Simple sines and variants:
- unix.stackexchange.com/questions/82112/stereo-tone-generator-for-linux/536860#536860
- stackoverflow.com/questions/5109038/linux-sine-wave-audio-generator/57610684#57610684
- superuser.com/questions/724391/how-to-generate-a-sine-wave-with-ffmpeg
- stackoverflow.com/questions/59551013/how-to-generate-stereo-sine-wave-using-ffmpeg-with-different-frequencies-for-eac/77730492#77730492
This is a good first concrete example of a Lie algebra. Shown at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4.2 "How to linearize a Lie Group" has an example.
Every element with this parametrization has determinant 1:Furthermore, any element can be reached, because by independently settting , and , , and can have any value, and once those three are set, is fixed by the determinant.
Remembering that the Lie bracket of a matrix Lie group is really simple, we can then observe the following Lie bracket relations between them:
One key thing to note is that the specific matrices , and are not really fundamental: we could easily have had different matrices if we had chosen any other parametrization of the group.
TODO confirm: however, no matter which parametrization we choose, the Lie bracket relations between the three elements would always be the same, since it is the number of elements, and the definition of the Lie bracket, that is truly fundamental.
Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4.2 "How to linearize a Lie Group" then calculates the exponential map of the vector as:with:
TODO now the natural question is: can we cover the entire Lie group with this exponential? Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 7 "EXPonentiation" explains why not.
As mentioned at youtu.be/16BzIG0lrEs?t=397 from Video "Applied Materials by Asianometry (2021)", originally the companies fabs would make their own equipment. But eventually things got so complicated that it became worth it for separate companies to focus on equipment, which then then sell to the fabs.
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A bit like the classification of simple finite groups, they also have a few sporadic groups! Not as spectacular since as usual continuous problems are simpler than discrete ones, but still, not bad.
This is a tag for films that depict some sort of artificial general intelligence, but in which that is not the primary focus of the film.
This section is about companies that design semiconductors.
For companies that manufature semiconductors, see also: company with a semiconductor fabrication plant.
One of Satoshi's email addresses, it's how he made the First public announcement of Bitoin on first public announcement of Bitcoin on 2008-10-31.
At some point later on vistomail.com was discontinued and acquired by a super dodgy dude, Alex Elbanna, so it hasn't been Satoshi for a while.
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