. In the course of Polytechnique, we use one course to prove all the version of Picard theorem, the simple version, lip-version and C^k version. By this we derive the implicit function theorem and local inversion theorem, useful weapons in the later proof of stability. The Picard theorem in Banach space gives the famous Cauchy-Lip theorem, that is in Lip condition, the solution exists locally.
Although this is just the base, it is enough. The longtime evolution is just this basic point adding some extension theorem. In fact, the terminal of the evolution is necessarily blow-up and before this case, all the trace cannot intersect. By these two points, we can nearly decide all the evolution of ODE.
When talking about the asymptotic behaviors, the perturbation theory, the resolvant theory, and the idea from the geometry are so useful. In the case of periodic theory, generally we know that the period is stable even though there are errors and sometimes we can make the modification to make the exact periodic theorem. The study of the flot or resolvant, which generalize the evolution to study the diffimorphism between space has a high viewpoint mathematical. And also, we use the first return technique of Poincare. By this, we treat the pendulum problem totally, like in this exam. The period is determined mostly by the force but not its own period.
I believe that there are many other problems to answer, like the chaos theory and ergodic theory. I will continue some discovery in 3A.
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