🎨🤯 Ah, the gripping saga of "Abelian Sandpiles" that no one asked for, but everyone apparently needs! Because who could resist a deep dive into the riveting world of #grains #falling over on a grid?! Let's pretend we're all mesmerized by this mathematical #confetti. 🙄✨
https://eavan.blog/posts/beautiful-sandpiles.html #Abelian #Sandpiles #mathematics #grid #HackerNews #ngated

With the restriction to only this exponential, as shown by #Galois theory, only compositions of #Abelian extensions may be constructed, which suffices only for equations of 4 degree and below.

- All quadratic extensions, obtained by adjoining the roots of a quadratic polynomial, are #abelian

- the rank of #abelian group E(K) of points of E is the order of the zero of L(E, s) at s = 1, and the first non-zero coefficient in the Taylor expansion of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K

- The Gelfand–#Fourier transform is an isomorphism between the group C*-algebra C*(G) and C0(Ĝ). The theorem is essentially the dual statement for states of the two #abelian C*-algebras

It's unit circle vs #abelian group for #Fourier analyses
When #grouptheory matters

- if G is a finite group of order { p^{a}q^{b}} where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. Hence each non-#Abelian finite simple group has order divisible by at least three distinct #primes.
Theorem

Because 3,4 are relatively prime for Z #abelian group :
Z_12= Z_3 * Z_4

Because 3,4 are relatively prime for Z #abelian group :
Z_12= Z_3 * Z_4