1. Statistical Mechanics:
Note: this formula is used in Quantum Physics dealing with infinity n (although it cancels out each other in subsequent calculations)
Tau Special Function:
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).
“Extra constraints ”
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 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:
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.