Moonshine Monster Group & Fourier

Moonshine 196,883
What object can exist in 196,883 dimension ?

Simon Norton: "I can explain to you what Moonshine (Monster Group) is in one sentence.
"It is the voice of God."

While all the 5 mathematicians have been fully exhausted after 15 years of effort to categorize all Simple Groups in the Universe, only Simon Norton remains lonely in the search of the largest Monster Group (Moonshine).

Conway, who migrated from Cambridge to Princeton, does not want to touch anything on Group now, said, "Simon is the only person on earth who knows Moonshine".

Monster Group M, order |M|=
$latex 2^{46}. 3^{20}. 5^{9}. 11^{2}. 13^{3}. 17.19.23.29.31.41.47.59.71$

Moonshine Monster Group dimensions (dj) & relationship with Fourier expansion of coefficients (cj) in Modular Function:
$latex x^{-1} + 744+196,884x + 21,493,760 x^{2} + 864,229,970x^{3} +\dots $
$latex c_n= c_1+c_2+...c_{n-1} + d_{n}$
where
$latex d_1 = 196,883$
$latex d_2 = 21,296,876$
$latex d_3 = 842,609,326$
and
$latex c_1 = 1+ d_1 = 196,884$
$latex c_2 = c_1+d_2 = 21,493,760$
$latex c_3 = c_1 + c_2 + d_3 = 864,229,970$

What a coincidence! no wonder Conway said this discovery was the most exciting event in his life.

French Curve

The French method of drawing curves is very systematic:

"Pratique de l'etude d'une fonction"

Let f be the function represented by the curve C

Steps:

1. Simplify f(x). Determine the Domain of definition (D) of f;
2. Determine the sub-domain E of D, taking into account of the periodicity (eg. cos, sin, etc) and symmetry of f;
3. Study the Continuity of f;
4. Study the derivative of f and determine f'(x);
5. Find the limits of f within the boundary of the intervals in E;
6. Construct the Table of Variation;
7. Study the infinite branches;
8. Study the remarkable points: point of inflection, intersection points with the X and Y axes;
9. Draw the representative curve C.

Example:

$latex \displaystyle\text{f: } x \mapsto \frac{2x^{3}+27}{2x^2}$
Step 1: Determine the Domain of Definition D
D = R* = R - {0}

Step 2: There is no Periodicity and Symmetry of f
E = D = R*

[See Note below for Periodic and Symmetric example]

Step 3: Continuity of f
The function f is the quotient of 2 polynomial functions, therefore f is differentiable
=> f is continuous in $latex ]-\infty,0[ \cup ]0,+\infty[ $
[See previous post CID Relation]

Step 4: Determine f'
$latex \displaystyle\forall x \in R^{\star}, f'(x) = \frac{6x^{2}.2x^{2} - 4x (2x^{3}+27)}{4x^{4}} = \frac{4x^{4}-4.27x}{4x^{4}} = \frac{4x(x^{3}-27)}{4x^{4}}$
$latex \forall x \in R^{\star}, (x^{3} - 27 >0) \iff (x>3)$
Therefore f' has the same sign as $latex x \mapsto x(x-3)$

$latex \begin{cases} \forall x \in ]-\infty,0[ \cup ]3,+\infty[, & f'(x)>0 \\
\forall x \in ]0,3[ , & f'(x)<0
\end{cases}$

Step 5a: Limit at x=0

$latex \displaystyle\lim_{x\to 0}(2x^{3}+27) = 27$
$latex \displaystyle\lim_{x\to 0} 2x^{2} = 0 , (\forall x \in R^{\star}, x^{2} >0)$
Therefore, $latex \displaystyle\lim_{x\to 0}f(x) = + \infty$

Step 5b: Limit at $latex x= + \infty$
$latex \displaystyle\lim_{x\to +\infty} f(x) =\lim_{x\to +\infty} \frac{2x^{3}+27}{2x^{2}}=\lim_{x\to +\infty} \frac{2x^{3}}{2x^{2}} = \lim_{x\to +\infty} x = +\infty$

Step 5c: Limit at $latex x= - \infty$
Similarly,
$latex \displaystyle\lim_{x\to -\infty} f(x) = \lim_{x\to -\infty} x = -\infty$

Step 6: Construct the Table of Variation

$latex \begin{array}{|l|l|l|}
\hline
x & - \infty \rightarrow \: \: \: \: 0 & 0 \:\:\:\:\: \rightarrow \:\:3 \rightarrow \:\:\: +\infty \\
\hline
f'(x) & \:\: \: \: \:\: \: + & \:\:\:\: - \:\:\:\:\:\:\:\:\: 0 \:\:\:\:\:\:\: + \\
\hline
f(x) & -\infty \nearrow +\infty & +\infty \searrow \: \frac{9}{2} \nearrow +\infty\\
\hline
\end{array}$

Step 7: Study the infinite branches

7a) $latex \displaystyle\lim_{x\to 0}f(x) = + \infty$
=> y-axis is the asymptote

7b) $latex \displaystyle\forall x \in R^{\star}, f(x) = \frac{2x^{3}+27}{2x^{2}}= x+\frac{27}{2x^{2}}$
$latex \displaystyle\lim_{x\to +\infty}\frac{27}{2x^{2}} = 0$ , $latex \displaystyle\lim_{x\to -\infty}\frac{27}{2x^{2}} = 0$
=>
$latex \displaystyle\lim_{x\to +\infty}f(x) = x$ , $latex \displaystyle\lim_{x\to -\infty}f(x) = x$
=> y= x is another asymptote
$latex \forall x \in R^{\star}, \frac{27}{2x^{2}} >0$
=> The curve C is above the asymptote y=x

Step 8: Study the remarkable points: intersection points with x-axis
$latex \forall x \in R^{\star},(2x^{3}+27 =0)
\iff (x^{3}=-\frac{27}{2})
\iff (x=-\frac{3}{\sqrt[3]{2}}) = -2.38$

Step 9: Draw the representative curve C of f.

[caption id="attachment_2564" align="alignnone" width="500"] french curve[/caption]

Note:
$latex \displaystyle\text{Let g: } x \mapsto \frac{sin x}{2- cos^{2}x}$
D = R
g(x) is periodic of 2π => E = [0 , 2π]
$latex \displaystyle\forall x \in R, g(-x)= \frac{sin (-x)}{2-cos^{2}(-x)}=-\frac{sin x}{2-cos^{2}x}=-g(x) $
=> g(x) is symmetric with respect to the origin point O

We can restrict our study of g(x) in E = [0,π]

$latex \displaystyle\forall x \in R, g(\pi-x)= \frac{sin (\pi-x)}{2-cos^{2}(\pi-x)}=\frac{sin x}{2-cos^{2}x}=g(x) $
=> g(x) is symmetric w.r.t. to the equation x= π/2

Finally, we can further restrict our study of g(x) in E = [0, π/2]

TeX Math Editor

1989 Stanford Professor Donald Knuth published the first version of TeX mathematical software.

Subsequent versions follow π:
3.14,
3.141,
3.1415,
....
3.1415926 (current version)

La Ligne Directe du Dieu

Cédric Villani (Médaille Fields 2010) "Théorème Vivant":

"La fameuse ligne directe, quand vous recevez un coup de fil du dieu de la mathématique, et qu'une voix résonne dans votre tête. C'est très rare, il faut l'avouer!"

"The famous direct line, when you receive a 'telephone call' from the God of the Mathematic, and that a voice resonates in your head. It is very rare, one has to admit."

QuYuan 屈原 Symmetry

屈原 QuYuan (343–278 BCE) Symmetry:
http://en.wikipedia.org/wiki/Qu_Yuan

离騷《天问》
1. "九天之际, 安放安属,
隅隈多有, 谁知其数 ?"
=> 天 (Sky) 和 地 (Earth) must be 2 symmetric spheres.

If 地 (Earth) were flat, then there would be (隅隈) edges and angles at the 天 (Sky) & 地 (Earth) boundary (九天之际).

2. "东西南北, 其修孰多,
南北顺橢, 其衍几何。"
=> 南北顺橢 = The Earth is ellipse (橢), with north-south (南北) slightly flatten.

几何 = Geometry

3. How did QuYuan know this advanced astronomy & geometry in ~ 300 BCE?

[caption id="attachment_2393" align="alignnone" width="201"] 屈原[/caption]

墨子 Mozi & Force

[caption id="attachment_2380" align="alignnone" width="166"] Mozi[/caption]

墨子 Mozi (468 BCE~ 376 BCE), 2000 years earlier than Newton

"墨子" :  “力, 行之所以奋也。”

行: Moving
奋: Acceleration
力: Force is due to acceleration by the moving object.

F ∝ a
F = m.a

εδ Confusion in Limit & Continuity

1. Basic:
|y|= 0 or > 0 for all y

2. Limit: $latex \displaystyle\lim_{x\to a}f(x) = L$ ; x≠a
|x-a|≠0 and always >0
hence
$latex \displaystyle\lim_{x\to a}f(x) = L$
$latex \iff $
For all ε >0, there exists δ >0 such that
$latex \boxed{0<|x-a|<\delta}$
$latex \implies |f(x)-L|< \epsilon$

3. Continuity: f(x) continuous at x=a
Case x=a: |x-a|=0
=> |f(a)-f(a)|= 0 <ε (automatically)
So by default we can remove (x=a) case.

Also from 1) it is understood: |x-a|>0
Hence suffice to write only:
$latex |x-a|<\delta$

f(x) is continuous at point x = a
$latex \iff $
For all ε >0, there exists δ >0 such that
$latex \boxed{|x-a|<\delta}$
$latex \implies |f(x)-f(a)|< \epsilon$

Yin-Yang of Hopf Algebra

Prof Heinz Hopf invented this new Hopf Algebra during WW II, it remained unnoticed for years after he & his wife's tragic death by the Nazi.

The essence of Hopf Algebra is a weird 'Yin-Yang' view of every algebra concept.
Examples:
Associative co-Associative
Commutative co-Commutative
Algebra co-Algebra
Multiplication co-Multiplication
Unit co-Unit
Homomorphism co-Homomorphism
Monoïd co-Monoïd
...
the 'co-' notion represents the 'Yin' opposite of the 'Yang' notion.

[Analogy: Set & coSet 倍集， l'ensemble à gauche ou à droite]

Hopf Algebra is applied in Quantum Group for Quantum Physics, Yang-Baxtor equation, Combinatorics and Topology, etc.

If you find a structure which has a combined properties of Group, Ring and Vector Space, then it could be a Hopf Algebra.

Applied it in Graphs, we can re-discover the Euler's (E-V+F = 2 ) formula, Möbius function, and Symmetric Group (Sn).

Note: Excellent 30-lecture series by Prof Federico Ardila (San Francisco University) on Hopf Algebra:

http://math.sfsu.edu/federico/Clase/Hopf/lectures.html

French Baccalaureat Math

Below are two French Baccalaureat Math textbooks (Volume 1 & 2) for Grade 12th (or Junior College Year 2, GCE A Level, Lycée: Terminale C) students in the late 1970s.

There was a strong Bourbaki style of influence in this Math teaching, which was later criticized as being too abstract and less applied for 18-year-old students.

Nevertheless, the syllabus proved to be excellent for those Math-inclined young minds who later entered the special 2-year Classe Préparatoire aux Grandes Ecoles - Preparatory class (equivalent to Bachelor of Science) for graduate schools in crème de la crème top universities (Grandes Écoles ) such as Ecole Normale Supérieure or Ecole Polytechnique , where the solid Preparatory Math trainings (Maths Supérieures, Maths Spéciales) are extremely rigorous and high standard, incubating most of the French great mathematicians, Fields Medalists and Nobel Prize Scientists.

Voici les 2 tomes de Mathematiques Terminales C et E
Bulletin Officiel du 24 Juin 1971
par M. Monge, M.-C. audouim -Egoriff F. Lemaire-Body
LIBRAIRIE BELIN, 1974 -4 ISBN 2-7011-0224-3

Tome 1: Algèbre et Géométrie

Chapitre 1: Structures Algebriques
1. Groupes-Anneaux -Corps
2. Homomorphisms
3. Espaces vectoriels
4. Applications linéaires et matrices

Chapitre 2: Les nombres complexes

Chapitre 3: Sous espaces vectoriels d'un espace vectoriel réel

Chapitre 4: Espace affines et sous-espace affines

Chapitre 5: Application linéaires

Chapitre 6: Application affines

Chapitre 7: Transformation orthogonales
1. Orthoganalité dans un espace vectoriel
2. Endomomorphismes orthogonaux
3. Transformations orthogonales de E2
4. Transformations orthogonales de E3
5. Orientation d'un espace vectoriel

Chapitre 8: Angles. Product vectoriel

Chapitre 9: Isométrie affines

Chapitre 10: Similitudes

Chapitre 11: Coniques

Chapitre 12: Geometrie descriptive

TOME 2: Arithmétique, Analyse et Probabilités
Programs (B.O. du 24.6.1971 et du 19.7.1973)

Chapitre 1: Entiers naturels. Numération.
1. Ensembles ordonnés
2. Ensemble N des Entiers naturels
3. Numeration

Chapitre 2: Entiers rationnels. Entiers modulo n.
1. Construction de l'ensemble Z
2. Sous-groupes de (Z,+)
3. Congruences dans Z

Chapitre 3: PGCD PPCM Nombres premiers.

Chapitre 4: Nombres réels. Suites

Chapitre 5: Continuité. Limites
1. Continuité en un point
2. Continuité sur un intervalle
3. Limites

Chapitre 6: Fonction réciproque d'une fonction continue strictement monotone

Chapitre 7: Dérivation

Chapitre 8: Pratique de l'étude d'une fonction

Chapitre 9: Fonction vectorielle d'une variable réelle

Chapitre 10: Calcul intégral

Chapitre 11: Fonction logarithmiques. Fonction exponentielles

Chapitre 12: Probabilités sur un ensemble fini

Annexé: Équations différentielles
1. Équations: y'= ay
2. Équations: y'' + ω²y = 0

Note: The 2012 French Baccalaureat (Terminale S) Syllabus is modified simpler to suit the majority of science students with more applied math and lesser abstract math:
http://www.maths-france.fr/Terminale/TerminaleS/ProgrammeOfficiel_TerminaleS_2012.pdf

Old Mathematicians Live Young

J.J. Sylvester, who coined the term Matrix, pointed out that Leibniz, Newton, Euler, Lagrange, Laplace, Gauss, Plato, Archimedes, and Pythagoras all were productive until their 70s or 80s.

"The mathematician lives long and lives young," he wrote.

"The wings of the soul do not early drop off, nor do its pores become clogged with the earthly particles blown from the dusty highways of vulgar life."

Sylvester himself was his 82nd year, in 1896, when he "found a new enthusiasm and blazed up again over the theory of compound partitions and Goldbach's conjecture."

Another mathematician Harold S. M. Coxeter (9/2/1907 - 31/3/2003) attributed his longevity to love of mathematics.

Hilbert's Problem Solving

David Hilbert was a most concrete, intuitive mathematician who invented, and very consciously used, a principle: namely, if you want to solve a problem first strip the problem of everything that is not essential. Simplify it, specialize it as much as you can without sacrificing its core. Thus it becomes simple, as simple as it can be made, without losing any of its punch, and then you solve it. The generalization is a triviality which you don't have to pay too much attention to.

Contemporary Math

Bourbaki started as rebels against the established modes of thought in French mathematics in the early 20th century.

But with new discoveries and the increasing important interaction between physics and math, their exclusionist (from Physics etc) approach lost its effectiveness.

Contemporary math is more multifaceted, it includes more varied theoretical and applied approaches.

Algorithm For Love

Each heart represents a person.

The numbers indicate how well they answered questions based on the other person's wants.

The result is a mathematical expression of how happy you could be.



- By Sam Yagan (co-founder of online dating site OkCupid)

Differentiating under integral

Prove: (Euler Gamma Γ Function)
$latex \displaystyle n! = \int_{0}^{\infty}{x^{n}.e^{-x}dx}$

Proof:
∀ a>0
Integrate by parts:

$latex \displaystyle\int_{0}^{\infty}{e^{-ax}dx}=-\frac{1}{a}e^{-ax}\Bigr|_{0}^{\infty}=\frac{1}{a}$

∀ a>0
$latex \displaystyle\int_{0}^{\infty}{e^{-ax}dx}=\frac{1}{a}$ ...[1]

Feynman trick: differentiating under integral => d/da left side of [1]

$latex \displaystyle\frac{d}{da}\displaystyle\int_{0}^{\infty}e^{-ax}dx= \int_{0}^{\infty}\frac{d}{da}(e^{-ax})dx=\int_{0}^{\infty} -xe^{-ax}dx$

Differentiate the right side of [1]:
$latex \displaystyle\frac{d}{da}(\frac{1}{a}) = -\frac{1}{a^2}$
=>
$latex a^{-2}=\int_{0}^{\infty}xe^{-ax}dx$

Continue to differentiate with respect to 'a':
$latex -2a^{-3} =\int_{0}^{\infty}-x^{2}e^{-ax}dx$
$latex 2a^{-3} =\int_{0}^{\infty}x^{2}e^{-ax}dx$
$latex \frac{d}{da} \text{ both sides}$
$latex 2.3a^{-4} =\int_{0}^{\infty}x^{3}e^{-ax}dx$
...
...
$latex 2.3.4\dots n.a^{-(n+1)} =\int_{0}^{\infty}x^{n}e^{-ax}dx$
Set a = 1
$latex \boxed{n!=\int_{0}^{\infty}x^{n}e^{-x}dx}$ [QED]

Another Example using "Feynman Integration":

$latex \displaystyle \text{Evaluate }\int_{0}^{1}\frac{x^{2}-1}{ln x} dx$

$latex \displaystyle \text{Let I(b)} = \int_{0}^{1}\frac{x^{b}-1}{ln x} dx$ ; for b > -1

$latex \displaystyle \text{I'(b)} = \frac{d}{db}\int_{0}^{1}\frac{x^{b}-1}{ln x} dx = \int_{0}^{1}\frac{d}{db}(\frac{x^{b}-1}{ln x}) dx$

$latex x^{b} = e^{ln x^{b}} = e^{b.ln x} $

$latex \frac{d}{db}(x^{b}) = \frac{d}{db}e^{b.ln x}=e^{b.ln x}.{ln x}= e^{ln x^{b}}.{ln x}=x^{b}.{ln x}$

$latex \text{I'(b)}=\int_{0}^{1} x^{b} dx=\frac{x^{b+1}}{b+1}\Bigr|_{0}^{1} = \frac{1}{b+1}$
=>
$latex \text {I(b)} = ln (b+1) + C$

Let b=0
I(0) = 0= ln (1) + C = 0+C => C=0

$latex \boxed{I(b)=ln(b+1)}$

Let b= 2
$latex \displaystyle\int_{0}^{1}\frac{x^{2}-1}{ln x} dx = I(2) = ln (3)$ [QED]

Surjective Proof

Proof Surjective (or Onto 满射 )

Let f: R² -> R²
(x,y) -> (x+1, y+2)
Prove f onto?

Proof:
By defn of f, f(R²) ⊆ R²
To be onto: f(R²) = R²
=> we must prove:
f(R²) ⊇ R²

∀ (x’, y’) ∈ R² ...(1)
We must show that
(x’, y’) ∈ f(R²),
∃ (x, y) in domian R²
s.t. f(x,y)=(x’,y’)
x’=x+1, y’=y+2
x=x’-1, y=y’-2
With these values (x,y)∈R² and f(x,y)=(x’,y’) as required,
=> (x’, y’) ∈ f(R²)...(2)
Thus R² ⊆ f(R²)

f(R²) ⊆ R² and R² ⊆ f(R²)
=> f(R²) = R²= codomain
=> f is onto
[QED]

九章算术 & Fermat Little Theorem

500 BCE Confucius Period 孔子时代，Chinese ancient Math text "Nine Chapters of Arithmetic" 中国 (九章算术) already recorded the following:
$latex p \mid (2^{p} - 2) \text { for p prime}$
$latex \iff 2^{p} \equiv 2 \mod {p} $

Fermat Little Theorem (17th century):
$Latex m^{p} \equiv m \mod {p}$

Note: Computer Cryptography is based on two math theorems: Chinese Remainder Theorem and Fermat Little Theorem.

What is Infinity

What is Infinity?
(Dedekind’s definition)

$Latex 1+ \aleph_0 = \aleph_0 $
$latex \text {where } \aleph_0 \text{ is Aleph 0} $

The Peano axiom was influenced by Dedekind.

Z Risk Ratio

Z risk ratio defined as:

Z = 0.012A + 0.014B + 0.033C+
0.006D + 0.010E

Let T= Total Asset
A= Net Current Asset / T
B= Retained Earning /T
C= Profit b4 interest before tax / T
D= Market Cap / T
E= Sales / T

Z < 1.8 => company bankrupt
Z > 3 => company healthy


Accuracy:
bankruptcy (3 years 95%) (2 yrs 70%)

Ramanujan-Hardy number

Ramanujan-Hardy number

1729 = 1.1.1 + 12.12.12
1729 = 9.9.9 + 10.10.10

Invention Formula

InVention =
(Passion + Perseverance) / (Knowledge + Experience + Inspiration)

$latex V = \frac{2P}{KEI}$

1's Association

1's Association
= x² + y² [Radius of unity circle]
= sin² X + cos² X
= log 10
= x²/a² + y²/b² [oval]
= lim sin x / x ; x-> 0
= nCn
= (-1/2 ± i√3 /2)³ [cubit root of unity : x³ = 1]

Talent vs Genius

Talent is doing easily what others find difficult, but that genius is doing easily what others find impossible.

Homomorphism History

1830 Group Homomorphism
(1831 Galois)

1870 Field Homomorphism
(1870 Camile Jordan Group Isomorphism)
(1870 Dedekind: Automorphism Groups of Field)

1920 Ring Homomorphism
(1927 Noether)

Set Theory in John 1:1

"In the beginning was the Word, and the Word was with God, and the Word was God. " (John 1:1)

太初有道。道与神同在，道就是神。

Mathematically in Set Theory:
Let W = Word, G = God

∃ W, (In the beginning was the Word)
and
W⊂ G, W⊃ G (the Word was with God)

=> W=G (the Word was God)

(John 14:11 NIV)
"Believe me when I say that I am in the Father and the Father is in me; or at least believe on the evidence of the works themselves."

Proof: by Set Theory
⊂: in, include, inside

Jesus ⊂ Father (I am in the Father )
Father ⊂ Jesus (the Father is in me)
=> Jesus = Father (= God)
[QED]

What is “sin A”

What is “sin A” concretely ?

1. Draw a circle (diameter 1)
2. Connect any 3 points on the circle to form a triangle of angles A, B, C.
3. The length of sides opposite A, B, C are sin A, sin B, sin C, respectively.

Proof:
By Sine Rule:

$latex \frac{a}{sin A} = \frac{b}{sin B} =\frac{c}{sin C} = 2R = 1$
where sides a,b,c opposite angles A, B, C respectively.
a = sin A
b = sin B
c = sin C

杨振宁欣赏数学

1938 杨振宁高二（JC1) 时，父亲杨武之（华罗庚，陈省身的老师 ）借他二本书：
Hardy's "Pure Mathematics",
E.T. Bell's "Men of Mathematics".
并讨论 Set Theory, The Continuum Hypothesis.

40年后杨振宁说：
"我的物理学界同事们大多对数学采取功利主义态度。也许受父亲的影响，我较为欣赏数学，其优美和力量：它有战术上的机巧与灵活，又有战略上的雄才远虑。而且，奇迹的奇迹，它的一些美妙概念竟是支配物理世界的基本结构。＂

陈省身: 数学=美

陈省身 数学=美

“宇宙世界万象纷呈, 都暗含着一种伟大的秩序, 這种秩序, 可以被数学家发现, 用数学公式表示出来; 而公式的演算, 应该做到由繁就简, 最后的形式最简洁, 才最漂亮。”

New Geometry 新几何

New Geometry (新几何) invented by Zhang JingZhong (張景中) derived from 2 basic theorems:

1) Triangles internal angles =180º

2) Triangle Area = ½ base * height
=> derive all geometry
=> trigonometry
=> algebra
(These 3 maths are linked, unlike current syllabus taught separately)

The powerful Area (Δ) Proof Techniques:

1) Common Height:
Line AMB, P outside line
Δ PAM / Δ PBM = AM/BM

2) Common 1 Side (PQ):
Lines AB and PQ meet at M
Δ APQ /Δ BPQ = AM/BM

3) Common 1 Angle:
∠ABC=∠XYZ (or ∠ABC+∠XYZ = ∏ )
Δ ABC /Δ XYZ= AB.BC /XY.YZ

These 3 theorems can prove Butterfly and tough IMO problems.

Coset

The powerful notion of Coset was invented by Galois (l'ensemble à gauche ou à droit), but only named as Coset after 150+ yrs later by G.A. Miller in 1910.

Prove:
Coset * Coset = Coset
=> Normal Subgroup

[Hint] Proof technique: use
1) $latex a^{-1}$
2) e

Proof:
1) For any a ∈G, H subgroup of G,

$latex (Ha)(Ha^{-1})= H.(aHa^{-1})$

2) Given $latex H.(aHa^{-1}) $ is right coset,
Choose $latex (aHa^{-1}) = e \in G$
$Latex H.(aHa^{-1})= He = H$
=> $latex aHa^{-1} \subset H$
=> H Normal subgroup

Lord of the Ring

Lord of the "Ring":
The term Ring first introduced by David Hilbert (1862-1943) for Z and Polynomial.
The fully abstract axiomatic theory of commutative rings by his student Emmy Noether in her paper "Ideal Theory in Rings" @1921.

eg. 3 Classical Rings:
1. Matrices over Field
2. Integer Z
3. Polynomial over Field.

Ring Confusions
Assume all Rings with 1 for * operation.

Ring has operation + forms an Abelian group, operation * forms a semi group (Close, Associative).

1) Ever ask why must be Abelian + group ?
Apply Distributive Axioms below:
(a+b).(1+1) = a.(1+1) + b.(1+1)
= a + a + b + b ...[1]

Or,
(a+b).(1+1) = (a+b).1 + (a+b).1
= a + b + a + b ...[2]

[1]=[2]:
a + (a + b) + b = a + (b + a) +b
=> a + b = b + a

Therefore, + must be Abelian in order for Ring's * to comply with distributive axiom wrt +.

2). Subring
Z/6Z ={0,1,2,3,4,5}
3.4=0 => 3, 4 zero divisor

has subrings: {0,2,4},{0,3}

3). Identity 1 and Units of Ring

Z/6Z has identity 1
but 2 subrings do not have 1 as identity.
subrings {0,2,4}:
0.4=0
2.4=2,
4.4=4 => identity is 4
4 is also a unit.

Units: Ring R with 1.
∀a ∈ R ∃b ∈ R s.t.
a.b=b.a = 1
=> a is unit
and b its inverse a^-1

Z/6Z: identity for * is 1
5.5 = 1
5 is Unit besides 1 which is also unit. (1.1=1)

Group Action: Orbit

Orbit demystified
Let G= {rotation around center O }
Let X= point {x,y } ∈ R²
Orbit (X) = subset of points X on the concentric circles of different radius, centered at point O.
In fact,
Orbit = Coset
Hence is equivalent relation
=> can partition set X
=> X = Uj Orbit (Xj)

French Flag Colors

National Flag of France has 3 color stripes: red,white, blue = (r,w,b)
If we permute the 3 colors, how many flags are there ?

Eg. (w,b,r) = (r,b,w) since they are same if flip over the flag.
Eg. (b,b,b),(r,w,r)... repeat color allowed.
[Ans: 18]
Hint: Apply Group action on the Set (r,w,b)


French Flag:
Let Group G = {g1,g2 }
g1={e}= {(1)(2)(3)}
g2={(1 3)(2)}
|G|=order = 2
Apply Group Action on 3 colors set X {r,w,b}:
g1.X=e.{r,w,b} => Fix 3 stripes each with 3 colors
=> |Fix(g1)|= 3³ ways

(1 3)= Fix 1st & 3rd stripe as 1 stripe, x 3 colors => 3 ways
(2) => Fix 2nd stripe x 3 colors = 3 ways

g2.X=(1 3)(2).X=> 3x3
=> |Fix(g2)|= 3²

By Counting Theorem:
Total Flags = Orbit(X)
= ∑|Fix(g) | /|G|
= (3³ + 3² ) / 2
= 18



Exercise:
Chessboard (8x8=64) colored arbitrarily in black or white.
How many different patterns are there ?

[Consider 2 patterns as the same if a rotation (90, 180, 270 degrees) takes one to another ?]

Mystery of i

$latex \sqrt {-1} = i $ took 300 Years to be accepted.

1. 17AD - Cardano: (stole it from other) for cubic equation.

2. 18AD - Lebniz: i is like "Holy Spirit" between 'there' and 'not there'.

3. 19AD - Gauss: i as Geometry
Point (a,b) in plane RxR :
z=a+ib
a on x-axis, b on i-axis

4. 20AD - Hamilton:
i as ordered-pair (a,b) with unique operations:
(a, b)+(c,d)=(a+c, b+d)
(a, b).(c,d)=(ac-bd, bc+ad)

Langlands' Program

Langlands' Program:

Langland wrote to André Weil in 1967:
Analysis & Algebra linked up by L-function, which converts algebraic data from Galois theory into Analytic functions in complex numbers.

He goes beyong Modular Form to the Automorphic Forms (complex functions whose symmetries are described by larger matrices).

Key concepts on Algebra marry up with those from Analysis.

Elliptic Curve

Elliptic Curve (E):

$latex y^{2}= x^{3} + Ax + B$

1. Every E(Q) over Q is Modular

2. Δ = -16(4A³ + 27B²)

Modular Form

Modular Form (MF):
Is a function which takes Complex numbers from the upper half-plane as inputs and gives Complex numbers as outputs.

MF are notable for their high level of symmetry, determined not by a single number (2π for sine) but by 2x2 Matrices of Complex numbers.

Uses:
1. Proof of the FLT
2. Investigation of Monster Group.
3. Elliptic curve = MF
4. L-function provides dictionary for translating between Analysis and Number Theory.

Pedigree

Mathematician Pedigree: 名师出高徒
1. Littlewood-> Peter Swinnerton-Dyer
2. GH Hardy -> Ramajunian / Hua Luogeng
3. Richard -> Evariste Galois, Charles Hermite
4. Gauss -> Dedekind, Riemann
5. Dirichlet -> Dedekind, Riemann
6. Charles Hermite -> Lindermann
7. Hua Luogeng-> Chen Jin-run
8. SS Chern -> Yau ST
9. Camile Jordan -> Felix Klein, Sophus Lie

Princeton IAS

Princeton

Fine Hall = Institute of Adanced Study

"Raffiniert ist der Herr Gott, aber boshaft ist Er nicht."

The Lord God is subtle, but malicious he is not.

Group Representation

Group Representation
Faithful: Isomorphic (1 to 1)
Unfaithful: homomorphic (m to 1)

Group of Equilateral Triangle
Permute the 3 vertices.
Isomorphic to S3 (permute 1 2 3)
Faithful representation by matrices:
e = $latex
\begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
$

C₃= $latex
\begin{pmatrix}
-\frac {1}{2} & \frac{\sqrt{3}}{4} \\
-\frac{\sqrt{3}}{4} & -\frac {1}{2}
\end{pmatrix}
$

C₃² = $latex
\begin{pmatrix}
-\frac {1}{2} & -\frac{\sqrt{3}}{4} \\
\frac{\sqrt{3}}{4} & -\frac {1}{2}
\end{pmatrix}
$

σ₁ = $latex
\begin{pmatrix}
-1 & 0 \\
0 & 1
\end{pmatrix}
$

σ₂ = $latex
\begin{pmatrix}
\frac {1}{2} & \frac{\sqrt{3}}{4} \\
\frac{\sqrt{3}}{4} & -\frac {1}{2}
\end{pmatrix}
$


σ₃ = $latex
\begin{pmatrix}
\frac {1}{2} & -\frac{\sqrt{3}}{4} \\
-\frac{\sqrt{3}}{4} & -\frac {1}{2}
\end{pmatrix}
$

Group Theorems: Lagrange, Sylow, Cauchy

1. Lagrange Theorem:
Order of subgroup H divides order of Group G

Converse false:
having h | g does not imply there exists a subgroup H of order h.
Example: Z3 = {0,1,2} is not subgroup of Z6
although o(Z3)= 3 which divides o(Z6)= 6

However,
if h = p (prime number),
=>
2. Cauchy Theorem: if p | g
then G contains an element x (so a subgroup) of order p.
ie.
$latex x^{p} = e $ ∀x∈ G

3. Sylow Theorem :
for p prime,
if p^n | g
=> G has a subgroup H of order p^n:
$latex h= p^{n}$

Conclusion: h | g
Lagrange (h) => Sylow (h=p^n) => Cauchy (h= p, n=1)


Trick to Remember:

g = kh (god =kind holy)
=> h | g
g : order of group G
h : order of subgroup H of G
k : index

Note:
Prime order Group is cyclic
(Z/pZ, +) order p is cyclic & commutative.

Order 4: Z4 not isomorphic to Z2xZ2

Order 6: only Z6 isomorphic Z2xZ3.
Z6 non-commutative

S3 = {1 2 3} ≈ D3 Not Abelian
(1 2)(1 3) = (1 3 2)

(1 3)(1 2) =(1 2 3)

Lagrange: |G|=6
=> order of subgroups in G = 1,2,3,6
6= 2x3
Cauchy : 2|6, 3|6 (2,3 prime)
=> order of elements in G
= 2, 3

Win Win Problem

Yoyo lost $3, Pierre found it and returned to Yoyo.
Yoyo wanted to give all $3 to Pierre as gift, but Pierre refused.

Joe is the judge. How to solve it ?

Solution:
Joe donates $1 to make total $4,
divide between Yoyo and Pierre，

=> Yoyo and Pierre each gets $2
Also Yoyo, Pierre and Joe all lose $1.

Note: "三方各损一两" Win-Win Problem

Aristotle & Plato

Aristotle said about his teacher:
"Plato is dear to me, but dearer still is truth."

吾爱吾师，吾更愛真理！
子曰：当仁不让于师

Poor-Rich Divide: GINI

GINI
0.459 in 2012 (Singapore)
(without govt. help: 0.478)
=>
Income of household at top 10% was 9.14 times that of the bottom 10% (after tax and transfer by government, becomes 7.87x)

OECD Method 1: (0.457)
First adult in family:1 point
Each additional adult 0.5 point
Each child 0.3 point
GINI = household income / sum of points

OECD Method 2:
GINI = square root of family size = 0.435

Note: Worsening GINI
GINI for USA : 0.43 (1990) to 0.47 (2010)
GINI for China: 0.35 (1990) to 0.47 (2010)
GINI for Singapore: 0.43 (1990) to 0.46 (2010)

Einstein & Riemann

Albert Einstein owed much to Riemann's conception, saying that if he had not been acquainted with them
"I never would have been able to develop the theory of relativity."

Manifold

Manifold introduced by Riemann
Whereby the measurement of Length, angle, curvature could vary from point to point, unlike a sphere where the curvature is the same all over.

Manifold: 流形 [文天祥：正气歌]

无用之用

周培源教授： 数学是＂无用之用＂

"Useless" Mathematics is "Useful".

Axiom

Axiom (Greek): meant request. The reader is requested to accept the axioms unquestioningly as the rules of the game.

Euclid's "Element" built the whole Geometry with only 5 axioms.
The 5th axiom "Parallel line" was not challenged for 3,000 years until 19th CE Gauss & Riemann developed the Non-Euclidian Geometry.

Richard Dedekind

Julius Wilhelmina Richard Dedekind (6 Oct 1831 - 12 Feb 1916)

- Last student of Gauss at Göttingen
- Student and closed friend of Dirichlet who influenced his Mathematical education
- Introduced the word Field (Körper)
- Gave the first university course on Galois Theory
- Developed Real Number 'Dedekind Cut' in 1872
- Accomplished musician
- Never married, lived with his unmarried sister until death
- "Whatever provable should be proved."
- By 1858: still yet established ?
$latex \sqrt{2}.\sqrt{3} = \sqrt{2.3} $
- Gave strong support to Cantor on Infinite Set.

http://www-history.mcs.st-and.ac.uk/Biographies/Dedekind.html

CID Relation

C Continuous
I Integrable
D Differentiable

The CID Relation:




$latex D \to C, C \to I$
=>
$latex D \to I$

What it means a curve (function) is :
1. Continuous = not broken curve
2. Differentiable = no pointed 'V' or 'W' shape curve
3. Integrable: can compute the area under the (unbroken) curve.

Visualize e^iπ = -1

Visualize by mapping Exp e (red and pink arrows):

$latex e^{i\pi} = -1$

What is i^i

$Latex i^{i } = 0.207879576...$
$latex i = \sqrt{-1}$

If a is algebraic and b is algebraic but irrational then $latex a^b $ is transcendental. (Gelfond-Schneider Theorem)

Since i is algebraic but irrational, the theorem applies.

1. We know
$latex e^{ix}= \cos x + i \sin x$

Let $latex x = \pi/2 $

2. $latex e^{i \pi/2} = \cos \pi/2 + i \sin \pi/2 $

$latex \cos \pi/2 = \cos 90^\circ = 0 $

$latex \sin 90^\circ = 1 $
$latex i \sin 90^\circ = (i)*(1) = i $

3. Therefore
$latex e^{i\pi/2} = i$
4. Take the ith power of both sides, the right side being $latex i^i $ and the left side =
$latex (e^{i\pi/2})^{i}= e^{-\pi/2} $
5. Therefore
$latex i^{i} = e^{-\pi/2} = .207879576...$

Transcendental Number

Transcendental numbers: e, Π, L...

What about $Latex e^{e}, \pi^{\pi} ,\pi^{e} $ ?

Aleksander (Alexis) Osipovich Gelfond (1906-68):

Gelfond-Schneider Theorem

$Latex a^ b $ transcendental if
a is algebraic, not 0 or 1
b irrational algebraic number

Examples:
$latex \sqrt{6}^{\sqrt{5}}, 3^{\sqrt{7}}$
Hilbert Number: $latex 2^{\sqrt {2}}$ (Hilbert Problem proven by Gelfond}

Is log 2 transcendental ?
[log = logarithm Base 10]

Proof:
$latex 10^{log 2} = 2$

1) Sufficient to prove log 2 irrational
Assume log 2 rational
log 2= p/q, p and q integers
$latex 10^ {log 2} = 2 = 10^ {p/q}$
raise power q
$latex 2^{q} = 10^{p} = (2.5)^{p}$
$latex 2^{q} = 2^{p}.5^{p}$

Case 1: p>q
$latex 1 = 2^{p-q}.5^{p}$
=> False

Case 2: q>p
$latex 2^{q-p}= 5^{p}$
Left is even : $latex 2^{m} \text { = even} $
Right is odd: $latex 5^{n} \text {= ....5} $
=> False

Therefore p,q do not exist,
=> log 2 irrational

Reference: Top 15 Transcendental Numbers: http://sprott.physics.wisc.edu/pickover/trans.html

Chicken & Rabbits

This is an ancient Math Puzzle before Algebra was invented, so the ancient intelligent Chinese used this funny "Method of Elimination" below:

Puzzle: In a cage housing some chicken and rabbits. There are total of  35 heads, and 94 legs. Find how many chicken and rabbits ?

Solution:

Imagine letting each animal raises up 2 legs, then there would be 35x2 = 70 legs raised.

94 - 70 = 24 legs

Note that all chicken raising 2 legs would have fallen on ground (no leg how to stand ?), so the 24 remaining legs still standing on the ground MUST belong ONLY to the rabbits standing on 2 legs (the other 2 legs raised up)

24 / 2 = 12 rabbits (Ans)

35 - 12 = 23 chicken (Ans)

Prime Secret: ζ(s)

Riemann intuitively found the Zeta Function ζ(s), but couldn't prove it. Computer 'tested' it correct up to billion numbers.

$latex \zeta(s)=1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\dots$

Or equivalently (see note *)

$latex \frac {1}{\zeta(s)} =(1-\frac{1}{2^{s}})(1-\frac{1}{3^{s}})(1-\frac{1}{5^{s}})(1-\frac{1}{p^{s}})\dots$

ζ(1) = Harmonic series (Pythagorean music notes) -> diverge to infinity
(See note #)

ζ(2) = Π²/6 [Euler]

ζ(3) = not Rational number.

1. The Riemann Hypothesis:
All non-trivial zeros of the zeta function have real part one-half.

ie ζ(s)= 0 where s= ½ + bi

Trivial zeroes are s= {- even Z}:
s(-2) = 0 =s(-4) =s(-6) =s(-8)...

You might ask why Re(s)=1/2 has to do with Prime number ?

There is another Prime Number Theorem (PNT) conjectured by Gauss and proved by Hadamard and Poussin:

π(Ν) ~ N / log N
ε = π(Ν) - N / log N
The error ε hides in the Riemann Zeta Function's non-trivial zeroes, which all lie on the Critical line = 1/2 :

All non-trivial zeroes of ζ(s) are in Complex number between ]0,1[ along real line x=1/2

2. David Hilbert:

'If I were to awaken after 500 yrs, my 1st question would be: Has Riemann been proven?'

It will be proven in future by a young man. 'uncorrupted' by today's math.

Note (*):

$latex \zeta(s)=1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\dots = \sum \frac {1}{n^{s}}$ ...[1]

$latex \frac {1}{2^{s}}\zeta(s) =
\frac{1}{2^{s}}(1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\dots) $

$latex \frac {1}{2^{s}}\zeta(s) =
\frac {1}{2^{s}}+ \frac{1}{4^{s}} + \frac{1}{6^{s}} + \frac{1}{8^s} +\dots$ ... [2]

[1]-[2]:

$latex (1- \frac{1}{2^{s}})\zeta(s) = 1+ \frac{1}{3^{s}} + \frac{1}{5^{s}} + \dots + \frac{1}{p^{s}} +\dots $

$latex \text {Repeat with} (1-\frac{1}{3^s}) \text { both sides:} $

$latex (1- \frac{1}{3^{s}})(1- \frac{1}{2^{s}})\zeta(s) = 1+ \frac{1}{5^{s}} + \frac{1}{7^{s}} + \dots + \frac{1}{p^{s}} +\dots $

Finally,

$latex (1- \frac{1}{p^{s}}) \dots (1- \frac{1}{5^{s}})(1- \frac{1}{3^{s}})(1- \frac{1}{2^{s}})\zeta(s) = 1$

Or

$latex \zeta(s) = \prod \frac {1}
{1- \frac{1}{p^{s}}}= \sum \frac {1}{n^{s}}$

Note #:
$latex \zeta(s) = \prod \frac {1}
{1- \frac{1}{p^{s}}}= \sum \frac {1}{n^{s}}$

Let s=1
RHS: Harmonic series diverge to infinity
LHS:
$latex \prod \frac {1}{1- \frac{1}{p}}= \prod \frac{p}{p-1}$
Diverge to infinity => there are infinitely many primes p

[caption id="" align="alignright" width="300"] English: Zero-free region for the Riemann_zeta_function (Photo credit: Wikipedia)[/caption]

Artin Field Extension

Emile Artin's very unique book "Galois Theory" (1971) on "Finite Field Extension" interpreted by Vector Space.

Let H a Field with subfield G
F is G's subfield:
H ⊃ G ⊃ F

Example:
Let
F = Q = Rational Field
G = Q(√2) = Larger Extended Field Q with irrational root √2
H = Q(√2, √3) = Largest Extended Field Q with irrational roots (√2 & 3)

{1, √2} forms basis of Q(√2) over Q

{1, √3} basis of Q(√2, √3) over Q(√2)
[since √3 ≠ p+ q√2 , ∀p,q ∈ Q]

=> {1,√2, √3, √6} basis of Q(√2, √3) over Q
=> Q(√2, √3) is a 4-dimensional Vector Space over Q.

Isomorphism (≌)

Q(√2) ≌ Q[x] / {x² - 2}

Read as:

Q(√2) isomorphic to the quotient of the Polynomial ring Q[x] modulo the Principal Ideal {x² - 2}

Q[x] the Polynomial Ring

{x² - 2} is the Principal Ideal

Complex Number (C)

C = R[x] / {x² + 1}

R[x] the Polynomial Ring with coefficients in the Field R

{x² + 1} is the Principal Ideal

Questions:

Since R[x] / {x² + 1} is the Field C

Why below are not Fields ?
R[x] / {x³ + 1}
R[x] / {x^4 + 1}
C[x] / {x² + 1}

Hint: they are not irreducible in that particular Field, not a Principal Ideal.

Note: C[x] the Polynomial Ring with coefficients in the Field C

Google Symmetries

Google = 10 ^ 100

1000 = 10³ = 2³ x 5³
10^google = (10³)³³. 10
= (2³)³³ x (5³)³³. (2x5)
= 2¹ºº x 5¹ºº

=> 100 flips (2) , 100 rotations of pentagons (5) = Google symmetries.

Irrational 'e'

In secondary school, we know how to prove √2 is irrational, how about e ?

e= 1 + 1/1!  +1/2!  + 1/3!  + 1/4!  +...

Prove by Reductio ad absurdum (contradiction):

Assume e= p/q as rational
multiply both sides by q!
LHS: e. q!= (p/q) .q! = p.(q-1)!  => integer

RHS:  q!+q! + (3.4...q)+ (4.5...q) +...1 + 1/(q+1) +.... => fraction

Contradiction !

Therefore e is irrational.

Tie shoe lace in 1 sec

How to tie shoelace in 1 second:

http://www.youtube.com/watch?v=wMuNjnNyaiA

Origamatic: Fold T-Shirt

Origami + Mathematics = Origamatic
http://www.langorigami.com/science/math/math.php

Watch this video on useful trick to fold T-shirt:

http://www.youtube.com/watch?v=BAxhr0j0thY

Abelian Group

This interesting example is like solving simultaneous equations in Group, using only one Group property tool (Inverse => cancellation law)

Let Group G, ∀a,b ∈G,
for any 3 consecutive integers i,
$latex (a.b)^{i}= a^{i}.b^{i} $
Prove：G is abelian?
[Herstein: i, i+1,i+2]

Proof:
(a.b)ⁿ= aⁿ.bⁿ ...(1)
(a.b)ⁿ⁺¹= aⁿ⁺¹.bⁿ⁺¹ ...(2)
(a.b)ⁿ⁺² = aⁿ⁺².bⁿ⁺² ...(3)
Take inverse (1):
(a.b)⁻ⁿ = (aⁿ.bⁿ)⁻¹ = b⁻ⁿ.a⁻ⁿ ...(4)

Left * (4) to (2)
(a.b)ⁿ⁺¹(a.b)⁻ⁿ =ab

{Right *} (4) to (2):

ab = (aⁿ⁺¹.bⁿ⁺¹).(b⁻ⁿ.a⁻ⁿ) = aⁿ⁺¹.b.a⁻ⁿ
{Left *} a⁻¹
=> a⁻¹(ab) = b = (a⁻¹aⁿ⁺¹).b.a⁻ⁿ = aⁿ.ba⁻ⁿ
Right x aⁿ
=> b.aⁿ = aⁿ.b(a⁻ⁿaⁿ) = aⁿ.b ...(5)
Take inverse of (2):
(a.b)⁻ⁿ⁻¹= b⁻ⁿ⁻¹.a⁻ⁿ⁻¹ ...(6)
Right x (6) to (3)
(a.b) = aⁿ⁺².bⁿ⁺².(b⁻ⁿ⁻¹.a⁻ⁿ⁻¹)
ab = aⁿ⁺².b.a⁻ⁿ⁻¹ ...(7)
Right x aⁿ⁺¹
abaⁿ⁺¹ = aⁿ⁺².b
Left x a⁻¹
baⁿ⁺¹ = aⁿ⁺¹.b ...(8)
(b.aⁿ).a = (aⁿ.b).a from (5) b.aⁿ = aⁿ.b
aⁿ.b.a = aⁿ⁺¹.b
cancellation law:
=> ba= ab
=> G is abelian [QED]

Note:

Trick is inverse (1) then x to (2):

b.aⁿ = aⁿ.b ...(5)
Similarly inverse (2) then x (3):
baⁿ⁺¹ = aⁿ⁺¹.b ...(8)
Solve (5) & (8): by cancellation law
aⁿ.b.a = aⁿ⁺¹.b
=> ba= ab

Get Rich Rule 72

Rule 72: Compound Interest

Let P= Principal at interest rate i after n years:

Compound Interest Formula:

$latex P'=P.(1+i)^{n}$

$latex (1+i)^{n} = 1 + ni + \frac {1}{2} (ni)^{2} + \dots $

if ni=0.72 =72%

$latex (1+i)^{n} = 2$

=> P'=2 P

ie ni=0.72, P double

so if i = ROI (Buffett recommended) = 15% => n=72/15= 4.8 ~ 5 years

=> every 5 yrs x2

=> every 10 yrs x4

If you wish to double many times to get 1 million after 10 years, reverse the calculation:

Invest $ 1 m / 2 / 2 = $250,000 now

Buffett advised to put a  MoS (Margin of Safety=50%) to buffer mistake buys and have bigger ROI,

=> invest 250K - 50% = $125 K (now)

Spread it into a portfolio of max. 5 stocks at any time => S$25 K/ stock.

If the companies are good one, they will increase equity, or ultimately acquired, and you will be paid handsomely with higher stocks. eg. Google, DELL 10 years ago.

π in 1King 7:23

King Solomon - The Temple's Furnishing:

"He made the Sea of cast metal, circular in shape,
measuring 10 cubits from rim to rim and 5 cubits
high. It took a line of 30 cubits to measure around
it." - 1King 7: 23

[10 cubits = 15 feet,  30 cubits = 45 feet]

So mathematically by today's primary school geometry:

Circumference = 30 = 2 *π *r
Diameter = 10 = 2 * r


π = 30/10 = 3

Note: Solomon's period is equivalent to China's Zhou 周 dynasty of the famous Prince "Zhou Gong" 周公

German Terms

1. The electron orbits: first 4 orbits from atom

s, p, d, f

s = Sharfe (Sharp)

p =prinzipielle (principle)

d = diffusiv (diffuse)

f= fundamentale (fundamental)

2. eigen (special)

eigenvector

eigenvalue

eigenfunction

eigenfrequency

3. Math:

e = neutral element (I=Identity)

K=Korps (Fields)

Z = Zahl (Integer)

4. Physics:

F-center = Color Center  (F=Farbe=color)

Umklapp process = reverse process

Aufbau principle (quantum chemistry) = Building (bau) Up (Auf)  principle

Group is Symmetry

Landau's book "Symmetry" explains it as follow:

Automorphism = Congruence= 叠合 has
1). Proper 真叠合 (symmetry: left= left, right = right)
2). Improper 非真叠合 (non-symmetry: reflection: left changed to right, vice-versa).
Congruence => preserve size / length
=> Movement 运动 (translation 平移, rotation about O )
= Proper congruence (Symmetry)

In Space S, the Automorphism that preserves the structure of S forms a Group Aut(G).
=> Group Aut(G) describes the Symmetry of Space S.

Hence Group is the language to describe Symmetry.

 

Pawned Fields Medal

1936	Lars Valerian Ahlfors (Harvard University)

The first person in history to receive a Fields medal was a Finnish Ahlfors in 1936. At that time Finland was at war with England, no person could bring  more than 10 crowns out of Finland. Ahlfors was accepting a job offer from Harvard, so he pawned his Fields Medal for ticket to leave Finland, later got the medal back from the pawn shop.

[caption id="" align="alignright" width="300"] Lars Valerian Ahlfors (Photo credit: Wikipedia)[/caption]

Bumpy-free CD/DVD

When you drive on a bumpy road, your CD music plays normally and smoothly while your body bumps up and down in your car?

Why CD disk can be bumpy-free ?

Thanks to Galois, the 19-year-old French Math genius and tragic victim of the X Concours. Galois invented the whole Abstract Algebra with two revolutionary ideas: Group and Field.

Galois Group explains why symmetry is a Math structure which plays the role of Quintic equation with no radical solutions.

Galois Field is used in the cryptography coding for computers, digital music in CD / DVD.

--------------

Galois Field Theorem: the only one in Field Theory

There is EXACTLY ONE such system whenever the number of element is a power of any prime number p.

<=>

Fp contains EXACTLY 1 sub-Field with pⁿ elements. (n ≥1)

<=>

For p prime, n ≥1

|Fp | = pⁿ unique

Example:

p=2 n =1

=> Galois Field = {0,1} with pⁿ = 2 elements

Galois Field {0,1}
1+1 = 0 = 0+0
1+0 = 1 = 0 + 1

Reed-Solomon codes for CD, DVD

(1960) using binary Galois Field {0,1}

Error correcting codes based on algebraic properties of Polynomials, whose coefficients are taken from a Galois Field.

255 bytes block = 223 + 32

223 = encode the signal

32 bytes = parity codes

When the car bumps and so the CD player's laser beam on the CD disk, digital music code errors can be auto-corrected using Reed-Solomon code Algorithm, so you don't hear any jerking disruption.

Continuous Monster

Dirichlet Monster Function (reverse of Monster 3): $latex D(x) = \begin{cases} 1/b, & \text{if }x=a/b\in\mathbb{Q}\text{ rational} \\ 0, & \text{if }x\text{ irrational} \end{cases} $ Weirstrass proved by 'ε-δ' that D(x) is continuous at any irrational x. Proof: Take any irrational xo =√2 Let x= 13/10 (the smallest rational x=a/b nearest to xo) |x-xo|=|13/10 - √2|= 0.114 < δ Take δ =0.2 => |D(x) - D(xo)|= |D(13/10)-D(√2)| = 1/10 - 0 = 0.1 < ε=δ=0.2 For all ε, there is δ=ε such that |x-xo|< δ => |D(x) - D(xo)|< ε

Hence,
D(x) is continuous at irrational xo.

[QED]
---------------------------

Note:

Monster 3:

$latex
f(x) =
\begin{cases}
1, & \text{if }x \text{ is irrational} \\
0, & \text{if }x\text{ rational}
\end{cases}
$

Black-Scholes Financial Crisis

Black-Scholes Equation (1997 Nobel Economics)

Use: Pricing Derivatives (Options): calculate the value of an option before it matures.

1/2 (σS)².∂²V/∂S² + rS.∂V/∂S + (∂V/∂T - rV) = 0

Without last 2 terms=> heat equation !

Time T
Price S of the commodity
Price V of the derivative
Risk free interest r (govt bond)
Volatility = σ of the stock = standard deviation

Assumptions: (Arbitrage Pricing Theory)
No transaction costs
No limit on short-selling
Possible to borrow/lend at risk-free rate

Market prices behave like Brownian motion: constant in rate of drift and market volatility

Put option: right to sell at a specific time for an agreed price if you wish.
Call option: right to buy at a specific time for an agreed price if you wish.

One Black-Sholes formula each for Put and Call respectively.

Derivative was invented in 1900 by Mr. Bachelier, a French PhD student of Poincaré, the Mathematics of Financial prediction on stock market, based on Brownian Motion Theory of gas from Einstein. Poincaré was not impressed, gave the thesis only mention honorable (credit) , not trés honorable (distinction).

The problem of this Black-Sholes formula is the assumption being too 'academic', assume no financial 'Fat Tail', but sadly everybody believes in it and uses it blindly, which causes the recent Financial Crisis.

Self-Study Math Master

Hua Luogeng (华罗庚) urged using the daily 10-20 mins intervals while waiting for buses, queues, idle times, make it at least 1 hour a day to read Math books which you carry along with you.
Hua advised on speedy self-learning Math :
1) Choose the Best book on the Topic written by the Master (say, Abstract Algebra), read completely and do the exercises.
2) Read other reference books. Read only those new topics not covered in 1).
If not much new things, return them to bookshelf. This way speed up reading many books in short time.
3) Then read International renown Math Journals.
Beware 90% are copy-cats or rubbish by University lecturers to meet their yearly publishing quota. Only < 10% are masterpieces.
4) Pick one topic to do your independent research.
5) Discuss with friends with better knowledge in the field.
This way you can be a Master in the topic in 4 to 5 years.

Song of Simple Group

http://www.youtube.com/watch?v=lJhmJJ_q6Go&NR=1

The lyrics to the song can be found in "Contemporary Abstract Algebra" (Seventh Edition) by Joseph A. Gallian

(ISBN 978-0-547-16509-7) pages 422 & 423.

Think Abstractness

Why some students can learn abstract Math easily while most can't ?

Education Psychologists help to reduce abstraction level when learning
Abstract Math Concepts.

There are 3 abstract levels:

1) Quality of the relationship between the object of thought and the students:
The opposite of abstract is concrete.
Some students can relate the abstract math objects to concrete
familiar concepts, the closer the relationship the more concrete the
objects are.
eg. Relate abstract 'Ring' to familiar concrete object 'Integer Z'.

2). Process conception and Object conception:
The mental process that leads from process conception to object
conception is called "Reflective Abstraction". (Piaget).

eg. Quotient Group = G/H = {Hg = gH | g ∈ G}
Process Conception (Canonical procedure ): take all elements from H,
multiply them on the right with some element from G. Similarly for
multiplication on the left.
Objection Conception: Partition called Quotient Group.

3). Degree of Complexity of the concept:
eg. Group of prime p, called p-iadic group is more complex than any group.

Epidemic Equation

Let p the portion of  population infected by the contagious disease
(like SARS) at time t.

The rate of infection is known empirically and historically
proportional to p(t).

$latex \frac {dp}{dt}=k.p$
where k is constant.

Solving the differential equation by A-level math,
$latex p=p_0.e^{kt}$
where $latex p_0$  is p at t=0 (initial infected population).
=> the infection growth rate is exponential, and multiplied by a factor $latex p_0$.

That is why there is a need to contain $latex p_0$ at the beginning of the epidemic by:
1. Isolate all $latex p_0$;
2. Destroy all dead $latex p_0$ by burning, etc.
3. For flu (H1N1), put on masks by the sick...

Math saves our life !

By taking measure to reduce $latex p_0 $ to very small population, say, $latex p_0 \to 0$

$latex p=p_0.e^{kt} \to 0$

The epidemic will die off over time, although there is still no
medical cure for it (eg. SARS).

Landau's Beautiful Proofs

Landau's beautiful proofs:
1= cos 0 = cos (x-x)

Opening cos (x-x):
1 = cos x.cos (-x) - sin x.sin (-x)
=> 1= cos² x + sin² x
[QED]

Let cos x= b/c, sin x = a/c
1= (b/c)² + (a/c)²
c² = b² + a²
=> Pythagoras Theorem
[QED]

Landau (1877-1938) was the successor of Minkowski at the Gottingen University (Math) before WW II.

Graduate Student Tips

Graduate student tips:
1. Choose the right advisor.

2. Read a lot of papers, work out the details, learn the body of math. Learn as much as possible of applied math, physics & biology.

3. Identify a field of interest and learn it in depth. Very important to choose the questions to study. You should have the big questions in mind. The smaller questions you work on should have a bearing on the big ones.
- Source: "NUS IMS Visiting Professors"  (Pg 273) Slaman & Woodin

Exchange Currency

Which bank is better for you as the traveller to:

1. exchange home currency H$ to a foreign currency F$, or
2. exchange F$ back to H$ ?

1) Bank 1 rate:
Sell 1.6021 (1.5821)
Buy 1.4954 (1.5039)

2) Bank  2 rate:
Sell 1.5642 (1.5525)
Buy 1.5218 (1.5234)

Note:
Transact in home currency H$,
Sell: the shop sells F$ to you
Buy: the shop buys F$ from you
In Bracket means (Travel Cheque exchange rate)

Answer:

Calculate diff% = (sell - buy) / avg  %
avg= (sell + buy)/2

If diff% < 3% => bank rate reasonable.

1/2 diff%  = means one-way transaction fee
(either buy or sell) that you will lose in exchange

Bank 1:  diff%=6.88% > 3%
=> 1/2.diff% = 3.44%
Every $100 (from H to F or vice versa) lose 3.44.
If buy F then convert back to H immediately  (double conversions) => lose 3.44 x 2 = $6.88.

Bank2: diff% = 2.78% < 3%
=> good exchange rate.

Note: Traveller check Buy rate always higher than cash Buy rate,
because bank does not have to hold the actual cash. It is costly for
the bank to ship out the cash when a large amount has been collected.

Euler on Math Education

Euler was invited by Peter I of Russia in 1727 to work in the
Petersburg Academy of Sciences. He introduced the fundamental math
disciplines in school math education:
1. Arithmetic
2. Geometry
3. Trigonometry
4. Algebra
these 4 are taught as separate and specific subjects, versus 19 duplicated disciplines in Europe.

Euler influenced not only in Russia schools, but in schools worldwide today.

Source: Russian Mathematics Education
Vol 1: History and world significance
Vol 2: Programs and practices
(Publisher: World Scientific)

Russian Math VZMSh

Israel Gelfand, the student of Kolmogorov (the Russian equivalent of
Gauss), created in 1964 the famous VZMSh, a national Math Correspondence School.

He wrote: "4 important traits which are common to Math, Music, and
other arts and sciences:
1st Beauty
2nd Simplicity
3rd Precision
4th Crazy ideas.
"
The Russian mathematicians also built special Math-Physics schools:
Moscow School #7, #2, #57 (one of the best high school in the world, www.sch57.msk.ru) Leningrad Schools #30, #38, #239 (Perelman studied here)

Puzzle: Switches and Bulbs

Solve this real-life problem:
Two totally sealed rooms with no windows, each with a single door.

Room A: 3 light bulbs
Room B: 3 switches, each switch corresponds to one bulb in Room A.

Rule: you are allowed only 1 visit to each Room (after which the door is shut).

Identify:  which switch corresponds to which bulb?

Hint: no mathematical solution, but soluble by physical approach.

[By Arthur Ekert - Quantum Computer]

Solution:

This method below is heuristic, or fuzzy logic, which is impossible to
solve with discrete deterministic math logic:

1. Enter room B (with switches):
Switch on any switch (X) for 5 mins then switch it off,
then switch on another switch(Y).
The 3rd switch (Z) untouched.

2. Enter room A (with bulbs):
The bulb with light on => switch Y
The bulb is still warm but no light => switch X
The cold bulb, no light => switch Z

Abstract Algebra compulsory

Prof S.S. Chern 陈省身 (the “Gauss No. 2" in Differential Geometry) retired from Berkeley University , he went back to his Chinese Alma mata Nankai University 南开大学 in Tianjing city. (Nankai produced the first prime minister Zhou Enlai 周恩来 and the recent Prime Minister Wen Jiabao 溫家宝).

Knowing the important role of Abstract Algebra in linking all branches of Math and all sciences, Prof Chern  made "Abstract Algebra" compulsory with effect 2001 for all students of Maths, Science, Engineering, IT, Finance:

19 weeks of 3 hrs per week = 57 hrs.

Syllabus:

Group, Ring, Field, Galois Theory.

Note: The French Classe Preparatoire for Grande Ecoles already implemented Abstract Algebra for more than 100 years.

Shimura Memoire on André Weil

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 to throw away your wrong idea.”

2. According to him, one of the best way to learn French or any foreign language was to see the same movie in that language again and again, staying in the same seat in the same movie theatre.

3. A French gentleman’s ideal is to have three concurrent loves: the first one, whom he cares about at present; the second, a potential one, whom he has his eye on with the hope that she will eventually be his principal love; the third, the past one, with whom he hasn’t completely cut off his relations. Then he observed: “It’s a good idea for a mathematician to have three mathematical loves in the same sense.”

4. As to Fields medals, he said: “It’s a kind of lottery. There are so many eligible candidates, and the whole selection process is a matter of chance. Therefore the prize could be given to any of them as in a lottery.”

5. He used to say that a good mathematician must have two good ideas. “It is possible for someone to have a really good idea, but it may be just a fluke. Once the person has a second good idea, then there is a good chance for him to develop into a better mathematician.”

6. In the summer of 1970 after the Nice Congress, I was talking with him somewhere in the Institute about French mathematicians. He observed that there were three young mathematicians in Paris who started brilliantly, and so there were high expectations for them. He mentioned three well-known names and said, “What happened to them? They utterly failed to produce anything great.” After around 1975 he expressed, more than once, his pessimistic view that French mathematics had been declining for some time.

{Note: This recorded memoire of Shimura made the French very unhappy, for which Shimura refused to delete it from the book }

7. Weil told me several anecdotes about Hardy. “Hardy’s opinion that mathematics is a young man’s game is nonsense,” Weil said.

8. When I prodded the guests to tell their ambitions in their next lives... “I want to be a Chinese scholar studying Chinese poems,” said Weil. After visiting China twice, he had been reading English translations of Chinese standard literature like ＂The Dream of the Red Chamber＂(红楼梦).

{Note: Weil met Hua LuoGeng 华罗庚 and remarked if every Chinese is like Hua, very soon in future the Westerners will have to learn Math in Chinese}

9. Weil said, "I would like to see the Riemann hypothesis settled before I die, but that is unlikely.” Weil died at 90+ in 1996.

Modelling sans Algebra

Singapore PSLE Modelling Math (for 11 year-old pupils)

Problem:

1) 20% of John saving is $120 more than 30% of Brian savings.

2) After John spent 5/6 of his saving and Brian spent 1/2 of his saving ,
John has $20 more than Brian.

Find the saving of John at first ?

Solution:

(Technique: work backwards from the End.)

Before 2nd condition:

J: 60 units+ $120

[1][2].....[9][10] [$20]

[1][2].....[9][10] [$20]

[1][2].....[9][10] [$20]

[1][2].....[9][10] [$20]

[1][2].....[9][10] [$20]

[1][2].....[9][10] [$20]

B: 20 units

[1][2].....[9][10]

[1][2].....[9][10]

After 2nd condition:

J: [1][2].....[9][10] [$20]

B: [1][2].....[9][10]

20% of J

= 0.2 x 60 units + 0.2 x $120

= 12 units + $24

30% of B

= 0.3 x 20 units

= 6 units

From 1st condition:

(12 units+$24)- 6 units

->$120

6 units +$24 ->$120

6 units ->$96

1 unit ->$16

J: 60 units+ $120

J= 60x$16+ $120= $1080

[QED]

Number Theory is Discrete

Why Number Theory is called Discrete Math ?

because whole Number N, generator is 1, so the gap increment is 1 => discrete.

Opposite is Analysis (or Calculus), small increments ε,δ between [0,1]=> continuous, not discrete.

New branch of "Analytical Number Theory": to prove Riemann Hypothesis.

Virus Magic Number

Magic Number in Virus Structure

(For the future Medicine Nobel Prize wannabe)

If you inject extra protein unit to the virus,  fool its Magic Number structure, wouldn't the virus collapse and killed by itself ?

Magic Number limitation for Virus:

Magic numbers: 12, 32, 42, 72,92,122,132,162,192,212,252, 272, 362...

These Magic numbers play a limiting factor to the structure of viruses (which take the symmetric Platonic solid forms of dodecahedral or Icosahedral).

A similar role the Fibonacci number plays in the plant structure.

芜菁黄嵌纹病毒:  32 units of proteins

Human Wart virus (人类多瘤性病毒)，BK，Rabbit Papillioma (兔子乳头瘤病毒) 72

REO 92

Herpes Simplex Virus 单纯泡疹(Cold Sore virus 口唇泡疹)162

Chicken Adenovirus 鸡腺病毒252

Infectious Canine hepatitis virus 犬类传染性肝炎病毒362

Magic Sequence : $latex 10.n^{2} + 2$

https://oeis.org/search?q=12%2C32%2C42%2C72%2C92&language=english&go=Search

12, 32, 42, 72, 92, 122, 132, 162, 192, 212, 252, 272, 282, 312, 362, 372, 392, 432, 482, 492, 522, 572, 612, 632, 642, 672, 732, 752, 762, 792, 812, 842, 912, 932, 972, 1002, 1032, 1082, 1092, 1112, 1122, 1172, 1212, 1242, 1272, 1292, 1332, 1392, 1442, 1472 (list; graph; refs; listen; history; internal format)

OFFSET

1,1

COMMENTS

Refers to the capsomere count in virus architectural structures. Since the Loeschian numbers (A003136) of the form x^2 + xy + y^2 reduces to squares (A000290) when either x or y equals 0,a(n) contains A005901which is 10*n^2 + 2.

REFERENCES

I. Stewart, "Game, Set and Math", Chapter 6, Table 6.1 pp. 81 Penguin London 1991.

LINKS

K. Urner, Synergetica:A case of Deja Vu

FORMULA

a(n)=10*A003136(n) + 2.

KEYWORD

nonn

AUTHOR

Lekraj Beedassy (blekraj(AT)yahoo.com), Oct 30 2003

Intuitive Math

Intuitive Math: (By observation)

Let Set I = {(x,y)| x,y∈ R}

Set O = {(x,y)| (y-3 / x-2) = 1}

Set E = {(x,y)| y ≠ x+1}

Let A = O U E (union of O and E)

Find: Ā ?

Solution:

By Intuition:

Let I be a blackboard

O be the straight line (x≠2, y≠3):

y-3 / x-2 = 1

y -3 = x -2

y= x+1 exclude point (2,3)

E the straight line y=x+1

Ō = remove the straight line but keep (2,3)

Ē = remove the straight line

Ā = Ō ∩ Ē  = {(2,3)}

[QED]

Verify:

O={(x,y) | x≠2, y≠3,  y= x+1}

=> Ō= {(x,y) | x=2,y=3}

Ē={(x,y)| y= x+1}

Ā= Ō ∩ Ē  = {(2,3)}    [QED]

Klein's Geometry in Group

This is the "New Geometry" introduced by Klein 200 years ago in his Erlangan Program (his PhD Thesis).

Rigid Motion is defined by 3 components: Translation, Rotation and Reflection.

If fixed at one point (origin), there is no translation, only Rotation ρ(θ) and Reflection r(θ) are possible around that fixed point.

We can prove r(θ) and ρ(θ) form a Group O2, namely Orthogonal Group with this property:

$latex A^{T}. A = A. A^{T} = I $

where A can be any of the 2 matrices represented by ρ(θ)

or r(θ),

$latex A^{T} $ is the transpose of A (columns => rows, rows => columns).

1. Rotation

ρ(θ)=

(cos θ  -sin θ)

(sin θ   cos θ)

2. Reflection

r(θ) =

(cos θ   sin θ)

(sin θ   -cos θ)

when θ =0,

r0 =

(1 0)

(0 -1)

=> r(θ) = ρ(θ).r0

Change of Reference Axis:

 Make a shift from fixed origin A to another fixed original A' by a translation t(α), the first Orthogonal Group O at A and the second Orthogonal Group O' at A' are related by:

O' = t(α).O.t^-1(α)
ρ'(θ) = t(α).ρ(θ).t^-1(α)
r'(θ) = t(α).r(θ).t^-1(α)

Note: this looks analogous to Conjugate groups (Normal Subgroups).

Einstein Relativity interpreted by Rigid Motion (M4).

If first origin A is the Earth, second origin A' is the spaceship traveling at speed of light, ie t(α) = c

O' = t(α).O.t^-1(α) ; O & O' ∈ O4

<=> O'.t(α) = t(α).O

Why French excel in math ?

Mathematics and quantitative finance, France

Since 1990, there have been 22 winners of the Fields Medal, widely regarded as the Nobel Prize of mathematics. Thirteen came from just two countries, Russia and France. Russia has more winners (seven), but more than twice the population, so the honours go to France, with six winners.

Cédric Villani, the 2010 Fields Medallist, cited national character. “Maths is an abstract way of looking at the world, which fits well with the French mentality. We apply algebra to everything.” Elite institutions help too. France’s brightest school leavers progress to the grandes écoles, which traditionally educate top scientists, administrators and presidents. For maths, you want Monsieur Villani’s alma mater, the École Normale Supérieure (ENS). All 10 French Fields Medallists learnt there. At ENS, no teacher can stay longer than 10 years. Instead of ancient dons, students get tutors at the forefront of mathematics. Many try, but only 40 mathematicians a year enter the ENS.

The French have applied their maths genius to the money markets too. The Financial Times business schools rankings suggest France leads the world in producing “financial engineering” experts, with six institutions in the top 10 masters courses in finance. France can thus claim to dominate quantitative finance, the highly mathematical specialism involved in about half of all financial trades.

They should thank Michel Crouhy. In 1986, at the École des Hautes Études Commerciales (EHESS) in Paris, he devised the world’s first masters course in financial engineering. “The business school students didn’t have good enough maths, so I said ‘Let’s take only maths graduates, engineers. I won’t have to spend forever explaining the equations.’ It worked; the EHESS still offers the world’s best finance masters course, according to the Financial Times.

“Americans told me they wanted to start a course like ours but they weren’t allowed,” says Crouhy. “Because US MBA programmes were so strong, the universities worried a finance masters would compete with their MBA and destroy the MBA’s franchise.” America’s hesitation seems to have cost them.
--

From Durian to Group Theory

Durian & Group

The Nature applies Group Theory to the King of fruits : Durian.

Look at the kernels, there are more than one, each kernel partitions the Durian Group into several similar sections (which you can pull them apart ).

Those durians which have no kernel (jiu-jee) but meat are excellent - they are SIMPLE.

Eating one kernel (Normal Subgroup) is enough to know whether the Durian (Group) is D24 or D18 type.

Bon appétit !

Knowing the kernel 核of a fruit will allow biologists to understand the whole fruit.

In Group, a kernel of group homomorphism is a Normal subgroup, hence will let us know the whole group.

Normal subgroup is the important essence revealing the whole group.

http://www.rowan.edu/open/depts/math/howe/BookChapters/Normal%20Subgroups.pdf

First, you must realize what a Group is? Group is a set with an operation (Transformation) acting on its elements such that

"CAN I" -

C: closed

A: Associative

N: Neutral (Identity)

I: Inverse

Within a Group G, it has some subgroups other than two trivial subgroups (Identity and itself G). These subgroups obey CANI within itself like G.

Among the subgroups, there is one type called "invariant" (Galois's discovered it Invariant, or Conjugate subgroup, Americans call it Normal subgroup), which keeps the "symmetry" (meaning, no change after transformation, eg. rotate a circle around its center, the circle is not changed by the Rotation).

The Normal subgroups N reveal the "symmetry" of the Group - that is what Group is for studying Symmetry, so the importance of knowing N.

With the Normal Subgroup found, we can do the followings:

1) Partition the entire group G into Quotient Group (G/N), a structural category or template of G.

2) When compare two Groups G and G' by a mapping f (called Group Homomorphism, ie see if they are 'similar'),

f : (G,e) -> (G',e')

Group Theorem tells us the Kernel of G (Ker f) is a Normal Subgroup of G

ie ∀x ∈ Kerf, f(x) = e'

3) Normal subgroup can have a smaller Normal subgroup, which in turn have its smaller Normal Subgroup, and so on.

When there is no more Normal Subgroup (Simple Group), then the Group is not Solvable (or Soluable) ie no Solution. This was Galois proof of Quintic Equations have no radical Solution because Alternating Group A5 is Simple Group.

Game of Go

Chinese invented Go (圍棋 \weiqi) since antiquity. The Japanese called it 碁 (Go) and introduced it to the West.

Go is the game of wisdom of Universe and Life.

There are 19 x 19 grids = 361
Each dot can have 3 ways= black, white or empty
So total combinations = 3^361

Take a look at this number: 3^361
is the approximate between 3^360 and 3^365

In a Lunar calendar a year = 360 days.
In Solar calendar a year = 365 days

3 represents the Trinity: 天地人 (God, Earth, Man)
(In Christianity : God, Son and Holy Spirit)

Elegant Proof: Fermat Little Theorem

Let’s construct a necklace with p (p being prime) beads chosen from m distinct colors.
There will be m^p permutations of necklaces
less:
1. m permutations of necklaces with beads of same color;
2. Joining the 2 ends of the necklace to form a loop. For prime p beads, there will be p cyclic permutations of beads which are the same.

Total distinct necklaces = (m^p – m) / p  …[*]
which is an integer, ie p divides (m^p – m)
m^p - m ≡ 0 (mod p)
or m^p  ≡ m (mod p) if p prime  ∎

From [*],  m.[m^(p-1) – 1] / p
If m and p are relatively prime, i.e. (m, p) = 1
then p divides (m^(p-1) – 1),
m^(p-1)  – 1≡ 0 (mod p)
or
m^(p-1) ≡ 1 (mod p)  if  p prime & (m, p) = 1 ∎

"Turn-off" School Math

"...There's a long history of high caliber mathematicians finding their experiences with school mathematics alienating or irrelevant. "
Read here:
http://lesswrong.com/r/discussion/lw/2uz/fields_medalists_on_school_mathematics/

In Récoltes et Semailles Fields Medalist Alexander Grothendieck describes an experience of the type that Alain Connes mentions:

I can still recall the first "mathematics essay" (math test, or Composition Mathématique) , and that the teacher gave it a bad mark. It was to be a proof of "three cases in which triangles were congruent." My proof wasn't the official one in the textbook he followed religiously. All the same, I already knew that my proof was neither more nor less convincing than the one in the book, and that it was in accord with the traditional spirit of "gliding this figure over that one." It was self-evident that this man was unable or unwilling to think for himself in judging the worth of a train of reasoning. He needed to lean on some authority, that of a book which he held in his hand. It must have made quite an impression on me that I can now recall it so clearly.

Erdös Number

Erdös Number:
0: Erdös himself
1: direct collaborator with Erdös
2: collaborate with Erdös's collaborator
...
If you don't have a finite Erdös number, then your Erdös number is infinity.

Notes:
Einstein's Erdös Number is 2 through his assistant who worked with Erdös.

My Erdös Number is 4. My collaborator's lecturer whose professor worked with Erdös.

PISA Test 2009

Programme for International Student Assessment (PISA) for 15-year-old students in 60 countries.

The top 5 countries are Shanghai, Korea, Finland, HK, Singapore.

Shanghai scored 1st in all categories: Reading, Math, and Science. Singapore scored 5th, 2nd and 4th, respectively, in these categories.

Zooming in the elite students % in the test population (5000 students in each country representatives), Singapore elites are not far off (within 1%) than the Shanghai elites in Reading and Science, but lose out in Math with a big gap of 11% (26.6% -15.6%).
That means Singapore top Math students in Maths are far off than Chinese top Math students.

Why ?

Dr. James Li from Fudan University - its form teacher of the 1st batch gifted class - compares the Math education syllabi in Primary schools (P1 to P6), which build the Math foundation of the Shanghai and Singapore students.

The good points of the Chinese Math syllabus (and the lack of in Singapore's):

1. Chinese P3 uses calculator (P5 in Singapore);
2. P4 learn rules of operation like commutative, associative, distributive laws (lack of in Singapore).
3. Geometry: emphasis on drawing and mapping (inadequate in Singapore Geometry class)
4. P5 learn Algebra & equation (P6 only algebra but no equation).
5. Basic (non-competition) Math Olympiad skill selectively included in Chinese textbooks (lack of in Singapore)
6. Incorporate Mathematician stories like 祖冲之 in being the first person in the world to calculate pi up to 7 decimals. ( lack of in Singapore).
7. Incorporate computing tools and games in Math teaching materials. (Lack of in Singapore).

Tortoises & Hares

"... the most profound contributions to mathematics are often made by tortoises rather than hares. "

http://lesswrong.com/lw/2v1/great_mathematicians_on_math_competitions_and/

Yijing & Orchestra Conductor

Problem:
The Orchestra conductor has been asked to conduct a series of concert for 2 consecutive months.
The Orchestra has to play few pieces of classical music with the following conditions requested by the organising committee:

1) Play at least 1 piece of music a day;
2) Avoid playing exactly the same music in 2 consecutive days (although partially is allowed)
3) Due to resource constraint, it is impossible to play all pieces of music within a single day.

Question:
At least how many (N) pieces of music the conductor has to prepare in order to fulfill these conditions?

Solution:
Using Yijing （易经）notation:
* a piece of music being played in a day is denoted by
__ (solid line);
while not being played is denoted by
- - (broken line)
* 2 consecutive months in any year have maximum 62 (= 31+31) days.
* A bagua (8 diagramme 八卦) has 64 gua（卦), starting from 6 solid lines stacked on each other (Qian 乾gua) to 6 broken lines stacked on each other (Kun 坤gua), each line (broken or solid line) is called a Yao (爻). 6 Yao makes a gua.
There are total of 64 ($latex = 2^6 $) gua.
* Condition 1) eliminates Kun gua (all broken lines, ie all N pieces unplayed) since at least 1 piece of music to be played in a day.
* condition 3) eliminates Qian gua (all solid lines, sice all pieces can't be played in a single day.
* 64 gua minus 2 above-eliminations left 62 gua.
* these 62 gua have mixture of 6 solid/broken lines, representing 6 played/unplayed combination pieces of
music, with each gua being unique.

=> Answer: The conductor needs to prepare MINIMUM 6 pieces of music.

Google PageRank Search

Google: "A page is good insofar as good pages link to it"
=> PageRank invented by Larry Page while a graduate Math student in Stanford University, under the supervision of Prof Tony Chan (now the President and Professor of Mathematics & Computer Science and Engineering of Hong Kong University of Science and Technology).

Below is a simplified 3 web pages of PageRanks x, y, z respectively.

x page receives link from z :
(x=z)

y page receives 1/2 link from x:
(y= 1/2 x)

z page receives 1/2 link from x and 1 link from y:
(z = 1/2 x + y)

All PageRank scores = 1 (Conservation Law of PageRank):
(x + y + z = 1)

Solve PageRank Simultaneous Equation:
x = z
y = 1/2 x
z = 1/2 x + y
x + y + z = 1

=> x = 2y = z
=> x = 2/5, y = 1/5, z = 2/5

Visually from below, you can see x and z pages have more links than y.

So the search result should return x,z pages in front of y page.

Using Linear Algebra to compute the eigenvalues and the eigenvectors of large matrix of billion variables, Google applies mathematics and the power of clustering 50,000 cheap CPU to search the Internet, making billion dollars.

Computers are powerful only when they are 'told' with great algorithm, which is based on Math. The other example is RSA Encryption with big prime number factorization.

Probability by 2 Great Friends

Today Probability is a "money" Math, used in Actuarial Science, Derivatives (Options) in Black-Scholes Formula.

In the beginning it was "A Priori" Probability by Pascal (1623-1662), then Fermat (1601-1665) invented today's "A Posteriori" Probability.

"A Priori" assumes every thing is naturally "like that": eg. Each coin has 1/2 chance for head, 1/2 for tail. Each dice has 1/6 equal chance for each face (1-6).

"A Posteriori" by Fermat, then later the exile Protestant French mathematician De Moivre (who discovered Normal Distribution), is based on observation of "already happened" statistic data.

Cardano (1501-1576) born 150 years earlier than Pascal and Fermat, himself a weird genius in Medicine, Math and an addictive gambler, found the rule of + and x for chances (he did not know the name 'Probability' then ):

Addition + Rule: throw a dice, chance to get a "1 and 2" faces:
1/6 +1/6 = 2/6 = 1/3
(Correct: 1 & 2 out of other six faces)

Multiplication x Rule: throw two dices, chance to get a "1 followed by a 2 " faces : 1/6 x 1/6 = 1/36
(Assume 1st throw does not affect or influence the 2nd throw: independent events)


Pascal never met Fermat personally, only through correspondences (like emails today), but Pascal regarded with respect Fermat as superior in Math than himself.

Fermat and Descartes were not so. Descartes openly criticized Fermat as a second class mathematician. Both independently discovered Analytical Geometry, but Descartes scored the credit in 'Cartesian' coordinates.

Fermat never published any books in Math. As a successful judge in Toulous, he spent his free time as an amateur mathematician, especially in Number Theory. He showed his Math discoveries to friends in letters but never provided proofs. Hence the Fermat's Last Theorem made the world mathematicians after him (Gauss, Euler, Kummer, Sophie Germaine, Andrew Wiles...) busy for 380 years until 1994.

Fermat died 3 years after Pascal. Another Modern Mathematics was being born - Calculus - in UK by Newton and Germany by Leibniz. Probability was put in the back seat over-taken by Calculus.