今天历时一个学期的EA答辩结束了，随手写几点吧。
1.对一个方向特别感兴趣，和真正钻进去研究和学习还是很不一样的。有时候我们满足于自己在某些方面有点见解，可是真正投入到一个方向上，那绝对是一个全新的战场。自己是菜鸟，而其他人都是遍地老手。这个时候才是检验是否是真爱吧？
2.我就是这样一个例子。想着要做随机几何想了好久，终于有机会上手试试了，上述就是我的一些真实感受。然而，当中有那么一段时间全身心投入，拼了命想折腾点东西的劲头还是感动了自己。以及最后写报告时发现好些证明似是而非，只能一一自己补全，也算是一种科研锻炼吧。
3.终了，还是非常喜欢这个方向。有机会让我再去画些奇奇怪怪的东西。以后还要加油干呢。导师也说现在既然已经略窥门径了，要再接再厉啊。比如说做个什么问题或者证明吧，不能说是等着人家告诉你能不能证明和怎么证明，要是这样岂不是变成了DM了么？
4.答辩的时候，PPT要少弄一点。老师说一分钟看一张，快了大家就不开心了……哦不开心了额。
5.以后读论文，粗读一遍看大意，略读一遍掌握思路，然后必须要精读一遍（如果是钻研论文的话）验算过程啊！！！
6.毕竟后来没有拍照合影。我觉得我还是需要一个自己的成果才能填满欲望。定个小目标，3A结束前写出一篇论文吧。
就说这么多了，加油！
2016年12月4日星期日
Erlang and Jackson Network
This is a note for reviewing the MAP554 and some points about network.
M/M/1, M/M/$\infty$, birth and death
The basic model of queue theory. M/M/1 has just one server and has an invariant measure like geometric law,
$$
\pi(n) = p^n(1-p), p = \frac{\lambda}{\mu}
$$
M/M/$\infty$ has infinite server and Poisson law
$$
\pi(n) = e^{-p} \frac{p^n}{n!}, p = \frac{\lambda}{\mu}
$$
where $\lambda$ is the rate of arrival and $\mu$ the rate of waiting. A more general case can be done like change of power.
$$
\pi(n) = p^n(1-p), p = \frac{\lambda}{\mu}
$$
M/M/$\infty$ has infinite server and Poisson law
$$
\pi(n) = e^{-p} \frac{p^n}{n!}, p = \frac{\lambda}{\mu}
$$
where $\lambda$ is the rate of arrival and $\mu$ the rate of waiting. A more general case can be done like change of power.
Erlang network:
This is just an application for truncated technique. That is if we have already a network with reversible invariant measure, we can generate a new by changing the power of that part. That is
\begin{eqnarray*}\tilde{q}(x,y) &=& C q(x,y), \forall x \in \mathcal{A}, y \in \mathcal{S} - \mathcal{A}\\
\tilde{q}(x,y) &=& q(x,y) , \text{ otherwise }
\end{eqnarray*}
Then the new invariant measure becomes
\begin{eqnarray*}
\tilde{\pi}(x) &=& K\pi(x) , \forall x \in \mathcal{A} \\
\tilde{\pi}(y) &=& KC\pi(y) , \forall y \in \mathcal{S} -\mathcal{A}\\
K &=& \frac{1}{\pi(\mathcal{A}) + \pi(\mathcal{S} - \mathcal{A})}
\end{eqnarray*}
The application is that we make $C = 0$ then the network is defined in just the part $\mathcal{A}$. For example, in the network of route with restriction $\mathcal{R}$, we can just do the case without restriction to get $\pi$, which is just the case of several M/M/1 independent, then we do restriction and normalization.
$$
\tilde{\pi}(x) = K\pi(x) , \forall x \in \mathcal{A}, K = \frac{1}{\sum_{x \in \mathcal{R}} \pi(x)}
$$
Jackson network
A more general model of network is like that. Each station has rate $\lambda_i$ of arrival and $\phi_i(n_i)\mu_i$ rate to tackling the service. Here $\phi_i(n_i)$ can be considered as the power of server, in the case M/M/1 it is always 1 and M/M/$\infty$ it is always $\phi_i(n_i) = n_i$. However, the difference is that after each service of station $i$, it has possibility $r_{ij}$ to go to the station jThe key is to find a equivalent $\tilde{\lambda}_i$ which satisfies that
$$
\tilde{\lambda}_i = \lambda_i + \sum_j \tilde{\lambda}_j r_{ji}
$$
then the station looks like independent and has the invariant measure
$$
\pi(n) = \Pi_i \frac{\tilde{p}_i^{n_i}}{\Pi_{m=1}^{n_i}\phi_i(m)}, \tilde{p}_i = \frac{\tilde{\lambda}_i}{\mu_i}
$$
2016年12月2日星期五
Levy characterization, representation of martingale and change of probability
I am preparing for the final, so I write some notes for the course maths finance.
Levy characterization for Brownian motion:
If $\phi_s^T \phi_s = Id$, then
If $\phi_s^T \phi_s = Id$, then
$B_t = \int^t_0 \phi_s dW_s$
is a standard Brownian motion
This theorem is very useful and it describes the nature that after a con-formal transform, the BM keeps its properties.
Representation of martingale:
This theorem has different version. The most general version is that for a $\mathcal{F}_t$ adapted local martingale $M_t$, it can be written as
$M_t = \mathbb{E}[M_t] + \int_0^t H_s dWs$
where $H_t \in \mathbb{H}^2_{loc}$.
This is a mathematical version of perfect duplication theorem. The proof starts from the case $L^2 \text{martingale} \rightarrow L^1 \text{martingale} \rightarrow \text{local martingale}$. It has many application in the stochastic calculus.
Change of probability:
First we define $Z_T = \exp{(\int_0^T \phi_s dW_s - \frac{1}{2}\phi_s^2 ds)}$. Generally, it's only a local martingale and if it satisfies $\mathbb{E}[Z_T] = 1$, we can define a change of probability
$\frac{d\mathbb{Q}}{d\mathbb{P}} = Z_T$
then under the new probability $\mathbb{Q}$, we can define a new BM in the form
$\tilde{B}_t = B_t - \int_0^t \phi_s ds$
We remark that in the case $\phi$ is deterministic, then the there is no problem since in this case, $Z_T$ is well defined of expectation 1. Otherwise, the expectation is not so clear but there is a theorem Novikov, says that if $\exp{(\int_0^T \frac{1}{2}\phi_s^2 ds)} < \infty$, then all the condition is satisfied.
The change of probability can simplify the formula and has important applications on Monte-Carlo algorithms.
2016年11月25日星期五
聊聊大选
今天我也说一段大选的事情。这个大选不是美国的大选，是我们学校学生会的大选。这不是一个故事，但其实我是想从这一连串的故事讲一个问题。
我们就先说说这个主线故事吧。X学校的每年学生会选举，两组人马全上全下，方式就是大家公投，三年级的票权只有一票但二年级相对权重大有两票（毕竟他们是学校主体，三年级不久就毕业了）。裁判标准就是一周的好吃好喝活动表演。
今年的不同之处是我的直系学妹在某一队里，好多年了终于又有中国同学参选了。学妹人缘应该算是不坏：办活动也积极，人也热心。传统上，中国同学中很热心去当干部的，会被大多数人觉得有点官腔，或者觉得就是过度热心政治，毕竟大学里靠这些活动攒资历的也是有的。不过学妹不管怎么说，办事的时候还是一板一眼大家还算认可的。我自己觉得嘛，不管怎么说选个中国同学到学生会里也算是一件大事，大家至少应该是要上点心的。
末了开票了，学妹的那一队没赢，而且最后知道其实输的也不多，输了10票。因为票数实在接近(667:657)，最后还复查了好多遍甚至推迟了开票时间。
这个10票，被我不幸言中了，连学妹自己最后都承认，还不如输了40票痛快呢。
原因就是这个票，不是免费的。说起来就是公民和人民的关系，要有投票权么平时也要支持学生会工作，大多数人每月大约都会交上个20欧的样子，被戏称为“保护费”吧。而时间拨回一年前，好多现在三年级的中国学生，当年觉得这个费交的不值，因为自己参加活动不多么，所以就没有交。好一笔经济账啊！
我不想就经济账和政治账这两件事情多展开了。本来么，也没人说非要把咱们中国代表投上去对吧？我这动不动上纲上线什么民族荣誉感，肯定有人觉得不舒服（不过如果要定个性，我也毫无疑问给他们扣个没有民族荣誉感的帽子）。即使不谈这些，连话语权都不想要了吗？我当然可以接受说，大家拿着票不投自己同学，觉得那一组不够好，可是放弃投票权又是怎么一回事？这不是大家羡慕了好久的自由民主吗？直接就不要了？说真的，我从去年到今年，就觉得为了这两票的民主，“保护费”还是值得的，哪来没有成本的民主呢？
我要说的第二个故事是关于去找实习的。快到实习季了么，大家都忙。我早早定了实习也就给大家创造机会来个顺水人情，组织一趟去华为数学所参观的活动。我就偶然碰上他们所长的，聊数学聊得开心，然后就提出来了这么一个想法。报名期间拖过ddl的，集合迟到的，这些我就不提了，算大家和我熟也就无所谓了。参观到最后，还有最后一个实验的时候，走了一半人。我当时在想，说好的要来找实习呢？华为现在的水平世界上也排得上号了啊，最后再看一个实验会很久吗？为什么就要抢着那么几分钟就离开了呢？
我要说的第三个故事，关于我们这里一位同学的，当年来法时拿的埃菲尔奖学金，这个奖学金非常丰厚，说起来平日里大家都是很羡慕他们的。我们这位同学今年没有办公交卡，因为觉得今年出门次数不多，平日就管我借吧。我也没说什么，借就借呗。今天这位同学在路上后来被查了，得罚钱，她挺不乐意的，僵了好一会儿，后来在同学提醒下想起卡是我的毕竟还要有交代，才勉强交了钱。
当我讲到第三个故事的时候，我真的已经不知道对于很多人而言，生活除了实习、工作、收入、房子、车子、结婚、生子还有别的吗？
我听过很多故事，比如“人生不能只有眼前的苟且，还有诗和远方”。
比如我初中那年，两位语文老师分别和我们读过“沉默的大多数”和“集体无意识”两个故事，前者说的是文革，后者说的是南京大屠杀。
比如我大学毕业那年，老师和我说出国了要做三件事：读好书，学习了解当地文化和语言，做好组织工作（大家要团结，不能窝里斗）。
时下有文章抨击当下制度培养的所谓精英，大多只是一群“精致的利己主义者”，说更多人似乎得了“空心病”，纯当我是一个不幸的人在异国他乡这么几天里碰上了诸多极端事件。只盼着，这一切都只是妄言，而大多数年轻人如新闻联播里描述的那样，满怀理想和朝气，将时代精神、民族复兴和个人追求结合在一起奋发图强吧。
2016年11月24日星期四
How to get out of the prison?
[Question]: A and B are jailed in two room and throw a coin independently. Then they give a guess, if at least one of them get the right answer, then they can get out, Do they have a strategy to get out of the prison.
[Answer]: Yes. If we let A give the answer as it is, and B reverse the answer, we have four situations: (A, B) = (1,0), (1,1), (0,0), (0,1) and their answer (F(B), F(A)) = (1, 1), (0,1), (1,0), (0,0), at least one will be right.
If we analyse a little, we will find no matter what happens, the expectation of right answer is just 1, what we need to do is to reduce the 0. So we need to make their prediction as two random variable correlated. This is the strategy and we make some try to get the right answer.
[Answer]: Yes. If we let A give the answer as it is, and B reverse the answer, we have four situations: (A, B) = (1,0), (1,1), (0,0), (0,1) and their answer (F(B), F(A)) = (1, 1), (0,1), (1,0), (0,0), at least one will be right.
If we analyse a little, we will find no matter what happens, the expectation of right answer is just 1, what we need to do is to reduce the 0. So we need to make their prediction as two random variable correlated. This is the strategy and we make some try to get the right answer.
2016年11月13日星期日
Random Graph : Erdos-Renyi Graph
OK, Finally I finish finding the way to add maths equation into the blog and this will be my first blog with the beautiful equations.
Recently, in the course of MAP554, we talk about different types of random graphs, a branch very active in maths and its applications since the appearance of the social network. In fact, there exists various types of random graphs to simulate different situations, we will start one by one and in this post we concentrate on Erdos-Renyi Graph
We consider a concrete question: Now the ads is diffused, what's the probability that all the people are influenced? What's the possible size of the largest connected component?
$$
A_k = A_{k-1} + D_{k} - 1
$$
where $D_k$ follows the distribution of $Bin(n - A_{k-1} - k + 1)$.
This description is the most important way and we will repeat that many and many times in the proof. The estimation about the size of Watson-Galton is very standard by the Chernoff inequality:
$$
\mathbb{P}(C > k) \leq \mathbb{P}(A_k > 0) = \mathbb{P}(D_1 + D_2 + \dots D_k > k)
\leq e^{-\theta k } \mathbb{E}[e^{\theta(D_1 + D_2 + \dots D_k )}]
$$
Then, we use the technique to minimize the right term.
We declare directly the conclusion : In the case sub-critical, $np < 1$, the largest connected component is of the order $\log{n}$ with a large probability. In the case super-critical, $np > 1$, with large probability, the largest connected component is of the order $O(n)$, but the second one is of the order $\log{n}$.
Recently, in the course of MAP554, we talk about different types of random graphs, a branch very active in maths and its applications since the appearance of the social network. In fact, there exists various types of random graphs to simulate different situations, we will start one by one and in this post we concentrate on Erdos-Renyi Graph
Definition of Erdos-Renyi graph
The definition of Erdos-Renyi graph is very simple. In graph $G(n, p)$, each edge between vertex $u,v$ has probability $p$ to appear, and for different edges, they are independent. This model simulates well the situation the propagation of virus in the society, or the diffusion of message in a small environment, where every one is same and no limit of distance.We consider a concrete question: Now the ads is diffused, what's the probability that all the people are influenced? What's the possible size of the largest connected component?
The emergence of great connected component
We simulate the propagation of the information in the way like the random walk associated with Watson-Galton branching movement. Each step, we deactiver a vertex and add all the vertex non-explored who has connection with that one. If we note $A_k$ the number active and $D_k$ the number explored in the k-th time. That is$$
A_k = A_{k-1} + D_{k} - 1
$$
where $D_k$ follows the distribution of $Bin(n - A_{k-1} - k + 1)$.
This description is the most important way and we will repeat that many and many times in the proof. The estimation about the size of Watson-Galton is very standard by the Chernoff inequality:
$$
\mathbb{P}(C > k) \leq \mathbb{P}(A_k > 0) = \mathbb{P}(D_1 + D_2 + \dots D_k > k)
\leq e^{-\theta k } \mathbb{E}[e^{\theta(D_1 + D_2 + \dots D_k )}]
$$
Then, we use the technique to minimize the right term.
We declare directly the conclusion : In the case sub-critical, $np < 1$, the largest connected component is of the order $\log{n}$ with a large probability. In the case super-critical, $np > 1$, with large probability, the largest connected component is of the order $O(n)$, but the second one is of the order $\log{n}$.
The probability of a connected graph
The conclusion: $\lim_{n \rightarrow \infty} np - \log{n} = c$, then the probability that the graph is connected is $e^{-e^{-c}}$.
The proof of this theorem uses the first and second moment method, the approximation of Poisson distribution (little number theorem). Something amazing is that in fact, this probability is very close to that of non isolated vertex in the graph. That is to say, when $n$ is big enough, we can consider the two equivalent.
Conclusion: The Erdos-Renyi graph is the simplest model of random graph, since there are no geometric structures, which makes it simple and accessible. When the probability is so small, there are high probability that the graph isn't connected and when it rises, the probability of connectivity climbs, the biggest connected component will also grow from the $\log{n}$ to $O(n)$. Moreover, in some more complicated model like Ising model or percolation, there are also the similar conclusion like sub-critical and super-critical.
Phase transition, the critical point, the asymptotic behavior, scaling limit and continuous model associated, the random walk on the random graph, the mixing time of the former...These will compose my future research. Exciting!
The proof of this theorem uses the first and second moment method, the approximation of Poisson distribution (little number theorem). Something amazing is that in fact, this probability is very close to that of non isolated vertex in the graph. That is to say, when $n$ is big enough, we can consider the two equivalent.
Conclusion: The Erdos-Renyi graph is the simplest model of random graph, since there are no geometric structures, which makes it simple and accessible. When the probability is so small, there are high probability that the graph isn't connected and when it rises, the probability of connectivity climbs, the biggest connected component will also grow from the $\log{n}$ to $O(n)$. Moreover, in some more complicated model like Ising model or percolation, there are also the similar conclusion like sub-critical and super-critical.
Phase transition, the critical point, the asymptotic behavior, scaling limit and continuous model associated, the random walk on the random graph, the mixing time of the former...These will compose my future research. Exciting!
2016年11月9日星期三
BS formula, PDE and Mote-Carlo
In mathematical finance, a key problem is the price of the derivative, for example the price of the option. This semester, I take a standard master course in X, which gives me so much impression: How a system can train the mathematics financial engineer as quickly as possible. After all, there are beautiful maths in it. Here, I would like to take some point.
The answer is BSDE, an advanced method that give a backward method to solve the equation.
In addition, some other option can make the pricing totally into a maths game. The existence and uniqueness of the SDE has also some practical meaning: the count cannot be infinite in debt. So the finance and maths interact one with the other, just like my tutor Pierre does.
Preparation: Martingale, Brownian motion, Ito calculus
Before beginning the study of mathematical finance, some basic knowledge is necessary. Basic probability, large number theory and central limit is far from sufficient, but the the theory Martincale is also important. We can find a previous article in my blog. Then, the trajectory most common in maths-finance is the standard BM. Yes, we can generate some others like Poisson process, Markov process, Levy process and branching process, but the most basic one is the BM, the scaling limit of the simple random walk, who simulate the behavior of large number of investor in the market.
Personally, I believe the research about the random curve itself is interesting enough, but in maths-finance, what makes sense is the Ito calculus, which makes the integration along a random trace has sense. The integration can be defined for H^2 space, local martingale and even semi-martingale, considering adding one term of finite variation. The construction of these space and integration can be seen as a special application of real analysis and functional analysis.
Mathematics weapons: PDE vs Proba, Numeric vs Mote-Carlo, Feyman-Kac
A key formula who plays an important role and make connection of maths-finance with other domain is the Feyman-Kac formula. This formula tells us that, for calculating the expectation of some random variable, we can study a PDE associated using the generator. The inverse procedure is also correct: to treat a type of parabolic equation, we can also study the associated random process.
In practice, these two branches also lead to different realization: numerical solution or Mote-Carlo method. Personally, the Mote-Carlo method is easier to implement, but is it necessarily better? No idea, because after we will see other examples.
Feyman-Kac can also treat the Dirichlet question, which associate not a heat equation but a Poisson equation.
Application in the context of finance
The maths-finance is not a course pure probability, so we cannot ignore the part practical. For example, in the pricing, what's the principal? If we do not make it clear, those weapons will be abused.
There are two principle, one is no arbitrage and the other is perfect replication. The perfect replication is simple, there is no option in the nature, so when we vend one option, we have to prepare some strategy to produce these service so that we can give the money. However, the production procedure follows also one condition self-finance, from what we see the final PDE. Then, the no arbitrage principal makes sense. That is the price should be exact the expectation. If not, the other has arbitrage principal to make money.
When we draw conclusion, we always say that the price is the expectation after actualization under the risk neutral measure. This simple phrase uses the change of probability, the PDE and other knowledge. We pay attention that the price does not depend on the change of the stock! But just the volatility. The drift is in fact erased in the neutral risk. We know, the final object of pricing is that equilibrium but not profit!
Beyond the Feyman-Kac
From the technique part, but also the maths-financial part, the Feyman-Kac is not enough since the generator is always linear. But for some general case? For example, the optimization problem, what we do?The answer is BSDE, an advanced method that give a backward method to solve the equation.
In addition, some other option can make the pricing totally into a maths game. The existence and uniqueness of the SDE has also some practical meaning: the count cannot be infinite in debt. So the finance and maths interact one with the other, just like my tutor Pierre does.
2016年10月31日星期一
How the TCP works ?
In the course probabilistic algorithm, we talk about how the TCP works. In fact, what we do on the course is a reverse engineering, that is, we know the policy taken in our reality and then try to find the philosophy behind.
What the TCP does?
What TCP does is very simple. When we send the pocket on Internet, there are always limits for the bandwidth. The PC will increase the number sent if it receives a success signal in order to accelerate the speed, but divide by 2 to avoid the traffic problem if it receives a signal of failure. Intuitively, this is a good strategy, but what it achieves?
A general model
Inspired by the model in economics, we apply the model of optimization. In this model, we aim at a function of utility and some limits. In our problem, the limit is clearly the physical limit: the sum of flux cannot be greater than the admissible one. But the goal function, in fact, can be very different. There are max-min, proportional function, alpha weight function. The TCP is just one specific case of alpha weighted function.
Technique part 1: primal algorithm
The next problem is that what can we do when facing a general optimization problem. We ca ignore the condition and just treat the relaxed question and apply the primal algorithm. The Lyapunov function and ODE theory assures that this method converges.Technique part 2: Lagrange multiplier and dual method
A second method which treat the problem directly is the theory of optimization. Maybe we know always the method of Lagrange multiplier method, it is correct and always correct given a good function. However, the most profound theory is Kuhn-Tucker theory and a dual problem. If we would like to minimiser the function, we can minimiser given the parameter, and then maximiser the parameter. This procedure always works if we find the Kuhn-Tucker vector, this is true when it has control from the lower bound. So this idea propose the dual algorithm for our question.
2016年10月27日星期四
Greedy algorithm for viral marketing is good engough
The viral marketing is a very mathematical question. We assume that each client has some probabilities to distribute the information to other places, then we need to calculate the minimum number to ensure that all the network is infected after some time. We can just simplify the question such that we choose a community and then consider its neighborhood. Then a naive method is to find the vertex who will add the most number into the community. Is this method smart enough?
The answer is amazing, Yes! If we do like this, we can achieve a solution that infected the network at least constant times of the optimal solution.
In the following, we list the main steps in the proof. We denote F(A) the number of vertex connected. Then a key formula is
F(A \union B) + F(A \intersection B) <= F(A) + F(B)
Then, in the stream of greedy algorithm C_i, the increment is not so bad. That is
F(C_{i+1}) - F(C_i) >= 1\k * (F(C) - F(C_i))
If we change a little this formula, we will get
F(C) - F(C_i) <= (1 - 1\k)^k * (F(C) - F(0))
So the greedy algorithm attends at least 1 - 1/e = 0.632 times the optimal solution, a result good enough in certain situation.
OK, greedy algorithm is not always the best but not bad.
2016年10月26日星期三
To be or not to be? 何去何从？
我就十分偶尔地写一篇文章，来讲讲今天的种种，关于我，关于我的职业，和我的未来。
第一个故事是我和Y老师的日常谈话，她问我为什么数学这么好？我说父母也都没有接受过高等教育，就是小时候去妈妈办公桌上玩玩计算器，然后后来偶然一次学校里竞赛考到了第三名，感觉在人生中的一无是处里找到了那么一点点属于自己的天地。之后的故事也有起起伏伏，终究这么过来了，瓶颈也是一次次地突破过去的，谁说没有撞天花板呢。而且现在感觉也是状态最好的时候，每天都在学习新的知识，不说一日千里吧至少节节高，感觉以后还是能做得不错呢。
第二个故事是看到USNEWS我旦数学再创新高。说实在的，现在对排名这个东西已经无感了：就是个牌子找找工作用的。真的内行人还会有很多标准来衡量水平，这年头骗不了人，而事实上我自己也并没有能力去衡量各位教授的工作，最多从自己喜好和专业出发说上那么一两点。P大同学说他们学校现在有点停滞，祝我们早日冲上前十，我自然客气的说咱们可以携手共进把哈佛普林拉下马啊！但这肯定要靠我们一代啦！
然后我就被第三个故事啪啪啪打脸了。
第三个故事是关于北京拼娃的，无非又是老三样：奥数难，学生苦，然而学校背地里依然以此作为选拔标准。我当时随手转发，评价道：就是清北复交都进了世界前十依旧解决不了这个供需关系啊。
我觉得我的背景对这个问题是颇有发言权的：首先理论上我是竞赛的受益者，但事实上普通教育我又一门没落下的全收了。现在数学也在慢慢转变成职业。我觉得选拔人才无可厚非，但把大家都逼上绝境人人自危这就不对了。大家都在说教育资源不公平，可是那些教竞赛的老师应该水平都不错吧？为什么没有补充到不充裕的教育资源里去呢？如果均贫富，或者至少资源相对平衡，那么不至于发展到这个地步啊。
如果所有人都把稀缺资源作为一种最终变现的手段，那么最终只剩下恶性竞争和内耗而没有真正的发展和进步了。
但故事到了自己身上，又真的不是那么容易了。
第四个故事是今天下午去听报告，挺学长讲了讲博士经历。忽然让我觉得现在正一个劲的想进入的领域，做了那么多工作，会不会也有一天因为掉到物理大坑里，然后大家发现所做的一切都是徒劳？又或者看着科研领域还是重重风险，是不是要及时撤出，找一份体面的工作，领一份丰厚的收入？
说着人到了最后，还是不免会想要自身利益最大化吧？好嘲讽啊。
末了回来的时候，我看到好多同学给我各种留意，又趴在电脑前自我折腾了好一会儿，我想我应该还是不会放弃的吧。
人们说数学的好处的时候，多半会说可以转行啊，教育、计算机、金融、数据科学、通信等等。他们大概也不会知道，在真正选择做数学的时候，也得把这些外在的物质上的诱惑一一抛弃了……
如果说读博士是个高风险，低收益的行当，我们为什么会选择？
人有很多状态，有些状态是人的状态，喜怒哀乐，斤斤计较，当我问这个问题的时候，显然也就是在这个状态。
可是我知道我还有另外一种状态：虽然世界很复杂，人生很艰难，但此时此刻，我只需要专心做一件事情，就是思考。然后就只剩下我和问题，还有为了真理的决斗！
我想，至少读个博，可以有充足的时间，去体验这样一种状态吧。
至于之后呢？我还是相信，在千锤百炼之后，大脑会更加灵光，或许也会洞察很多世间的规律吧，毕竟数学也源于生活么。
到那一天，我应该有足够的能力去改变这个世界吧。至少，让娃娃们小时候能有个快乐的童年，等长大了，咳咳，我教他们真正的数学。
P.S: 新赛季开始了，我最爱的小卡又涨球了。你说篮球打得好也不见得玩得了排球，小前打得好也不见得干得过中锋，前锋位置上历史上还有呆呆老詹，后卫位置上追一辈子帮主唠嗑也未必够得上，努力是为什么呢？可是我还是希望小卡能成为最棒的啊！
Voila, 一样的道理么。数学家练到顶级也不见得能填掉其他方向的坑，分析概率练到化境估计还是不懂代数，可是我依然觉得能把喜欢的事情做到顶级是件很棒的事情！不要拦着我！
P.S: 新赛季开始了，我最爱的小卡又涨球了。你说篮球打得好也不见得玩得了排球，小前打得好也不见得干得过中锋，前锋位置上历史上还有呆呆老詹，后卫位置上追一辈子帮主唠嗑也未必够得上，努力是为什么呢？可是我还是希望小卡能成为最棒的啊！
Voila, 一样的道理么。数学家练到顶级也不见得能填掉其他方向的坑，分析概率练到化境估计还是不懂代数，可是我依然觉得能把喜欢的事情做到顶级是件很棒的事情！不要拦着我！
2016年10月25日星期二
Complex analysis: a review
Recently, I pay two days to review the most basic one variable complex analysis. I have to say that, in fact, I have learned this topic in 2012 (OK, I have learned system in 2011 even earlier but I still have to review them in this year), but in that time, I find that difficult to understand this topic and I got even the worst result during my undergraduate in this course. However, for most people, complex analysis is easier than real analysis. Really?
I think that maybe the difficulty is that, the real analysis is more quantitative but complex analysis has more structure. If we have well understood the differential calculus and some topology, this topic will be obvious. I recommend this Note as a reference.
0. Angle, multiple value function, logarithm
The first difficulty is to define the logarithm. The angle of a complex is clear, but why we need complex logarithm to make it sense by striping one line? In fact, we always hope to make a function continuous, but since the angle is a multiple value function, to achieve we have to pay something. Then the log is well defined and so is the power.
1. Differential, analytic, confromal, holomorphic function
This four words represent in fact the same sense in complex analysis. The derivative can be understood as the form on C and if we know the sense in Banach space, we know that it is just a linear approximation in distance. But in complex situation, it has other senses, like the Cauchy-Riemann equation, the conformal map keeps the angle and the real part and imaginary part are all harmonic, so we have all the property from harmonic function.
Thanks to the Cauchy integral, we pass the differential to analytic. So the regularity is not C^1, C^2 but C^infinity. What an amazing result! So we would like to know how to prove that a function is analytic? A strong weapon is Motel lemma, who says that if the integral along any triangle is 0, this function is analytic. We can develop other methods from this, like passing the limit from a series of functions.
2. Singularities and Laurent series
The opposite side of analytic is singularity. What the other side of analytic? There are three type, removable singularity, pole and essential singularity. The removable singularity says that we can redefine the value on the singularity. The pole is the case that limit infinity so its inverse is 0. The essential singularity says the value around it is dense in C! Notice that all the case discussed above is the isolated singularity. Well, to define something in any form makes no sense.
Then the Laurent series give a criteria for these singularities. If it has only positive part, it is analytic. If it has some negative part, it is pole. But if it has infinite negative term, it is an essential singularity.
3. More about integral - homotopy and residue theorem
A theorem, which is always correct says that along two homotopic trajectories the integral is the same. This strong property makes the connection between the complex analysis and topology. So we have a stronger version of Cauchy integral and even the residue theorem. That is when we make the integral, we do this only in the part around the singularity.
We apply this technique also in counting the zeros and the argument of angle.
4. Reimann mapping theorem
Last but not the least, a powerful theorem, any simply connected domain have a conformal mapping to disk.
Why I revise complex analysis? Recently, the probability theorem has a strong connection with the complex analysis. I believe that these basic elements will help me understand those delicate theory.
I think that maybe the difficulty is that, the real analysis is more quantitative but complex analysis has more structure. If we have well understood the differential calculus and some topology, this topic will be obvious. I recommend this Note as a reference.
0. Angle, multiple value function, logarithm
The first difficulty is to define the logarithm. The angle of a complex is clear, but why we need complex logarithm to make it sense by striping one line? In fact, we always hope to make a function continuous, but since the angle is a multiple value function, to achieve we have to pay something. Then the log is well defined and so is the power.
1. Differential, analytic, confromal, holomorphic function
This four words represent in fact the same sense in complex analysis. The derivative can be understood as the form on C and if we know the sense in Banach space, we know that it is just a linear approximation in distance. But in complex situation, it has other senses, like the Cauchy-Riemann equation, the conformal map keeps the angle and the real part and imaginary part are all harmonic, so we have all the property from harmonic function.
Thanks to the Cauchy integral, we pass the differential to analytic. So the regularity is not C^1, C^2 but C^infinity. What an amazing result! So we would like to know how to prove that a function is analytic? A strong weapon is Motel lemma, who says that if the integral along any triangle is 0, this function is analytic. We can develop other methods from this, like passing the limit from a series of functions.
2. Singularities and Laurent series
The opposite side of analytic is singularity. What the other side of analytic? There are three type, removable singularity, pole and essential singularity. The removable singularity says that we can redefine the value on the singularity. The pole is the case that limit infinity so its inverse is 0. The essential singularity says the value around it is dense in C! Notice that all the case discussed above is the isolated singularity. Well, to define something in any form makes no sense.
Then the Laurent series give a criteria for these singularities. If it has only positive part, it is analytic. If it has some negative part, it is pole. But if it has infinite negative term, it is an essential singularity.
3. More about integral - homotopy and residue theorem
A theorem, which is always correct says that along two homotopic trajectories the integral is the same. This strong property makes the connection between the complex analysis and topology. So we have a stronger version of Cauchy integral and even the residue theorem. That is when we make the integral, we do this only in the part around the singularity.
We apply this technique also in counting the zeros and the argument of angle.
4. Reimann mapping theorem
Last but not the least, a powerful theorem, any simply connected domain have a conformal mapping to disk.
Why I revise complex analysis? Recently, the probability theorem has a strong connection with the complex analysis. I believe that these basic elements will help me understand those delicate theory.
2016年9月20日星期二
Independence of random variable
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.
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.
2016年8月23日星期二
再看《龙门客栈》
上回说到新作龙门飞甲，这次会说旧作龙门客栈。因为之前一文中已经多处对两作进行比较了，加上本作实在太有名，好的影评解读可能也不少，这里就力求能写点新的东西了。
龙门客栈的故事细节线索繁多，主线并不复杂，就是周淮安一伙救了兵部尚书两个孩子，从客栈准备出关，和东厂三大档头斗智斗勇，同时双方极力拉拢金镶玉的故事。末了三方火并，引来大boss曹少钦大决战的故事。
重看旧作，其实也有诸多不足之处。开篇其实一样有一段邱莫言救孩子的武戏，其夸张程度不下于龙门飞甲，而金镶玉一把飞镖杀掉一波人的情景也不在少数。黑旗箭队的武器也是极度YY。所有故事压缩在客栈与其说是结构漂亮，也不排除有条件制约因素。换言之，当年的团队如果有现在的资金技术，电影玩得更浮夸也不是没可能的。
但就是这么碰巧，在当年的制约条件下，他们把一个武侠片拍成了舞台剧并用上了很多戏剧的要素（这个我不是专业，就有这么点感觉）。而其实最棒的就是这部电影里的细节了，比如开始大家滞留客栈，一个原因就是天气。某天晚上雨下一整晚，说起来比龙卷风现实多了，但理由就很充分。比如电影里的对话，“龙门山风雨”好似一个黑话语录大全，一群人一而再再而三的使用着。
人物方面各人性格比后传更加丰富。这里我们还是要说东厂三大档头，整个故事好都因为他们演技太高拉开续作几条街。毕竟，正面角色大多是相似的，但坏人可以各有各的坏法。常言笑是个急性子，最容易发火。小川则是贾廷宠幸，看出来是个gay，他对金镶玉最反感，一句刁不遇”比我们东厂还狠“不幸言中。贾廷则是演技担当，他做的是坏人的事，表面却能不露声色，甚至和正面角色侃侃而谈，如果不说他是要抓周淮安，并看不出来这是个歹角。他看手相本质是想确定是不是周淮安，他做主婚先是一副尴尬，随即爽朗一笑，说自己本无妻无子，身处塞外却忽然能帮金童玉女主婚实乃幸事，这语气中三分客套，却有七分好似由衷而发。有一段千户来访的戏，他和小川说”坏了，捉贼不成反成贼了“，随即小川拿出钱贿赂千户。整个一出戏，周淮安的隐忍，莫言的率直（装东厂只能糊弄一时），金镶玉为人，千户只看钱，常言笑的冲动，还有贾廷小川的手段都一览无余。
从结果看，到了最后其实还是考武力解决了问题，虽然如果没有刁不遇奇兵相助大家就都GG了，但周淮安之前种种为了出关的举动看起来其实意义并不大。中间最复杂的一段，的确看出了周淮安工于心计和金镶玉计算利益，可是两个聪明人算了半天都算不明白东厂终究是不会放过他们，迂回和周旋只是一时之计罢了最终还是要开战的。从这点上说，周淮安对不起莫言，甚至是贺虎铁竹，他的计划最后就是扯淡（两个土匪的担心其实都是正确的）。大婚那一幕我们看到莫言也换上了女装，这个细节其实已经表明了她的态度，可惜周还幻想着他的计划。所以最后邱莫言的死也是带给周淮安和金镶玉很多思考，多少都是两个人自以为聪明的算计让目的最单纯的人买了单。
龙门客栈和龙门飞甲电影倒还真是能无缝对接的。回到上次的话题，如果周淮安和金镶玉对各自行为足够反思，他们后来三年的变化倒也是非常好理解的，正邪对抗其实就是针锋相对地干啊！只是从电影角度，这样的做法，略显单薄了。如果是这样我们要部分原谅龙门飞甲：并非人设不好而是人物发展自然成了这个样子。
影片结束，周淮安的表现对得起无情无义四个字，而金镶玉却依旧念念不忘，这种羁绊和纠缠自然是要到龙卷风之后才算明了啦。
龙门客栈的故事细节线索繁多，主线并不复杂，就是周淮安一伙救了兵部尚书两个孩子，从客栈准备出关，和东厂三大档头斗智斗勇，同时双方极力拉拢金镶玉的故事。末了三方火并，引来大boss曹少钦大决战的故事。
重看旧作，其实也有诸多不足之处。开篇其实一样有一段邱莫言救孩子的武戏，其夸张程度不下于龙门飞甲，而金镶玉一把飞镖杀掉一波人的情景也不在少数。黑旗箭队的武器也是极度YY。所有故事压缩在客栈与其说是结构漂亮，也不排除有条件制约因素。换言之，当年的团队如果有现在的资金技术，电影玩得更浮夸也不是没可能的。
但就是这么碰巧，在当年的制约条件下，他们把一个武侠片拍成了舞台剧并用上了很多戏剧的要素（这个我不是专业，就有这么点感觉）。而其实最棒的就是这部电影里的细节了，比如开始大家滞留客栈，一个原因就是天气。某天晚上雨下一整晚，说起来比龙卷风现实多了，但理由就很充分。比如电影里的对话，“龙门山风雨”好似一个黑话语录大全，一群人一而再再而三的使用着。
人物方面各人性格比后传更加丰富。这里我们还是要说东厂三大档头，整个故事好都因为他们演技太高拉开续作几条街。毕竟，正面角色大多是相似的，但坏人可以各有各的坏法。常言笑是个急性子，最容易发火。小川则是贾廷宠幸，看出来是个gay，他对金镶玉最反感，一句刁不遇”比我们东厂还狠“不幸言中。贾廷则是演技担当，他做的是坏人的事，表面却能不露声色，甚至和正面角色侃侃而谈，如果不说他是要抓周淮安，并看不出来这是个歹角。他看手相本质是想确定是不是周淮安，他做主婚先是一副尴尬，随即爽朗一笑，说自己本无妻无子，身处塞外却忽然能帮金童玉女主婚实乃幸事，这语气中三分客套，却有七分好似由衷而发。有一段千户来访的戏，他和小川说”坏了，捉贼不成反成贼了“，随即小川拿出钱贿赂千户。整个一出戏，周淮安的隐忍，莫言的率直（装东厂只能糊弄一时），金镶玉为人，千户只看钱，常言笑的冲动，还有贾廷小川的手段都一览无余。
从结果看，到了最后其实还是考武力解决了问题，虽然如果没有刁不遇奇兵相助大家就都GG了，但周淮安之前种种为了出关的举动看起来其实意义并不大。中间最复杂的一段，的确看出了周淮安工于心计和金镶玉计算利益，可是两个聪明人算了半天都算不明白东厂终究是不会放过他们，迂回和周旋只是一时之计罢了最终还是要开战的。从这点上说，周淮安对不起莫言，甚至是贺虎铁竹，他的计划最后就是扯淡（两个土匪的担心其实都是正确的）。大婚那一幕我们看到莫言也换上了女装，这个细节其实已经表明了她的态度，可惜周还幻想着他的计划。所以最后邱莫言的死也是带给周淮安和金镶玉很多思考，多少都是两个人自以为聪明的算计让目的最单纯的人买了单。
龙门客栈和龙门飞甲电影倒还真是能无缝对接的。回到上次的话题，如果周淮安和金镶玉对各自行为足够反思，他们后来三年的变化倒也是非常好理解的，正邪对抗其实就是针锋相对地干啊！只是从电影角度，这样的做法，略显单薄了。如果是这样我们要部分原谅龙门飞甲：并非人设不好而是人物发展自然成了这个样子。
影片结束，周淮安的表现对得起无情无义四个字，而金镶玉却依旧念念不忘，这种羁绊和纠缠自然是要到龙卷风之后才算明了啦。
2016年8月20日星期六
说部今天看的惊悚片The shallows
今天和同学去影院随手挑了部电影看的，之前在YouTube上看过预告片，大概是讲人和鲨鱼搏斗的故事，那么大家大概知道这类电影最终人还是会获得胜利的，只不过看个过程有多惊悚罢了吧。
这部电影节奏把握都非常棒，故事基本是从女主游泳、受伤，爬上小礁作为第一阶段，这个时候女主还处在受伤和惊恐中。之后自己治疗、等待流浪汉救援、等待游泳同伴救援属于第二阶段，此时女主基本有了自保能力，却依旧还期待着外界的帮助。等到这些队友都不幸丧生后，她拍摄了视频，这是一个标志：誓死一搏爬上灯塔。然后就是第三个阶段，在最后的挣扎无解后女主选择和鲨鱼搏杀并最终干掉鲨鱼脱险。
作为一个曾经和同学盲目爬山差点死在小山丘上的人，对女主临终拍摄视频这一手法非常深有体会。拿个时候的绝望就是想着录下点什么，好留给自己亲人，又好像有千言万语，和以前伤害过的人来个和解，和家人道句对不起等等。然后自然就是孤注一掷求生！
女主有两处表现十分惊艳：一个是她的另外队友看到他划过来的时候，这时她第一反应是让他们赶紧走不要过来，因为鲨鱼危险。看的出来即使醉汉没有帮她，她依然对醉汉的死有点愧疚所以看到好队友第一反应是非常善意的。第二处是最后斗鲨鱼，这段很燃。她发射两枚信号弹毛线用都没有，然后就把心一横索性把枪射向鲨鱼来个鱼死网破，这个时候的女主已经不只是女主了而是十足的复仇女神，手上各种道具玩得飞起拈花飞叶皆可伤人。
人的精神力量是强大的，我们在生活中面对的深渊、大海、鲨鱼不见得是自然界的，但挑战未必更小，此时也要咬紧牙关奋力一搏吧。当然，我想故事除了歌颂人类顽强的求生意志，还是想告诉大家提高安全意识no zuo no die。人类是地球主宰只在工具齐全的前提下，跑去荒山野岭分分钟被教做人，何况面对鲨鱼这种自然界进化的高级杀戮机器？
2016年8月19日星期五
龙门飞甲——武侠与维数无关
这个标题是用来呼应当年《龙门飞甲》号称首部3D武侠巨制。昨日看了黄飞鸿就想到刘洵老爷子，想到他在龙门客栈里的表现，然后想起这部电影至今还没看，就YouTube扫了一遍。整体说来这就是一个搞着噱头、带着情怀的续作，它在艺术性和故事性上都比不上原作，但好在最后还保留了那么一丝丝精髓以及毕竟大制作带入了些许 亮点，所以即使是原作粉，也还能把它姑且视作是一部续作吧。
【情节篇】
首先本片虽然还打着《龙门客栈》续作的名号，并且赵怀安和凌雁秋分明就是当年的周淮安和金镶玉，但整个故事中客栈已经不是主体了。当年龙门客栈在短暂的背景交代后，东厂、武林侠士、金镶玉三伙人甚至可能还包括千户，就在一个狭隘的空间里面展开了角逐。在龙门飞甲中，我们依然还看到客栈，可是因为没有了金镶玉在各方势力中的周旋，客栈的意义早已不复当年了。所以故事基本已经单薄的只剩下正邪大对抗。
这里不得不说整个电影的前三十分钟，几乎就是各种正反派人物一一登场亮相然后捉对杀一遍。对于喜欢直来直去的观众这是很好的，因为交代的清楚么，小学生都看明白了谁好谁坏。相反的，譬如很多年前的我看《龙门客栈》开始是没看出正反派的，因为前半段一直在压在积蓄力量，可是细细品味《客栈》里的每一句台词和每一个动作都有信息量，所以即使不大打出手，看着比动手了还过瘾，而到后半段爆发出来的时候才更显张力。现在《飞甲》上来就打打打，还打得没什么道理，直接就没有了布局和铺垫。我这么理解，这是为了在秀3D，告诉大家“我们的故事是真3D不是伪3D”，所以雨化田和赵怀安的前三十分钟属于智商下线为打而打（当然他们整部影片智商也没怎么上线过）
到了中间一段，此时故事终于回到了客栈，也有不少桥段致敬前作。可惜致敬就已经说明了无法超越前作了，风里刀算是一个惊喜，也是唯一稍微体现了一点点当年《客栈》的情节，可是假扮的桥段怎么看也觉得水平不是那么高，以及最后效果也不是很明显。
最后一段扯到寻宝夺宝，我倒觉得这一段还不如多展开多铺开一些，因为此处“各有各的小算盘”才是当年《龙门客栈》的精髓所在，而之前大家轻轻松松被赵怀安统一起来和西厂干架完全没有逻辑。
所以我们总结一下，这个故事的第三方势力是欠缺的。寻宝党们固然是第三方，在抉择上本身和两方独立，又主观受制于赵怀安，和当年金镶玉左右徘徊最后选择上是有那么些差距的。
【人物篇】
说到人物篇，老一辈的两大角色周淮安和金镶玉是“崩坏”的，前者直接热血起来要日天日地，全没有了当年的刚柔并济，后者也玩起了江湖女侠，除了黑话已经再也没有了从前的印记。在故事的前半段，我们看这样的“崩坏”是有点不适应，但最后的一点点蛛丝马迹至少让我接受了这样的“崩坏”，这个我们留待最后一个章节说。
大反派雨化田只剩下武功高强这一个设定了，其实这倒也无关，以前曹少钦除了武功还让我们记得其他了么？手下几个其他角色在塑造上也完全不如刘洵、熊欣欣、吴启华那么让人印象深刻。
所以唯一的亮点落到了新一代三人组顾少棠、常小文、风里刀三个人身上。三人是寻宝党，顾使大刀会暗器，常用毒药环形刀，风里刀则直接只剩嘴皮子。这个组合是有意义的。我们概念中老一辈侠客是个什么形象？一袭长衫，剑胆琴心，国仇家恨，儿女情长，金庸古龙梁羽生笔下的形象多多少少都得沾那么一点点吧？用陈平原先生的说法，这是“士”这个阶级在文学作品中的理想投射。而寻宝三人组从兵器看，就已经直接是实用主义当家了，精通个暗器毒药就能横行江湖了何苦闭关练功呢？挑拨离间狐假虎威就能全身而退，这更加是孙子兵法的上上策啊。再看标的，“只谈买卖不谈感情”，“买不到的不稀罕”则直接摒弃了从前的大侠枷锁。
按以前的小说路数，这样的角色能成为主角之友就不错了，但偏偏在本片大放光彩。素慧容捅刀子，顾少棠直接赏她一把飞镖，简直大快人心。这是时代的产物，和平年代的年轻人哪还理解什么天下社稷呢？摆平老板攒够钱大把花可能才是大多数人的理想吧，所以新时代的侠客就这么大大咧咧应运而生了。
【都说武侠是个情义，而不是线性空间维数、拓扑维数或者Haussdorff维数】
武侠披上3D其实意义不大或者效果相反。武功再高，能够飞天遁地吗？能够脱离地球引力吗？这不是美国科幻电影（何况漫威那套基本是软科幻、真奇幻、伪科学），所以拼命加特效是没有意义的，倒不如实实在在把那些武打套路练好。《龙门飞甲》也是中了这个毒，我们姑且就宽容认为徐老怪是在练兵吧。就客观而言，这烧钱的特效烧出来效果不及之前的《七剑》、低成本的《绣春刀》。雨化田武功高强的3D渲染比不上甄子丹一把长剑密不透风的狠辣，特效龙卷风把大家圈上天也没有邱莫言一点点陷入沙海看得揪心。
所以说武侠和维数没有关系，讲的还是个情义。
本片失败之处，是没有了这个“义”字。《龙门客栈》侠士千里护送忠臣遗孤体现的是个“义”，这也是当时全片的主轴，而不是正邪杀杀杀。《龙门飞甲》换成了护送宫女，且不说最后发现大家被耍了这是个圈套，就真是个怀孕宫女，好像也的确是触犯条规的吧？又或者这个“义”字体现在何处呢？没道理的拔刀相助就是有点点莽撞和冲动了。
而本片最后能让我们认可为续作的地方，却恰恰还有那么一点“情”在里头。这也是凌雁秋作为老江湖，一再想告诉年轻一辈的。我们之前说寻宝三人组的性格，有那么一点点年轻时候金镶玉和周淮安的意思，可性格里唯独缺了邱莫言那一道。夺宝组是年轻人“不知者无谓”，邱莫言虽然有时候也有些许天真的一面，但更有一种“明知不可为而为之”的想法在里头。
所以当我说到周淮安和金镶玉人设“崩坏”了却最终接受，是发现在两个人对话中始终还是绕不开邱莫言。金拿着她的笛子，穿上她的行头，努力活成她的样子；周则是扔掉城府，不计代价，用邱的方式做一回侠客。
赵怀安让凌雁秋远走他方，说该去哪逍遥就去哪，凌雁秋回来和他并肩作战。
凌雁秋说自己远走他方逍遥自在去了，赵怀安醒来立刻追过去。
英雄气短，儿女情长，什么相濡以沫不如两忘于江湖都是骗人的，放不下就是放不下，做完好事杀掉boss，为什么不可以在一起长相厮守？
（P.S.素慧容再次证明聪明人装不来，而傻白甜都是骗人的，心机婊的很，远离远离……）
【情节篇】
首先本片虽然还打着《龙门客栈》续作的名号，并且赵怀安和凌雁秋分明就是当年的周淮安和金镶玉，但整个故事中客栈已经不是主体了。当年龙门客栈在短暂的背景交代后，东厂、武林侠士、金镶玉三伙人甚至可能还包括千户，就在一个狭隘的空间里面展开了角逐。在龙门飞甲中，我们依然还看到客栈，可是因为没有了金镶玉在各方势力中的周旋，客栈的意义早已不复当年了。所以故事基本已经单薄的只剩下正邪大对抗。
这里不得不说整个电影的前三十分钟，几乎就是各种正反派人物一一登场亮相然后捉对杀一遍。对于喜欢直来直去的观众这是很好的，因为交代的清楚么，小学生都看明白了谁好谁坏。相反的，譬如很多年前的我看《龙门客栈》开始是没看出正反派的，因为前半段一直在压在积蓄力量，可是细细品味《客栈》里的每一句台词和每一个动作都有信息量，所以即使不大打出手，看着比动手了还过瘾，而到后半段爆发出来的时候才更显张力。现在《飞甲》上来就打打打，还打得没什么道理，直接就没有了布局和铺垫。我这么理解，这是为了在秀3D，告诉大家“我们的故事是真3D不是伪3D”，所以雨化田和赵怀安的前三十分钟属于智商下线为打而打（当然他们整部影片智商也没怎么上线过）
到了中间一段，此时故事终于回到了客栈，也有不少桥段致敬前作。可惜致敬就已经说明了无法超越前作了，风里刀算是一个惊喜，也是唯一稍微体现了一点点当年《客栈》的情节，可是假扮的桥段怎么看也觉得水平不是那么高，以及最后效果也不是很明显。
最后一段扯到寻宝夺宝，我倒觉得这一段还不如多展开多铺开一些，因为此处“各有各的小算盘”才是当年《龙门客栈》的精髓所在，而之前大家轻轻松松被赵怀安统一起来和西厂干架完全没有逻辑。
所以我们总结一下，这个故事的第三方势力是欠缺的。寻宝党们固然是第三方，在抉择上本身和两方独立，又主观受制于赵怀安，和当年金镶玉左右徘徊最后选择上是有那么些差距的。
【人物篇】
说到人物篇，老一辈的两大角色周淮安和金镶玉是“崩坏”的，前者直接热血起来要日天日地，全没有了当年的刚柔并济，后者也玩起了江湖女侠，除了黑话已经再也没有了从前的印记。在故事的前半段，我们看这样的“崩坏”是有点不适应，但最后的一点点蛛丝马迹至少让我接受了这样的“崩坏”，这个我们留待最后一个章节说。
大反派雨化田只剩下武功高强这一个设定了，其实这倒也无关，以前曹少钦除了武功还让我们记得其他了么？手下几个其他角色在塑造上也完全不如刘洵、熊欣欣、吴启华那么让人印象深刻。
所以唯一的亮点落到了新一代三人组顾少棠、常小文、风里刀三个人身上。三人是寻宝党，顾使大刀会暗器，常用毒药环形刀，风里刀则直接只剩嘴皮子。这个组合是有意义的。我们概念中老一辈侠客是个什么形象？一袭长衫，剑胆琴心，国仇家恨，儿女情长，金庸古龙梁羽生笔下的形象多多少少都得沾那么一点点吧？用陈平原先生的说法，这是“士”这个阶级在文学作品中的理想投射。而寻宝三人组从兵器看，就已经直接是实用主义当家了，精通个暗器毒药就能横行江湖了何苦闭关练功呢？挑拨离间狐假虎威就能全身而退，这更加是孙子兵法的上上策啊。再看标的，“只谈买卖不谈感情”，“买不到的不稀罕”则直接摒弃了从前的大侠枷锁。
按以前的小说路数，这样的角色能成为主角之友就不错了，但偏偏在本片大放光彩。素慧容捅刀子，顾少棠直接赏她一把飞镖，简直大快人心。这是时代的产物，和平年代的年轻人哪还理解什么天下社稷呢？摆平老板攒够钱大把花可能才是大多数人的理想吧，所以新时代的侠客就这么大大咧咧应运而生了。
【都说武侠是个情义，而不是线性空间维数、拓扑维数或者Haussdorff维数】
武侠披上3D其实意义不大或者效果相反。武功再高，能够飞天遁地吗？能够脱离地球引力吗？这不是美国科幻电影（何况漫威那套基本是软科幻、真奇幻、伪科学），所以拼命加特效是没有意义的，倒不如实实在在把那些武打套路练好。《龙门飞甲》也是中了这个毒，我们姑且就宽容认为徐老怪是在练兵吧。就客观而言，这烧钱的特效烧出来效果不及之前的《七剑》、低成本的《绣春刀》。雨化田武功高强的3D渲染比不上甄子丹一把长剑密不透风的狠辣，特效龙卷风把大家圈上天也没有邱莫言一点点陷入沙海看得揪心。
所以说武侠和维数没有关系，讲的还是个情义。
本片失败之处，是没有了这个“义”字。《龙门客栈》侠士千里护送忠臣遗孤体现的是个“义”，这也是当时全片的主轴，而不是正邪杀杀杀。《龙门飞甲》换成了护送宫女，且不说最后发现大家被耍了这是个圈套，就真是个怀孕宫女，好像也的确是触犯条规的吧？又或者这个“义”字体现在何处呢？没道理的拔刀相助就是有点点莽撞和冲动了。
而本片最后能让我们认可为续作的地方，却恰恰还有那么一点“情”在里头。这也是凌雁秋作为老江湖，一再想告诉年轻一辈的。我们之前说寻宝三人组的性格，有那么一点点年轻时候金镶玉和周淮安的意思，可性格里唯独缺了邱莫言那一道。夺宝组是年轻人“不知者无谓”，邱莫言虽然有时候也有些许天真的一面，但更有一种“明知不可为而为之”的想法在里头。
所以当我说到周淮安和金镶玉人设“崩坏”了却最终接受，是发现在两个人对话中始终还是绕不开邱莫言。金拿着她的笛子，穿上她的行头，努力活成她的样子；周则是扔掉城府，不计代价，用邱的方式做一回侠客。
赵怀安让凌雁秋远走他方，说该去哪逍遥就去哪，凌雁秋回来和他并肩作战。
凌雁秋说自己远走他方逍遥自在去了，赵怀安醒来立刻追过去。
英雄气短，儿女情长，什么相濡以沫不如两忘于江湖都是骗人的，放不下就是放不下，做完好事杀掉boss，为什么不可以在一起长相厮守？
（P.S.素慧容再次证明聪明人装不来，而傻白甜都是骗人的，心机婊的很，远离远离……）
2016年8月18日星期四
黄飞鸿系列影评3
黄飞鸿三狮王争霸是三部曲的最后一部，之后的四五都由赵文卓扮演，而第六部则更像一个番外篇，所以通常我们也把这一部看作是这个系列的终结。
虽然断断续续看过好多遍这一部，但从还真是到了今天才从头到尾细细品味一番。严格说来这一部但从故事角度是不如上一部的：诚然它一如既往的加入了历史人物李鸿章和慈禧，诚然这一部加入了拉风的鬼脚七、德高望重颇有威严的黄麒英，诚然这一部情感戏很有看点，但闹得京城满城风雨的狮王争霸在立意上看起来还是差了第二部不少。
这里我们再探讨一下电影故事年份的问题，在此也为上一篇影评勘误一下。孙中山离开广州、陆皓东就义、公车上书说的都是同一年1895，而且那时确实就有了青天白日旗，所以第二部整个故事是基本严格严谨的。狮王争霸的故事只会在此之后而不在此之前，如此参照第四部时间线八国联军侵华1900年，所以这个故事应该在这5年之间。开篇慈禧说道战事应该是说甲午海战战败的事情。如此情况下再办狮王争霸彰显国威奋发上进看起来也有些不妥，私以为这个应该是甲午战败前大家还沉醉在洋务运动的成果里毕竟合适。之后这几年间最大的事情在于变法，所以狮王争霸的主题怎么看都不是那个年代的主旋律。
不过好吧，也许那时大家对洋务又失望了，转向又回到国粹上也不是不可能。这一点，上一部对西洋事物不了解、但仍愿意学习的黄师傅，到了这一部反而保守到对蒸汽机、电影全方面的排斥上，又或者是阵痛中。虽然我们知道，这当中或多或少因为他个人情感的介入，也因为探讨的对象不是孙先生那么有魅力。但整体上黄师傅在接受新鲜事物这一问题上在本片中是排在倒数的。这或许是导演觉得，黄师傅身上已经有了传统的各种美德了，文武双全、医者仁心、以理服人，如果再加上国际眼光那么也太开挂了。所以如果有人要在本片里充当中西文化碰撞的角色，不妨就让这个传统文化的理想主义者来承担吧。
这一集比较出彩的人物是黄的父亲黄麒英。开篇登场告诉大家这是一代宗师，之后再从他购置机器流水线做药告诉大家虽然人年纪大，但眼光和胸怀比黄师傅高的不知道哪里去了。老人家一路大多数时间都在卖萌装逗比，但接受起新鲜事物比谁都快。黄飞鸿的不伦之恋在他这也就是一句“老狮头是该扔了”。向老黄师傅致敬！
另一个有进步的人物是梁宽。我们知道梁宽是全片搞笑担当，他喜欢出风头、拍马屁、凑热闹，喜欢十三姨。上一部的最后他也和师傅一块经历了一次革命事业，这一集虽然疯癫依旧，却疯癫地比他师傅多了几分洒脱。他对赵天霸恨得彻底，和鬼脚七过招玩得聪明。要知道开篇到此处，所有人和鬼脚七较量都没讨过便宜，偏偏一物降一物阿宽来了个“策马奔腾”，顺带还打停了混乱的狮队。之后看电影时他和鬼脚七的互动是个非常有趣的细节：鬼脚七刚刚伤愈复出看大家看电影热闹也凑过去看，此时他的脸上是难得的傻笑。梁宽呢？先是给他脸色看，末了扔了他一个瓜子。意思是“哥确实不喜欢你，但你既然改过了那之前就一笔勾销以后都是自家兄弟了”。这个环节很生动，关键是不做作，很好诠释了阿宽为啥能当大师兄，因为统战工作做的漂亮啊。
十三姨没有过多可说的，每一集里的外国友人都是她的朋友或老同学，最有国际眼光和未来视角的女主角。这集讲了很多她和摄像机的故事，而摄像机也扮演了重要角色。电影起源于1895年，所以某种意义上说的确是当时最先进的产物，应该和现在大家玩VR一个新鲜劲吧。为了电影机，她和黄师傅吵了至少两到三次，然而最后在狮群里再度相聚，当黄师傅都有意为了讨好她去扶摄像机时，她的眼里却只有黄飞鸿一个人，这一刻我们相信十三姨不只是一个普通的留洋青年，而是一个真正培养了知识、价值观和国际视野的姑娘。
我说了这么多好话，所以电影为什么不好呢？
原因还就在狮王争霸和赵天霸身上。倒不是说他演的不好，赵天霸这个角色演得很好、非常好、特别好。可就是他演得太好了，把这段故事看得憋屈。赵是京城恶霸，家里有钱有势力，为了狮王拿冠军可以不择手段可以兴师动众。黄师傅在和他的过招中其实一直处在下风，因为人家不讲道理，因为人家没什么在乎，可就是要胡搅蛮缠。结果就是，黄师傅整集里面只能没完没了地和那些叫不上名的罗罗打来打去，看得让人憋屈。观众想看的不是这般和小角色打来打去的群架斗殴，而是和纳兰的高手对决，这点上我们颇为纳兰喊冤。
可是这个故事拍的很现实。它在讲述着某种现实的无奈。黄飞鸿和纳兰过招，讲的还是革命党和保皇党的争斗，讲的还是民主共和和君主立宪在历史进程上的争斗。黄飞鸿打赵天霸那是为了什么？本质上说，赵天霸才是历史进程最大的障碍。那是既得利益者，他们不在乎历史的未来，他们只要声色犬马，只要自己开心，但另一面又养了一把鹰犬，乐意时参合一脚。更重要的是，他们可以欺世盗名，可以粉饰自己的行为。如狮王争霸，如果没有黄飞鸿最后的参与，大家会怎样评价赵天霸？不知情人或许以为他也是一代大侠吧？然而呢？
所以如果一个落后的国家只有纳兰那也不坏，至少他在思考未来，输赢看历史进程。但太多的赵天霸这就是历史的阻力——甭管这个方向朝哪，都是阻力。可怜的是，即使百年过去了，赵天霸依旧还有，大家在讨厌着他们的时候是不是很多人也在羡慕着他们或者希望成为他们？或者再其次，最后无奈成为了他们的手下，跟着霸爷混口饭吃。
毕竟，只手打出一片天、风里来雨里去、此身何惧的大侠实在太少了。
虽然断断续续看过好多遍这一部，但从还真是到了今天才从头到尾细细品味一番。严格说来这一部但从故事角度是不如上一部的：诚然它一如既往的加入了历史人物李鸿章和慈禧，诚然这一部加入了拉风的鬼脚七、德高望重颇有威严的黄麒英，诚然这一部情感戏很有看点，但闹得京城满城风雨的狮王争霸在立意上看起来还是差了第二部不少。
这里我们再探讨一下电影故事年份的问题，在此也为上一篇影评勘误一下。孙中山离开广州、陆皓东就义、公车上书说的都是同一年1895，而且那时确实就有了青天白日旗，所以第二部整个故事是基本严格严谨的。狮王争霸的故事只会在此之后而不在此之前，如此参照第四部时间线八国联军侵华1900年，所以这个故事应该在这5年之间。开篇慈禧说道战事应该是说甲午海战战败的事情。如此情况下再办狮王争霸彰显国威奋发上进看起来也有些不妥，私以为这个应该是甲午战败前大家还沉醉在洋务运动的成果里毕竟合适。之后这几年间最大的事情在于变法，所以狮王争霸的主题怎么看都不是那个年代的主旋律。
不过好吧，也许那时大家对洋务又失望了，转向又回到国粹上也不是不可能。这一点，上一部对西洋事物不了解、但仍愿意学习的黄师傅，到了这一部反而保守到对蒸汽机、电影全方面的排斥上，又或者是阵痛中。虽然我们知道，这当中或多或少因为他个人情感的介入，也因为探讨的对象不是孙先生那么有魅力。但整体上黄师傅在接受新鲜事物这一问题上在本片中是排在倒数的。这或许是导演觉得，黄师傅身上已经有了传统的各种美德了，文武双全、医者仁心、以理服人，如果再加上国际眼光那么也太开挂了。所以如果有人要在本片里充当中西文化碰撞的角色，不妨就让这个传统文化的理想主义者来承担吧。
这一集比较出彩的人物是黄的父亲黄麒英。开篇登场告诉大家这是一代宗师，之后再从他购置机器流水线做药告诉大家虽然人年纪大，但眼光和胸怀比黄师傅高的不知道哪里去了。老人家一路大多数时间都在卖萌装逗比，但接受起新鲜事物比谁都快。黄飞鸿的不伦之恋在他这也就是一句“老狮头是该扔了”。向老黄师傅致敬！
另一个有进步的人物是梁宽。我们知道梁宽是全片搞笑担当，他喜欢出风头、拍马屁、凑热闹，喜欢十三姨。上一部的最后他也和师傅一块经历了一次革命事业，这一集虽然疯癫依旧，却疯癫地比他师傅多了几分洒脱。他对赵天霸恨得彻底，和鬼脚七过招玩得聪明。要知道开篇到此处，所有人和鬼脚七较量都没讨过便宜，偏偏一物降一物阿宽来了个“策马奔腾”，顺带还打停了混乱的狮队。之后看电影时他和鬼脚七的互动是个非常有趣的细节：鬼脚七刚刚伤愈复出看大家看电影热闹也凑过去看，此时他的脸上是难得的傻笑。梁宽呢？先是给他脸色看，末了扔了他一个瓜子。意思是“哥确实不喜欢你，但你既然改过了那之前就一笔勾销以后都是自家兄弟了”。这个环节很生动，关键是不做作，很好诠释了阿宽为啥能当大师兄，因为统战工作做的漂亮啊。
十三姨没有过多可说的，每一集里的外国友人都是她的朋友或老同学，最有国际眼光和未来视角的女主角。这集讲了很多她和摄像机的故事，而摄像机也扮演了重要角色。电影起源于1895年，所以某种意义上说的确是当时最先进的产物，应该和现在大家玩VR一个新鲜劲吧。为了电影机，她和黄师傅吵了至少两到三次，然而最后在狮群里再度相聚，当黄师傅都有意为了讨好她去扶摄像机时，她的眼里却只有黄飞鸿一个人，这一刻我们相信十三姨不只是一个普通的留洋青年，而是一个真正培养了知识、价值观和国际视野的姑娘。
我说了这么多好话，所以电影为什么不好呢？
原因还就在狮王争霸和赵天霸身上。倒不是说他演的不好，赵天霸这个角色演得很好、非常好、特别好。可就是他演得太好了，把这段故事看得憋屈。赵是京城恶霸，家里有钱有势力，为了狮王拿冠军可以不择手段可以兴师动众。黄师傅在和他的过招中其实一直处在下风，因为人家不讲道理，因为人家没什么在乎，可就是要胡搅蛮缠。结果就是，黄师傅整集里面只能没完没了地和那些叫不上名的罗罗打来打去，看得让人憋屈。观众想看的不是这般和小角色打来打去的群架斗殴，而是和纳兰的高手对决，这点上我们颇为纳兰喊冤。
可是这个故事拍的很现实。它在讲述着某种现实的无奈。黄飞鸿和纳兰过招，讲的还是革命党和保皇党的争斗，讲的还是民主共和和君主立宪在历史进程上的争斗。黄飞鸿打赵天霸那是为了什么？本质上说，赵天霸才是历史进程最大的障碍。那是既得利益者，他们不在乎历史的未来，他们只要声色犬马，只要自己开心，但另一面又养了一把鹰犬，乐意时参合一脚。更重要的是，他们可以欺世盗名，可以粉饰自己的行为。如狮王争霸，如果没有黄飞鸿最后的参与，大家会怎样评价赵天霸？不知情人或许以为他也是一代大侠吧？然而呢？
所以如果一个落后的国家只有纳兰那也不坏，至少他在思考未来，输赢看历史进程。但太多的赵天霸这就是历史的阻力——甭管这个方向朝哪，都是阻力。可怜的是，即使百年过去了，赵天霸依旧还有，大家在讨厌着他们的时候是不是很多人也在羡慕着他们或者希望成为他们？或者再其次，最后无奈成为了他们的手下，跟着霸爷混口饭吃。
毕竟，只手打出一片天、风里来雨里去、此身何惧的大侠实在太少了。
2016年8月16日星期二
从乒乓球到数学——说说团队建设
最近奥运会打得火热，明天就是乒乓男团女团决赛了，预计两块金牌少不了。今年得益于网络的传播，对国家队的展示和奥运精神的诠释在舆论上都算是比较成功吧，大家茶余饭后调控大魔王小魔王爱酱乐此不疲。那么我也就来从乒乓球到数学谈谈团队建设吧。
【优秀团队的起源】
在更早的时候，咱们的乒乓球也是谈不上超级大国的。世界各国高手当年还是不虚中国运动员的，就是00年悉尼那会儿你看老瓦背后的瑞典乒乓球迷多么自信啊。然后再看比赛，我就觉得现在完全不要厚此薄彼，现在运动员整体技术水平绝对是比当年先进来着。
我也算看乒乓球好多年了，现在这个制度大家也都差不多明白。国家队是个大团体，大约二三十人，每年都有各色优秀选手选拔调拨进入，这当然考虑实力，也考虑各色选手的丰富程度用于模拟外国对手。每年大赛前，大循环小循环公开竞争+教练组推荐。这个制度保障了公开透明和流动性，也对国家队选手产生了最大的激励作用。往近里说，时下网络当红的张继科就是硬生生打出一条路最后火箭成为大满贯的。他在07年我们看少年赛时就崭露头角却也熬了好多年才从陪练打上来。
现在重要赛事当然是优中选优，常规赛事现在也常派出些年轻选手练兵或者一档非核心国手或老运动员，毕竟大家都不容易总要拿个冠军才有交代么。我们看乒乒乓乓的视频，大家就感慨了。如王励勤马琳这些老将在视频里真的是能看到岁月的痕迹，如改朝换代也明明白白写在那，到了某一季万能的主角张怡宁大魔王就不再回归了，如国家队来来往往也是好些人可能就出现在了那么一集后来也就没有出现了。去奥运会任务重得保险派绝对主力大家理解，可是其他人真的就没有冠军的实力吗？不见得。所以啊，多少都要给他们个机会去圆这么一个世界冠军的梦想。
在这个体系之外，当然还有前往各国的中国二队三队四队，也都是相当优秀的运动员了。
【什么是一个优秀的团队】
奥运征程前有张照片是马琳、王皓、王励勤送别国家队，当年三驾马车打法相互克制，结果谁都没集齐大满贯。时下张继科马龙倒是都在年纪轻轻收下大满贯，一个持续前进另一个则是爆发性超强。再说老一代的刘国梁和孔令辉如今都各种执掌教鞭了。我们江苏老乡陈玘单打一直差了一层纸，双打倒是百搭拿下各种双打冠军，他说退役了最想念的还是在国家队的日子。如今老队员都退役各自去地方队担任了领军人物，过几年恐怕又要在全运会俱乐部比赛里继续切磋过招了。
这字里行间其实都有着说不尽的故事，人生不过数十载寒暑，有那么一段经历能毕生铭记并以之为基础继续奋斗，也是相当难能可贵了。
咱们缺一个好的漫画家把这些好故事改编一下。
所以什么是一个好的团队呢？
共同的目标与人生实现的可能性
公开透明的流动机制
友好的合作与竞争氛围
荣誉感与归属感
伙伴与友谊
大众互动。
【推广与外延】
毫无疑问中国乒乓球队是一个好的团队。与之类似的还有什么呢？美国的篮球、欧洲足球的青训、法国的数学，参照这些一条条对照都对。
这个时候我们说到中国的数学，大概这几条大概都是还值得有待提高的，丘先生一己之力在努力推动前进其实也是很了不起了，但还需要更多人愿意为之奋斗来改善这个大环境。
【小小的愿景】
中国数学界这个范围太广，往常州这个小范围看，可不可能好一点点呢？
我们有个很好的范例样本，是曹文老师的计算机团队，但这个也是十多年功夫花下去了。数学这面倒不见得有这般几何级数的增长。虽然说起来这个社会需要的纯数学人才比之计算机可能是要少一点，哦，是比程序员是要少一点，但也是需求量挺旺盛的，还没到数学饱和的那一天。
我们曾经有个机会打造一个团队，可惜那时的我还不懂。我觉得现在说起来其实也不算晚吧。恽神得道成仙了，却只留下传说而没有太多可供后人参考。我道行还不够，不过怎么说也在修行的道上，还是挺愿意去分享一些想法的。
这件事说干就干起来，每年做一点，身体力行，过个好多年总会开花结果的吧。
2016年8月14日星期日
From discrete martingale to continuous martingale
Recently, I give a tutorial about the theory of martingale and Markov chain to my classmate, who has missed the exam, so I will write something about the martingale theory.
[Conditional expectation]
To understand the theory of martingale, I believe the conditional expectation is the most important part. According to my experience, I would like to learn some advanced topics in probability theory for long time, but I found it so difficult to study it without a good understanding of conditional expectation.
In fact, the sigma algebra can be seen as a kind of information. The conditional expectation is just a kind of approximation given a subset of the information. In the image process, the counterpart is multi-resolution analysis; in the language of functional analysis, it is the projection in subspace of Hilbert space. But the general existence is the density of intergrable function in L^2 space.
[Martingale: Its motivation and importance]
To define a martingale, we should have a family of filtration, i.e a series of information. The martingale is to say the best approximation of today is just the random variable yesterday.
Then is the Doob's theorem, a discrete version of integration. That is to say, if we do invest only depending on the old information, we cannot do better than just keep the mean 0.
More tricks can be found in the theory of martingale, specially when we add the role of stopping time. Perhaps, we can also say, the development of martingale is just for answering a lot questions about the stopping - - - all from optimal stopping time theorem. So, the profs always say, we know all about the random processes if only we find a martingale.
Last but not the least is the convergence of martingale, the most beautiful theorem in the martingale theorem because it uses nice inequality like Doob's maximal inequality and upcrossing inequality. We know generally, if the martingale is uniformly L^1 bounded, it converges almost sure. If the condition is stronger L^p, we get a L^p convergence. But for get L^1 convergence, the suitable condition is uniformly intergrable. In addition, this convergence, the most precise resolution can do approximation in different level to get the discrete martingale.
[Martingale: from discrete to continuous]
When we pass from discrete to continuous martingale, it's necessary to say something about the regularity about the random processes. Why? While people talk about the random processes, they sometimes say cadlag. a french word, continue à droite et limite à gauche. This condition makes sure that some martingale has sense, and the processes is more regular. In fact, we can change the processes in a 0 measure to make it cadlag.
And, only in this case, all the convergence theorem like L^P, p.s , L^1 works. The technique part is just like the discrete version but with more treat in the limit of all the time.
In conclusion, we give a short introduction about what the martingale is and why this tool is so useful. I agree that it is a little abstract and we can understand its sense only with the concrete example.
2016年7月26日星期二
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。
2016年7月12日星期二
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.
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.
2016年7月6日星期三
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.
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.
2016年6月27日星期一
读后感《给青年的十二封信》（上）
适逢暑假在校实习，每天工作下班回来之后都感觉特别疲惫，都是打代码惹得祸。因祸得福的是读书的习惯倒是在好久之后捡回来了——这说的不是看专业书籍啦。然后么，好处是学长还寄存在我这里一大箱书，正好给我每天晚上看看。
今天挑了一本，朱光潜先生的《给青年的十二封信》，开卷不能释手。大家都知道朱光潜先生是美学大家，我高中毕业那会儿附庸风雅也读过《中国美学十五讲》，当中说了不少他的观点和想法。这本书呢，虽然写于民国时期，然而说的事情时至今日依然适用。话说今时今日已经不是当年那个“危急存亡之秋”了，可是先生探讨的青年人的问题依旧还是时下青年人的问题。好吧，估计是传统文化一脉相承，青少年身心发展具有特征和代表性，这些都远超意识形态了吧。
既然开卷有益，就摘录一些我觉得写的非常好的话。
《谈读书》
兴味要在青年时设法培养，过了正当时节，便会萎谢。比方打网球，你在中学时喜欢打，你到老都喜欢大。假如你在中学时代错过机会，后来要发愿去学，比登天还要难十倍。养成读书习惯也是这样。
如果你每天能抽出半点中钟，你每天至少可以读三四页，每月可以读一百页，到了一年也就可以都四五本书了。何况你在假期中每天断不会只能读三四页呢？你能否在课外读书，不是你有没有时间的问题，是你有没有决心的问题。
《谈动》
流行语中又有“闲愁”的字样，闲人大半易于发愁，就因为闲时生机静止而不舒畅。青年人比老年人易于发愁些，因为青年人的生机比较强旺。
总之，愁生于郁，解愁的方法在泄，郁由于静止，求泄的方法在动。从前儒家讲心性的话，从近代心理学眼光来看，都很粗疏，只有孟子的“尽性”一个主张，含义非常深广。一些道德学说都不免肤浅，如果不从“尽性”的基点出发。如果把“尽性”两个字懂得透彻，我认为生活的目的在于词，生活方法也就在此。
《谈中学生与社会运动》
比分我欢喜谈国事，就蔑视你读书；你欢喜读书，就蔑视我谈国事。其实单方面锣鼓打不成闹台戏。要撑起中国场面，也要生旦净末丑角角俱全。
蔡孑民先生说，“读书不忘救国，救国不忘读书，”这两句话是青年人最稳妥的座右铭。
老实说，社会已经把你我们看成眼中钉了。这并非完全是社会的过处。现在一般学生，有几个配谈革命？吞剥捐款聚赌宿娼的是否曾充过代表，赴过大会？勾结绅士政客以捣乱学校的是否没曾谈过教育寺严？向日本政府立誓感恩以分润庚子赔款的，是否没曾喊过打倒帝国主义？
《谈十字街头》
一种社会所最可怕的不是民众浮浅顽劣，因为民众通常都是浮浅顽劣的。它最可怕的是没有在浮浅卑劣的环境中而能不浮浅不卑劣的人。比方英国民众就是很沉滞顽劣的，然而在这种沉滞顽劣的社会中，偶尔跳出一二个性坚强的人，如雪莱，卡莱尔，罗素等，其特立独行的胆于识，却非其他民族所可多得。这是英国人力量所在的地方。
《谈多元宇宙》
在恋爱的状态中，两人的脉搏一起一落，两人心灵一往一复，都恰能契合无间。在这种境界，如果身家财产、学业名誉道德等观念渗入一分，则恋爱真纯的程度须减少一分。真能恋爱的人只能因为恋爱而恋爱，恋爱以外，不复另有宇宙。
《谈升学与选课》
此全篇我都极为推荐，故不再摘抄。即使在今日看来，此文依然字字珠玑，不禁令人感叹历史总在重复着昨天的故事啊。
在第一个升学的问题上，先生就说到当时的学生重文凭，轻学术。只求速成拿上证书，好去社会上谋个生路。（时下也是）还有各种崇洋媚外情结，想去留学镀金。就是在这样的大环境下，作为十里洋场的上海挂牌办的大学竟然超过了英法之和，还有各种美国注册的学校，就是为了迎合大家的需求么。
然后到了专业，大众也是只挑热门而忽视兴趣。
最后说到选课的问题，由于有欧洲留学经历，朱先生说到真正的学者在“术业有专攻”之前都是有一个非常扎实而宽泛的根基的。比如做文学的应该要懂各国文学，做哲学的应该要懂些历史。而国人常常想造空中楼阁，结果做出的文章都惹人笑话。
本文写于1929年，差不多百年过去了，好像在文化积淀这一刻仍然还没有迈出一大步。其中战乱、斗争，过去的也就过去了。现在三、四十年重新来过，当年的老家当说实话也没留下多少，甚是可惜。时下办学热出国热依然高涨，好在信息交流加快了，大家也趋于理智了。然而何时才能办好自己的大学？这也是我们这代人的事情了。
2016年6月17日星期五
2A - 还有三周
说说过去的P3情况
【P3】
【马尔科夫链和随机过程】
花心思很多的一门课。拿到书的时候发现两块内容基本都自学过，然而还是又好好读了书，做了习题，帮老师查了好多打印错误，拓展着看了Yuval的书可惜还没有看完。
可以认为Markov作为离散动力系统中的一环，随着计算机的发展已经变得和连续动力系统中的微分方程一样重要了。嗯，其实我们也常用Markov来模拟拉普拉斯方程的对吧？今天还读到关于Mixing Time的科普性文章，这个我非常喜欢。
简单说，Markov就是一个只和当下有关而不在乎过去的随机过程。我们关心周期性、渐进行为、渐进速度Mixing time、截断情况Cutoff。还有很多技巧，例如Couple 这些都是值得研究的。包括从纯数学角度，研究群结构意义下的随机游动也帮助我们知道群的结构。
离散鞅论就是连续鞅论的前奏，赌场股市必备神技的入门篇。当然概念来的很抽象啦，不过由于他特殊的结构，使得收敛证明其他条件减弱了。在信号方面他也有另一套说法，这些我也和同学介绍过。
【变分PDE】
变分类的PDE，核心就是Lax-Milgram定理，所用的其实是泛函中的表示定理。然后各种提条件方法，一些空间的介绍。
变分类问题也算数学上一大类了，时下机器学习盛行，很多时候也是优化一个目标函数。能用到变分方法的机会也还是不少的。这门课就是告诉大家：数学上，你要找到的上界就TMD是最大值。
还有计算特征值的方法（研究谱），也是一大类了，本课讲了最基本的观点。
这门课最后一个大的projet我们做了流传已久的“能不能从频率听出鼓形状”这样一个课题。用了Freefem，老实说到现在也不太会真正使用这个软件，就凑合着写了点边界条件吧。
【统计物理1】
统计物理1主要还是在说热力学的内容。统计物理的研究对象是大规模相同粒子的整体行为，但这个整体行为同时又是千千万万个体行为的综合表示。这个学科特别有意义，在于现实中很多事情都可以用这种类似的模型去类比，比如金融市场。
统计物理很多时候会和概率论扯在一块，这也是时下一个研究方向。la fonction de repartition 就是概率论中的统计物理1中粒子大多都是idd的。这也是理想气体的假设。
【Modal】
最后是Modal，又是NS方程，这个方程在本科时候已经和老师学过并且做了关于Onsager猜想的研究了。现在重新学了一遍，把之前不了解的经典理论补上了。
NS方程最经典的理论来自于Leray，他的方法导出了目前为止最主要的一些成果。通常我们在H^1空间中研究这个问题，比如NS方程在二维情况存在、唯一、正则性都是有的，但在3维忠就不对，因为空间嵌入不够好。
Leray的研究方法说起来就是对函数先阶段，然后微分算子在频率有界情况下就是有界算子了，那么ODE的存在唯一性都能用上（这里体现了ODE和PDE的差别）。对于一族解，再利用泛函里面的一些逼近性质就得到了解。
之前说了在三维中，因为空间嵌入不够好，就没有唯一性和正则性了，怎么办呢？我们还是有一些结论的，比如在初值小的情况下，正则性还是有的。比如我们可以研究换个函数空间这样的。
最后一个结论，在有科里奥利力的作用下，三维的力本质和二维差不多，也可以得到一些关于正则性的结论。这是最新的研究成果了，我们也就读到这里。
【P3】
【马尔科夫链和随机过程】
花心思很多的一门课。拿到书的时候发现两块内容基本都自学过，然而还是又好好读了书，做了习题，帮老师查了好多打印错误，拓展着看了Yuval的书可惜还没有看完。
可以认为Markov作为离散动力系统中的一环，随着计算机的发展已经变得和连续动力系统中的微分方程一样重要了。嗯，其实我们也常用Markov来模拟拉普拉斯方程的对吧？今天还读到关于Mixing Time的科普性文章，这个我非常喜欢。
简单说，Markov就是一个只和当下有关而不在乎过去的随机过程。我们关心周期性、渐进行为、渐进速度Mixing time、截断情况Cutoff。还有很多技巧，例如Couple 这些都是值得研究的。包括从纯数学角度，研究群结构意义下的随机游动也帮助我们知道群的结构。
离散鞅论就是连续鞅论的前奏，赌场股市必备神技的入门篇。当然概念来的很抽象啦，不过由于他特殊的结构，使得收敛证明其他条件减弱了。在信号方面他也有另一套说法，这些我也和同学介绍过。
【变分PDE】
变分类的PDE，核心就是Lax-Milgram定理，所用的其实是泛函中的表示定理。然后各种提条件方法，一些空间的介绍。
变分类问题也算数学上一大类了，时下机器学习盛行，很多时候也是优化一个目标函数。能用到变分方法的机会也还是不少的。这门课就是告诉大家：数学上，你要找到的上界就TMD是最大值。
还有计算特征值的方法（研究谱），也是一大类了，本课讲了最基本的观点。
这门课最后一个大的projet我们做了流传已久的“能不能从频率听出鼓形状”这样一个课题。用了Freefem，老实说到现在也不太会真正使用这个软件，就凑合着写了点边界条件吧。
【统计物理1】
统计物理1主要还是在说热力学的内容。统计物理的研究对象是大规模相同粒子的整体行为，但这个整体行为同时又是千千万万个体行为的综合表示。这个学科特别有意义，在于现实中很多事情都可以用这种类似的模型去类比，比如金融市场。
统计物理很多时候会和概率论扯在一块，这也是时下一个研究方向。la fonction de repartition 就是概率论中的统计物理1中粒子大多都是idd的。这也是理想气体的假设。
【Modal】
最后是Modal，又是NS方程，这个方程在本科时候已经和老师学过并且做了关于Onsager猜想的研究了。现在重新学了一遍，把之前不了解的经典理论补上了。
NS方程最经典的理论来自于Leray，他的方法导出了目前为止最主要的一些成果。通常我们在H^1空间中研究这个问题，比如NS方程在二维情况存在、唯一、正则性都是有的，但在3维忠就不对，因为空间嵌入不够好。
Leray的研究方法说起来就是对函数先阶段，然后微分算子在频率有界情况下就是有界算子了，那么ODE的存在唯一性都能用上（这里体现了ODE和PDE的差别）。对于一族解，再利用泛函里面的一些逼近性质就得到了解。
之前说了在三维中，因为空间嵌入不够好，就没有唯一性和正则性了，怎么办呢？我们还是有一些结论的，比如在初值小的情况下，正则性还是有的。比如我们可以研究换个函数空间这样的。
最后一个结论，在有科里奥利力的作用下，三维的力本质和二维差不多，也可以得到一些关于正则性的结论。这是最新的研究成果了，我们也就读到这里。
2016年5月14日星期六
Galton Waston Tree - base and description by random walk
Motivation
Galton Waston Tree is a basic model in probability theory and sometimes we call it branching process. This model can be taught in the introductory course of probability in Polytechnique during Tronc Commun, but its background is so profound that we can find it in so much domain in applied mathematics and pure mathematics, some examples for the former is the biological process like the gene and, however, some example for the later is the recent breakthrough in maths and physics like random mapping theory. In fact, we start from the BGW model and develop the continuous random tree, a continuous version as the convergence of the tree, and at last is the most fashionable object.As I have passion to continue the study in this field, I will devote a series of blog in this domain. This first introduction comes from the talk given by Igor Kortchmeski last week.
Definition
The BGM process is can be defined as a tree. In each generation, the parent gives birth to its children and the law of production follows a random variable uniformly and independently. Usually we study the situation that the expectation of the production is finite and it is logical in reality.The first problem comes from the biology: when the population will distinct? The answer is that when the expectation of production is less than 1, the population will die necessarily but even though this expectation is larger than 1, there is still a probability that the population die out. Then precise study of this problem relied the study by the generated function, who is in form of iterations.
Simulation of uniform tree
The second problem is how many generated tree without label? A famous formula of Cayley is n^(n-2). This can also be solved by BGW model and in fact our BGW model can simulate this process given certain probability of production. If the probability of production is like a geometry law, this gives a uniform ordered and root fixed tree, but if it follow the law of Poisson, it is the simulation of uniform non-labelled but root fixed tree. The technique detail will be given in the next section.Characterization
In the lecture, Igor introduced two method to code the tree: one by the function of contour and the other by the random walk. The first one, each step is the height of the vertex in the tree in order of the depth first search and in the second one, we assign each move the law of production. Then the process of production becomes the process of random walk. This is nature since we can always consider the number of population in a given and then the number denote the random walk. The second method by random walk has incredible power: since we understand well the random walk and it can calculate many property like the number of non-labelled tree, which we have introduced int the previous section.
Some further study like the asymptotic growth of number of tree can be approximated by the local limit theorem, a extended version of central limit theorem. Until here we have presented almost all the content in this lecture and I will write some notes by simple description of these beautiful maths.
2016年1月23日星期六
2A 1/3 passé
这周结束了2A前两个小学期也就是一个大学期的考试，随便写点什么回顾一下学了什么以及后面的展望。
【P1】 = 统计 + 量子力学2 + INF411 + 动力系统（sup)
【统计】第一次给了我一个感觉统计也可以非常非常理论。应该说这个统计课就是告诉我们，在什么样的假设下，我们脑洞打开设计的方法是有道理的（estimateur）, 或者随机变量之间应该有怎样的一种依赖性（regression），以及我们设计了一个方法，然后根据这个方法找变量的关系,这个方法的准确性是多少（test）。稍显美中不足的，过于理论了，甚至都没有拿这些方法做过什么。当下统计学习如此火热，我们当然要懂些原理，可是连一点点直觉都没有好像不太好。
【量子力学2】继续上学期的内容加强深化了些。物理毕竟外行没有学精，就记得好像我们推出了Pauli不相容原理，这个挺让我映像深刻的。
【INF411】继续打程序……链表二叉树深度优先广度优先最短路动态规划都一个个做了遍，原来算法计算机真的不难。
【动力系统】因为最后没有参加考试所以后来劲就送了，只能等到来年再来过了。不过我个人觉得这门课内容对我而言是有提升的。本科对ODE的了解就停留在解解方程了，而事实上呢？法国整个动力系统的框架完全是建立在更一般抽象的体系上的么。当然基石还是Cauchy Lipthiz.还有整个Flot的想法应该是渗透到了力学还有PDE当中去的。
不过最后因为和Section疯玩了一把，期末复习相当不到位，中间也没有抓的特别紧。说实话是要检讨的。成绩考出来也没有那么理想（但也不至于毁灭），想到未来成绩还是有用的，所以P2果断端正态度……
【P2】 = Distribution + 数值入门 + INF421 + 狭义相对论（sup）
【Distribution】在Coursera上已经上过了课拿了证书，再看老师上课的内容说实话还不如当时电子课。整个课程进度甚至比电子课更慢，留下了之后一大本关于PDE的内容，当中还有不少是很重要的，比如关于Laplacian的正则性。不过还好这门课开课的时候终于又捡回了TC的状态做掉了绝大部分习题。要给自己打分至少是个80分吧。
内容：分布理论就是把内积结构的一种推广。分布，紧支撑分布，缓增分布，Hopf方程，基本解和正则性。卷积一块是是比国内说的细的多，紧支撑分布可以各种和其他玩意儿做卷积。
（在此再次感谢当年李老师和雷老师教诲，X工程师教育再好，预科制度再棒，流水线生产比不过讨论班手把手拆招练出来的本事……）
【数值入门】整个学期最花精力的一门课。一方面因为刚刚上手觉得不花功夫难以学好，另一方面是这个里面的方程降低比上一门还要多些。事实也证明最后这门课学的得心应手。反思之处：还要多花时间写写数值解的程序啊。然后这门课虽然入门，但是体系里面该说的也说了特别多了。
内容：差分格式，变分和有限元，优化和算法。差分主要做热传导和波动，变分有限元则什么都能做。优化中说了些定性描述，优化算法还是有意思的。
当然目前的数学工具用的不多，可以感觉是各种数分习题的实践版本，我觉得一定有把小波这些高级工具用上的炫酷进阶版。
【INF421】stabe matching, DP, 各种最短路最小生成树，TPS，做了个Projet TTP，还有胡来的随机算法和启发式算法。
课程还不错无奈德国老师英语听着真累。反正每次我都带头做DM……当中还想出过一个关于随机树联通性的证明，也是佩服自己。
【狭义相对论】又当科普课上了，总算会用lorenz 变换写各种光啊波啊等等。
【小结】：说真的不是每一门课都那么喜欢，但是也强迫着自己拿出至少85分热情去面对了所以学着还挺好（？）吧。后面有概率模型，统计物理，BIG Data，还有NS方程。以及我们不能把PSC就这样落下啊。最后两个学期，是体现120分热情的时候了，小宇宙烧起来~
【P1】 = 统计 + 量子力学2 + INF411 + 动力系统（sup)
【统计】第一次给了我一个感觉统计也可以非常非常理论。应该说这个统计课就是告诉我们，在什么样的假设下，我们脑洞打开设计的方法是有道理的（estimateur）, 或者随机变量之间应该有怎样的一种依赖性（regression），以及我们设计了一个方法，然后根据这个方法找变量的关系,这个方法的准确性是多少（test）。稍显美中不足的，过于理论了，甚至都没有拿这些方法做过什么。当下统计学习如此火热，我们当然要懂些原理，可是连一点点直觉都没有好像不太好。
【量子力学2】继续上学期的内容加强深化了些。物理毕竟外行没有学精，就记得好像我们推出了Pauli不相容原理，这个挺让我映像深刻的。
【INF411】继续打程序……链表二叉树深度优先广度优先最短路动态规划都一个个做了遍，原来算法计算机真的不难。
【动力系统】因为最后没有参加考试所以后来劲就送了，只能等到来年再来过了。不过我个人觉得这门课内容对我而言是有提升的。本科对ODE的了解就停留在解解方程了，而事实上呢？法国整个动力系统的框架完全是建立在更一般抽象的体系上的么。当然基石还是Cauchy Lipthiz.还有整个Flot的想法应该是渗透到了力学还有PDE当中去的。
不过最后因为和Section疯玩了一把，期末复习相当不到位，中间也没有抓的特别紧。说实话是要检讨的。成绩考出来也没有那么理想（但也不至于毁灭），想到未来成绩还是有用的，所以P2果断端正态度……
【P2】 = Distribution + 数值入门 + INF421 + 狭义相对论（sup）
【Distribution】在Coursera上已经上过了课拿了证书，再看老师上课的内容说实话还不如当时电子课。整个课程进度甚至比电子课更慢，留下了之后一大本关于PDE的内容，当中还有不少是很重要的，比如关于Laplacian的正则性。不过还好这门课开课的时候终于又捡回了TC的状态做掉了绝大部分习题。要给自己打分至少是个80分吧。
内容：分布理论就是把内积结构的一种推广。分布，紧支撑分布，缓增分布，Hopf方程，基本解和正则性。卷积一块是是比国内说的细的多，紧支撑分布可以各种和其他玩意儿做卷积。
（在此再次感谢当年李老师和雷老师教诲，X工程师教育再好，预科制度再棒，流水线生产比不过讨论班手把手拆招练出来的本事……）
【数值入门】整个学期最花精力的一门课。一方面因为刚刚上手觉得不花功夫难以学好，另一方面是这个里面的方程降低比上一门还要多些。事实也证明最后这门课学的得心应手。反思之处：还要多花时间写写数值解的程序啊。然后这门课虽然入门，但是体系里面该说的也说了特别多了。
内容：差分格式，变分和有限元，优化和算法。差分主要做热传导和波动，变分有限元则什么都能做。优化中说了些定性描述，优化算法还是有意思的。
当然目前的数学工具用的不多，可以感觉是各种数分习题的实践版本，我觉得一定有把小波这些高级工具用上的炫酷进阶版。
【INF421】stabe matching, DP, 各种最短路最小生成树，TPS，做了个Projet TTP，还有胡来的随机算法和启发式算法。
课程还不错无奈德国老师英语听着真累。反正每次我都带头做DM……当中还想出过一个关于随机树联通性的证明，也是佩服自己。
【狭义相对论】又当科普课上了，总算会用lorenz 变换写各种光啊波啊等等。
【小结】：说真的不是每一门课都那么喜欢，但是也强迫着自己拿出至少85分热情去面对了所以学着还挺好（？）吧。后面有概率模型，统计物理，BIG Data，还有NS方程。以及我们不能把PSC就这样落下啊。最后两个学期，是体现120分热情的时候了，小宇宙烧起来~
Three interesting questions
Yesterday, we talk about three interesting questions in the cuisine and luckily I have found the answers by myself for this three enigma. Now, I share them with you.
[Probability and a series of number]
A series of integers are coming but we don't know the total number of these integers. We have a fix number of memory. Try to design a way to give one number of them randomly.
Idea: Because we don't know the number, so in fact we keep the random during all the process. We start from the simple case. We we given one number, we just keep it. But once we have two, we must update it with the prob 0.5. That is the key. Once we know the total number until now, we have the method to update the data randomly.So two memory is OK.
[Find the polynomial]
We are given a polynomial of degree integer positive. We have two chance to test the polynomial by value. How we find it?
Idea: Intuitively, this is not so logical because in maths we know we can construct a polynomial of certain zeros. But pay attention to the positive integer coefficient. The addition of one polynomial cannot always be correct.
The correct way to think by maths: if we can separate the coefficient, we get it. But to separate the coefficient we have to know the range, so the first chance we can use it to test the range of coefficient. For example, we put 1 in it. We get the range. We put 100000~0 to it, then we get the coefficient.
RMK: It is said that this method can be used to attack the HashCode.
[The most frequent number]
Once again, we are given a series of integer and we know that there is one number which appears more than half of it. We don't know the number, try to find it.
Idea: this is one in which I use the most of time. I got the answer by intuition and then prove it by maths. We get one number, count the number from it, count the number it occurs. If the frequency is lower than half, erase all the data and re gain the number.
A better arrangement can reduce the memory to two. But the heart of this idea is that if we do partition of the interval, at least from one point, the frequency of this number is always above 0.5.
[Probability and a series of number]
A series of integers are coming but we don't know the total number of these integers. We have a fix number of memory. Try to design a way to give one number of them randomly.
Idea: Because we don't know the number, so in fact we keep the random during all the process. We start from the simple case. We we given one number, we just keep it. But once we have two, we must update it with the prob 0.5. That is the key. Once we know the total number until now, we have the method to update the data randomly.So two memory is OK.
[Find the polynomial]
We are given a polynomial of degree integer positive. We have two chance to test the polynomial by value. How we find it?
Idea: Intuitively, this is not so logical because in maths we know we can construct a polynomial of certain zeros. But pay attention to the positive integer coefficient. The addition of one polynomial cannot always be correct.
The correct way to think by maths: if we can separate the coefficient, we get it. But to separate the coefficient we have to know the range, so the first chance we can use it to test the range of coefficient. For example, we put 1 in it. We get the range. We put 100000~0 to it, then we get the coefficient.
RMK: It is said that this method can be used to attack the HashCode.
[The most frequent number]
Once again, we are given a series of integer and we know that there is one number which appears more than half of it. We don't know the number, try to find it.
Idea: this is one in which I use the most of time. I got the answer by intuition and then prove it by maths. We get one number, count the number from it, count the number it occurs. If the frequency is lower than half, erase all the data and re gain the number.
A better arrangement can reduce the memory to two. But the heart of this idea is that if we do partition of the interval, at least from one point, the frequency of this number is always above 0.5.
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