“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.

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。

An excellent article on the Future of world economy.  A view from Stanford University….

3 Keywords: Clean Energy, Robotics,  3D-Printing

Governments, businesses, and economists have all been caught off guard by the geopolitical shifts that happened with the crash of oil prices and the slowdown of China’s economy. Most believe that the price of oil will recover and that China will continue its rise. They are mistaken. Instead of worrying about the rise of China, we need to fear its fall; and while oil prices may oscillate over the next four or five years, the fossil-fuel industry is headed the way of the dinosaur. The global balance of power will shift as a result.

LED light bulbs, improved heating and cooling systems, and software systems in automobiles have gradually been increasing fuel efficiency over the past decades. But the big shock to the energy industry came with

Excellent Category Theory lectures by retired Prof Steven Roman  from Uni. California: he used pen and A4 – paper with iPhone camera. Simple & good. (Only lighting could be brighter.)

Category Theory is one level higher abstraction, above the Abstract Algebra (Group, Ring, Field, Vector Space, Set…). It is the “Math of Math”, to make difficult math easy by studying the ‘relationship’ (or Morphism).

This year my 61st birthday cake is going to be cut mathematically !

This is the correct way to teach Math to children, but it makes parents crazy ! 🙂

Why anything multiplying by 0 gives 0 ?

Proof:

0.x + 0.x = (0 + 0).x [distribution law]
                 = .x       [definition of 0]

Subtracting 0.x from both sides,
0.x + 0.x = 0.x
we get
$latex boxed{0.{x} = 0 }$ [QED]

Parents teach children the intuitive way:
“nothing times anything is nothing”
but this is not rigorous.

Later in life, the children would learn that when doing “zero times” of any thing (eg. bad behaviors: fighting, stealing…) produces “something” (good) 🙂

In the USA and UK, many parents grow up feeling great antipathy towards Math, whereas in Asia like China, Japan, Korea, HK, Singapore, and Europe (eg. France), we grow up fearing Math, probably because of how Math was taught in school. I remember during schools (and…

Hi all! Today I’ll introduce simplicial sets, and talk about how they relate to higher categories. Simplicial sets are basically combinatorial models for topological things; in fact, a particular kind of simplicial set (a Kan complex) is essentially equivalent to a topological space! So more general kinds of simplicial sets should model something more general than a topological space. Exploring this will be the topic of this post. The reader should note that these are directly from the (rather terse) notes I took while preparing for the Intel International Science and Engineering Fair last year (so this blog post is currently also serving as the notes for preparing for the Intel Science Talent Institute), so there may be small errors.

We will begin by talking about some combinatorial constructions. Let $latex [n]&s=1$ denote the set $latex {0,…,n}&s=1$ equipped with the linear ordering. Define $latex mathbf{Delta}&s=1$ to be the category whose objects are the sets $latex…

To the outsiders of the USA,  we can understand the “Common Core Math Program” is good for American children education, which is supported by Bill Gates with generous $6-billion donation.

The American parents who resist and oppose the ‘Common Core’ Math were like the Singaporean parents who opposed the “Singapore Math Program” in the 1980s. The Singapore parents had to go through remedial classes before they could coach their kids in the ‘new way’. Result proved that after 20 years, Singapore is ranked 2nd in the PISA Test World Ranking in Math (after Shanghai of China).

All good medicines are bitter to swallow. 良药苦口 !

Here are 4 methods of subtraction:

1st : Traditional (aka Abacus) “Carry” Method
2nd: Mental Calculation
3rd : “Singapore Math” (no Carry)
4th: Common Core: “Count Up” (French Method), with which French supermart cashiers usually perform changes at counter — much…