Author: tomcircle

Maths Spéciales (2nd Year French Engineering University Math ”Linear Algebra”): The French way of treating Matrices is very general and abstract. Advantage is it studies Matrices at a theoretical high-level, disadvantage is it ignores on applications.

This young French prof explained in 30 mins at great length of what is simply the Characteristic Polynomial Equation:

$latex boxed {det (A – lambda.I) = 0 }&fg=aa0000&s=3$

Compare it with the more practical (but less theoretical – 知其然而不知其所以然) American teaching below from the famous MIT Prof Gilbert Strang for the same Diagonalization of Matrices:

Excellent video for the curious minds! Who cares about Topology such as Torus (aka donut) or Mobius Strip ? They can be used to prove difficult math such as the unsolved problem “Inscribed square/rectangle inside any closed loop”.

To understand the Topology on Loops, please view the lecture here : Homotopyand the fundamental group of surface.

NJ Wildberger AlgTop24: The fundamental group

Homotopy 同伦: When playing skip rope, the 2 ends of the rope are held by 2 persons while a 3rd person jumping over the “swings of rope” – these swings at any instant arehomotopic.

1) Row reduction, row-echelon form and reduced row-echelon form

2) Rank of Matrix = Rank (A)

An autobiography of a Chinese PhD student  (France, University  Paris-11) in Number Theory and Algebraic  Geometry.
His journey from 武汉大学 4 years undergrad to Beijing, learn mostly from the French-educated Chinese Math professors (Ecole Normale  Superieure, Polytechniques).

来源：梁永祺的日志（转载请注明出处）2012-12-16 06:52

“趁着一丝冲动记下一些经历，也趁着现在还能想起来，写写路途中遇到的人和碰到的一些书。”
http://blog.sina.cn/dpool/blog/s/blog_4ee63ce90102ea2r.html

Ecole Normale Supérieure (Paris) (E.N.S.): 巴黎高等师范大学 (Galois 的母校但他因参加法国革命被开除, 现在是法国/世界 Fields Medals 的摇篮, 出Bourbaki 学派的大师André Weil, Cartan, Dieudoné, …医学细菌发现者 Pasteur是排班上最后一名的劣等生)。E.N.S.训练未来的教授 (文, 数, 理), 每年只收全法国前50位精英学生, 培养成博士。后来演变成研究院, 出了不少 Nobel Prize (Science, Literature) 和Fields Medalists.

Paris University 11 = Paris Sud (Paris South). 欧洲最古老的巴黎大学(Sorbonne 索尔本), 继承十世纪阿拉伯人创办的大学制度 (Bachelor, Licencié, Baccalaureate, “Chair” of department…)。出科学家居里夫人 (Madame Curie)。现在有13分校, 其中第11分校是数学研究的重镇。

Ecole Polytechnique (aka X): 巴黎综合理工大学 (拿破仑建的工程军校, 出很多科学家, 数学家: Hertz, Ampere, Fourier, Cauchy, Poincaré, Louiville,…偏偏天才Galois 入学连续考2年Concours不及格, 学弟 Charles Hermite 入学考最后一名, 第二年又被踢出门)。
新加坡30年来至今有4位数学顶尖学生考进 “X”。最近一位(2012)林恩隆 (公教/南洋GEP小学/ RI 中学/ RJC 高中 / Lycée St. Geneviève @ Versailles )是全法(外国考生 ‘Concours’ 工校”科举”入考)第一名, 他同时也是 E.N.S.全法(纯法国人的Concours)第15名, 后生可畏!! (用法文考数理化和法国哲学, 不公平的竞争, 却难不倒华人学子)!

What is “Motif” (Motive) ?

Recommended Books:
1. Jean-Pierre Serre: 《Cours d’Arithmetique》

“Arithmetic”…

Column Space and Null Space of a matrix

Create your own Homology: (Important lecture in Applied Algebraic Topology)

http://slideplayer.com/slide/9473277/

Euler Characteristicχis an invariant in Topology.

Interesting to note that a tree with no cycle (loop),χ =1

Lecture 3: Modular Arithmetic (easy)https://youtu.be/5YEu-vB2dD8

Lecture 4: Addition and Free Vector Spaceshttps://youtu.be/9qL003DG7Og

https://anthonybonato.com/2016/08/31/algebraic-topology-and-the-brain/

The Classe Prépa Math for Grandes Écoles is uniquely French pedagogy – very rigorous based on solid abstract theories.

In this lecture the young French professor demonstrates how to teach students the rigorous Math à la Française:

$latex displaystyle {lim_{ntoinfty} bigl( 1 + frac{1}{n} bigr)^{n} = e}&fg=00bb00&s=3$

This French Classic (8 series lectures) on “Introduction to Modern Maths” made in 1969 is still valid today for Modern Maths students who need to be “initialized” for abstract, rigorous math, different from high-school maths.

Partie 1 : ensembles et parties

Cours math sup, math spé, BCPST.

The French University (engineering) 1st & 2nd year Prépa Math: “Vector Space” (向量空间), aka Linear Algebra (线性代数), used in Google Search Engine. The French treats the subject abstractly, very theoretical, while the USA and UK (except Math majors) are more applied (directly using matrices).

Note: First year French (Engineering) University “Classe Prépa”: Math Sup (superior); 2nd year Math Spéc (special).

Part 2:

Applications Lineaires (Linear Algebra):

The lower in the score the better : Trump (4.1) beats Hillary (7.7) who beats Sanders (10.1)

Trump defied most expectation from the world to win the 2017 President of the USA. His victory over the much highly educated Ivy-league Yale lawyer-trained Hilary Clinton who speaks sophisticated English is “SIMPLE English”:
1-syllable words mostly: eg.dead, die, point, harm,…

2-syllable words to emphasize: eg.pro-blem, service, root cause, …

3-syllable words to repeat : eg. tre-men-dous

His speech is of Grade-4 level, reaching out to most lower-class blue-collar workers who can resonate with him. That is a powerful political skill of reaching to the mass. Hilary Clinton’s strength of posh English is her ‘fatal’ weakness vis-a-vis connecting to the mass.

In election time, it is common to see candidates who win the heart of voters by using the local dialects of the mass, never mind they are discouraged in…

1. Statistical Mechanics: $latex e^ {- Ht} $

Quantum Mechanics: $latex e^{iHt}$

2. Ramanujian:

$latex 1 +2 + 3 + …+ n = -frac {1}{12} $

Tau Special Function:

$latex boxed {displaystyle sum_{n=1}^{infty}tau (n) x^{n} = x {(1-x)(1-x^{2})(1-x^{3})… }^{24}}$

3. Boolean Algebra: George Boole (1847 in 《The Mathematical Analysis of Logic》) used Symbolic variables (not numbers) for Logic, inspired by Galois (1832 in Groups & Finite Fields), Hamilton’s quaternion algebra (1843),

“AND” $latex boxed {x.y}&fg=00bb00&s=3$

“NOT” $latex boxed {1-x}&fg=00bb00&s=3$

“XOR” $latex boxed {x+y-2x.y}&fg=00bb00&s=3$

“Extra constraints ” $latex boxed {x^{2}=x}&fg=00bb00&s=3$

4. Solomon Golomb, Sol: “Linear Feedback Shift Register” (LFSR) – shift left the first register, fill in the back register with XOR of certain “Taps” (eg.chosen the 1st, 6th, 7th registers)

Maximal Length = The shift register of size n will repeat every $latex 2^{n}-1$ steps (exclude all ‘0’ sequence).

Which arrangement…

Key Motivations for Category Theory 范畴 :
1. Programming is Math.
2. Object-Oriented is based on Set Theory which has 2 weaknesses:
◇ Set has contradiction: The “Russell’s Paradox”.
◇ Data Immutability for Concurrent Processing : OO can’t control the mutable state of objects, making debugging impossible.

Category (“cat“) has 3 properties:
1. Objects
eg. Set, List, Group, anything…
2. Arrows (“Morphism”, between Objects) which are Associative
eg. functions etc
3. Identity Object

Note: If the Identity is “0” or “Nothing”, then it is called Free Category.

Extensions :
1. “Cat” = Category of categories, is also a category.
2. Functor 函子 = Arrows between Categories.
3. Monoid = A Category with ONLY 1 Object.

Monoid (么半群) is a very powerful concept (used in Natural Language Processing) — basically it is a Group with No Inverse (Mo‘No‘-‘I‘-d)

Math has been taught wrongly since young, either is boring, or scary, or mechanically (calculating). This lecture by Queen Mary College (U. London) Prof Cameron is one of the rare Mathematician changing that pedagogy. Math is a “Universal Language of Truths” with unambiguous, logical syntax which transcends over eternity.

I like the brilliant idea of making the rigorous Math foundation compulsory for all S.T.E.M. (Science, Technology, Engineering, Math) undergraduate students. Prof S.S. Chern (Wolf Prize) after retirement in Nankai University (China) also made basic “Abstract Algebra” course compulsory for all Chinese S.T.E.M. undergraduates.

The foundations Prof Cameron teaches are centered around 4 Math Objects:

1. SETS
– Founding block of the 20th century modern math, made into world’s university textbooks which were influenced by the French “Bourbaki” school after WW1.

2. FUNCTIONS
– A vision first proposed by the German Gottingen School’s greatest Math Educator Felix Klein, who said Function…

$latex displaystylelim_{xto a}f(x) = L
iff$
$latex forall varepsilon >0, exists delta >0 $ such that
$latex boxed{0<|x-a|<delta}
implies |f(x)-L|< varepsilon $

The above scary ‘epsilon-delta’ definition of “Limit” by the French mathematician Cauchy in 19th century is the standard rigorous definition in today’s Analysis textbooks.

It was not taught in my Cambridge GCE A-Level Pure Math in 1970s (still true today), but every French Baccalaureate Math student (Terminale,  equivalent to JC 2 or Pre-U 2) knows it by heart. A Cornel University Math Dean recalled how he was told by his high-school teacher to memorise it — even though he did not fully understand — the “epsilon-delta” definition by “chanting”:

“for all epsilon, there is a delta ….”

(French: Quelque soit epsilon, il existe un delta …)

In this video, I am glad someone like Prof N. Wildberger recognised its “flaws”  albeit rigorous, by suggesting another more intuitive…

MathFoundations (all videos): http://www.youtube.com/playlist?list=PL5A714C94D40392AB

All the Math we learn are taught as such by teachers and professors, but why so? what are the foundations ? These 200 videos answer them !

Good for students to appreciate Math and, hopefully, they will love the Math subject after viewing most of these 200 great videos.

Video 1: Natural Number This should be taught in kindergartens to 3-year-old kids.

◇ What is number ? (strings of 1s),
◇ Equal, bigger, smaller concepts are “pairing up” (1-to-1 mapping) two strings of 1s.
◇ Don’t teach the kids how to write first 12345…, without prior building these mathematical foundational concepts.
.
.
.
Video 106: What is a Limit ?

$latex displaystylelim_{xto a}f(x) = L
iff$
$latex forall varepsilon >0, exists delta >0 $ such that
$latex boxed{0<|x-a|<delta}
implies |f(x)-L|< varepsilon $

The above scary ‘epsilon-delta’ definition of “Limit” by the French mathematician Cauchy in…

Affine Line: $latex {mathbb {A}^1}&s=3$

Six Representations of a Circle: $latex {mathbb {S}^1}&s=3$
1) Euclidean Geometry
Unit Circle : $latex x^2 + y^2 = 1$

2) Curve:
Transcendental Parameterization :
$latex boxed { e(theta) = (cos theta, sin theta) qquad
0 leq theta leq 2pi }&fg=aa0000&s=3
$

Rational Parameterisation :
$latex boxed {
e(h) = left(frac {1-h^2} {1+h^2} : , : frac {2h} {1+h^2}right) quad text { h any number or } infty
} &fg=aa0000&s=2
$

3) Affine Plane $latex {mathbb {A}^2}&s=3$
1-dim sub-Space = Lines thru Origin

4) Polygonal Representation

5) Identifying Intervals: (closed loop)

6) $latex text {Translation } (tau, {tau}^{-1}) text { on a Line } $ $latex {mathbb {A}^1}&s=3$

$latex boxed {
{mathbb {S}^1} = {mathbb {A}^1 } Big/ { langle tau , {tau}^{-1} rangle}
}&fg=aa0000&s=3
$
$latex {mathbb {S}^1} = text { Space of all orbits} $

Simplices:
0-dim (Point) $latex triangle_0 $
1-dim (Line) $latex triangle_1 $
2-dim (Triangle) $latex triangle_2 $
3-dim (Tetrahedron) $latex triangle_3 $

Simplicial Complex: built by various Simplices under some rules.

Definitions of Simplex (S)
Face
Orientation
Boundary ($latex delta $)
$latex displaystyle boxed {
delta(S) = sum_{i=0}^{n} (-1)^i (v_0 …hat v_i …v_n)}&fg=aa0000&s=3
$

Theorem: $latex boxed { delta ^2 (S) = 0}&fg=00bb00&s=4 $

Abstract Algebra is scary but not with this pretty lecturer!

1. Group
◇ Definition
◇ Symmetric Group
◇ Cayley Table

2. Linear Groups:
◇ GLn (R)
◇ SLn(R) — Determinant = 1

3.  Relationship between Group Structures:
◇ Homomorphism,
◇ Isomorphism, 
◇Kernel

4.Ring Definition

5. Field Definition

Year	Ver	Code Name	Description
1995	1.0	Java	Applets
1977	1.1	Java	Event, Beans, Internationalization
Dec 1978	1.2	Java 2	J2SE, J2EE, J2ME, Java Card
2000	1.3	Java 2	J2SE 1.3
2002	1.4	Java 2	J2SE 1.4
2004	1.5	Java 5	J2SE 1.5
Nov 2006	6	Java 6	Open-Source Java SE 6. “Multithreading” by Doug Lea
May 2007	OpenJDK free software
2010	Oracle acquired Sun
Jul 2011	7	Java 7	“Dolphin”
Mar 2014	8	Java 8	Lambda Function

Javac: Java Compiler

Java Distributions:

1. JDK (Java Developer Kit )
◇JRE & Javac & tools

2. JRE (Java Runtime Environment)
◇ JVM & core class libraries
◇ Windows / Mac / Linux

Java is Object-Oriented Programming (OOP):
1. Class
public class Employee {

public int age;
public double salary;

public Employee () { [<– constructor with no arg]
}

public Employee (int ageValue, double salaryValue) { [<- constructor with args]

age…

The French Emperor Napoleon Bonaparte was himself a Mathematician. He posed a question to his mathematician friends Monge, Laplace, Fourier etc:

How to divide a circle equally in 4 sections using only a compass ?

Note: He was more stringent than the ancient Greeks who allowed a compass and a non-marked ruler.

3 个日本人
1 个中国(香港): 丘成桐
1 个澳洲华人 : Terence  Tao 陶轩哲
1 个印度人: Manjul Bhargava
1 个越南人 : 吴宝珠

中国人? 新加坡人 ? 韩国人 ? 泰国人? ….加油!!

List of Fields Medalists:

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/

Homology: 同调 (江泽涵 教授)
Homotopy : 同伦 (江泽涵 教授)
Manifold : 流形 (源于文天祥的《正气歌》: 天地有正气，杂然赋流形)
Geometry: 几何 (明朝 徐启光 / Italy Jesuit 利马窦 )
Topology: 拓朴 (杜甫诗)

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)!

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

http://isomorphism.es/post/615614573/category-theory

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)

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…

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.

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)…

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…

Blog by Jeremy Kun (Ph.D in Math 2016)

Homology Theory — A Primer

Prerequisites :
◇ The Fundamental Group : A Primer

Gilbert Strang 35 lectures on Linear Algebra (MIT): http://www.youtube.com/playlist?list=PL49CF3715CB9EF31D

Highly Recommended Prof Strang’s Amazon book: (4th Edition)

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…

《Harvests and Sowings》

http://www.fermentmagazine.org/rands/recoltes1.html

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…

[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。

“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).

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.

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}…

Lecture 1.1 : Overview

Lecture 1.2 : Maps

[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:

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…

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

Homology (同调) is easier to compute than Homotopy (同伦) , hence Computational Topology uses the Algebraic Topology of Homology.