Curious Thoughts in Math & Science 

1. Statistical Mechanics: e^ {- Ht}

Quantum Mechanics: e^{iHt}

2. Ramanujian:

1 +2 + 3 + …+ n = –  1/12

Note: this formula is used in Quantum Physics dealing with infinity n (although it cancels out each other in subsequent calculations)

Tau Special Function:

\boxed {\displaystyle \sum_{n=1}^{\infty}\tau (n) x^{n} = x \{(1-x)(1-x^{2})(1-x^{3})... \}^{24}}

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

AND\boxed {x.y} 

NOT\boxed {1-x} 

XOR\boxed {x+y-2x.y} 

Extra constraints ”  \boxed {x^{2}=x} 

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

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

Which arrangement of “Taps” will produce the maximal length ? 

Solomon applied Pure Math : represent the above sequence of  registers algebraically  by: 

\boxed {x^{7}+x^{6}+ 1} 

in reducible modulo 2 (prime in polynomial, ie can’t be factored).

=> the sequence is Maximal length

LFSR Applications: Telecommunications, 3G/4G/5G, CDMA, Wifi, computers, network, signal transmission error-correction CD/DVD, Astrology : Venus-Earth distance,  etc.

Stop Teaching Calculating, Start Learning Maths!

The British Wolfram company is the maker of the Math Computer Tool “Methematica” used for university students to prove theorems besides computing. It is written in Lisp functional language.

Conrad Wolfram provoked the new idea of Computer-Based Math education:

Teach the ‘Why’ of Maths, leave the ‘How’ to the computer.

‘How’: solve quadratic equation, simultaneous equations, differentiation, integration….

He mentioned Singapore is interested in this new approach of teaching Math ? The O & A Level students can now use scientific calculator in Exams.

Conrad Wolfram comes from the British angle, telling us the English Math is too computational, or Applied. It is like a fast-food chef without knowing much about the food science, the temperature and the art of color 色, taste 香, smell 味. The French Math is, au contraire, theoretical, like the ‘haute cuisine’ chef, more “Why” than “How”, slowly cook and taste the finest quality of food (dégustation 品尝).

Having taken GCE A-level Math and French Classe Prepas Math, I experienced the strengths and weaknesses of both Maths. Giving an Integration question to an English educated student, he would immediately plunge into all sorts of techniques (by substitution, by parts…) and get the answer; give it to a French student, he would first study the ‘domain of definition’ of the function, continuity at which intervals, … before attempting to integrate if it is “integritable” — meaningful to integrate or has any solution — in the first place !!

A best Math education is the combination of both English and French. The computer can be used to calculate faster in the (English) Applied Math, to verify / prove the (French ) Theoretical Math.

It is like the 少林巭 Shaolin Kungfu (English Applied Math ) versus 武当太极 Wudang Taiji (French Pure Math). You can have both !

Stephen Wolfram: Computing a theory of everything

Stephen Wolfram: Founder & CEO of Mathematica (UK)

Wolfram Alpha: Knowledge-base Computing using public data on the net and private information.

Example: Calculate : For Year 2014
Spain GDP / Google Revenue

Mathematica: Math tool using Symbolic Functional Language (LISP)

New Kind of Science: Cell Automata

Physics: From Computing World to find new Physical World