他被誉为天才少年，李修福的得意弟子

他是国际奥林匹克数学竞赛满分金牌得主

他麻省理工学院博士毕业后，如今是浙大数学讲师

对于别人说他是否成功，他有自己的态度

东北网5月10日讯 7日，《“奥数教父”李修福的反思》刊发后，李修福的纠结、个体反思，引发了家长、老师及社会各界的群体性反思。你的孩子在为奥数纠结吗？奥数到底学不学，怎么学？当教育变成应试，成材的标准又是什么？连续两届国际奥林匹克数学竞赛满分金牌得主、李修福的得意弟子罗炜，当年的天才少年，他认为自己成材了吗？昨天，记者带着诸多问题，连线罗炜。

　现在大量时间在做数学研究

罗炜话少，冷静得像在分析他的数学问题。

从哈师大附中毕业后，罗炜被保送到北京大学数学系，大学毕业之后到麻省理工学院读博士。他说：“要进行学术研究的话，都要申请读博士。”

回国后，罗炜到浙江大学数学中心当了一名老师，今年38岁了。

“现在是教授了吧？”记者问。

“没有，讲师。”罗炜淡淡地回答。

他简单地介绍了一下自己的研究领域，那是关于数学、物理有关圆的问题，还有应用的一些问题。不是只做一个领域，和组合、计算机图形有关的。

对于记者在内的圈外人来说，这个实在太专业了。至于在学术成果上，他说自己发表论文的刊物比较普通，没有TOP(顶尖)的。

记者试问：“你是不是太谦虚了？”

“不是，可以查得出来的。”和李修福老师一样，他在用1+1=2的数学思维，说着自己的话。

作为大学老师，罗炜有自己的学生，一般来说，一星期有五六节课。剩下的大量时间，他都用来做研究了。浙大的数学中心，也是一个研究所。

对于为什么没回到母校北大工作，罗炜称没有机会，另外，回北大也不一定很好。一个好学校，在某种程度上能给学生很大的自信心，在文化方面的培养可能也有好处。而对于工作的话，要考虑工作的自由程度，另一个是待遇。

辉煌已成过去现在很低调

提到哈尔滨人仍关注着他，他还是很多人的偶像，罗炜好像很意外：“大家很关心吗？”仿佛说的不是他自己。

他说自己不太在意这个，在意了是会有压力的，所以反而喜欢低调，没人注意。

当然，他承认自己人生最辉煌的是得金牌的那段日子。只不过大学以后开始低调了，是刻意的低调。比如说，不爱说话，不主动发表意见。

至于周围人对他的看法，是否觉得他傲气，他称自己不知道，没人跟他说，他也不太在意别人的看法。即使别人有那样的错觉也是临时的。如果是自己认为有意思的问题，他愿意真诚交流。

他说，自己现在接触的人很多都是喜欢数学的。这些人不在意人对人的态度，这是他们这个圈子里人的特点。

不强求女儿学习奥数

为了升学，目前很多家长逼着孩子学奥数。

对此，罗炜直言，奥数还是比较难，如果没什么兴趣，碰到难题的话，对自信心、学习兴趣的影响都很大。可能就比较讨厌了。连带着可能其他的学科也学不好。因此，学奥数如果孩子很喜欢，就好，如果不喜欢就没什么意思。

当然，很多孩子学奥数是为了考试。因为学校也希望找到比较聪明的学生，学数学确实能看到这方面的领悟力。如果要上好学校，就要证明自己。

“你的孩子学奥数吗？”记者问。

罗炜说，女儿很小，才一岁。但她长大后，他会尽量引导她的兴趣。但不会强逼着她学，她有她的自由，不学坏就行。

我不在意别人怎么看我

兴趣，是孩子成材最重要的。罗炜执此观点。

“我觉得一个小孩不喜欢的话，不可能成材。大部分还是看他是不是把时间都充分地利用起来，去做喜欢的事情。老师讲得好也是次要的。老师只是告诉你有一个好的东西，需要你去探索、去学习。而靠老师手把手地教，不会有出色的学生。”

罗炜说，成材本身就是一个社会标准，到底是别人觉得你的孩子很好，还是自己很开心、很满足，每个人的看法不一样。

曾经的金牌，曾经的第一，罗炜又是怎样看待现在的自己？他说：“其实自己不在意别人的评价，成材不成材是自己对自己是否满意的事情，别人的标准都可以忽略。我的思维方式可能和别人不一样。”

正因为这样，所以罗炜对自己的现在很满意，觉得挺好的。比较重要的是，自己现在比较自由，有自己的时间做喜欢的数学研究。

记者随后在他的QQ空间看到，他的空间也叫做“数学的无限空间”，在那里，你会发现有几百篇的专业数学证明题，也有很多对此有兴趣的网友参与其中。

“包括恩师李修福在内，人们对你的期望值很高。”记者又扔出了一枚“砖块”。

罗炜犹豫了一下：“是吧。我没考虑过这个问题，我觉得不需要完成任何人对我的期望，因为这是我自己的事情。”

“李修福老师说了，捡破烂儿也得捡第一呀。”记者用李修福老师的话反问罗炜。

“这个价值每个人看法不同，争第一是一种，提高自己也是，不争也可以。”罗炜这样看。

前两年曾回过哈尔滨

对于故乡哈尔滨，罗炜很少回来，因为父母都在北京。父亲是电机学教授。上高二时，父亲工作调动到北京，一家人迁居北京。罗炜在清华附中借读。

在北京定居的父母，也经常来杭州看儿子，现在还要看的是一岁多的小孙女——可爱的歆艺。罗炜说现在在家都不太上网了，女儿淘气，不时过来捣乱。

回哈尔滨，只是前两年有一次，是师大附中找他讲课。当然还是讲奥数的事。据他讲，现场的效果挺好，同学们注意力很集中，还是愿意听他讲。

30多岁，正是忙事业、忙家庭的时候，罗炜也不能例外，如今只有一个初中同学一直联系，其他的都没联系了。由于高二下半学期转到清华附中借读，和哈尔滨的高中同学在一起时间都不长。他的高中校友也有一个学数学的，只记得好像在北京大学数学系。

罗炜的时间大部分是属于数学的，他告诉记者马上要去搞科研了，匆匆道别。记者截稿时，罗炜肯定还在自己的研究室里忘我地工作。

不管他将来是不是华罗庚、陈景润，他在做自己喜欢的事，在自己的领域里创造价值。沙子里的金子是极少的，更多的还是普通的沙子，这样才有浪漫美丽的海滩吧。

超前学习不如打好基础

奥赛冠军怎么学数学？罗炜介绍，他基本上就是看书、做题。那时候有几本奥数习题书，拿来尽量看、做，不会的，就看答案，分析人家是怎么解答的。

因为自己有兴趣，所以花了很多时间在数学上，也影响了一些其他学科的学习，有点偏科。语文在初中一般80多分，高中70多分(按满分100)。其他的理科也都能打90多分，所以整体上没影响升学。

对于现在很多初中生超前学习，罗炜很不赞成。他称，虽然是出于中考的需要，但这不算是很好的学习方法。学数学要按顺序一步一步来，逻辑关系、难易程度、思维方式，都是有台阶的。假如基础打不好，后面很吃力，越学越难。学数学总要做题的，学太快理解得不够深。

很多学生到高中跟不上。是初中的学习方法没让他适应高中的学习，还有很多高中考得好，大学也学不好的。

“把你看过的要弄懂，要能用来做题，这是基本方法。”罗炜说，做题，不是题海战术，是要把你学的概念能应用。

他的一个有用的方法是：做题后要总结自己的方法，或者看到的方法，做题中比较关键的步骤是什么，这样才会对今后的学习有帮助。而不是一种题型做好多遍。

“一种题型会做了，知道怎么做了，也知道结果怎样，实际上就没必要再做同样的了。”他说，我知道怎么做的，就跳过了，做更难的题。

如果初中没打好基础，到了高中发现不会了，别人学新的东西，你还要补旧的，你就得再不停地补。

原文链接：

http://heilongjiang.dbw.cn/system/2013/05/10/054758366.shtml?from=singlemessage&isappinstalled=0

PHD的生涯

［转载］关于如何写正式英文email的总结

November 25, 2014 zr9558 Leave a comment

这里说的是比较正式的email，如：写邮件联系国外教授要资料，或者商业email….

感觉自己还有些地方不是很懂..尤其是一些礼貌,句子语气之类的.

说来有些惭愧,学了这样多年英语居然不会用英语跟人交流.只会看看英文电影,读写纸面英语…教育的悲哀?

查资料的过程中发现国外的老师是会辅导大一的学生跟授课的老师(professor)之类的如何联系的…

其实国外的人比我们规矩多……小总结下,希望对某些人有用.如有谬误欢迎鸡蛋.

一：常识：

1．要用教育网邮箱，避免被过滤掉。地址格式直接用wi@cqu.edu.cn>避免其他格式某些系统不支持。

2．地址栏只有一个地址，不要同时发给几个人，尤其是不要让一个接收者知道。就是避免在接收者收到的邮件里面看到正文前面一串邮箱地址：（1）不安全，把某些人的邮箱传给了不认识的人。（有时候居然发现有些人写满自己个人信息群发邮件。。）（2）一个问题发给两个人，两个人如果都觉得对方会回答，你的问题就没人回答了。

3．写完，发之前要通读几遍，找错误，最好找别人读一下，看会不会被误解。尤其是要核对后面“句型和句子的注意事项”中的问题，确保在邮件中你显得比较专业，比较认真。。尤其是在找人帮忙或问问题的时候。

4．如果问问题或索要资料，要交代清楚背景，避免收件人不知所问。

二：整体格式框架注意事项：

1 . 称呼和正文之间，段落之间，正文和信尾客套话之间一般空一行，开头无须空格。有多个要点时,分项列举或分行以便于阅读。（参考最后的例子）；

2. 介绍自己。

联系关系比较淡，或陌生人时要在开头简单介绍下自己的主要信息，姓名，职业，等。

3. 不要把某个词全部大写，这样常会被认为是在吼叫或骂人（很不礼貌）。如果要强调某些词语或句子用底线,斜字,粗体就可以了. 如：MUST change to OS immediately. 外国人就觉得不礼貌和喝令人一样. 要强调的话。

4. 不要用简写和笑脸等符号 J，不要用长词和不常用的词语。简单的单词便于理解。

（其实估计我们也只会常用的词语）。。。

5. 整个email不要太长，每个段落不要太长，每个句子不要太长，不要用结构复杂的句子。

大家不喜欢读长的，太长可能人家就不读了。整个email不要超过一个屏幕.即:不需要scroll就可以看完。一个段落大概只由一到三个句子组成。一个句子不超过15-20个词。

三：标题：

1. 要写一个 meaningful 标题。不要太宽泛，不要含糊不清,不要太长，一般不要超过35个字母，只需要将位于句首的单词和专有名词的首字母大写.

比较下面几个：

“Product A information “ is good than “product information”.

“News about the meeting” vs “Tomorrow’s meeting canceled”.

professional trainees from sister company should abide by rule of local company（太长）
可以写出自己的主要目的

“Could I please get the assignment for next Wednesday?”

2. 视信的内容是否重要，还可以开头加上URGENT或者FYI（For Your Information，供参考），如：URGENT：Submit your report today!
但别乱用，否则就成了喊狼来了的小孩子了.

四：email头称呼：

Dear Professor Sneedlewood 即：Dear Professor +lastname, 千万不要用first name称呼。千万不要写错名字,头衔（会很让人反感），有的人有荣誉学位就不喜欢用一般头衔.

Dear Committee Member: 注意:冒号是可以用的.

不要用词给人鲁莽感觉的词语.(老实说还真不知道英文里面那些词鲁莽除了fuck,shit,bullshit).如果文中要用he ,she,madam,……要查收件人性别，确保无失误。

五：段落：

1. 段落开头写重要的和要强调的事情.同样重要的事情要写在句子首.

（1）Because he was unable to attend the meeting personally, he forwarded his congratulations on cassette tape.
（2）He forwarded his congratulations on cassette tape because he was unable to attend the meeting personally. 两者强调的事情就有分别了.

2. 轻重有分，同等重要的用and来连接，较轻放在次要的句子里.

六：句型和句子的注意事项.(结合了一些商业email和联系教授email的混杂例子)

1. 用主动句型而不是被动。

“We will process your order today” is more personal than “your order will be processed today”. 后者用太多，sounds unnecessarily formal.

2. 那么I和you呢?好烦好烦.一般来说,收信人的利益比较重要,名义上都要这样想.给人尊重的语气就一般不会错了. 多用you有时会有隔阂的感觉.
You will be pleased to learn that you have been selected to serve on our advisory board. Your prompt response will be appreciated. (好像欠你一样)
I am pleased that our board has selected you as the best qualified candidate to serve on our advisory board. I hope you’ll agree to serve. (这就友善多了)
Your book was well written and comprehensive. (不用你来判断我呀~~)
I thoroughly enjoyed your book and found an answer to every one of my questions about performance appraisals. (客气一点,人家受落)
总之,语气和宾词的运用得当能决定你的礼貌程度.

3. 亲切,口语化是比较受欢迎! 用宾词和主动的词,让人家受落.
例如: （1）This information will be sincerely appreciated.”
（2） We sincerely appreciate your information. 明显地,我们会喜欢第2句.

4. 要求人做事要有客气Ask politely.

“Could you please email me the page numbers for the next reading? Thanks!” sits better than “I need the assignment. Please send it.” It’s a Golden Rule kind of thing, right?

尽量减少词数，如：on a regular basis =====regularly.

5. 不可主客不分或模糊.

Deciding to rescind the earlier estimate, our report was updated to include $40,000 for new equipment.” 应改为：Deciding to rescind our earlier estimate, we have updated our report to include $40,000 for new equipment. (We决定呀, 不是report.)

6. 结构对称,令人容易理解. The owner questioned the occupant’s lease intentions and the fact that the contract had been altered with ink markings.
应改为: The owner questioned the occupant’s lease intentions and ink alterations of the contract.

7. 单数复数问题:
例如: An authorized person must show that they have security clearance.

8. 动词主词要呼应. 想想这两个分别:
1）This is one of the public-relations functions that is underbudgeted.
2）This is one of the public-relations functions, which are underbudgeted.

9. 时态和语气不要转变太多.看商务英语已经是苦事,不要浪费人家的精力啊.

10. 标点要准确.
例如: He did not make repairs, however, he continued to monitor the equipment.
改为: He did not make repairs; however, he continued to monitor the equipment. (没想到老外也用分号啊 !!);

11. 选词正确. 好像affect和effect, operative和operational等等就要弄清楚才好用啦.

12. 拼字正确. 有电脑拼字检查功能后,就更加不能偷懒.

13. 意思转接词要留神. 例如: but (相反), therefore (结论), also (增添), for example (阐明). 分不清furthermore和moreover就不要用啦.

14. 修饰词的位置要小心,
例如: He could only reimburse the cost after July 15.
应为 He could reimburse the cost only after July 15.

15. 用语要肯定准确.切忌含糊.
例如:The figures show a significant increase.” 怎样significant呀,
改为: The figures show an increase of 19%.

16. 立场观点一致. 少用被动语.
例如: Partial data should be submitted by April.
改为: You should submit partial data by April.就很好了.

17. 求人做事，最后写:thanks 即可.

七：结尾

要写自己全名，比较详细的东西。。。

1. 格式问题，

Ken Green

Vice president, Unicom China.

下面这个不如上面的

Ken Green

Vice President of Unicom China （其实就多个”of”,但没办法要专业…要专业.）

2. 书信的结尾致意要留意,弄清大家的关系才选择用词,例子:
（1） Very Formal非常正规的(例如给政府官员的)
Respectfully yours, Yours respectfully,
（2） Formal正规的(例如客户公司之间啦)
Very truly yours, Yours very truly, Yours truly,
（3） Less Formal不太正规的(例如客户)
Sincerely yours, Yours sincerely, Sincerely, Cordially yours, Yours cordially, Cordially,
（4）Informal非正规的(例如朋友,同事之类)
Regards, Warm regards, With kindest regards, With my best regards, My best, Give my best to Mary, Fondly, Thanks, See you next week!

八: 获得别人回复要再回复:Thanks .

九: 如果没有回复，一周后可以再发个提醒一下。

例子:

To: Carmine Prioli <prioli@social.chass.ncsu.edu>

Subject: Will be absent next Wednesday. Could I get the assignment?

_______________________________________________________________

Hi, Dr. Prioli-

I will be playing my cello for a friend’s conference performance in San Antonio, TX next Wednesday, Thursday, and Friday (November 16, 17, & 18).

I am afraid my Wednesday flight leaves before your Colonial Literature class. Could I please get the assignments for that day so I can prepare for Monday?

Thanks!

Susanna Branyon
Colonial Lit, MW 1-3pm

一个外国老师说的如何写英文邮件联系教授——老外的规矩比我们多多了。

How to write clear emails to your professor (or, why I currently think my undergrad students are rockstars)

we talk about how to address faculty and staff (a hint: call them “Dr” or “Professor” as a default, and they may tell you to call them something else if you’re lucky).

Together we came up with the following guidelines:

Write a clear subject line that actually summarizes what the question is and what it might be connected to in the course.

Address me in the email, and remember to call me “Dr.” or “Prof.”

Give me some context for the question, situating it in the particular assignment or activity you’re working on.

Punctuate. Capitalize appropriately. Use complete words and sentences; this is not texting. Check your spelling.

Be specific and detailed about what the difficulty or challenge is regarding.

Ask an actual question, rather than leaving it up to me to infer what you don’t understand.

Be nice and thank me for answering.

Sign your full name and give what ever institutional markings might be helpful for me to keep this in context.

例子:

Dear Dr. Pawley,
Our lab group was working on the class project for ENGR 126 and we didn’t understand one of the requirements (#4). Can you please clarify for us what you mean by “what the experts say”?
Thanks
Astu Dent, Team 4

一个微软的人写的seek job 电子邮件写法,抱怨了一大堆不过感觉很有道理;

Sounds simple but so many people screw it up. Seth talks about writing a personal email. But I see these mistakes in the emails I get from job seekers and people trying to get my attention for some other professional reason. Listen up staffing tools vendors, agency recruiters and the people that want me to introduce them to some nameless person within Microsoft that they can contact about their business idea/product concept, etc. Yeah, let me get right on that. I have a ton of extra time and absolutely no priorities. And I definitely would not rather be spending my time doing something else. Ooh, snarky.

I think the worst offenders are the folks that contact me through LinkedIn looking for a job. I can’t even tell you how many of these rules have been broken. But let’s just say that cut and paste isn’t always your friend. If you don’t take the time to craft an email that, say, addresses me by name and/or references my company, then can you really expect that I will take the time to review your resume and forward it along to the recruiters here? Really?

I get a lot of mail from people that don’t really know what they want to do at Microsoft. Oh yes, I actually do. If they don’t know, how the hell am I supposed to know? I always send them to our career site to find some positions that they could be interested in. I would like to believe that they are just experiencing a momentary lapse of reason and are not expecting me to wade through all of our open positions in order to find the ones that would be a fit for their background and that they would personally enjoy. You know, because I know them personally.

The thing is that I actually want to help people. But not if it’s a waste of my time. And helping people that don’t have the good sense to not spam a bunch of staffing folks or do a little research so they know who they are asking for help is definitely wasting time. So here are some of my rules for sending a job search email:

1) Address me personally. If you don’t, I know….KNOW that you are cutting and pasting. And if that is the case, I know that you think that your job search is a numbers game. Knock on enough doors, etc. That makes me think that you are not a sought after prospect. Or, it makes me think you are lazy. Either way… not good.

2) When and if you do address me by name, make sure it’s my name. We all know what mail merge is. Refer to #1 above. And on the same note, that whole “Sir/Madam” thing? Come on! Even if you are not from the US, you have access to the same interwebs I do and can identify “Heather” as a female name. Nobody has ever called me “madam” to my face…ever! Or “sir” for that matter.

3) You don’t have to send me a long email with a narrative of your professional life. It’s best to tell me where you work and what you do plus a little about any previous work that is relevant to the position that you are looking for (“I am currently working as an account manager at XYZ and previously worked at ABC in tech support.”), plus any experience with specific markets (“My experience is primarily in the healthcare and biomedical industries.”) and what you are looking to do (“I would like to get back into a role where I can utilize both my account management and technical expertise. I noticed a position open at Microsoft for a Technical Account Manager, focused in healthcare and feel I would be a good fit.”). The goal is to get the recruiters to view your resume, not to restate the resume. The email is, at most, a teaser.

4) If you are open to relocation, state it up front. It’s one of the first questions we will ask you.

5) If you reference specific positions or groups, include a job code from our career site. You should spend time on our career site looking regardless. Including job codes helps me get your resume to the right person. It also shows me that you are serious.

6) Don’t tell me you are willing to “do anything.” Wow, that is a red flag! OK, well first, nobody is qualified to do any/every job. So it’s not smart. And it sounds desperate. I know that it’s hard if you are out of work; that is probably an understatement. But despite this fact, you want to make employers feel that they would be fortunate to get you. Because you got skillz.

7) Don’t tell me about your personal life. There is some stuff that I am more comfortable not knowing. If you are sending an email to inquire about open positions, include only information that is relevant to the position. I know that people ask for advice and include a little personal info, and that is fine. But if you are reaching out to me about a position, I don’t need to know that the reason you want to relocate is that your mother-in-law is living with you and you’d like to leave her behind because she chews loudly. Just sayin’.

8) Attach your resume from the beginning. I’ll look at it and forward it along to any appropriate recruiters. It’s how I roll. So withholding it and asking me to tell you more about the position is just going to result in extra emails.

9) You can ask me to spend some time talking to you about a position or group, but it’s not going to happen. Of course we all want that. It might be reasonable if you are reaching out to a recruiter and you have all of the requirements of an open position (be honest with yourself about that too), but consider whether the person you are reaching out to is the recruiter for the open positions or even a recruiter at all. And to that end…

10) When you are reaching out to someone at a company, especially when you are asking for something, take a little time to research them. Just search on their name (might I recommend that you Bing them?). It might inform how you engage that person. For example, if someone did a search on my name, they would find that I am not currently a recruiter but I do work in Staffing, that I am female (picture frequently accompanying my contact info), that I am a blogger, that I am open to forwarding resumes and that I provided a list of how to write an effective job inquiry email.

I don’t mean to be overly critical. Any one of these things is not a deal-breaker but most of it seems like common sense. You obviously want to make a good impression and get your resume in the right place ASAP. So yeah, consider this a little email tough love.

原文链接如下：

http://www.cqumzh.cn/uchome/space.php?uid=102519&do=blog&id=278386

PHD的生涯

［转载］写英文 Email 要特别留心这些点，一些原则私信也适用

November 25, 2014 zr9558 Leave a comment

今天的笔记主要是日常的 Email 写作，只适用于理工科范围内，对于商业、法律、艺术等等领域可能均不适用。

Email 要以收信人为主题

写 Email 的一个原则，应该是以收信人为主题。信件的内容，应该是跟收信人相关的，或者至少，不能让收信人觉得这是一封群发邮件。

我在知乎收到过很多私信，绝大多数我都不想回，也没法回，知道为什么吗？这些私信篇幅都很长，有的长达好几百字，内容全部是关于写信人自己，几乎从来没有出现过收信人的信息。“我是什么什么学校的学生……我要转专业……”、“我是什么什么情况……我该考研还是工作……”、“我在哪里哪里工作了几年，我要不要转行……”

事实上，这些信换一个收信人的名字就可以原封不动的再发给别人，就像春节的时候大家讨厌的那些群发短信一样。而且很多时候，他们问的很多问题我都有过类似的回答，哪怕他们花三分钟的时间点开我的主页，就能找到相关的信息。

你说我该怎么回复呢？请问他们写的这些信跟我有关系吗？请问我说过我可以提供免费的教育就业咨询吗？当然，我很乐意帮助别人，但是前提是，你得让我知道我能怎么帮。

比如说，你是一位教授，收到一位学生的套磁邮件，打开一看，满篇都是“我特别特别优秀，我 985 大学专业第一……”、“我 GRE 多少多少分”、“我怎么怎么牛逼……”。看到这样的信，你会怎么想？可能绝大多数情况，你会觉得，跟我有什么关系呢，这信换个收信人的名字一样能用。

反过来，如果信件里提到的都是你的研究内容、写信人对你的研究内容的看法、写信人自己的经历跟你的这些研究的相关程度，至少，如果我是这个收信的教授，我不会把这封信拖进垃圾箱。

同样的道理，你看一下好的广告，侧重点并不是“我的手机跑分超级高”、“我的车跑的特别快”……而是“你用我的手机可以享受便捷生活”、“你开我的车可以享受驾乘体验”…… 同样，必须是 YOU-oriented，而不是通篇的“我怎么怎么样”。

最最简单的判断方法，数一数你的 Email 里面，是 you/your 出现的次数多，还是 I/my 出现的次数多。如果通篇都是 ”I……“、”My……“，那么问一下自己，我写的这些东西跟收信人有关吗？收信人会对我的这些信息感兴趣吗？

Email 要简洁明了

既然是以收信人为主题，那就不要说那么多废话。对收信人最大的尊重，不是说“感谢你抽时间看我的信”，而是信写得简洁明了，一点不浪费收信人的时间。

除了极其正式的信件，一般来说，第一句话就应该直入主题。这封信是问问题、交作业、约时间、反映情况、提建议、提交简历，收信人读到第一句话就应该知道。

即使是比较正式的信件，比如求职信、套磁信，第一句话也应该说明主题，比如 “I’m very interested in the XX position currently open in your company.” 或者 “I have a very strong interest in the XX position that you have advertised on your website.” 不要以 “My name is ……” 或者 “I am s student of ……” 开头，一定要让收信人最快时间知道你想干嘛。

一般来说，没有必要在正文里介绍你的名字、地址、联系方式之类的，这些东西应该在信件的专门位置，如果收信人感兴趣，想要回信或者以其它方式联系你，自然会去找。如果是求职信或者套磁信，可以在第一句说明目的之后，紧接着突出重点的介绍自己，比如 “I am currently a PhD/master’s student in XX at XX University with a focus on XXXX. I expect to graduate in XX/XXXX.”

Email 要避免套话空话

既然要简洁明了，一个有效的办法就是避免套话空话。什么叫套话空话呢？就是没有任何意义的话。如果删去这些句子或者这些表达，一点都不影响信件的意思，那这样的句子就是套话空话，写这样的句子就是浪费收信人的时间。

应该尽量避免什么样的套话空话呢？

过于客气的表达。Please do not hesitate to contact me at your earliest convenience 或者 Very truly yours 。假想一下收信人就坐在你的面前，你直接跟收信人面对面谈话，你还会用这些表达吗？你不觉得别扭吗？信件无非就是用书面的形式传递这些谈话，怎么说，就怎么写。过多的使用这些 cliche 或者 canned speech 会让你的信显得不真诚，给人一种油嘴滑舌的感觉。

最最常见的几个过分客气的表达包括：

at your earliest convenience （直接说 when you can 或者 soon）
please find enclosed 或者 I have forwarded （直接说 I have sent you）
please do not hesitate （不需要说，直接删掉）
过于书面的表达。比如 herewith、aforementioned、hereby、herein 这样的词语，或者非常老派的表达方式，让人感觉你是从一本出版于 60 年代的语法书上抄来的。
过于口语化的表达。这个对于非英语母语的人来说，写出过于口语化的英语来其实是不太可能的。比如说，用 Let’s touch bases next week 代替 Let‘s talk next week，用 bottom line、team player、square one 等等类似的口语化词汇，这些都属于所谓的 slang 或者 buzzword。这样的表达会让人觉得你不太职业，给人不信任的感觉。其实，现在的中文里有大量这样的例子，“碉堡”“么么嗒”“蛮拼的”之类的最好不要用在除了非常私人的信件以外的任何 Email 里。

Email 要注意性别指代

在英文语境里，这其实是非常重要的。在写 Email 之前，你应该尽量弄清楚收信人的性别，然后用正确的前缀。现在有 Google、LinkedIn、Facebook 等等，很容易就能确定收信人的性别。

在信件的内容里，也要注意避免性别指代的用法。其实不只是信件，这也适用于绝大多数英语写作。

不要说 Each employee must show his identification，请用 Employees must show their identification.
不要说 By the age of three, a child should be able to feed and dress himself，请用 By the age of three, a child should be able to eat and get dressed without help.
不要说 Although a nurse often comes to the job without computer experience, she can easily be trained to use the hospital software，请用 Although a nurse often comes to the job without computer experience, this person can easily be trained to use the hospital software.
不要说 actress，请用 actor。（可以注意好莱坞的采访或者发言，大多数女演员会说 as an actor… 医生都是 doctor，没有 doctress，所以演员都是 actor，越来越少的人用 actress 这个词）
不要说 businessman，请用 businessperson。
不要说 fireman，请用 firefighter。
不要说 maid，请用 housekeeper。
不要说 policeman，请用 police officer。
不要说 mailman，请用 mail carrier。
不要说 salesman，请用 sales representative 或者 sales agent。
不要说 waiter 或者 waitress，请用 server。
不要说 mankind，请用 humankind。

Email 的格式和字体

Email 的段落应该尽量短小，尽量避免一整段的长篇大论，尤其是第一段更应该紧抓重点。段落首行不需要缩进，单倍行距，段落之间空一行。

注意选用合适的字号和字体。字号不要过大或者过小，字体不要用太花哨的艺术字体，不要用花花绿绿的颜色，尽量避免太多的粗体和斜体。

原文链接如下：

http://mp.weixin.qq.com/s?__biz=MjM5MTAxNjMwMA==&mid=201786861&idx=1&sn=3a5116e49dc0d64c2a3fadbd6163fdd4&key=a96d0bed9b36e45d0559ce641e0b01de38e4dc9cadf68761b9d95ccb2dca0fca9a18ec70ed08b12f9dd7cc639eda0a72&ascene=14&uin=NDIzMzY5MzIw&devicetype=iPhone+OS8.0&version=16000114&pass_ticket=jY2IBlEpeIJYJ0DwooHqYpGrm9RmG1kAr0U4%2Bm5EO9Y1xcWLKHDTishNUjl2DR%2F9

PHD的生涯

［转载］英文Email

November 21, 2014 zr9558 Leave a comment

这样写英文Email，老外会感觉你很有礼貌、很有风度，很想帮助你

作者：白羽轩❤Queenie

需要写的英文邮件多了，就觉得很吃力，尤其是当需要经常写给同一个人时。希望邮件的开头、结尾、一些客套的话能有不同的表达~~

邮件的开头：感谢读者是邮件开场白的好办法。感谢您的读者能让对方感到高兴，特别是之后你有事相求的情况下会很有帮助。

Thank you for contacting us.如果有人写信来询问公司的服务，就可以使用这句句子开头。向他们对公司的兴趣表示感谢。

Thank you for your prompt reply.当一个客户或是同事很快就回复了你的邮件，一定记得要感谢他们。如果回复并不及时，只要将“prompt”除去即可，你还可以说，“Thank you for getting back to me.”

Thank you for providing the requested information.如果你询问某人一些信息，他们花了点时间才发送给你，那就用这句句子表示你仍然对他们的付出表示感激。

Thank you for all your assistance.如果有人给了你特别的帮助，那一定要感谢他们！如果你想对他们表示特别的感激，就用这个句子，“I truly appreciate … your help in resolving the problem.”Thank you raising your concerns.

就算某个客户或是经理写邮件给你对你的工作提出了一定的质疑，你还是要感谢他们。这样你能表现出你对他们的认真态度表示尊重及感激。同时，你也可以使用，“Thank you for your feedback.”

在邮件的结尾：在邮件开头表示感谢一般是表示对对方过去付出的感谢，而在邮件结尾处表示感谢是对将来的帮助表示感谢。事先表示感谢，能让对方在行动时更主动更乐意。

Thank you for your kind cooperation.如果你需要读者帮助你做某事，那就先得表示感谢。

Thank you for your attention to this matter.与以上的类似，本句包含了你对对方将来可能的帮助表示感谢。

Thank you for your understanding.如果你写到任何会对读者产生负面影响的内容那就使用这句句子吧。

Thank you for your consideration.如果您是在寻求机会或是福利，例如你在求职的话，就用这封邮件结尾。

Thank you again for everything you’ve done.这句句子可以用在结尾，和以上有所不同。如果你在邮件开头已经谢过了读者，你就可以使用这句话，但是因为他们的帮助，你可以着重再次感谢你们的付出。

十种场合的表达

1. Greeting message 祝福

Hope you have a good trip back. 祝旅途愉快。

How are you? 你好吗?

How is the project going? 项目进行顺利吗?

2. Initiate a meeting 发起会议

I suggest we have a call tonight at 9:30pm (China Time) with you and Brown. Please let me know if the time is okay for you and Ben.

我建议我们今晚九点半和Brown小聚一下,你和Ben有没有空?

I would like to hold a meeting in the afternoon about our development planning for the project A.

今天下午我建议我们就A项目的发展计划开会讨论一下。

We’d like to have the meeting on Thu Oct 30. Same time.

十月三十号(周四),老时间,开会。

Let’s make a meeting next Monday at 5:30 PM SLC time.

下周一盐湖城时区下午五点半开会。

I want to talk to you over the phone regarding issues about report development and the XXX project.

我想跟你电话讨论下报告进展和XXX项目的情况。

3. Seeking for more information/feedbacks/suggestions 咨询信息/反馈/建议

Should you have any problem accessing the folders, please let me know.

如果存取文件有任何问题请和我联系。

Thank you and look forward to having your opinion on the estimation and schedule.

谢谢你,希望能听到更多你对评估和日程计划的建议。

Look forward to your feedbacks and suggestions soon.

期待您的反馈建议!

What is your opinion on the schedule and next steps we proposed?

你对计划方面有什么想法?下一步我们应该怎么做?

What do you think about this?

这个你怎么想?

Feel free to give your comments.

请随意提出您的建议。

Any question, please don’t hesitate to let me know.

有任何问题,欢迎和我们联系。

Any question, please let me know.

有任何问题,欢迎和我们联系。

Please contact me if you have any questions.

有任何问题,欢迎和我们联系。

Please let me know if you have any question on this.

有任何问题,欢迎和我联系。

Your comments and suggestions are welcome!

欢迎您的评论和建议!

Please let me know what you think?

欢迎您的评论和建议!

Do you have any idea about this?

对于这个您有什么建议吗?

It would be nice if you could provide a bit more information on the user’s behavior.

您若是能够就用户行为方面提供更多的信息就太感激了!

At your convenience, I would really appreciate you looking into this matter/issue.

如果可以,我希望你能负责这件事情。

4. Give feedback 意见反馈

Please see comments below.

请看下面的评论。

My answers are in blue below.

我的回答已标蓝。

5. Attachment 附件

I enclose the evaluation report for your reference.

我附加了评估报告供您阅读。

Attached please find today’s meeting notes.

今天的会议记录在附件里。

Attach is the design document, please review it.

设计文档在附件里,请评阅。

For other known issues related to individual features, please see attached release notes.

其他个人特征方面的信息请见附件。

6. Point listing 列表

Today we would like to finish following tasks by the end of today:1…….2…….

今天我们要完成的任务:1…….2…….

Some known issues in this release:1…….2…….

声明中涉及的一些问题:1…….2…….

Our team here reviewed the newest SCM policy and has following concerns:1…….2…….

我们阅读了最新的供应链管理政策,做出如下考虑:1…….2…….

Here are some more questions/issues for your team:1…….2…….

以下是对你们团队的一些问题:1…….2…….

The current status is as following: 1……2……

目前数据如下: 1……2……

Some items need your attention:1…….2…….

以下方面需提请注意:1…….2…….

7. Raise question 提出问题

I have some questions about the report XX-XXX

我对XX-XXX报告有一些疑问。

For the assignment ABC, I have the following questions:…

就ABC协议,我有以下几个问题:……

8. Proposal 提议

For the next step of platform implementation, I am proposing…

关于平台启动的下一步计划,我有一个提议……

I suggest we can have a weekly project meeting over the phone call in the near future.

我建议我们就一周项目开一个电话会议。

Achievo team suggest to adopt option A to solve outstanding issue……

Achievo团队建议应对突出问题采用A办法。

9. Thanks note 感谢信

Thank you so much for the cooperation感谢你的合作!

Thanks for the information

谢谢您提供的信息!

I really appreciate the effort you all made for this sudden and tight project.

对如此紧急的项目您做出的努力我表示十分感谢。

Thank you for your attention!

Thanks to your attention!

谢谢关心!

Your kind assistance on this are very much appreciated.

我们对您的协助表示感谢。

Really appreciate your help!

非常感谢您的帮助!

10. Apology 道歉

I sincerely apologize for this misunderstanding!

对造成的误解我真诚道歉!

I apologize for the late asking but we want to make sure the correctness of our implementation ASAP.

很抱歉现在才进行询问,但是我们需要尽快核实执行信息。

数学科普

Fractals – A Very Short Introduction

November 20, 2014 zr9558 Leave a comment

Excerpt From: Falconer, Kenneth. “Fractals: A Very Short Introduction (Very Short Introductions).” iBooks.

Chapter 7
A little history

Geometry, with its highly visual and practical nature, is one of the oldest branches of mathematics. Its development through the ages has paralleled its increasingly sophisticated applications. Construction, crafts, and astronomy practised by ancient civilizations led to the need to record and analyse the shapes, sizes, and positions of objects. Notions of angles, areas, and volumes developed with the need for surveying and building. Two shapes were especially important: the straight line and the circle, which occurred naturally in many settings but also underlay the design of many artefacts. As well as fulfilling practical needs, philosophers were motivated by aesthetic aspects of geometry and sought simplicity in geometric structures and their applications. This reached its peak with the Greek School, notably with Plato (c 428–348 BC) and Euclid (c 325–265 BC), for whom constructions using a straight edge and compass, corresponding to line and circle, were the essence of geometric perfection.

As time progressed, ways were found to express and solve geometrical problems using algebra. A major advance was the introduction by René Descartes (1596–1650) of the Cartesian coordinate system which enabled shapes to be expressed concisely in terms of equations. This was a necessary precursor to the calculus, developed independently by Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1714) in the late 17th century. The calculus provided a mathematical procedure for finding tangent lines that touched smooth curves as well as a method for computing areas and volumes of an enormous variety of geometrical objects. Alongside this, more sophisticated geometric figures were being observed in nature and explained mathematically. For example, using Tycho Brahe’s observations, Johannes Kepler proposed that planets moved around ellipses, and this was substantiated as a mathematical consequence of Newton’s laws of motion and gravitation.

The tools and methods were now available for tremendous advances in mathematics and the sciences. All manner of geometrical shapes could be analysed. Using the laws of motion together with the calculus, one could calculate the trajectories of projectiles, the motion of celestial bodies, and, using differential equations which developed from the calculus, more complex motions such as fluid flows. Although the calculus underlay Graph of a Brownian process8I to think of all these applications, its foundations remained intuitive rather than rigorous until the 19th century when a number of leading mathematicians including Augustin Cauchy (1789–1857), Bernhard Riemann (1826–66), and Karl Weierstrass (1815–97) formalized the notions of continuity and limits. In particular, they developed a precise definition for a curve to be ‘differentiable’, that is for there to be a tangent line touching the curve at a point. Many mathematicians worked on the assumption that all curves worthy of attention were nice and smooth so had tangents at all their points, enabling application of the calculus and its many consequences. It was a surprise when, in 1872, Karl Weierstrass constructed a ‘curve’ that was so irregular that at no point at all was it possible to draw a tangent line. The Weierstrass graph might be regarded as the first formally defined fractal, and indeed it has been shown to have fractal dimension greater than 1.

In 1883, the German Georg Cantor (1845–1918) wrote a paper introducing the middle-third Cantor set, obtained by repeatedly removing the middle thirds of intervals (see Figure 44). The Cantor set is perhaps the most basic self-similar fractal, made up of 2 scale copies of itself, although of more immediate interest to Cantor were its topological and set theoretic properties, such as it being totally disconnected, rather than its geometry. (Several other mathematicians studied sets of a similar form around the same time, including the Oxford mathematician Henry Smith (1826–83) in an article in 1874.) In 1904, Helge von Koch introduced his curve, as a simpler construction than Weierstrass’s example of a curve without any tangents. Then, in 1915, the Polish mathematician Wacław Sierpiński (1882–1969) introduced his triangle and, in 1916, the Sierpiński carpet. His main interest in the carpet was that it was a ‘universal’ set, in that it contains continuously deformed copies of all sets of ‘topological dimension’ 1. Although such objects have in recent years become the best-known fractals, at the time properties such as self-similarity were almost irrelevant, their main use being to provide specific examples or counter-examples in topology and calculus.

It was in 1918 that Felix Hausdorff proposed a natural way of ‘measuring’ the middle-third Cantor set and related sets, utilizing a general approach due to Constantin Carathéodory (1873–1950). Hausdorff showed that the middle-third Cantor set had dimension of log2/log3 = 0.631, and also found the dimensions of other self-similar sets. This was the first occurrence of an explicit notion of fractional dimension. Now termed ‘Hausdorff dimension’, his definition of dimension is the one most commonly used by mathematicians today. (Hausdorff, who did foundational work in several other areas of mathematics and philosophy, was a German Jew who tragically committed suicide in 1942 to avoid being sent to a concentration camp.) Box-dimension, which in many ways is rather simpler than Hausdorff dimension, appeared in a 1928 paper by Georges Bouligand (1889–1979), though the idea underlying an equivalent definition had been mentioned rather earlier by Hermann Minkowski (1864–1909), a Polish mathematician known especially for his work on relativity.

For many years, few mathematicians were very interested in fractional dimensions, with highly irregular sets continuing to be regarded as pathological curiosities. One notable exception was Abram Besicovitch (1891–1970), a Russian mathematician who held a professorship in Cambridge for many years. He, along with a few pupils, investigated the dimension of a range of fractals as well as investigating some of their geometric properties.

Excerpt From: Falconer, Kenneth. “Fractals: A Very Short Introduction (Very Short Introductions).” iBooks.

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Excerpts from the Grothendieck-Serre Correspondence

November 15, 2014 zr9558 Leave a comment

Matt Baker's Math Blog

Like many fellow mathematicians, I was very sad to hear the news that Alexander Grothendieck passed away yesterday. The word “genius” is overused; or rather, does not possess sufficiently fine gradations. I know quite a few mathematical geniuses, but Grothendieck was a singularity. His ideas were so original, so profound, and so revolutionary – and he had so many of them! – that I will not even attempt to summarize his contributions to mathematics here. Rather, I thought that I would share some of my favorite passages from the fascinating Grothendieck-Serre Correspondence, published in a bilingual edition by the AMS and SMF. They illuminate in brief flashes what made Grothendieck so extraordinary — but also human. They also illustrate how influential Serre was on Grothendieck’s mathematical development. Before I begin, here is a quote from another wonderful book, Alexander Grothendieck: A Mathematical Portrait, edited by Leila Schneps:

…the…

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MA 1505 Mathematics I

MA1505 and MA1506 Help Sheet

November 15, 2014 zr9558 Leave a comment

MA1505 and MA1506 Help Sheet.

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254A announcement: Analytic prime number theory

November 15, 2014 zr9558 Leave a comment

What's new

In the winter quarter (starting January 5) I will be teaching a graduate topics course entitled “An introduction to analytic prime number theory“. As the name suggests, this is a course covering many of the analytic number theory techniques used to study the distribution of the prime numbers $latex {{mathcal P} = {2,3,5,7,11,dots}}&fg=000000$. I will list the topics I intend to cover in this course below the fold. As with my previous courses, I will place lecture notes online on my blog in advance of the physical lectures.

The type of results about primes that one aspires to prove here is well captured by Landau’s classical list of problems:

Even Goldbach conjecture: every even number $latex {N}&fg=000000$ greater than two is expressible as the sum of two primes.
Twin prime conjecture: there are infinitely many pairs $latex {n,n+2}&fg=000000$ which are simultaneously prime.
Legendre’s conjecture:…

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MA 1505 Mathematics I

MA 1505 Formula List

November 13, 2014 zr9558 Leave a comment

https://zh.scribd.com/doc/114003589/MA1505-Summary

MA1505 Summary

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Published by: Khor Shi-Jie on Nov 21, 2012

PHD的生涯, 心理学

战胜拖延—–让PHD达成每天必要的工作时间

November 8, 2014 zr9558 10 Comments

作为一个PHD，每次被老板问道每周工作多少时间，总是支支吾吾回答不出，于是老板质问：你每天工作一个小时还是两个小时！瞬间就会让自己在当天晚上感到非常的焦虑和恐慌，但是第二天醒来之后继续忘记前一天的恐慌，依旧保持着一种非常拖延的状态。白天依旧无所事事，深陷PHD的时间陷阱而不能自拔；每到深夜，就会为自己在白天的虚度光阴而悔恨。每天，每周，每个月，甚至一整年都处于一种非常拖延的状态，科研依旧毫无进展，每次看到别的PHD一年写出几篇文章，都会觉得自己的压力越来越大。即使在这种情况下，拖延症还是有可能继续困扰着每一个PHD，甚至会一直的影响下去。对于一个患有严重拖延症的PHD而言，要保证一天好几个小时的科研时间，简直就是天方夜谭。而且有一段时间，患者的学习的状态时好时坏，科研一直停滞不前，就像这样：白天起床—焦虑—瞎忙—挫败—焦虑—放松—发现自己之前做错了—挫败—拖延—忧愁。那段时间，别说一天能够工作8,9个小时，哪怕能够克服心理障碍并且高效的工作一个小时，都已经是非常不容易的了。

后来到了高年级，面临着毕业的压力，科研必须有进展，痛定思痛，于是就开始想办法克服自己多年的拖延。据说连续两周让自己保持一种状态就可以形成习惯，要突然改变自己拖延的状态谈何容易。苦海无涯，回头是岸。但是对于一个高年级的PHD而言，苦海无涯是真，回头是岸是假。猛然回头看一眼，已经没有退路，只能竭尽全力游到对岸。同时也会觉得，身边的人能做到的事情，自己也没啥做不到的。于是思考许久，就想到用GOOGLE日历记录时间的方法来看看自己每天都干了啥。

首先第一步需要做的，就是弄清楚自己每天必须要做的事情有哪些。作为一个PHD，当然科研是我们必须要做的，但是我们出门在外，远离家乡，肯定还有很多生活上面的琐事需要我们来处理。同时，学校也会要求每一个PHD每个学期完成一定数量的助教任务。做这些事情都肯定会占用我们的时间。这个时候，就需要我们先提前一周把这些杂事先在GOOGLE日历上面标记出来，表示这些时间段是没有弹性的，是我们必须处理的。标记这类时间的时候，一定要做到尽量精细。比方说：坐车的时间，花在路上大概需要多少时间，吃饭的时候，休息的时间，运动的时间，社交活动的时间，诸如此类。

标记完这些时间之后，就可以很清楚的看到自己每天，每周，甚至每个月能够在科研上面最多投入多少时间。于是剩下的就是开始执行科研这项事情。科研毕竟不是上班，上班重复的工作多，需要创造的时间少。但是科研恰好相反，需要自己创造的时候是非常多的。由于博士论文是需要PHD自己独立完成一个项目或者一个课题，这个时候就非常容易给人带来一种挫败感。能够选择读PHD的，虽然不全是特别聪明的人，但是至少不是傻瓜，至少在读本科的时候都得到过老师们赞许的人。对于学习这件事，只要在自己的能力范围内规定一天看多少书，基本上还是能够按时解决。但是科研这种事情恰好不同，规定一个PHD在几天甚至几周内搞定一个博士课题，几乎是不可能的事情。就算天天不睡觉，天天想问题，在这段时间内也未必有新的想法来解决手上的问题。如果是一个完美主义者，在科研的过程中就很容易产生一种挫败感，因为科研的道路并不像自己所想象的那样一帆风顺，都是在曲折中不停地往前走。这个时候就一定要放弃那种所谓的完美主义，有的事情做到就好，不需要完美，只要自己在不停的做这件事情就可以了。为了保证自己每天都有一定的时间投入在科研上，就必须要时刻记录好自己的工作时间。如果是严重的拖延症患者，一开始工作的时间不能够太长，就以一个小时，甚至半个小时作为最佳的时间。这个就是所谓的30分钟工作法。每当自己专心的投入科研半个小时，就可以在日历上面记录自己工作了半个小时。如果三心二意的在看书，就不要记录这段时间。每天投入的时间也不需要太长，一定要放弃那种一天能够工作10个小时的想法，一开始的时候每天投入三个小时即可。只要自己做完了这三个小时的科研，剩下的21个小时就可以自己做自己的事情，吃饭睡觉，想做啥做啥。这个30分钟工作法的目的就在于减轻PHD的时间焦虑，改进自己的工作学习流程，精确的保质保量。把一天，一周，甚至一个月分成可以控制的时间片段，通过不断地积累来促进科研的进展。俗话说：不积跬步，无以至千里；不积小流，无以成江海。要对抗拖延，就必须选择一个小的，可以操作的目标，集中注意力做30分钟，甚至一个小时。此时不需要惦记着一个很宏大的目标，只需要着重于脚下的路。

看到这里，也许有人会说，每天3小时算什么，每天有24小时呢。当然，对于每天能够持续思考科研难题8,9个小时的人来说，3个小时确实少了许多。但是，对于普通人来说，科研与学习有本质的区别，在学习的过程中，通过自己的聪明才智能够持续不断地做出书后面的习题，从而刺激自己每天不停地学习下去。在科研过程中，几乎没有任何一个合格的博士课题是能够让一个PHD在几天，几周，甚至几个月之内完成的。在一个人持续几个月没有新结果新想法的时候，就会没有动力来刺激自己继续进行这项工作，继续做这项工作，就很有可能给自己带来一种更大的挫败感。在这个时候，就需要把自己的注意力从最终的结果转移到过程上面，只需要关注自己每天，每周，甚至每个月工作了多少时间即可，不要一直想着自己什么时候能够做出来结果，也不要期望着自己努力了几天就能够解决最终的科研问题。每天需要做的就是进行一次不完美，但是完全符合人性的努力。其实每天3小时的工作量虽然看上去不够多，但是几年累积下来，就是一个非常大的工作量，一年的平均科研时间已经达到了1000小时。不要以读完一本书，写完一篇论文，或者连续工作4个小时作为自己的奋斗目标，而要以30-60分钟的高质量专注工作为目标。

最后，对于一个PHD来说，科研中的失败肯定是家常便饭，如果一直怀着一种完美主义的心态，就不会愿意去冒险，不会愿意去采取行动。当你在为了自己而找各种各样借口的时候，就是一种退缩；而怀着一种成长的心态，就会乐于采取行动去解决问题，即使这件事情看上去很难，看上去是多么的遥不可及，或者说不是很喜欢去做它。与其相信自己找的各种借口，让它们带着你进入泥沼，不如不去理睬这些借口，直接采取行动去解决问题。

One Dimensional Dynamical System

Perron-Frobenius Operator

November 5, 2014 zr9558 Leave a comment

Perron-Frobenius Operator

Consider a map $f$ which possibly has a finite (or countable) number of discontinuities or points where possibly the derivative does not exist. We assume that there are points

$\displaystyle q_{0}<q_{1}<\cdot\cdot\cdot <q_{k}$ or $q_{0}<q_{1}<\cdot\cdot\cdot<q_{\infty}<\infty$

such that $f$ restricted to each open interval $A_{j}=(q_{j-1},q_{j})$ is $C^{2}$ , with a bound on the first and the second derivatives. Assume that the interval $[q_{0},q_{k}]$ ( or $[q_{0},q_{\infty}]$ ) is positive invariant, so $f(x)\in [q_{0},q_{k}]$ for all $x\in [q_{0}, q_{k}]$ ( or $f(x)\in [q_{0},q_{\infty}]$ for all $x\in[q_{0},q_{\infty}]$ ).

For such a map, we want a construction of a sequence of density functions that converge to a density function of an invariant measure. Starting with $\rho_{0}(x)\equiv(q_{k}-q_{0})^{-1}$ ( or $\rho_{0}(x)\equiv(q_{\infty}-q_{0})^{-1}$ ),assume that we have defined densities up to $\rho_{n}(x)$ , then define define $\rho_{n+1}(x)$ as follows

$\displaystyle \rho_{n+1}(x)=P(\rho_{n})(x)=\sum_{y\in f^{-1}(x)}\frac{\rho_{n}(y)}{|Df(y)|}.$

This operator $P$ , which takes one density function to another function, is called the Perron-Frobenius operator. The limit of the first $n$ density functions converges to a density function $\rho^{*}(x)$ ,

$\displaystyle \rho^{*}(x)=\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\rho_{n}(x).$

The construction guarantees that $\rho^{*}(x)$ is the density function for an invariant measure $\mu_{\rho^{*}}$ .

Example 1. Let

$\displaystyle f(x)= \begin{cases} x &\mbox{if } x\in(0,\frac{1}{2}), \\ 2x &\mbox{if } x\in(\frac{1}{2},1). \end{cases}$

We construct the first few density functions by applying the Perron-Frobenius operator, which indicates the form of the invariant density function.
Take $\rho_{0}(x)\equiv1$ on $[0,1]$ . From the definition of $f(x)$ , the slope on $(0,\frac{1}{2})$ and $(\frac{1}{2},1)$ are 1 and 2, respectively. If $x\in (\frac{1}{2},1)$ , then it has only one pre-image on $(\frac{1}{2},1)$ ; else if $x\in(0,\frac{1}{2})$ , then it has two pre-images, one is $x^{'}$ in $(0,\frac{1}{2})$ , the other one is $x^{''}$ in $(\frac{1}{2},1)$ . Therefore,

$\rho_{1}(x)= \begin{cases} \frac{1}{1}+\frac{1}{2} &\mbox{if } x\in(0,\frac{1}{2}), \\ \frac{1}{2} &\mbox{if } x\in(\frac{1}{2},1). \end{cases}$

By similar considerations,

$\displaystyle \rho_{2}(x)=\begin{cases}1+\frac{1}{2}+\frac{1}{2^{2}} &\mbox{if } x\in(0,\frac{1}{2}), \\ \frac{1}{2^{2}} &\mbox{if } x\in(\frac{1}{2},1).\end{cases}$

By induction, we get

$\displaystyle \rho_{n}(x)=\begin{cases}1+\frac{1}{2}+\cdot\cdot\cdot+\frac{1}{2^{n}} &\mbox{if } x\in(0,\frac{1}{2}), \\ \frac{1}{2^{n}} &\mbox{if } x\in(\frac{1}{2},1).\end{cases}$

Now, we begin to calculate the density function $\rho^{*}(x)$ . If $x\in(0,\frac{1}{2})$ , then
$\displaystyle \rho^{*}(x)=\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\rho_{n}(x) =\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1} \sum_{m=0}^{n}\frac{1}{2^{m}} =\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\left(2-\frac{1}{2^{n}}\right)=2.$
If $x\in(\frac{1}{2},1)$ , then
$\displaystyle \rho^{*}(x)=\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\rho_{n}(x) =\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\frac{1}{2^{n}} =\lim_{k\rightarrow \infty}\frac{1}{k}\left(2-\frac{1}{2^{k}}\right)=0.$
i.e.

$\displaystyle \rho^{*}(x)= \begin{cases} 2 &\mbox{if } x\in(0,\frac{1}{2}), \\ 0 &\mbox{if } x\in(\frac{1}{2},1). \end{cases}$

Example 2. Let

$\displaystyle f(x)=\begin{cases} 2x &\mbox{if } x\in(0,\frac{1}{2}), \\ 2x-1 &\mbox{if } x\in(\frac{1}{2},1). \end{cases}$

Take $\rho_{0}(x)\equiv1$ on $(0,1)$ . By induction, $\rho_{n}(x)\equiv1$ on $(0,1)$ for all $n\geq 0$ . Therefore, $\rho^{*}(x)\equiv1$ on $(0,1)$ .

Example 3. Let

$\displaystyle f(x)=\begin{cases} x &\mbox{if } x\in(0,\frac{1}{2}), \\ 2^{n+1}\cdot\left(x-\left(1-\frac{1}{2^{n}}\right)\right) &\mbox{if } x\in\left(1-\frac{1}{2^{n}},1-\frac{1}{2^{n+1}}\right) \text{ for all } n\geq 1.\end{cases}$

Take $\rho_{0}(x)\equiv1$ on $(0,1)$ . Assume

$\displaystyle \rho_{n}(x)= \begin{cases} a_{n} &\mbox{if } x\in(0,\frac{1}{2}), \\ b_{n} &\mbox{if } x\in(\frac{1}{2},1). \end{cases}$

for all $n\geq 0$ . It is obviously that $a_{0}=b_{0}=1$ . By similar considerations,
$\displaystyle \rho_{n+1}(x)= \begin{cases} \frac{a_{n}}{1}+\frac{b_{n}}{4}+\frac{b_{n}}{8}+\frac{b_{n}}{16}+\cdot\cdot\cdot= a_{n}+\frac{b_{n}}{2} &\mbox{if } x\in(0,\frac{1}{2}), \\ \frac{b_{n}}{4}+\frac{b_{n}}{8}+\frac{b_{n}}{16}+\cdot\cdot\cdot = \frac{b_{n}}{2} &\mbox{if } x\in(\frac{1}{2},1). \end{cases}$
That means

$\displaystyle \left( \begin{array}{ccc} a_{n+1} \\ b_{n+1} \end{array} \right) =\left( \begin{array}{ccc} a_{n}+\frac{1}{2}b_{n} \\ \frac{1}{2}b_{n} \end{array} \right) = \left( \begin{array}{ccc} 1 & \frac{1}{2} \\ 0 & 1 \end{array} \right) \left( \begin{array}{ccc} a_{n} \\ b_{n} \end{array} \right)$

for all $n\geq 0$ . From direct calculation, $\displaystyle a_{n}=2-\frac{1}{2^{n}}$ and $\displaystyle b_{n}=\frac{1}{2^{n}}$ for all $n\geq 0$ . Therefore,

$\displaystyle \rho^{*}(x)=\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\rho_{n}(x)=\begin{cases} 2 &\mbox{if } x\in (0,\frac{1}{2}), \\ 0 &\mbox{if } x\in (\frac{1}{2},1). \end{cases}$

Example 4. Let

$\displaystyle f(x)=\begin{cases} 1.5 x &\mbox{if } x\in(0,\frac{1}{2}), \\ 2^{n+1}\cdot\left(x-\left(1-\frac{1}{2^{n}}\right)\right) &\mbox{if } x\in\left(1-\frac{1}{2^{n}},1-\frac{1}{2^{n+1}}\right) \text{ for all } n\geq 1.\end{cases}$

Take $\rho_{0}(x)\equiv1$ on $(0,1)$ . Assume

$\displaystyle \rho_{n}(x)= \begin{cases} a_{n} &\mbox{if } x\in(0,\frac{3}{4}), \\ b_{n} &\mbox{if } x\in(\frac{3}{4},1). \end{cases}$

for all $n\geq 0$ . It is obviously that $a_{0}=b_{0}=1$ . By similar considerations,

$\displaystyle \left( \begin{array}{ccc} a_{n+1} \\ b_{n+1} \end{array} \right) =\left( \begin{array}{ccc} \frac{11}{12}a_{n}+\frac{1}{4}b_{n} \\ \frac{1}{4}a_{n}+\frac{1}{4}b_{n} \end{array} \right) = \left( \begin{array}{ccc} \frac{11}{12} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{4} \end{array} \right) \left( \begin{array}{ccc} a_{n} \\ b_{n} \end{array} \right)$

for all $n\geq 0$ . From matrix diagonalization , $\displaystyle a_{n}=\frac{6}{5}-\frac{1}{5}\cdot\frac{1}{6^{n}}$ and $\displaystyle b_{n}=\frac{2}{5}+\frac{3}{5}\cdot\frac{1}{6^{n}}$ for all $n\geq 0$ .

Therefore,

$\displaystyle \rho^{*}(x)=\lim_{k\rightarrow \infty}\frac{1}{k}\sum_{n=0}^{k-1}\rho_{n}(x)=\begin{cases} \frac{6}{5} &\mbox{if } x\in (0,\frac{3}{4}), \\ \frac{2}{5} &\mbox{if } x\in (\frac{3}{4},1). \end{cases}$

Perron-Frobenius Theory

Definition. Let $A=[a_{ij}]$ be a $k\times k$ matrix. We say $A$ is non-negative if $a_{ij}\geq 0$ for all $i,j$ . Such a matrix is called irreducible if for any pair $i,j$ there exists some $n>0$ such that $a_{ij}^{(n)}>0$ where $a_{ij}^{(n)}$ is the $(i,j)-$ th element of $A^{n}$ . The matrix $A$ is irreducible and aperiodic if there exists $n>0$ such that $a_{ij}^{(n)}>0$ for all $i,j$ .

Perron-Frobenius Theorem Let $A=[a_{ij}]$ be a non-negative $k\times k$ matrix.

(i) There is a non-negative eigenvalue $\lambda$ such that no eigenvalue of $A$ has absolute value greater than $\lambda$ .

(ii) We have $\min_{i}(\sum_{j=1}^{k}a_{ij})\leq \lambda\leq \max_{i}(\sum_{j=1}^{k}a_{ij})$ .

(iii) Corresponding to the eigenvalue $\lambda$ there is a non-negative left (row) eigenvector $u=(u_{1},\cdot\cdot\cdot, u_{k})$ and a non-negative right (column) eigenvector $v=(v_{1},\cdot\cdot\cdot, v_{k})^{T}$ .

(iv) If $A$ is irreducible then $\lambda$ is a simple eigenvalue and the corresponding eigenvectors are strictly positive (i.e. $u_{i}>0$ , $v_{i}>0$ all $i$ ).

(v) If $A$ is irreducible then $\lambda$ is the only eigenvalue of $A$ with a non-negative eigenvector.

Theorem.
Let $A$ be an irreducible and aperiodic non-negative matrix. Let $u=(u_{1},\cdot\cdot\cdot, u_{k})$ and $v=(v_{1},\cdot\cdot\cdot, v_{k})^{T}$ be the strictly positive eigenvectors corresponding to the largest eigenvalue $\lambda$ as in the previous theorem. Then for each pair $i,j$ , $\lim_{n\rightarrow \infty} \lambda^{-n}a_{ij}^{(n)}=u_{j}v_{i}$ .

Now, let us see previous examples, again. The matrix $A$ is irreducible and aperiodic non-negative matrix, and $\lambda=1$ has the largest absolute value in the set of all eigenvalues of $A$ . From Perron-Frobenius Theorem, $u_{i}, v_{j}>0$ for all pairs $i,j$ . Then for each pari $i,j$ ,
$\lim_{n\rightarrow \infty}a_{ij}^{(n)}=u_{j}v_{i}$ . That means $\lim_{n\rightarrow \infty}A^{(n)}$ is a strictly positive $k\times k$ matrix.

Markov Maps

Definition of Markov Maps. Let $N$ be a compact interval. A $C^{1}$ map $f:N\rightarrow N$ is called Markov if there exists a finite or countable family $I_{i}$ of disjoint open intervals in $N$ such that

(a) $N\setminus \cup_{i}I_{i}$ has Lebesgue measure zero and there exist $C>0$ and $\gamma>0$ such that for each $n\in \mathbb{N}$ and each interval $I$ such that $f^{j}(I)$ is contained in one of the intervals $I_{i}$ for each $j=0,1,...,n$ one has

$\displaystyle \left| \frac{Df^{n}(x)}{Df^{n}(y)}-1 \right| \leq C\cdot |f^{n}(x)-f^{n}(y)|^{\gamma} \text{ for all } x,y\in I;$

(b) if $f(I_{k})\cap I_{j}\neq \emptyset$ , then $f(I_{k})\supseteq I_{j}$ ;

As usual, let $\lambda$ be the Lebesgue measure on $N$ . We may assume that $\lambda$ is a probability measure, i.e., $\lambda(N)=1$ . Usually, we will denote the Lebesgue measure of a Borel set $A$ by $|A|$ .

Theorem. Let $f:N\rightarrow N$ be a Markov map and let $\cup_{i}I_{i}$ be corresponding partition. Then there exists a $f-$ invariant probability measure $\mu$ on the Borel sets of $N$ which is absolutely continuous with respect to the Lebesgue measure $\lambda$ . This measure satisfies the following properties:

(a) its density $\frac{d\mu}{d\lambda}$ is uniformly bounded and Holder continuous. Moreover, for each $i$ the density is either zero on $I_{i}$ or uniformly bounded away from zero.

If for every $i$ and $j$ one has $f^{n}(I_{j})\supseteq I_{i}$ for some $n\geq 1$ then

(b) the measure is unique and its density $\frac{d\mu}{d\lambda}$ is strictly positive;

(d) $\lim_{n\rightarrow \infty} |f^{-n}(A)|=\mu(A)$ for every Borel set $A\subseteq N$ .

If $f(I_{i})=N$ for each interval $I_{i}$ , then

(e) the density of $\mu$ is also uniformly bounded from below.

数学科普

Notes on Shape of Inner Space

November 1, 2014 zr9558 Leave a comment

Shape of Inner Space

shing-tung_yau_nadis_s._the_shape_of_inner_space

String Theory and the Geometry of the Universe’s Hidden Dimensions

Shing-Tung YAU and Steve NADIS

Chapter 3: P.39

My personal involvement in this area began in 1969, during my first semester of graduate studies at Berkeley. I needed a book to read during Chrismas break. Rather than selecting Portnoy’s Complaint, The Godfather, The Love Machine, or The Andromeda Strain-four top-selling books of that year-I opted for a less popular title, Morse Theory, by the American mathematician John Milnor. I was especially intrigued by Milnor’s section on topology and curvature, which explored the notion that local curvature has a great influence on geometry and topology. This is a theme I’ve pursued ever since, because the local curvature of a surface is determined by taking the derivatives of that surface, which is another way of saying it is based on analysis. Studying how that curvature influences geometry, therefore, goes to the heart of geometric analysis.

Having no office, I practically lived in Berkeley’s math library in those days. Rumor has it that the first thing I did upon arriving in the United States was visiting that library, rather than, say, explore San Francisco as other might have done. While I can’t remember exactly what I did, forty years hence, I have no reason to doubt the veracity of that rumor. I wandered around the library, as was my habit, reading every journal I could get my hands on. In the course of rummaging through the reference section during winter break, I came across a 1968 article by Milnor, whose book I was still reading. That article, in turn, little else to do at the time (with most people away for the holiday), I tried to see if I could prove something related to Preissman’s theorem.

Chapter 4: P.80

From this sprang the work I’ve become most famous for. One might say it was my calling. No matter what our station, we’d all like to find our true calling in life-that special thing we were put on this earth to do. For an actor, it might be playing Stanley Kowalski in A Streetcar Named Desire. Or the lead role in Hamlet. For a firefighter, it could mean putting out a ten-alarm blaze. For a crime-fighter, it could mean capturing Public Enemy Number One. And in mathematics, it might come down to finding that one problem you’re destined to work on. Or maybe destiny has nothing to do with it. Maybe it’s just a question of finding a problem you can get lucky with.

To be perfectly honest, I never think about “destiny” when choosing a problem to work on, as I tend to be a bit more pragmatic. I try to seek out a new direction that could bring to light new mathematical problems, some of which might prove interesting in themselves. Or I might pick an existing problem that offers the hope that in the course of trying to understand it better, we will be led to a new horizon.

The Calabi conjecture, having been around a couple of decades, fell into the latter category. I latched on to this problem during my first year of graduate school, though sometimes it seemed as if the problem latched on to me. It caught my interest in a way that no other problem had before or has since, as I sensed that it could open a door to a new branch of mathematics. While the conjecture was vaguely related to Poincare’s classic problem, it struck me as more general because if Calabi’s hunch were true, it would lead to a large class of mathematical surfaces and spaces that we didn’t know anything about-and perhaps a new understanding of space-time. For me the conjecture was almost inescapable: Just about every road I pursued in my early investigations of curvature led to it.

Chapter 5: P.104

A mathematical proof is a bit like climbing a mountain. The first stage, of course, is discovering a mountain worth climbing. Imagine a remote wilderness area yet to be explored. It takes some wit just to find such an area, let alone to know whether something worthwhile might be found there. The mountaineer then devises a strategy for getting to the top-a plan that appears flawless, at least on paper. After acquiring the necessary tools and equipment, as well as mastering the necessary skills, the adventurer mounts an ascent, only to be stopped by unexpected difficulties. But others follow in their predecessor’s footsteps, using the successful strategies, while also pursuing different avenues-thereby reaching new heights in the process. Finally someone comes along who not only has a good plan of attack that avoids the pitfalls of the past but also has the fortitude and determination to reach the summit, perhaps planting a flag there to mark his or her presence. The risks to life and limb are not so great in math, and the adventure may not be so apparent to the outsider. And at the end of a long proof, the scholar does not plant a flag. He or she types in a period. Or a footnote. Or a technical appendix. Nevertheless, in our field there are thrill as well as perils to be had in the pursuit, and success still rewards those of us who’ve gained new views into nature’s hidden recesses.

ZHANG RONG

Monthly Archives: November 2014

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