# 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.