# 用Latex写中英文简历，CV

moderncv 的笔记（支持中文）

TeX 中有很多文档类、宏包可以用来制作 CV

Writing the curriculum vitae with LaTeX

Moderncv LaTeX package. A really easy way to create a modern CV

The MacTeX-2014 Distribution

moderncv – A modern curriculum vitae class

ModernCV and Cover Letter

LaTeX模板

# ［转载］昔日“奥数天才”不逼女儿学奥数 大量时间研究数学

现在大量时间在做数学研究

“现在是教授了吧？”记者问。

“没有，讲师。”罗炜淡淡地回答。

“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。

“你的孩子学奥数吗？”记者问。

“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。”

“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”。

“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。

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。

“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。

“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。

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

# ［转载］关于如何写正式英文email的总结

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。

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。

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。

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.

“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）

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

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.

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.

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)

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.

（2） We sincerely appreciate your information. 明显地,我们会喜欢第2句.

“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?

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.

7. 单数复数问题:

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.

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,

15. 用语要肯定准确.切忌含糊.

16. 立场观点一致. 少用被动语.

17. 求人做事，最后写:thanks 即可.

1. 格式问题，

Ken Green

Vice president, Unicom China.

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!

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

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.

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

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.

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

# ［转载］写英文 Email 要特别留心这些点，一些原则私信也适用

Email 要以收信人为主题

Email 要简洁明了

Email 要避免套话空话

• 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

• 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 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。

# ［转载］英文Email

### 作者：白羽轩❤Queenie

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.

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 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。

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.

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

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

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

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

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

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

Look forward to your feedbacks and suggestions soon.

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

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

Any question, please let me know.

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

Please let me know what you think?

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

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

4. Give feedback 意见反馈

My answers are in blue below.

5. Attachment 附件

I enclose the evaluation report for your reference.

Attached please find today’s meeting notes.

Attach is the design document, please review it.

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

6. Point listing 列表

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

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

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

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

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

7. Raise question 提出问题

I have some questions about the report XX-XXX

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

8. Proposal 提议

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

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

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

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

Your kind assistance on this are very much appreciated.

10. Apology 道歉

I sincerely apologize for this misunderstanding!

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

# Fractals – A Very Short Introduction

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.

# Excerpts from the Grothendieck-Serre Correspondence

Like many fellow mathematicians, I was very sad to hear the news that Alexander Grothendieckpassed 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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# 254A announcement: Analytic prime number theory

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:

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

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# MA1505 Summary

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Categories:Types, School Work

# Perron-Frobenius Operator

## 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} or $q_{0}

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 $f(I_{k})\supseteq I_{j}$;

(c) there exists $r>0$ such that $|f(I_{i})|\geq r$ for each $i$.

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;

(c) $f$ is exact with respect to $\mu$;

(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.

# Shape of Inner Space

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

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.