Curious Thoughts in Math & Science 

1. Statistical Mechanics: $latex e^ {- Ht} $

Quantum Mechanics: $latex e^{iHt}$

2. Ramanujian:

$latex 1 +2 + 3 + …+ n = -frac {1}{12} $

Tau Special Function:

$latex boxed {displaystyle sum_{n=1}^{infty}tau (n) x^{n} = x {(1-x)(1-x^{2})(1-x^{3})… }^{24}}$

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” $latex boxed {x.y}&fg=00bb00&s=3$

“NOT” $latex boxed {1-x}&fg=00bb00&s=3$

“XOR” $latex boxed {x+y-2x.y}&fg=00bb00&s=3$

“Extra constraints ” $latex boxed {x^{2}=x}&fg=00bb00&s=3$

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 $latex 2^{n}-1$ steps (exclude all ‘0’ sequence).

Which arrangement…