Category: math

A Chinese-English Mathematics Primer

Link: http://maths.anu.edu.au/research/cma-proceedings/chinese-english-mathematics-primer

Interesting book on how to read mathematical texts in Chinese. Even students fluent in Chinese everyday language may have difficulty translating the mathematical terms from Chinese to English (and vice versa), hence this book is a very useful one.

My own interest in the work of Chinese mathematicans arises from their significant contributions to the qualitative theory of ordinary differential equations and, in particular, of plane quadratic systems. However, Chinese mathematics covers a very wide range. This is hardly surprising, since a quarter of the world’s population is Chinese. It is predictable that during the next quarter century the importance of Chinese mathematics will increase greatly.

How to prepare for Changes to PSLE grading

Source: http://www.straitstimes.com/singapore/education/changes-to-psle-grading-what-could-be-in-store

The latest update is that the Primary School Leaving Examination (PSLE) aggregate score is soon going to be scrapped and replaced with simple grade bands such as A, B and C. What effects will there be and how to prepare in advance for it?

Do leave your comments below!

My opinion is that there will be a few crucial changes that parents would have to prepare for as soon as possible:

Chinese (or Mother Tongue) is a top priority. In the past, many students from English-speaking families can get low marks for Chinese, high marks for Math/Science/English and still get a very good PSLE score (e.g. >250). This is unfortunately not possible anymore in the new system. A low Chinese grade will drag down the entire performance. Also read more about the benefits of studying Chinese.

DSA (Direct School Admission) GAT (General Ability Test) and CCA becomes more important. The effect of simple grade bands is that many students will get the perfect score of all As. However, the top schools have limited vacancies and thus will have to use other criteria like DSA and CCA to differentiate students. Check out this previous post on DSA.

The new system benefits all-rounders who are good (but not necessarily excellent) at all subjects, including CCA. All-rounders will manage to get ‘A’s in all subjects. However, the new system is unfortunately not good for those who are excellent in one single subject, but average in others.

This quote from the article is very good:

“The focus should not be on how one performs relative to others, but how well the person himself performs in the exam.”

DR TIMOTHY CHAN, director of SIM Global Education’s academic division, on the use of grade-banding to reflect pupils’ abilities.

Limit 极限

tomcircle's avatarMath Online Tom Circle

Mathematical Rigour:

“Domain of Definition” MUST be always considered first prior to tackling :
1) Continuity 连续性
2) Differentiability 可微性
3) Integrability 可积性
4) Limit 极限

Mathematics is linked to Philiosophy! In this life (Domain of Definition ) we have a limit of lifespan (120 years = 2 x 60 years = 2个甲子).

In this same “Domain of Definition” our life is Continuous unless interrupted by unforseen circumstances (accident, diseases, war, …). At certain junctures of life we Differentiate ourselves by having sharp turns of event (eg. graduation from schools and university, National Service in military, marriage, children, jobs, honours/promotions, as well as failures …). It is only in this life you can Integrate these fruits of labor. Beyond this “Domain of Definition” life is meaniningless because we shall return to soil with nothing ….

https://frankliou.wordpress.com/2013/04/25/微積分極限的一個概念/

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一位台大数学教授和学生的对话

tomcircle's avatarMath Online Tom Circle

https://frankliou.wordpress.com/2013/04/25/轉錄-陳金次老師訪談/ (click here):

1. 读书要以兴趣为导向, 决定未来的方向:

◇ 有兴趣不怕慢开窍。很多大数学家都是进大学后才发觉数学兴趣, 尤其美国学生, 特别用功, 勤能补拙。
◇ 高中前是吸收学问; 大学时是追求学问。
◇ 丘成桐 (华人第一个Fields Medalist)不能考进名校”香港大学”, 入”中文大学”, 反而在那里遇”贵人”美国籍教授, 推荐他去美国 Berkeley University 跟随世界第一流大师 陈省身 (SS Chern) 读博士。

丘成桐講的話是很有境界的，他說：「我對數學就是對宇宙真理的探求」，你們讀書要有這樣的氣概。

◇ 周华健是台大数学系的, 因兴趣而改行唱歌, 闯出名堂!

2. 爱情观: 「物以羣分，芳以類聚」

3. 大学生的社团活动: “君子以务为本, 本立而道生” , 应该以不影响学业为前提 …

4. 读 “Time & Space Invariant” 不朽的经典书 : 圣经, 论语, 孟子, 佛经, 都是哲理相通, 给人智慧, 培养你的”Value System”, 就不会被世俗污秽所诱惑。

5. 人生的读书”黄金时期“: 30岁以前!

台大陈金次教授:

Note:
高等微积分 ( Advanced Calculus), 就是美国 / 法国 更准确的名称 “Analysis” (分析), 是数学二大学派分嶺之一, 探讨”微观”(Micro)概念 eg. Differential Equation, Calculus, Topology,..。另一学派是代数 (Algebra), 研究”宏观”(Macro) eg. Abstract Algebra, Algebraic Structures, Category, …。近几十年来, 数学两派已混为一体, “你中有我, 我中有你” (eg. Algebraic Topology, Arithmetic Analysis, …)。所以 Mathematic (Math) 是单数 (singular)!

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Mathematicians prove the triviality of english?

tomcircle's avatarMath Online Tom Circle

All languages with Homophones (同音词, same sound but different words) can be  reduced to 1.
Eg. A = 1, B= 1, C = 1, …., Z = 1

mathematicians-prove-the-triviality-of-english? :

https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/oct/29/mathematicians-prove-the-triviality-of-english?CMP=fb_gu

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New Math Recommended Book: Mathematics without Apologies: Portrait of a Problematic Vocation (Science Essentials)

Mathematics without Apologies: Portrait of a Problematic Vocation (Science Essentials)

What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers–for the sake of truth, beauty, and practical applications–this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources.

Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party?

Disarmingly candid, relentlessly intelligent, and richly entertaining,Mathematics without Apologies takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.

Hahn-Banach Theorem: Crown Jewel of Functional Analysis

Hahn-Banach Theorem is called the Crown Jewel of Functional Analysis, and has many different versions.

There is a Chinese quote “实变函数学十遍，泛函分析心犯寒”, which means one needs to study real function theory ten times before understanding, and the heart can go cold when studying functional analysis, which shows how deep is this subject.

The following is one version of Hahn-Banach Theorem that I find quite useful:

(Hahn-Banach, Version) If V is a normal vector space with linear subspace U (not necessarily closed) and if z is an element of V not in the closure of U, then there exists a continuous linear map \psi:V\to K with \psi(x)=0 for all x\in U, \psi(z)=1, and \|\psi\|=\text{dist}(z,U)^{-1}.

Brief sketch of proof: Define \phi:U+\text{span}\{z\}\to K, \phi(u+\lambda z)=\lambda, and use the Hahn-Banach (Extension version).

Effective Homotopy Method

The main idea of the effective homotopy method is the following: given some Kan simplicial sets K_1,\dots,K_n, a topological constructor \Phi produces a new simplicial set K. If solutions for the homotopical problems of the spaces K_1,\dots,K_n are known, then one should be able to build a solution for the homotopical problem of K, and this construction would allow us to compute the homotopy groups \pi_*(K).

Inspirational Story of Sir John Gurdon, Nobel Prize winner

Source: http://www.telegraph.co.uk/news/science/science-news/9594351/Sir-John-Gurdon-Nobel-Prize-winner-was-too-stupid-for-science-at-school.html

A British scientist whose schoolmasters told him he was too stupid to study the subject has been awarded the Nobel Prize in medicine or physiology for his pioneering work on cloning.

At the age of 15, Prof Sir John Gurdon ranked last out of the 250 boys in his Eton year group at biology, and was in the bottom set in every other science subject. Sixty-four years later he has been recognised as one of the finest minds of his generation after being awarded the £750,000 annual prize, which he shares with Japanese stem cell researcher Shinya Yamanaka. Speaking after learning of his award in London on Monday, Sir John revealed that his school report still sits above his desk at the Gurdon Institute in Cambridge, which is named in his honour.

nobel-gurdon_2363595c

Moral of the story: Teachers may not be always right!

Congrats to Professor Andrew Wiles

http://www.telegraph.co.uk/news/science/science-news/12195189/Oxford-professor-wins-500000-for-solving-300-year-old-mathematical-mystery.html

Oxford professor wins £500,000 for solving 300-year-old mathematical mystery

Sir Andrew Wiles’ proof of Fermat’s Last Theorem has been described as ‘an epochal moment for mathematics’

An Oxford University professor has won a £500,000 prize for solving a three-century-old mathematical mystery that was described as an “epochal moment” for academics.

Sir Andrew Wiles, 62, has been awarded the Abel Prize by the Norwegian Academy of Science and Letters – and almost half a million pounds – for his proof of Fermat’s Last Theorem, which he published in 1994.

Linear Algebra – A Good Primer

tomcircle's avatarMath Online Tom Circle

Linear Algebra – A Primer

Linear transformations are intuitively those maps of everyday space which preserve “linear” things. Specifically, they send lines to lines, planes to planes, etc., and they preserve the origin.
(One which does not preserve the origin is very similar but has a different name; see Affine Transformation)

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Ivy League University Myths

tomcircle's avatarMath Online Tom Circle

Excellent Talk:
“Where you go is not where you’ll be “

This is the universal anxiety for all parents and students in Asian countries where there are limited university places: 2,500+ universities for 7 million Chinese high school students, 5 universities for 13,582 (@2015) Singaporean A-level students, …

Even the USA reputed with world-class university education, the American parents too face the same stress when sending 17-year-old kids to Ivy league universities !

The Myths:
◇ 60% Cornell students (2nd and 3rd year) lamenting not getting into Harvard or Yale !

◇ Those admitted into top universities take things for granted by “coasting” in lectures.

◇ Majority 2/3 of Top Fortune 100 CEOs did not go to Ivy league universities.

Key Points:
It is not which elite university you go to, it is how you explore these opportunities in any university :
◇ Diversity: people from…

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Ultimate Mathematician

tomcircle's avatarMath Online Tom Circle

引用巴拿赫Banach的名言，体现举一反三的境界：

A mathematician is a person who can find analogies between theorems;

A better mathematician is one who can see analogies between proofs

and

The best mathematician can notice analogies between theories.

One can imagine that the ultimate mathematician is one who can see analogies between analogies.

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Google PageRanking Algorithm

tomcircle's avatarMath Online Tom Circle

Google Illustration:

The following WebPages (1) to (n=6) are linked in a network below:
eg.
Page (1) points to (4),
(2) & (3) points to (1)…

image

Let
$latex a_{ij} $ = Probability (PageRank *) from Page ( i ) linked to Page (j).

(*) PageRank: a measure of how relevant the page’s content to the topic of your query. This value is computed by the proprietary formula designed by the 2 Google Founders Larry Page & Sergey Brin, whose Stanford Math Thesis mentor was Prof Tony Chan (who knows the ‘secret ‘ to put his name always on Google 1st search list.)

The Markov Transition Matrix (A) is :

$latex A =
begin{pmatrix}
a_{11} & a_{12}& ldots & a_{1n}
a_{21} & a_{22} & ldots & a_{2n}
vdots & vdots & ddots & vdots
a_{n1} & a_{n2} &ldots & a_{nn}
end{pmatrix}
$

Assume we start surfing from Page…

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Do your best without comparing yourself to others and without fear or failure

Recently, the A Level results just came out, 93.1% score at least 3 H2 passes, best results since curriculum change in 2006. However, as most students know, 3 H2 passes (low pass e.g. 3 C’s) is far from enough to enter the 3 local universities. For those looking for a rank point calculator, check out my post on how to calculate JC Rank Points.

Just to share some motivational advice for those who may not have done as well for A levels. Source: http://www.kuenselonline.com/do-your-best-without-comparing-yourself-to-others-and-without-fear-or-failure/

In reality, the purpose of education should be to open your mind, gain life skills and help you develop your human qualities. It should not merely be considered as a gateway to a job.

Even if your main reason for studying is motivated by your future career, you still need to first consider what you really want to do before proceeding with your education plans. If, for example, you have a passion for cooking, then it might be better to enter a chef training programme rather than spend two more years in school. On the other hand, if you want to be a teacher then class 12 will be your route to achieve your goal.

Anyway, whatever you do, you should do it to the best of your ability. At the same time, you should have no expectations about the result.

Maybe this example will be helpful: Think of the seeds of a sunflower and a violet. A sunflower seed will produce a large, bright yellow flower, while the seed of a violet will produce a small, dark purple bloom. A violet seed can never produce a sunflower blossom no matter how hard it tries. Likewise, a sunflower seed cannot produce the flower of a violet. Neither flower is better or worse than the other. They are just different. However, it is important that the flowers open fully and are not ashamed whether they are small or large, bright or dark.

In the same way, you might discover that you are great at studies or you might find that you are not so great. Like the seeds, you cannot change your natural inclinations, but like the flowers you have to open fully. This means that you try your very best in every situation.

Practically, to do your best means that whatever assignment you are given, you aim to do it beautifully – not to get high grades, but for your own satisfaction. If you have to compose an essay, for example, write each letter and word clearly and in a way that is easy for others to read. Do the same with a math or science or any other assignment. Make each page of your notebook a work of art.

Most importantly is to have tried your very best. Very nice article!

Tough at home but teen perseveres and scores at A-level exams

Source: http://www.straitstimes.com/singapore/tough-at-home-but-teen-perseveres-and-scores-at-a-level-exams

Very inspirational!

Excerpt:

Richmond Tan does not have a study table in the one-room rental flat that he shares with his father, grandfather and brother.

There were times when he did not even have a roof over his head, after his family was temporarily chased out of the Queenstown flat as a result of spats between his father and the landlord.

Richmond had to sleep in void decks or at Changi Airport.

He said what motivated him to do well in his studies was his role model, Han Xin, one of the heroes of the early Han dynasty in China.

He learnt of Han after reading The Art Of War by Sun Tzu, which inspired him to take China Studies for his A levels.

“Han Xin was so poor he had to beg for food. His circumstances were even worse than mine, but he studied hard and became a capable minister,” he added.

Han Xin was also a mathematician, and one of the earliest to discover the secret of the Chinese Remainder Theorem, a key result in Number Theory. According to legend, he used it to calculate the number of soldiers in his army. See this post for more details: http://chinesetuition88.com/2015/04/25/chinese-remainder-theorem-history-%E9%9F%A9%E4%BF%A1%E7%82%B9%E5%85%B5/

Interview of Michael Atiyah (aged 86!)

Source: https://www.quantamagazine.org/20160303-michael-atiyahs-mathematical-dreams/

Inspirational interview by Michael Atiyah, winner of both Fields Medal and Abel Prize, currently age 86!

Excerpt from the interview:

Is there one big question that has always guided you? 

I always want to try to understand why things work. I’m not interested in getting a formula without knowing what it means. I always try to dig behind the scenes, so if I have a formula, I understand why it’s there. And understanding is a very difficult notion.

People think mathematics begins when you write down a theorem followed by a proof. That’s not the beginning, that’s the end. For me the creative place in mathematics comes before you start to put things down on paper, before you try to write a formula. You picture various things, you turn them over in your mind. You’re trying to create, just as a musician is trying to create music, or a poet. There are no rules laid down. You have to do it your own way. But at the end, just as a composer has to put it down on paper, you have to write things down. But the most important stage is understanding. A proof by itself doesn’t give you understanding. You can have a long proof and no idea at the end of why it works. But to understand why it works, you have to have a kind of gut reaction to the thing. You’ve got to feel it.

Interesting comment that “A proof by itself doesn’t give you understanding. You can have a long proof and no idea at the end of why it works.”. Sometimes, intuitive understanding is needed, along with formal proof.

One example in high school mathematics is proving \displaystyle \sum_{i=1}^n i^2=\frac 16n(n+1)(2n+1). It is possible to prove it by induction without actually understanding how the formula comes about!

Artin-Whaples Theorem

There seems to be another version of Artin-Whaples Theorem, called the Artin-Whaples Approximation theorem.

The theorem stated here is Artin-Whaples Theorem for central simple algebras.

Artin-Whaples Theorem: Let A be a central simple algebra over a field F. Let a_1,\dots,a_n\in A be linearly independent over F and let b_1,\dots,b_n be any elements in A. Then there exists a_i',a_i''\in A for i=1,\dots,m such that the F-linear map f:A\to A defined by f(x)=\sum_{r=1}^m a_r'xa_r'' satisfies f(a_j)=b_j for all j=1,\dots,n.

Very nice and useful theorem.

Simple Algebra does not imply Semisimple Algebra

The terminology “semisimple” algebra suggests a generalization of simple algebras, but in fact not all simple algebras are semisimple! (Exercises 1 & 5 in Richard Pierce’s book contain examples)

A simple module is a semisimple module is true though.

Proposition: For a simple algebra A, the following conditions are equivalent:

(i) A is semisimple;

(ii) A is right Artinian;

(iii) A has a minimal right ideal.

Thus to find a algebra that is simple but not semisimple, one can look for an example that is not right Artinian.

Amazon Associates / Affiliates Payment Options (Outside US)

For those using Amazon Associates (or other US based affiliate programs) but are outside USA, there is now an option to transfer the earnings directly to your bank account. Works in Singapore, and most other countries.

The way to do it is through Payoneer. (Direct link: https://share.payoneer.com/nav/FdReUzoZLHa355a95CrNv8uVU528u1dPTojBD2lcYbYvHYFDYT0WxkdmgqlJSsjRs75isVLfZi9fhax47Braxw2)

The other option of receiving cheque from Amazon is bad since the administrative fees are quite expensive (around 20 USD if I remember correctly).

Amazon Associates is quite a good affiliate program. If your blog has just around 100 visits per day, it can already translate to around $20 USD or more per month earnings depending on what you are promoting. This will most likely to cover the cost of hosting your blog. People with extremely popular blogs (see here) have earned more than half a million USD from Amazon Associates.

For those thinking of signing up Payoneer please sign up using the link above. It is a special “Refer a Friend” program where both you and I will get $25 USD upon you signing up. (* After your friend signs up and receives a total of $100, you both earn a $25 reward.)

Shimura Memoire on André Weil

tomcircle's avatarMath Online Tom Circle

Goro Shimura (志村 五郎 born 23 February 1930) is a Japanese mathematician, and currently a professor emeritus of mathematics (former Michael Henry Strater Chair) at Princeton University

Shimura is known to a wider public through the important Modularity Theorem (previously known as the Taniyama-Shimura conjecture before being proven in the 1990s); Kenneth Ribet has shown that the famous Fermat’s Last Theorem (FLT) follows from a special case of this theorem. Shimura dryly commented that his first reaction on hearing of 1994 Andrew Wiles’s proof of the semi-stable case of the FLT theorem was ‘I told you so’.

Shimura’s mémoire on the 20th century great French mathematician André Weil (Fields Medal, Founder of Bourbaki):

1. Weil advised us not to stick to a wrong idea too long. “At some point you must be able to tell whether your idea is right or wrong; then you must have the guts…

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Advice to a Young Mathematician

The official preview is available here at: http://press.princeton.edu/chapters/gowers/gowers_VIII_6.pdf

Excerpt:

The most important thing that a young mathematician needs to learn is of course mathematics. However, it can also be very valuable to learn from the experiences of other mathematicians. The five contributors to this article were asked to draw on their experiences of mathematical life and research, and to offer advice that they might have liked to receive when they were just setting out on their careers. (The title of this entry is a nod to Sir Peter Medawar’s well-known book, Advice to a Young Scientist.) The resulting contributions were every bit as interesting as we had expected; what was more surprising was that there was remarkably little overlap between the contributions. So here they are, five gems intended for young mathematicians but surely destined to be read and enjoyed by mathematicians of all ages.

The full book can be bought on Amazon:

The Princeton Companion to Mathematics

Homology (同调 ) in Geometry & Topology

tomcircle's avatarMath Online Tom Circle

https://frankliou.wordpress.com/2011/10/07/幾何與拓樸簡介/

https://frankliou.wordpress.com/2011/11/21/同調論/

“同胚” homomEorphism (eg. Donnut 和茶壶), 可以扭捏泥土从前者变后者。

同态 (同样形态homomOrphism), 就是Same-Shape-ism. eg. (相似) Similar Triangle.

如果是congruent (全等), 就是 Isomorphism (同构, 同样结构)。

所有新加坡人自己人批评自己人kiasu, 其实大家都kiasu, 因为是自同态 (自己同样态度kiasu), 自=”Endo”
=> Endomorphism.

如果猪八戒照镜子, 看到镜子里面的丑八怪, 还是他猪八戒,
=> Automorphism

这些构造(structure)在WW1后被当时Structurism思想影响, Bourbaki 法国师范大学一批学生 (犹太人 André Weil是领袖)把全部人类的数学重写, 以structure (Set, Group, Ring, Module, Field, Vector Space, Topology. .. )为基础 就是新(抽象)数学, 影响到今。
WW2 后, 美国人Sanders McLane 更上一层楼, 把Set/Group/Ring…等structure 再归类成Category (范畴), 研究其共通的性质 (Morphism 动态), 能够 举一反十。应用在IT 里, 其 Category 就是Functional programming, Types…

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张益唐：我的数学人生

tomcircle's avatarMath Online Tom Circle

[录音小声, 请用earphone耳机听更清楚。]

Key Points Take Away:

1. 身处逆境, 不是勇气, 是淡定。

2. 对目的要穷追不捨, 不要放弃。他从北大的Analytic Number Theory (解析数论)兴趣, 被”人为”的转道去搞博士论文Algebraic Geometry, 7年毕业却无业。从新回到” 解析数论”的跑道, 才得到大成就。

3. 如果2个不同领域的学问之间有些联系, 只要往里鑽, 必能发现新东西。

4. 人生低谷, 碰到3个贵人(2位北大校友, 一位美国系主任青睐)协助。

5. 太太不知他干何学问, 不给 他家庭经济压力, 才能安心于数学。

Q&A:
1. 对于天才儿童, 他劝家长不要 “压 “也不要”捧”, 只要多鼓励, 像Perleman 的(俄国数学家, 证明100年的Poincaré Conjecture)父母循循教导儿子

2. 希望能收PhD学生, 会对他们负责任, 不要有像他个人的悲剧发生 (指被教授利用做私人的项目, 误了学生的前途)。他手头有半’成品’和 3/4’成品’, 可让学生拿去参考, 继续完成当论文。

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Comment j’ai détesté les maths

French Math Education Movie, 7 episodes, made in collaboration with Cédric Villanni (2010 Fields Medal).
In both French and English languages.

tomcircle's avatarMath Online Tom Circle

“Why I hate Math” – (Comment j’ai détesté les maths)

Cédric Villanni – French Fields Medalist (2010)

The French Movies (7 episodes) :

http://www.youtube.com/playlist?list=PLYp_byFGNmhSIzeX-Eiee_4bZPfxNaHTP

Jim Simons (episode 7/7) : The billionaire Mathematician who cracked Wallstreet – a PhD Math student of SS Chern (陈省身) and the university colleague of Prof Frank Yang (杨振宁).

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“LISP” – The God’s Computer Language

Lisp family language, invented since 1958, is the second oldest computer language (after Fortran), by MIT Professor John McCarthy (who also pioneered in Artificial Intelligence).

Some of the Lisp dialects are Common Lisp anf Clojure (run on top of Java VM). They are called Functional Programming Language, versus Imperative language (C language…) or Object-Oriented languages (C++ / Java).

tomcircle's avatarMath Online Tom Circle

Lisp is a Functional programming language, a 1950s product created for symbolic computing in Mathematics, used popularly in 1980s for Artificial Intelligence.

Famous software “Mathematica” is written in Lisp.

The original Lisp language, as defined by John McCarthy as “Recursive Functions of Symbolic Expression and Their Computation by Machine.”, defined the entire language in terms of only 7 functions (atom, car, cdr, cond, cons, eq, quote) and 2 special forms (lambda, label). Through the composition of ONLY these 9 forms, McCarthy was able to describe the WHOLE of computation — it doesn’t get more beautiful than that.

Unfortunately, because of the memory hungry requirement — hence the unique Garbage Collection slow backend processes — Lisp lost its attractiveness in the PC-dominant era of 1990s and 2000s, replaced by the most polular language Java which was invented by James Gosling, a former ‘Lisper’ who had created the popular FranzLisp.

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Smart Algebraic Technique

tomcircle's avatarMath Online Tom Circle

Calculate:
$latex (3+1). (3^2 +1). (3^4 + 1)(3^8 +1)…. (3^{32} +1)
$

Let
$latex x = (3+1). (3^2 +1). (3^4 + 1)(3^8 +1)…. (3^{2n} +1)
$

Or:
$latex displaystyle
x = sum_{n=0}^{n}(3^{2n} + 1) $

Quite messy to expand out:

$latex displaystyle {
sum_{n=0}^{n} (3^{2n})
+
sum_{n=0}^{n}(1)
= ….
}
$

This 14-year-old vienamese student in Berlin – Hyyen Nguyen Thi Minh discovered a smart trick using the identity:
$latex displaystyle { (a -1).(a + 1) = a^{2} – 1}$
or more general,
$latex displaystyle boxed {
(a^{n} -1).(a^{n} + 1) = a^{2n} – 1
}$

He multiplies x by (3-1):

$latex
x. (3-1) = (3-1)(3+1). (3^{2} +1)… (3^{2n} + 1)
$
$latex 2x = (3^{2} -1). (3^{2} +1)…(3^{2n} + 1)
$

$latex 2x = (3^{4} -1).(3^{4} +1) … (3^{2n} + 1)
$
.
.
.

$latex 2x = (3^{4n} -1) $

$latex displaystyle boxed
{
x = (3^{4n}…

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Category Theory (Steven Roman) – (Part II)

tomcircle's avatarMath Online Tom Circle

[Continued from (Part I)…]

Category Theory (范畴学) is the “lingua franca” (通用语) of mathematicians, used commonly by the 2 different major Math branches : Algebra & Analysis.

比喻:
武术分”外家拳”少林派, “内家拳” 武当派。
两家的 lingua franca (通用语)是 “气” – 硬气功(少林), 柔气功(武当)。

In essence: A Category consists of
1. Objects
2. Relationship among objects (Morphism)
3. Structure: preserved by Morphism
4. An identity (Self)

Examples:
1. SET Category:
◇ Objects (Sets),
◇ Structure (Cardinality),
◇ Morphism (Set Functions: which preserve Set Structure)
◇ Identity (Set itself)

2. GROUP Category
◇ Objects (groups)
◇ Structure (Set, 1 closed binary operation)
◇ Morphism (group mapping)
◇ Identity (neutral element ‘e’)

3. SINGAPOREAN Category
◇ Objects (Singapore citizens)
◇ Structure (multi-racial)
◇ Morphism (kiasu-ism)
◇ Identity (I = ME = 令伯 ‘lim-Peh-ism’)

Lecture 2:
◇ Functor: morphism between Categories
◇ Diagrams: arrows
◇ Commute
◇ Special Types of Functors:

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How to use Kenzo (Algebraic Topology CAS) on Clozure CL

After some trials, finally got Kenzo to run on Mac:

First download it from (https://github.com/gheber/kenzo). Quicklisp needs to be installed and loaded first.

Type:

(ql:quickload :kenzo)

followed by

(in-package “CAT”)

followed by any Kenzo commands.

Screenshot:

Screen Shot 2016-02-18 at 10.28.28 AM

How to optimise your brain’s waste disposal system

Source: https://www.theguardian.com/science/neurophilosophy/2015/aug/22/how-to-optimise-your-brains-waste-disposal-system

Summary: Sleeping on the side seems to clear brain’s waste most efficiently.

New research suggests that body posture during sleep may affect the efficiency of the brain’s self-cleaning process

The human brain can be compared to something like a big, bustling city. It has workers, the neurons and glial cells which co-operate with each other to process information; it has offices, the clusters of cells that work together to achieve specific tasks; it has highways, the fibre bundles that transfer information across long distances; and it has centralised hubs, the densely interconnected nodes that integrate information from its distributed networks.

Like any big city, the brain also produces large amounts of waste products, which have to be cleared away so that they do not clog up its delicate moving parts. Until very recently, though, we knew very little about how this happens. The brain’s waste disposal system has now been identified. We now know that it operates while we sleep at night, just like the waste collectors in most big cities, and the latest research suggests that certain sleeping positions might make it more efficient.

Applied Math: Computational Topology

Quite easy to follow this “Mr.Bean” lecturer in Computational Topology. Pre-requisite is undergraduate elementary Abstract Algebra (Linear Algebra, Surjectivity, Injectivity, Isomorphism, Quotient Vector Space, Matrices, ….). He used Python to compute rather than Lisp.

tomcircle's avatarMath Online Tom Circle

This is a series of 6 lectures on Computational Topology : Applied Math using computer in Algebraic Topology. Computer tool languages used can be Phyton  (this lecture) or Common Lisp (the preferred Functional Programming like Lisp for its rich mathematical background in Lambda  Calculus — new main feature in next Java 8).

As explained by this professor in Lecture 1, Computational Toplogy begins with Algebraic Topology aided by the arrival of computers in 1950s. The role of Algebraic Topology is to study Topology (Geometric Spaces “Manifolds” (流形) with continuous functions) using algebra (mainly Advanced Linear Algebra).

Just like Google Search revolutionises the world in 2000s, using only the classical Linear Algebra; Algebraic Topology will revolutionise Big Data Analytics using the Advanced Linear Algebra — the next wave in Mobile Age.

In layman’s term, it means using this tool to analyse Big Data in a geometric picture form…

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Is the Abstract Mathematics of Topology Applicable to the Real World?

tomcircle's avatarMath Online Tom Circle

1st speaker :
◇ History: Riemann discovered Topology on his papers left behind after death. He told friend Betti.
◇ Betti Number: number of
– scissor cut to make a tree (in 2 dim),
– drill cutty make a disk (in 3 dim).
◇
2nd speaker:
◇ Evolution (bacteria) using Topology Barcoding.
3 speaker:
◇ Liquid Crystal: Homology

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阅读理解中常见典故解说—”千里马”和”伯乐” Common Literary Quotation In Reading Comprehension

chinesetuition88's avatarChinese Tuition Singapore

千里马指日行千里，善跑的骏马。

伯乐本名孙阳，是古代春秋时期秦穆公时人。他擅长相马，对马很有研究。

唐代文学家韩愈写过一篇文章《马说》。文章中写道”世有伯乐，然后有千里马。千里马常有，而伯乐不常有。”意思是：世界上有了伯乐才会有千里马。千里马经常会有，但是伯乐却很少。”

现在的”千里马”通常用来比喻人才，而”伯乐”则是用来比喻发现，推荐，培养和使用人才的人。

“千里马”，即人才有很多，但是能够赏识人才的”伯乐”却很少。如果人才不被发现和重用，那就和普通人没有区别。只有当伯乐发掘了他们，他们的价值才会被体现出来，才会发挥一个人才的价值。所以先有伯乐，才有千里马。

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