The orthogonal group has 2 connected components:
- one with determinant +1, which is itself a subgroup known as the special orthogonal group. These are pure rotations without a reflection.
- the other with determinant -1. This is not a subgroup as it does not contain the origin. It represents rotations with a reflection.
It is instructive to visualize how the looks like in :
- you take the first basis vector and move it to any other. You have therefore two angular parameters.
- you take the second one, and move it to be orthogonal to the first new vector. (you can choose a circle around the first new vector, and so you have another angular parameter.
- at last, for the last one, there are only two choices that are orthogonal to both previous ones, one in each direction. It is this directio, relative to the others, that determines the "has a reflection or not" thing
As a result it is isomorphic to the direct product of the special orthogonal group by the cyclic group of order 2:
A low dimensional example:because you can only do two things: to flip or not to flip the line around zero.
Rasselas Prince of Abyssinia CHAPTER VIII www.gutenberg.org/cache/epub/652/pg652-images.html:
Oppression is, in the Abyssinian dominions, neither frequent nor tolerated; but no form of government has been yet discovered by which cruelty can be wholly prevented. Subordination supposes power on one part and subjection on the other; and if power be in the hands of men it will sometimes be abused. The vigilance of the supreme magistrate may do much, but much will still remain undone. He can never know all the crimes that are committed, and can seldom punish all that he knows.
Related to Leela Zero, a Go engine
Can ISPs deanonymize Tor users based on timestamps of public posts? Updated 2025-01-10 +Created 1970-01-01
A DNA sequence that marks the start of a transcription area.
www.youtube.com/watch?v=0hfdSrK46EE How To Use A Minimoog Synthesizer by Doctor Mix (2020)
Software emulators:
The Neurokernel Project aims to build an open software platform for the emulation of the entire brain of the fruit fly Drosophila melanogaster on multiple Graphics Processing Units (GPUs).
Quantum Mechanics 12b - Dirac Equation II by ViaScience (2015)
Source. - youtu.be/tR6UebCvFqE?t=23 particle at rest
- youtu.be/tR6UebCvFqE?t=322 unidirectional movement without a potential
- youtu.be/tR6UebCvFqE?t=507 shows that observers in different frames of reference also see different spin. We are reminded of how magnetism is just a side effect of special-relativity.
- youtu.be/tR6UebCvFqE?t=549 Dirac equation solution for the hydrogen atom, final result only + mentions fine structure prediction.
Unlisted articles are being shown, click here to show only listed articles.