黄飞鸿系列影评2

    近日迷恋上一些旧物，比如老电影老电视剧，比如老版央视的水浒传，比如徐克的黄飞鸿。虽然小时候都看过，不过那会儿就是真是当动作片看看的（当然大多数人也就当动作片看看了），现在回过头，觉得那时电影电视剧的剧本、场景、改编都还是相当不错的吧。

    徐克的黄飞鸿系列共六部，比较出彩的一般认为也就是前三部。时间跨度大概是19世纪末到20世纪初，如果一气呵成应该能看出个整体脉络。我这次回顾就不按常理出牌了先从第二部开始看吧。

    第二部黄飞鸿虽然是主角，但其实一直在用一个旁观者的视角观察着那个变革的时代：黄飞鸿因学术交流机会前往省城，时值戊戌变法阶段（影片初提示公车上书），白莲教闹事，还有孙中山建立同盟会这些事件结合在一起。白莲教的扶清灭洋、清政府的纵横捭阖、革命党人的民主独立几股势力交织在一起。期间还夹杂了中西医学学术交流、留洋儿童、黄师傅的情感线等多条线索，整个故事都立显饱满。

    当然，熟读历史的我们应该知道，这些故事其实是不应该发生在黄师傅去省城参加学术会议这几天的。留洋儿童是洋务运动时1860-1890的故事，虽然后来还有，但是应该不至于如最初那种危急存亡的感觉了。孙中山离开广州是1895年，比公车上书其实还早三年。当然纳兰元述和革命党人的矛盾应该是真实的，纳兰基本就是开明的封建阶级，上限保皇党，最多接受民主专制即康有为那一套，孙文的理念肯定是与他相左的。

    黄师傅是哪一派呢？其实黄师傅活在小县城，看出来对这些新鲜事物接受能力也没那么强。但身为医生治病救人还是相信科学的，对怪力乱神的白莲教肯定是没好感的，毕竟只靠愚昧是不可能救国的，就算政治纲领提的再正确呢？对外国人，也倒没那么抵触，毕竟开放码头到那会儿都五六十年了，想来大家都习惯了。至于保皇还是民主？我觉得黄师傅是没想过那么多的，求振兴中华的念头肯定还是有的，可是政治觉悟有多高真还就没看出来。不过是因为孙文和他同是医生，感觉更对路些，纳兰虽然反白莲教，但专制精神太足做事不择手段，黄师傅所做一切基本只是出于义气加部分报恩，恰好顺应了历史潮流吧。

    比如假设纳兰收纳了那些孩子呢？黄师傅是不是就转手帮着消灭革命党人呢？不会，估计黄师傅和十三姨他们第二天直接火车回家了……

    最后说说影片为什么成功呢？都说侠客梦千年一叹，士人都有那么一个游侠梦。妙手仁心+武功卓绝，济世救人还能惩奸除恶，这就满足了大部分人的代入感了。至于爱国情结，其实大众也不在乎那些政治纲领，只是朴素的国富民强就好了么。但这一切都来的太理想主义，现实是只靠侠客那是改变不了民族命运的，还是要开化、要启发民智、要纲领、要科学、要手段。

    向那个风雨飘摇年代的先驱们致敬了。

Measure in metric space 1 : The structure of continuous function

Recently, I start to study the topic of random geometry, which deals the convergence of some interesting geometric objects in the space of probability. That is to say the value of the random variable is sometimes the geometric object and there is some space very interesting but also strange in the first glance like Gromov-Haussdorff space. But how to define the convergence in this sense? After all, we have to restart from the base.

Generally, we define the measure in metric space as the duality of the continuous and bounded function. To reach this point, at first we have to learn something about the structure of continuous function in metric space, or more generally the Haussdorff locally compact space.

Two theorem are the bases: the theorem of Urysohn and the theorem of Tietze. The theorem of Urysohn tells us that in the normal space X and two closed set E,F , we can define a continuous function who takes 1 in E and 0 in F. This generalizes the linear function or hat function in dimension 1. Then the Tietze theorem tells us, given a continuous function defined in the closed set E of X, we can extend it in the whole space X as a continuous function, which is so naive in R.

A power application of this two theorem is that, in fact, we can define the plateau function in metric space. I believe that if one has learned some modern analysis must know the importance of the plateau function in the analysis. The convolution and the technique like localisation all come from here. To prove it, we have to observe that: T2 + compact = T4. In locally compact Haussdorff space, we can always add one open set O between the compact set K and an open set U who contains the compact, moreover, the closure of O is also contained in U. Then, the Urysohn gives the plateau function support in the closure of O.

A tricky lemma about the possibility to divide the compact K in two compact K1, K2 which belongs to U1 and U2 respectively and the union of U1 and U2 covers K. The proof is a little tricky, but it leads the decomposition of unity in locally compact Haussdorff space and then a continuous support compact function can be decomposed in the finite sum of the function of same type. Moreover, in disjoint compact set, we can define a continuous function to joint the simple function, so continuous support compact function is dense in many norms.

That is the first step to understand a profound measure, it is long but I believe that it deserve the hard work to conquer it.

How to guess the bigger number in two hands?

Today morning, I received a question from one of my old friends

Alice writes respectively one number in two hands, you can see one of them, then is there a strategy to guess in which hand is placed the bigger number with a probability bigger than 50%?

His intuition tells my friend that nothing will change even though we know one number because the other has always possibility to change. But the maths tell me that it is not the fact. (Thanks to Polytechnique, I remember that I have done this question in PC but not in the form to try to get the design)

First, we recall that for a fair game, we should have the possibility to see left or right hand. If not, the "cheat strategy" is that Alice shows always the left hand with a smaller number but hides the bigger in the right, then we have no chance to win.

The strategy is simple: we require left hand or right hand randomly with possibility of 0.5, then if it is bigger than 10, we guess this number is bigger. Else we guess that it is small. We neglect the situation that the two number are equal. Moreover, 10 is not essential, we can use any number as a criteria. 

What happens? We note the two number as random variable X, Y. As we require the two number randomly, we can suppose that they have a symmetry distribution. i.e p(x,y) = p(y,x). The domain that our strategy does not work is {Y>X>10} and {10>X>Y}. But our strategy works is the area {X>Y>10},{10>Y>X},{X>10>Y},{Y>10>X}. The first two compensate the negative situation. The last two situation we win.

If you do not believe in it, a simple MATLAB simulation will show our proof is correct.

We can also prove by contradiction. If as what we suppose, nothing has changed. Then, the number showed should be the mean number of the distribution. Then, any number is the mean? This is obviously wrong.

In conclusion, to figure out this question, we should has a basic knowledge about the base of probability. That is, what is the experimental space and what is measurable, what is probability. This also underlies the significance of the information and conditional probability.

PS: If I play this game with my naught cousin and I hope to win as many as possible, what should I do? Firstly, I will use the statistic method to approximate the mean of the distribution, given that we believe the number in two hands are independent. Secondly, I will make the choice by two hands as randomly as possible, at least, he should not know how I guess, or I will lose the game.