The complex analogue of the special orthogonal group, i.e. the subgroup of the unitary group with determinant equals exactly 1 instead of an arbitrary complex number with absolute value equal 1 as is the case for the unitary group.
There are only boring examples of taking an uncomputable language and converting it into a number?
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convert -size 256x256 radial-gradient:white-black out.pngLight watch transverse to direction of motion. This case is interesting because it separates length contraction from time dilation completely.
Of course, as usual in special relativity, calling something "time dilation" leads us to mind boggling ideas of "symmetry breaking": if both frames have a light watch, how can both possibly observe the other to be time dilated?
And the answer to this, is the usual: in special relativity time and space are interwoven in a fucked up way, everything is just a spacetime event.
In this case, there are three spacetime events of interest: both clocks start at same position, your beam hits up at x=0, moving frame hits up at x>0.
Those two mentioned events are spacelike-separated events, and therefore even though they seem simultaneous to you, they are not going to be simultaneous to the moving observer!
If little clock one meter away from you tells you that at the time of some event (your light beam hit up) the moving light watch was only 50% up, this is just a number given by your one meter away watch!
Most promising approaches as of 2020:
Why Private Billions Are Flowing Into Fusion by Bloomberg (2022)
Source. - Joint European Torus
- General Fusion: compress with liquid metal. Intends to demo in JET site.
- Helion Energy: direct fusion to electricity conversion without steam, direct from magnetic field movements
- First Light: shoot microscopic objct at a target to crush it so much that fusion happens
This ultimately determines which Java is used by a bunch of tools.
TODO is there a way to update it sanely in Ubuntu: askubuntu.com/questions/175514/how-to-set-java-home-for-java to always match the default
java executable?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 2025-03-28 +Created 1970-01-01
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