# Zaikun's Blog — “I Want to Be a Mathematician”

This page is a collection of non-scientific writings on mathematics, science, research, life, dreams … I hope one day it could be called “A Mathematician's Apology”; for the moment, it is only “I Want to Be a Mathematician”. Proper rendering support is needed to display the Chinese characters here. Tous droits réservés; Alle Rechte vorbehalten ©.

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## 博士论文致谢(2012 年 4 月 7 日，北京)

Powell 教授、葡萄牙 Universidade de Coimbra 的 Luís Nunes Vicente 教授、美国 Louisiana State University 的张洪超博士 (我的师兄) 以及李庆娜师姐、孙聪师妹和姜波师弟审阅了我的第一篇论文并提出了宝贵的意见。感谢他们对我的无私帮助。

2012 年 4 月 7 日

## 中科院 SIAM 学生会的由来(2013 年 5 月 24 日，科英布拉)

### 序

2013 年 6 月 7 日是中科院 SIAM 学生会 (CAS SIAM Student Chapter) 两岁生日。在这之前的 5 月 25 日 (本周六)，学生会终于要举行第一次年会。由于种种原因 (主要是我的失职)，学生会现有会员乃至执行委员会成员并不了解学生会的由来。作为学生会的十四位创始会员之一，我有幸见证了她从开始酝酿的第一刻到 2012年 8 月的全部历史。我想，我应该写点什么，来纪念那段不一样的日子。

### 一. 一封邮件

2010 年 8 月 29 日是个很普通的星期天，朋友约我到家里吃午饭。刚坐下，我收到袁老师的一封邮件。袁老师说：“在坤，你能否在中国成立一个 SIAM 学生会？附 Nick Trefethen 教授的邮件供参考。”

SIAM 我是知道的, 全称是 “工业与应用数学会 (Society of Industrial and Applied Mathematicians)”。但 SIAM 学生会是什么？这位 Trefethen 教授又是谁呢? 当时不能上网，在手机上读不到邮件的后半部分，但我感觉到这会是一件很有意思的事情。

SIAM 学生会是 SIAM 的学生组织，其主旨是加强学生和工业与应用数学界的联系，鼓励学生用数学来解决实际问题，同时锻炼学生的领导和组织能力。对于每一个 SIAM 学生会，SIAM 会提供一定的活动经费，并且 SIAM 学生会的会员将免费自动成为 SIAM 的学生会员 (SIAM studnet member)。显然，这是非常有意义的事情。

(未完)

## 冯 · 诺伊曼(2013 年 10 月 12 日，科英布拉)

“太让我惊讶了，你这么快就算出来了。” 她说道。“大多数数学家都没能看出这里面的技巧，而是用无穷级数去计算，这花费了他们很长时间。”

“什么技巧？我也是用无穷级数算的。” 冯 · 诺伊曼回答道。

When he was given a problem while standing, Johnny at one stage would dance from foot to foot. Although this practice caused some spills at his crowded cocktail parties, it forms one of the first stories half against him: his reaction to the fly puzzle. Two cyclists are 20 miles apart and head toward each other at 10 miles per hour each. At the same time a fly traveling at a steady 15 miles per hour starts from the front wheel of the northbound bicycle. It lands on the front wheel of the southbound bicycle, and then instantly turns around and flies back, and after next landing instantly flies north again. Question: What total distance did the fly cover before it was crushed between the two front wheels?

The slow way of answering is to calculate the distance that the fly travels on its first trip to the southbound front wheel, then the distance it travels on its next trip to the northbound wheel, and finally to sum the infinite series so obtained. It is extraordinary how many mathematicians can be fooled into doing that long sum. The short way is to note that the bicycles will meet exactly one hour after starting, by which time the 15-miles-per-hour fly must have covered 15 miles. When the question was put to Johnny, he danced and answered immediately, “15 miles”. “Oh, you've heard the trick before,” said the disappointed questioner. “What trick?” asked the puzzled Johnny. “I simply summed the infinite series.” It is worth adding that, when ribbed on this later, Johnny said “the figures actually put to me were not so simple.”

@book{MacraeN_1992_JVN,
title = {John von Neumann: The Scientific Genius who Pioneered the Modern Computer,
Game Theory, Nuclear Deterrence, and Much More},
author = {Macrae, N.},
isbn = {9780821826768},
lccn = {99037303},
url = {http://goo.gl/bUoQti},
year = {1992},
publisher = {American Mathematical Society}
}


## 自由谈(2014 年 3 月 15 日，科英布拉)

Liberty means responsibility. That is why most men dread it.
—  George Bernard Shaw

Unless a man has talents to make something of himself, freedom is an irksome burden.
—  Eric Hoffer

“三观” 这个词眼下很流行。这原本是一个严肃的概念，现在也不可避免地被解构了。我说的是它的本意。一个人的三观是什么决定的？无非是她/他经历的人与事。正常情况下，能影响到三观的经历只占极少数。对于跟我一般固执的人，这个比例就更低。如果连中午吃馒头还是米饭都管用，这三观也忒不值钱了。

@book{RosenthalR_JacobsonL_1968_PYG,
title = {Pygmalion in the Classroom: Teacher Expectation and Pupils' Intellectual Development},
author = {Rosenthal, R. and Jacobson, L.},
lccn = {lc68019667},
year = {1968},
publisher = {Holt, Rinehart and Winston}
}


Windows 如何影响用户的行为我不清楚，而 Linux 对用户的影响倒是十分有意思。既然假定 “用户知道自己想要什么，也明白自己在做什么，并且会为自己的行为负责”， 如果发生毕马隆效应，那么相当一部分用户会慢慢学会用自己的脑袋思考，按自己的意志行事，并为自己的行为负责。这就是自由，也是 “free software” 之 “free” 的本意。

1. 用户自主。用户决定系统如何工作，而不是反之。任何提示用户 “正在安装更新，请不要关机” 的行为都跟不以结婚为目的的恋爱是一个性质。用户是机器的主人，操作系统只是用户请来的助手；用户知道自己想要什么，明白自己在做什么，也会为自己的行为负责；操作系统尽量少说话，更不要下命令。用户自主还蕴含系统的可定制性：用户可以对系统做任何配置或改造。

2. 用户知情。如有必要，用户可以获知系统工作的任何细节，而不是仅仅被告知 “正在处理一些事情” —— 这比不以结婚为目的的恋爱还让人无法接受。

3. 系统高效。在此系统下用户可以很方便地获得各种工具，并且不同工具能很容易地组装在一起完成复杂的工作。机器更多是用来干活的，所以这一点很重要。

## On scientific collaborations(Mar. 19, 2014, Coimbra)

Today Luis submitted our paper “Direct search based on probabilistic descent”, which was started on Mar. 18, 2013. It is the first co-authored paper in my research career, the co-authors being S. Gratton, C. W. Royer, and L. N. Vicente. To celebrate the submission of this special work, I was going to write something on scientific collaborations, about which I learned a lot during this joint research. But it did not take me very long to realize that it would be better to discuss this profound topic after acquiring more experience.

Therefore I will do nothing but quoting a dialog that I appreciate very much, for the memory of the twelve months that we worked together on this paper.

- Grandpa, were you a hero in the war?
- No… but I served in a company of heroes.

This dialog was between Sergeant Myron N. Ranney and his grandson. Ranney was a World War II veteran, and he served in Easy Company (known as the “Band of Brothers”), 2nd Battalion, 506th Parachute Infantry Regiment, in the 101st Airborne Division of the United States Army during the war. Ranney's life story was featured in the book A Company of Heroes: Personal Memories about the Real Band of Brothers and the Legacy They Left Us (Berkley Caliber, 2010) by Marcus Brotherton.

Update (Mar. 31, 2015, Toulouse): The paper is accepted by SIAM J. Optim.

## 答吴乐秦：从井盖到布拉死磕–勒贝格定理(2014 年 3 月 29 日，科英布拉)

Remark: This essay has an English version.

### 参考文献

[1] S. Rabinowitz, A polynomial curve of constant width, Missouri Journal of Mathematical Sciences, 9: 23–27, 1997

[2] M. Bardet and T. Bayen, On the degree of the polynomial defining a planar algebraic curves of constant width, arXiv:1312.4358v1 [math.AG], 2013

[3] E. Barbier, Note sur le problème de l’aiguille et le jeu du joint couvert, Journal de Mathématiques Pures et Appliquées, 2ème série, tome 5: 273–286, 1860

[4] W. Blaschke, Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts, Mathematische Annalen, 76: 504–513, 1915

[6] H. Lebesgue, Sur quelques questions des minimums, relatives aux courbes orbiformes, et sur les rapports avec le calcul de variations, Journal de Mathématiques Pures et Appliquées, 8ème série, tome 4: 67–96, 1921

## A joke(May 26, 2014, Toulouse)

- What is the distance between Tsinghua University and the best research institutions in the world?
- Not very long. Take Bus 333 at the east gate of Tsinghua, and you will find such an institution after two stops.

(Do not take it seriously. By the way, if you know the local geography of the Zhong-Guan-Cun area of Beijing, you will find that Bus 333 brings you to Academy of Mathematics and Systems Science, Chinese Academy of Sciences, after two stops from the east gate of Tsinghua.)

## 致中科院 SIAM 学生会全体会员的一封信(2014 年 6 月 13 日，图卢兹)

From: Zaikun ZHANG <zhangzk [&] lsec.cc.ac.cn>
Date: 2013-06-18 5:57 GMT+02:00
Subject: Re: Nominate a Candidate Team for the next executive committee
To: “CAS.SIAM.Chapter” <siamstuch [&] lsec.cc.ac.cn>

• 执行委员会应当选举产生；

• 已卸任执行委员无权干涉新任委员会工作；

• 学生会成员有权弹劾执行委员会。

## 关于电子邮件的建议：排版篇(2014 年 9 月 5 日，图卢兹)

From: Zaikun ZHANG <zhangzk [&] lsec.cc.ac.cn>
Date: 2014-09-05 19:28 GMT+02:00
Subject: 关于电子邮件的建议：排版篇
To: “CAS.SIAM.Chapter” <siamstuch [&] lsec.cc.ac.cn>

1. 纯文本格式 v.s. html 格式

A. 中文界面下：

B. 英文界面下：
Options >> Preferences >> Function >> Send message/Compose message >> Set writing format >> Text format

2. 行宽

3. 强调

A. 纯文本格式下：

B. html 格式下：

4. 标点与空格

5. 中西文混排与空格

6. 段落与缩进

Modified Block Style Business Letter (semi-formal)，
Intended or Semi-Block Style Business Letter (informal)。

7. 邮件中的网址

http://goo.gl
http://app.baidu.com/app/enter?appid=212313

8. 参考文献

[1] How to send and reply to email：
http://matt.might.net/articles/how-to-email

[2] Internet Message Format (一份关于电子邮件的标准)：
http://tools.ietf.org/html/rfc5322

[3] Dan's Mail Format Site (一个关于电子邮件的网站)：
http://mailformat.dan.info

[4] Type is Beautiful (一个专门讨论排版的网站)：
http://www.typeisbeautiful.com

[5] 中文 Web 阅读体验 (一个关于中文 Web 排版的网页)：
http://goo.gl/AGh1xC

## From manhole cover to Lebesgue-Blaschke theorem(June 16, 2015, Montréal)

What shape can you make a manhole cover so that it cannot fall down through the hole? This question is said to be an interview question of Google (or Microsoft? Who knows).

If we take this question as a serious mathematical problem, then its answer will be the shapes bounded by curves of constant width.

Consider a simple curve $$\gamma$$ in $$\mathbb{R}^2$$. According to Jordan curve theorem, $$\gamma$$ separates $$\mathbb{R}^2$$ into two parts, the interior region and the exterior one. If the interior region is convex, then $$\gamma$$ is called a convex curve. The supporting lines of a convex curve are defined to be those of its interior region. If $$\gamma$$ is a convex curve, and the distance between each pair of its distinct parallel supporting lines is a constant, then $$\gamma$$ is called a curve of constant width, and this constant is called its width.

Circles are trivial curves of constant width. Besides them, the most famous curve of constant width is the Reuleaux triangle, which is the “triangle” consisting of three unit-circle arcs of length $$\pi/3$$. More generally, it can be shown that all the “Reuleaux polygons” (the polygons consisting of an odd number of specific arcs) are of constant width.

Circles are algebraic curves, which means that they are the zero points of some polynomials. Although we have infinitely many Reuleaux polygons, none of them is algebraic (of course, except the circles, if we regard them as trivial Reuleaux “polygons”). A natural question is, do there exist any algebraic curves of constant width other than circles? The answer is positive. Rabinowitz [1] presents a bivariate polynomial of degree 8 whose zero points make a curve of constant width. Mathematicians are greedy, and they go further to ask what is the lowest possible degree of such curves. Bardet and Bayen [2] prove that 8 is already the minimum.

There are two other interesting theorems about curves of constant width. In 1860, French mathematician Barbier [3] proved that all the curves of constant width share the same length formula as circles, namely $$L = \pi d$$, where $$d$$ is the width. Therefore, by Isoperimetric inequality, with a given width, a circle is the curve of constant width that bounds the largest area. Which one bounds the smallest? It is the Reuleaux triangle, and this result is called Blaschke-Lebesgue theorem (Blaschke [4], Lebesgue [5, 6]). Its original proofs are long, but of course people work out simplified proofs after one hundred years. E. Harrell [7] gives an inspiring proof, which justifies the theorem by directly analyzing the variational problem underlying the area minimization. The proof does not rely on any a priori knowledge about the solution of the problem (i.e., the Reuleaux triangle), while most other proofs do.

Despite the mathematical interests, the Reuleaux triangle is primarily known for its engineering applications (Reuleaux was a German engineer; he was not the one who invented the Reuleaux triangle, but the first one who applied it to engineering). For example, it can be used to drill “square holes”, or more accurately, square holes with “rounded corners”. In fact, it is possible to drill precisely square holes with the help of a curve of constant width based upon the right angled isosceles triangle. Maybe the most “valuable” application of curves of constant width is to use them to design stylish coins, like what Britain and Canada do. See here for these applications and more.

Surface of constant width are also of interest.

### References

[1] S. Rabinowitz, A polynomial curve of constant width, Missouri Journal of Mathematical Sciences, 9: 23–27, 1997

[2] M. Bardet and T. Bayen, On the degree of the polynomial defining a planar algebraic curves of constant width, arXiv:1312.4358v1 [math.AG], 2013

[3] E. Barbier, Note sur le problème de l’aiguille et le jeu du joint couvert, Journal de Mathématiques Pures et Appliquées, 2ème série, tome 5: 273–286, 1860

[4] W. Blaschke, Konvexe Bereiche gegebener konstanter Breite und kleinsten Inhalts, Mathematische Annalen, 76: 504–513, 1915

[6] H. Lebesgue, Sur quelques questions des minimums, relatives aux courbes orbiformes, et sur les rapports avec le calcul de variations, Journal de Mathématiques Pures et Appliquées, 8ème série, tome 4: 67–96, 1921

[7] E. M. Harrell II, A direct proof of a theorem of Blaschke and Lebesgue, arXiv:math/0009137 [math.MG], 2000