A summary from the maths-learning group

十一月 12, 2009 at 4:55 下午 | In Maths（数学）, life（生活）, test | Leave a Comment

Right now I am very pleased for where my maths-learning group is going. In fact, I am very confident that I can finish my plan written several days ago. ( maybe with some changes.) I try to make small sub-plans for no more than ten days a time. These small plans are very specific and easy to realize. In this way I can do the work more efficiently and better.

Anothe point is the importance of manifestation. Once again, I hope people who want to learn mathematics extensively in the next a few months can join me at

http://groups.google.com/group/maths-learning

I just write a small summary  for sub-plan number one in the group, as well as the next subplan, as following:

I am pretty glad that it seems I can finish my sub-plan number one in time, though there is apparently no one else working with me this time. But I am still very confident, because there are dozens of sub-plans waiting ahead in the next a few months. The importance of manifestation can’t be emphasized enough. I believe that only when you try to write mathematics down can you truly understand it. I guess maybe Mr. Akhil Mathew will agree with me on this point, because what he does on his own blog recently need so much persistence. I send him my compliment here.

Since I have covered several chapters of Complex Analysis, I decide to do a qualifying exam on analysis of UCLA. I call this sub-plan number 2. It is very convenient that UCLA provides past several years’ qualifying examinations at the following URL:

http://www.math.ucla.edu/grad/handbook/hbquals.shtml

I have uploaded the one on spring 2009 to the group. Anyone can feel free to download it. I’m afraid that I am not sure how much time this will cost, for maybe what I have already learned is not enough to deal with this exam. But once I finished it, I will write a post on my own blog, just like before.

There are totally 12 problems, which, of course, should all be done, since we don’t have time limitation.

 

 

Three proofs of a identity

十一月 11, 2009 at 2:02 上午 | In Complex Analysis (复分析) | Leave a Comment

In this post, I will give three proofs for a important identity, which all come from Stein’s book:’real analysis’(both in the context and in the exercises):

\frac{\sin \pi z}{\pi}=z\prod_{n=1}^\infty(1-\frac{z^2}{n^2}) (1)

The first proof:

The first proof has more of a ‘complex analysis’ taste, since it uses the properties of singularities of an entire function.

We first prove the following

\pi\cot \pi z= \lim_{N\to \infty} \sum_{n=-N}^N\frac{1}{z+n}= \frac{1}{z}+\sum_{n=1}^\infty \frac{2z}{z^2-n^2} (2)

First we observe both the left-hand side and the right-hand side have the following three properties (stated as F(z)): Continue reading Three proofs of a identity…

Entire functions

十一月 9, 2009 at 11:53 下午 | In Complex Analysis (复分析) | Leave a Comment

I have just finished the first read of Chapter 5 of Stein’s book: complex analysis. Roughly put, entire functions are decided by there zeros,only with difference of a multiplier, which is a entire function at the whole complex plane without zeros. It can also be represented as the form e^{g(z)}, where g(z) is entire. If the number of zeros are finite, then the function can be constructed as followings

\prod_{n=1}^K(z-z_n)

But when it comes to the situation when the number of zeros are infinite, it is much more complicated. However, we are lucky enough to have the following theorem:

Theorem 4.1(weierstrass infinite products) Given any sequence \{a_n\} of complex numbers with |a_n|\to \infty as n\to \infty, there exists an entire function f that vanishes at all z=a_n and nowhere else. Any other such entire function is of the form f(z)e^{g(z)}, where g is entire.

To prove this theorem, we should observe the important identity first:

\frac{\sin \pi z}{\pi}=z \prod_{n=1}^\infty(1-\frac{z^2}{n^2})

There are lots of ways to prove this identity, which I will write another post to make a summary. Observe that the zeros of the right hand side are precisely all the integers.

Basically what we will do to construct our entire function is to make our \{a_n\} act just like these integers. A problem we should pay attention to is that the speed with which \{|a_n|\} grows. Till now we can see, some very detailed analysis must be done in order to get this problem solved, and that is what the whole story goes. So, there are lots of paragraphs dealing with the growing speed. The first section, namely Jensen’s formula is a devise to do this. With the help of Theorem 2.1, Hadamard’s factorization theorem goes even further, strengthening the weierstrass infinite products.

(updated on Nov.10: ‘only with difference of a multiplier, which is an entire function at the whole complex plane without zeros. It can also be represented as the form e^{g(z)}, where g(z) is entire.’added, adviced by Akhil Mathew)

A plan for maths studying in the next a few months

十一月 7, 2009 at 2:38 下午 | In Elementary number theory(初等数论), Ergodic Theory(遍历论), Maths（数学）, Nonlinear dispersive equations, Real Analysis(实分析), The IMO problems（国际数学奥林匹克竞赛）, combinatorics(组合), elemantary mathematics, graph theory(图论), math.AP（偏微分方程）, math.AT（代数拓扑） | 1 Comment

This is basically the same post I wrote several days ago in Chinese, which is a plan to learn maths in the next a few months. I hope more people can join me so I rewrite it in English. There are so many things that I want to learn. So the following plan looks a little challenging. I will only take two materials at the same time, and I haven’t decided the sequence of them. I also set up a google group for this. Anyone who is interested is welcomed to join it at

http://groups.google.com/group/maths-learning

Continue reading A plan for maths studying in the next a few months…

A comment to one post from Professer Laba

十一月 2, 2009 at 2:43 下午 | In life（生活） | Leave a Comment

The post ‘paint it red’ was written by Professor Laba on her blog.

Dear Professor Laba:

I never know you came from Poland, a country which looks just like China in so many ways. Your article reminds me some of my memories, and the stories that I heard from my parents. Some say China is already a better place and I can live with that argument, though I am not satisfied with the current situation. Yes, it is true that we generally don’t get punished to death simply because we have said or write something wrong, it is true we don’t send all the intellectuals into the country to receive ‘reeducation as a farmer’ simply because the party won the ‘revolution war’ relying on farmers. But, people’s minds are devastated, with little free will left. They already get used to the giving up the opportunity of thinking by themselves. They don’t even try any more. Most only follow orders of their ‘the big leaders’. So most social problems existing in China today can track back to several decades ago, when ‘FREEDOM IS SLAVERY’, and ‘IGNORANCE IS STRENGTH.’ (George Orwell nineteen eighty four) Continue reading A comment to one post from Professer Laba…

十一月至明年上半年的学习计划

十月 30, 2009 at 11:01 下午 | In Elementary number theory(初等数论), Ergodic Theory(遍历论), Maths（数学）, Real Analysis(实分析), The IMO problems（国际数学奥林匹克竞赛）, combinatorics(组合), elemantary mathematics, graph theory(图论), math.AT（代数拓扑） | 3 Comments

我一年之前还常常做计划，然后发现总是不能完成。原因是候我总是有自虐倾向。以往做计划存在两个比较大的问题，其一是没有把计划内容做到切实可行，选了过多过难的材料；另一个问题是只有比较长时间的计划，而没有短期到一周甚至一天两天的具体安排。当然，做出这样的安排也是很困难的事情，因为学习不比做工程。常常事先很难断定完成一个任务所需要花费的精力。数学的学习更是如此。

所以我需要的一个灵活的计划。能够及时的调整先后顺序，而且易于具体化的计划。我打算从十一月上旬开始，陆续完成下面这个很有挑战的学习计划。大致要同一时期一直做其中两个任务，但是，哪个先做，哪个后做，我此刻暂时不能确定下来。希望有跟我有相同兴趣的朋友来与我联系。我可以加以合理的调整。 Continue reading 十一月至明年上半年的学习计划…

为什么做不好做不成

十月 20, 2009 at 7:04 下午 | In life（生活）, reading(读书), 学习经历 | Leave a Comment

那些凄凄戚戚，一心只‘想要朋友’的人，从来结交不到朋友。——C.S.Lewis, ‘四种爱’

我对生活一直有一个困惑，就是我发现下面的事情十分的普遍。当人们集中他们所能集中的一切力量，排他的做一件事情的时候，往往结果是做不好，这跟他们的预期恰恰相反，跟他们的投入相比十分不对称。从我整个成长的过程中，我在身边便看到了很多很多这样的例子。

在上大学之前，我身边的几乎全部的同学都把那么多精力投入到准备高考中去，包括那些所谓的‘学习不好’的孩子。他们也每天花费大量的时间学习。甚至我还了解到一个更极端一些的例子，这个一个跟我差不多同龄的孩子。他因为高考失利而反复重考，有时候会读上几年，但是遇到不如意又走回头路。去年，他放弃一切重新参加高考，依然成绩不甚理想。而他的母亲一直采取陪读的策略。结果，数年的大好光阴全部浪费在准备一个无聊考试之上。这基本上是一个悲剧。我之前想这个问题的时候，很简单的把原因归结到学习效率低下和长期的逆反情绪。当然这些都是正确的原因，但是并不是很深层次的原因。

Continue reading 为什么做不好做不成…

The International Mathematical Olympiad 2007 problems

十月 3, 2009 at 3:38 下午 | In Maths（数学）, The IMO problems（国际数学奥林匹克竞赛）, elemantary mathematics | Leave a Comment

Problem 1. Real numbers a_1,\cdots,a_n are given. For each $latx i(1\leq i\leq n)$ define

d_i=\max\{a_j:1\leq j\leq i\}-\min\{a_j:1\leq j\leq i\}

and let

d=\max\{a_j:1\leq i\leq n\}

(a)Prove that, for any real numbers x_1\leq x_2\leq \cdots \leq x_n,

\max\{|x_i-a_i|:1\leq i\leq n\}\geq \frac{d}{2}  (*)

(b)Show that there are real numbers x_1\leq x_2\leq \cdots \leq x_n such that equality holds in (*) Continue reading The International Mathematical Olympiad 2007 problems…

’solving mathematical problems’

九月 28, 2009 at 3:48 下午 | In Maths（数学）, The IMO problems（国际数学奥林匹克竞赛）, elemantary mathematics | Leave a Comment

I spent several days on a book, ‘solving mathematical problems: a personal perspective’, written by Terence Tao, who was only 15 years old when he finished it. I find this book a quite interesting one. It does not contain tons of problems but rather the specific details about intuition and motivation, which probably will be met during the journey of thinking when one first deals with these problems. Since last year, I have been learning about Professor Tao’s lecture notes about several courses, from which I grasped lots of modern mathematics. From this little book, however, I think I know why Professor Tao can explain so clearly about so many modern deep mathematical theories. It is because he has already long been used to writing down ideas as clearly as he can. Even when he was a 15-years-old little boy, he intentionally practiced doing so.

In July, it was my first time to get interested in International Mathematical Olympiad. These problems, no matter how tricky they could be, don’t require lots of knowledge to solve. They actually provide a very good opportunity to improve our ability to think mathematically. At first, I usually spent more than 4 or 5 hours for each problem before I can finish them. Only after I did one set or two, I can do them much quicker. There is definitely some fort of experience in dealing with them. That is why some can be trained in my country to get the golden medals. I figured if I read some books on this subject, I could do these problems better. So I did.

In fact, the book ‘solving mathematical problems’ is just what I need in the first place. It is even better than I thought. I recommend it to people who will be potentially interested in Mathematical Competition, such as high school students, and those who are just interested in elementary mathematics, who do maths for fun. Anyway, this is not a difficult book. It can be read in leisure time. It also can be used as an introduction for mathematical competition. Also, it is written by Professor Tao, whose books usually could give people a hard time to go through. Among those books written by him, for most people, this one would be the first one that ever can be finished, I guess.

性别歧视及其他

九月 21, 2009 at 10:01 下午 | In life（生活）, movies（影视评论） | Leave a Comment

加拿大女数学家Izabella Laba于九月五日写了一篇帖子’gender dependent’，讨论了社会对女性的隐性的歧视以及她对此的一些观点。这位女数学家的数学研究方向我有兴趣，所以我一直在关注她的博客。我发现她表达政治观点时候有时候很是尖锐。像她这样的数学家，我至少没见过太多。前一段时间有个孩子说我也‘有点歧视女性’。我一惊，这可一定要改，这中思想千万别侵入我的脑袋。

我回头翻了翻以前写的东西。发现大约一年之前，我曾经不知深浅的写过自己一篇帖子，谈论女性为何总体不能在事业上成功。 这一年我读了一些书，看法改变不少。总的来说，不应该把长期的社会的不公平所造成的结果反过来作为批评人的原因。同样的道理也适用于人种。明明是因为社会长期的不公平，因此带来客观上黑人总体受教育程度低。忽视掉这个因素而苛责黑人群体本身，这个做法只是在给骨子里给歧视做借口。事实上，中国这样的事情甚至更多一些。只要观察农民工便很容易看到。一方面，社会对他们没有公平可言。他们受到教育底下，反过来又成了别人看不起他们的基础。但是中国还有一个更严重的问题，那便是，官方永远不公开把实情讲出来让大家讨论。 Continue reading 性别歧视及其他…

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