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There are two sexes because of mitochondria by 
Ciro Santilli 35 Updated 2025-01-04 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-04 +Created 1970-01-01One of the key thesis of Power, Sex, Suicide by Nick Lane (2006).
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How To Get Tenure at a Major Research University by Sean Carroll (2011) by Ciro Santilli 35 Updated 2025-01-04 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-04 +Created 1970-01-01The brutality of this game, the fact that you can get killed at any point, and all players accept that, is awesome.
Unifies both special relativity and gravity.
Not compatible with the Standard Model, and the 2020 unification attempts are called theory of everything.
One of the main motivations for it was likely having forces not be instantaneous, but rather mediated by field to maintain the principle of locality, just like electromagnetism did earlier.
Synthetic geometry of the real projective plane by Ciro Santilli 35 Updated 2025-01-04 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-04 +Created 1970-01-01It good to think about how Euclid's postulates look like in the real projective plane:
- two parallel lines on the plane meet at a point on the sphere!Since there is one point of infinity for each direction, there is one such point for every direction the two parallel lines might be at. The parallel postulate does not hold, and is replaced with a simpler more elegant version: every two lines meet at exactly one point.One thing to note however is that ther real projective plane does not have angles defined on it by definition. Those can be defined, forming elliptic geometry through the projective model of elliptic geometry, but we can interpret the "parallel lines" as "two lines that meet at a point at infinity"
- points in the real projective plane are lines in
- For every two projective points there is a single projective line that passes through them.Note however that not all lines in the real plane correspond to a projective line: only lines tangent to a circle at zero do.
Unlike the real projective line which is homotopic to the circle, the real projective plane is not homotopic to the sphere.
The topological difference bewteen the sphere and the real projective space is that for the sphere all those points in the x-y circle are identified to a single point.
One more generalized argument of this is the classification of closed surfaces, in which the real projective plane is a sphere with a hole cut and one Möbius strip glued in.
Ultimate explanation: math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups/3732426#3732426
Only normal subgroups can be used to form quotient groups: their key definition is that they plus their cosets form a group.
Intuition:
One key intuition is that "a normal subgroup is the kernel" of a group homomorphism, and the normal subgroup plus cosets are isomorphic to the image of the isomorphism, which is what the fundamental theorem on homomorphisms says.
Therefore "there aren't that many group homomorphism", and a normal subgroup it is a concrete and natural way to uniquely represent that homomorphism.
The best way to think about the, is to always think first: what is the homomorphism? And then work out everything else from there.
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