Zipf’s Law in Linguistic


In English, 3 most common words:

  1. the” : occurring 7% of the time;
  2. of” : 3.5% = 7/2
  3. and” : 2.8% ~ 7/3

=> “the” is 2x occurs more often than the 2ndof“, 3x than the 3rdand” …

Zipf’s Law : the frequency of the nth ranked word is proportional to 1/n.



Blockchains and Application in Bitcoins

Encryption & Decryption: ECC (Elliptic Curve Cryptography):

Sending End: Encryption

1) SHA algorithm generates “Digital Signature” ;

2) Generate random “Private Key”.


3) ECC encrypts the text with “Private Key”;

4) From the Private Key generates a “Public Key”;

5) Send out the “original message” and the “Public Key” with the “encrypted message” from 3);

Receiving End: Decryption

6) ECC with Public Key generates Digital Signature 1 (S1);

7) Use SHA algorithm on the original message generates Digital Signature 2 (S2);

8) If S1 = S2, then accept transaction, otherwise reject.

Abstract “Nonsenses” in Abstract Math make “Sense”

After 40 years of learning Abstract Algebra (aka Modern Math yet it is a 200-year-old Math since 19CE Galois invented Group Theory), through the axioms and theorems in math textbooks and lectures, then there is an Eureka “AHA!” revelation when one studies later the “Category Theory” (aka “Abstract Nonsense”) invented only in 1950s by 2 Harvard professors.

A good Abstract Math teacher is best to be a “non-mathematician” , who would be able to use ordinary common-sense concrete examples to explain the abstract concepts: …

Let me explain my points with the 4 Pillars of Abstract Algebra :

\boxed {\text {(1) Field (2) Ring (3) Group (4) Vector Space}}

Note: the above “1-2-3 & 4″ sequence is a natural intuitive learning sequence, but the didactical / pedagogical sequence is “3-2-1 & 4″, that explains why most students could not grasp the philosophical essence of Abstract Algebra, other than the “technical” axioms & theorems.

If a number system (Calculator arithmetic) has 4 operations (+ – * ÷ ), then it is a “Field” (域) – Eg. Real, Complex, Z/pZ (Integer mod Prime)…

If a number system with +, – and * (but no ÷), then it is a “Ring” (环).
eg1. Clock arithmetic {1,2, 3,…,12} = Z/12 (note: 12 is non-prime). [Note: the clock shape is like a ring, hence the German called this Clock number a “Ring”.]
eg2: Matrix (can’t ÷ matrices)
eg3. Polynomial is a ring (can’t ÷ 0 which is also a polynomial).

If a system (G) with 1 operation (○) and a set of elements {x y z w …} that is “closed” (kaki-lang 自己人, any 2 elements x ○ y = z still stay inside G ) , associative (ie bracket orderless) :(xy)z = x(yz), a neutral element (e) s.t. x+e = x = e+x, and inversible (x^{-1}, y^{-1} … still inside G), then G is a Group.
eg. {Integer, +}: 2’s inverse (-2), neutral 0, (2+3)+4=2+(3+4)
eg2. Triangle rotation 120 degree & flip about 3 inner axes.

If a non-empty system V ={v u w z …} that is “closed”if any of its 2 elements (called vectors v, u) v + u = w still in V,
AND if any vector multiply it by a scalar “λ” s.t. “λv” still in V, then V is a Vector Space (向量空间)。
eg1. Matrix (Vector) Space
eg2. Function (Vector) Space
eg3. Polynomial (Vector) Space

Summarise the above 4 or more systems into 1 Big System called “Category” (C) 范畴, then study relation (arrow or morphism) between f: C1 -> C2, this is “Category Theory“.

In any number system (aka algebraic structure), you can find the “Yin / Yang” (阴阳) duality : eg. “Algebra” [#] / “Co-Algebra”, Homology (同调) / Co-Homology (上同调)… if we find it difficult to solve a problem in the “Yang”-aspect. eg. In “Algebraic Topology”: “Homology” (ie “Holes”) with only “+” operation, then we could study its “Yin”-aspect Co-Homology in Ring structure, ie with the more powerful “*” multiplicative operation.

Note [#]: “Algebra” (an American invented structure) is a “Vector Space” plus multiplication between vectors. (Analogy in Physics : Cross Product of vectors).


Bill Gates Returns to Harvard to Talk : Math55

Bill Gates, a top Math student at Harvard entrance exams, recalled his first year Harvard “Math55” Course (Advanced Calculus & Linear Algebra) – the toughest at his time because 4 years of Math coursewares condensed into 1 year (2 semesters) !

Note: Harvard “Math55” is even tougher than the “notorious” French Classe Préparatoire, which is a 3-year Math undergraduate courseware squeezed in 2 years : 1st year (code-name “un-demi” or “1/2”) Mathématiques Supérieures; 2nd year (“trois-demi” or “3/2”) Mathématiques Spéciales.

Math55 Syllabus:
Though Math 55 bore the official title “Honors Advanced Calculus and Linear Algebra”, advanced topics in complex analysis, point set topology, group theory, and/or differential geometry could be covered in depth at the discretion of the instructor, in addition to single and multivariable real analysis and abstract linear algebra. In 1970, for example, students studied the differential geometry of Banach manifolds in the second semester of Math 55.[4]

Math55 Survivors :
Of those students who could handle the workload, some became math or physics professors,[4] including members of the Harvard Math Department such as Benedict Gross and Joe Harris; also, Harvard physics professor Lisa Randall ’84[15] and Harvard economics professor Andrei Shleifer ’82.[16]
In addition to these professors, past students of Math 55 include Bill Gates[18] and Richard Stallman.[4]

The Professors teaching Math55: include Siu Yum Tong 萧荫堂(China/HK).

Seven Fields Medalists

The 7 Fields Medalists are:

2014 – Maryam Mirzakhani (1977-2017) – 1st lady Fields medalist

2010 – Cédric Villani (1973- )

2006 – Grigori Perelman (1966- ) – 1st declined the award

1998 – Andrew Wiles (1953- ) [silver plaque] – Fermat’s Last Theorem

1990 – Edward Witten (1951- ) – Physicist won Fields medal

1982 – Alain Connes (1947- ) – Quantum Theory

1966 – Alexander Grothendieck (1928-2014) – Hermit mathematician

The Modular Form

Synopsis 概要:
A Modular Form (模型式) is a type of function studied in a field of mathematics called complex Analysis.

The study of complex analysis reveals that Modular Forms have something called ‘q-expansion,’ like a generalized polynomial. The coefficients of these expansions come in patterns (Monster Group). There is a relationship between Partition Theory and Modular Form. The number theorists regard Modern Form as a basic part of their toolkit in important applications eg. Proof of the 350-year-old Fermat’s Last Theorem by Prof Andrew Wiles in 1994

Form” : Function with special properties – eg.

  • Space Forms: manifolds with certain shape.
  • Quadratic Forms (of weight 2): x^2+3xy+7z^2
  • Cubic Forms (of weight 3): x^3+{x^2}y + y^3
  • Automorphic Forms (particular case: Modular Forms): auto (self), morphic (shape).

1. Non-Euclidean Geometry

1.1 Hyperbolic Plane : is the Upper-Half in Complex plane H (positive imaginary part) where :

  • Through point p there are 2 lines L1 & L2 (called “geodesic“) parallel to line L.
  • Distance between p & q in H: \boxed {\int_{L} \frac {ds}{y}}
    where L the “line” segment (the arc of the semicircle or the vertical segment) and ds^2 = dx^2+dy^2

1.2 Group of Non-Euclidean Motions:
f: H \rightarrow H

  1. Translation: z \rightarrow {z + b} \quad \forall b \in \mathbb {R}
  2. Dilation: z \rightarrow {az } \quad \forall a \in \mathbb {R^{+}}
  3. Inversion: z \rightarrow - \frac {1} {z} \quad \forall z \in H \implies z \neq 0
  4. Flip about axis (or line): z \rightarrow - \bar{z}

z = x + iy
\bar{z} = x - iy
-\bar{z} = -x + iy

Let = Group of the above 1 & 2 & 3 motions (exclude 4 since Flip is NOT complex-differentiable function of z)

\boxed {G^{0} = \{\gamma (z) = \frac {az+b}{cz+d} \quad \text {;} \quad ad - bc > 0\}}

Fractional Linear Transformation:

z \rightarrow \gamma(z)

\gamma = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

z \rightarrow \begin{bmatrix} a & b \\ c & d \end{bmatrix} (z)

2. Group of Matrix M_2 (A)

Revision: Group = “CAN I

Matrix (K) with entries (a, b, c, d) from Set A (eg. Z, R, C…):

K = \begin{bmatrix} a & b \\ c & d \end{bmatrix}

Determinant = det (K) = ad – bc

Provided det (K) \neq 0
\displaystyle { \begin{bmatrix} a & b \\ c & d \end{bmatrix}}^{-1}= {\frac {1}{ad - bc}} {\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}}

2.1 General Linear Group GL_2(A)

\boxed {GL_2(\mathbb {R}) = \{ K \in M_2 (\mathbb {R}) \: | \: det (K) \neq 0\}}

\boxed {GL{_2}^{+}(\mathbb {R}) = \{ K \in M_2 (\mathbb {R}) \: | \: det (K) > 0\}}

\boxed {GL_2(\mathbb {C}) = \{ K \in M_2 (\mathbb {C}) \: | \: det (K) \neq 0\}}

\boxed {GL_2(\mathbb {Z}) = \{ K \in M_2 (\mathbb {Z}) \: | \: det (K) = \pm 1\}}

2.2 Special Linear Group SL_2(\mathbb{Z}) \subset GL_2(\mathbb{Z})

\boxed {SL_2(\mathbb {Z}) = \{ K \in M_2 (\mathbb {Z}) \: | \: det (K) = +1\}}

The Group SL_{2}(\mathbb {Z})= \{S, T\} acts on the upper half-plane H

T = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \quad \boxed {T (z) = z+1}

S = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \quad \boxed {S (z) = -\frac {1}{z}}


S^2 = -I \implies S^{4} = I

T^{k}= \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix} \quad \forall k \in {\mathbb {Z}}

3. Modular Form : M_{k} is an Analytic Function of weight k (k being a nonnegative Even Integer) f : H \rightarrow C with 2 properties:

(1) Transformation property
\boxed {f(\gamma (z)) = (cz+d)^{k}f (z)}

(2) Growth property: possess a “q-expansion” of the form:
\boxed {f(z) = a_0 + a_{1}q +a_{2}q^{2}+... }
where all aj are constants, and

Cusp Form of weight k : S_{k}
\boxed{f(z) = a_{1}q +a_{2}q^{2}+... }

Note: S for Spitze (German: Cusp) – “尖点” (A pointed end where 2 curves meet.)

Note: q(z+1) = q(z) [hint:] e^{2i\pi} = 1
More generally, with an automorphy factor \phi (X)
g(X+1) = \phi {(X)}g(X)
eg. g(X) = e^{X} \implies g(X+1) = e^{X+1}=e.e^X = e.{g(X)} \text { ;} \quad \phi (X)=e

(Complex) Vector Spaces (V) = \{S_{k} \subset M_{k}\}
(V1) V is nonempty.
(V2) For any function v in V, and any complex number c, the function cv is also in V.
(V3) For any function v and w in V, the function v + w is also in V.

4. Congruence Groups (of Level N)

\boxed {\Gamma (N) = \{ \gamma \in SL_{2}(\mathbb {Z}) | \gamma \equiv I \: (mod \: N)\}}

\boxed {\Gamma_{0} (N) = \{ \gamma \in SL_{2}(\mathbb {Z}) | c \equiv 0 \: (mod \: N)\}}

\boxed {\Gamma_{1} (N) = \{ \gamma \in SL_{2}(\mathbb {Z}) | c \equiv {a - 1} \equiv {d - 1} \equiv 0 \: (mod \: N)\}}

Note: It is one of the mysteries, or facts, of the theory that the above 3 are the main Congruence Subgroups needed to do most of the work that number theorists demand from Modular Form.

5. Applications

5.1 L-Function: when 2 different objects have the same L-function, this can mean that there is a very profound and often very useful tight connection between them.

5.2 Elliptic Curve

y^2 = x^3 + ax^2 + bx + c

5.3 Galois Representation

\rho : G_{Q} \rightarrow GL_{n}(K) \, | \, \rho ({\sigma}{\tau})= \rho(\sigma) \rho (\tau)

5.4 Monstrous “Monshine” – largest Simple Group

j(z) = q^{-1} +744 + 196884q + 21493760q^{2} + ...

The connection between j-function and the Monster Group was discovered by Simon Norton and John Conway, fully explained by Richard Borcherds in 1992 for which (partly) he was awarded the Fields Medal.

5.5 Fermat’s Last Theorem

5.6 Sato-Tate Conjecture

Note: “Operator” is synonymous to “Function of functions” (eg. Hecke Operator), just like “Form” is synonym for “Function”

Reference: [National Library NLB # 512.7]

Amazon Review:

Other related links:

1. Shimura-Taniyama Weil Theorem

2.费马大定理 Fermat’s Last Theorem