It is said, that once upon a time, programmers used CSV and collaborated on SourceForge, and that everyone was happy.
These days, are however, long gone in the mists of time as of 2020, and beyond Ciro Santilli's programming birth.
Infinitely many SQL answers.
As mentioned at Ciro Santilli's Stack Overflow contributions, he just answers every semi-duplicate immediatly as it is asked, and is therefore able to overcome the Stack Overflow maximum 200 daily reputation limit by far. E.g. in 2018, Gordon reached 135k (archive), thus almost double the 73k yearly limit due to the 200 daily limit, all of that with accepts.
This is in contrast to Ciro Santilli's contribution style which is to only answer questions as he needs the subject, or generally important questions that aroused his interest.
2014 Blog post describing his activity: blog.data-miners.com/2014/08/an-achievement-on-stack-overflow.html, key quote:so that suggests his contributions also take a meditative value.
For a few months, I sporadically answered questions. Then, in the first week of May, my Mom's younger brother passed away. That meant lots of time hanging around family, planning the funeral, and the like. Answering questions on Stack Overflow turned out to be a good way to get away from things. So, I became more intent.
www.data-miners.com/linoff.htm mentions he's an SQL consultant that consulted for several big companies.
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.
Given the function :the operator can be written in Planck units as:often written without function arguments as:Note how this looks just like the Laplacian in Einstein notation, since the d'Alembert operator is just a generalization of the laplace operator to Minkowski space.
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