What they presented on richard Feynman's first seminar in 1941. Does not include quantum mechanics it seems.
Initially a phenomenological guess to explain the periodic table. Later it was apparently proven properly with the spin-statistics theorem, physics.stackexchange.com/questions/360140/theoretical-proof-of-paulis-exclusion-principle.
And it was understood more and more that basically this is what prevents solids from collapsing into a single nucleus, not electrical repulsion: electron degeneracy pressure!
Bibliography:
- www.youtube.com/watch?v=EK_6OzZAh5k How Electron Spin Makes Matter Possible by PBS Space Time (2021)
We can almost reach the Lie algebra of any isometry group in a single go. For every in the Lie algebra we must have:because has to be in the isometry group by definition as shown at Section "Lie algebra of a matrix Lie group".
Bibliography:
All indefinite orthogonal groups of matrices of equal metric signature are isomorphic by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16Following the definition of the indefinite orthogonal group, we want to show that only the metric signature matters.
First we can observe that the exact matrices are different. For example, taking the standard matrix of :and:both have the same metric signature. However, we notice that a rotation of 90 degrees, which preserves the first form, does not preserve the second one! E.g. consider the vector , then . But after a rotation of 90 degrees, it becomes , and now ! Therefore, we have to search for an isomorphism between the two sets of matrices.
For example, consider the orthogonal group, which can be defined as shown at the orthogonal group is the group of all matrices that preserve the dot product can be defined as:
Consider this is a study in failed computational number theory.
The approximation converges really slowly, and we can't easy go far enough to see that the ration converges to 1 with only awk and primes:Runs in 30 minutes tested on Ubuntu 22.10 and P51, producing:
sudo apt intsall bsdgames
cd prime-number-theorem
./main.py 100000000
. It is clear that the difference diverges, albeit very slowly.
. We just don't have enough points to clearly see that it is converging to 1.0, the convergence truly is very slow. The logarithm integral approximation is much much better, but we can't calculate it in awk, sadface.
But looking at: en.wikipedia.org/wiki/File:Prime_number_theorem_ratio_convergence.svg we see that it takes way longer to get closer to 1, even at it is still not super close. Inspecting the code there we see:so OK, it is not something doable on a personal computer just like that.
(* Supplement with larger known PrimePi values that are too large for \
Mathematica to compute *)
LargePiPrime = {{10^13, 346065536839}, {10^14, 3204941750802}, {10^15,
29844570422669}, {10^16, 279238341033925}, {10^17,
2623557157654233}, {10^18, 24739954287740860}, {10^19,
234057667276344607}, {10^20, 2220819602560918840}, {10^21,
21127269486018731928}, {10^22, 201467286689315906290}, {10^23,
1925320391606803968923}, {10^24, 18435599767349200867866}}; Pinned article: Introduction to the OurBigBook Project
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