Entropy


This is high temperature and chaotic, which is called Ising model.


Jij>0


This is like Big Bang.

You can also write this.


Expansion is high temperature.

Therefore, you can divide it.

The division is cooling.

Tr is trace.

∴

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The Sylvester-Gallai Theorem

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The deformation is the mathematical concept. The Sylvester-Gallai Theorem is that a finite set of points in the plane have the proper line through any two of them, and it passes through a third point of the set. Moreover, it must be on the same line. P is the the noncollinear finite set. S(P) is the set of connecting lines in P. p∈P, s∈S(P) p and s are perpendicular, so they are not the ordinary line. (s*,p*) is the smallest distance, and s* is in the ordinary line. l∩P={p} l must be the ordinary line. This is the contradiction. l∩S(P) There is the cyclically intersection points (p,x1). x1,・・・xk is on l. Moreover it crosses S. This must be ordinary. You also see the extra dimensions. i lines are determined by P.
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Hybrid orbital

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