Ciro Santilli @cirosantilli 34

They are the finite projective special linear groups.

This was the first infinite family of simple groups discovered after the simple cyclic groups and alternating groups. The first case discovered was $PSL(2,p)$ by Galois. You should understand that one first.

The City of London is an obscene thing. Its existence goes against the will of the greater part of society. All it takes is one glance to see how it is but a bunch of corruption. See e.g.: The Spiders' Web: Britain's Second Empire.

In mathematics, a "classification" means making a list of all possible objects of a given type.

Classification results are some of Ciro Santilli's favorite: Section "The beauty of mathematics".

Examples:

Head of the theoretical division at the Los Alamos Laboratory during the Manhattan Project.

Richard Feynman was working under him there, and was promoted to team lead by him because Richard impressed Hans.

He was also the person under which Freeman Dyson was originally under when he moved from the United Kingdom to the United States.

And Hans also impressed Feynman, both were problem solvers, and liked solving mental arithmetic and numerical analysis.

This relationship is what brought Feynman to Cornell University after World War II, Hans' institution, which is where Feynman did the main part of his Nobel prize winning work on quantum electrodynamics.

Hans must have been the perfect PhD advisor. He's always smiling, and he seemed so approachable. And he was incredibly capable, notably in his calculation skills, which were much more important in those pre-computer days.

Contains the first sporadic groups discovered by far: 11 and 12 in 1861, and 22, 23 and 24 in 1973. And therefore presumably the simplest! The next sporadic ones discovered were the Janko groups, only in 1965!

Each $M_n$ is a permutation group on $n$ elements. There isn't an obvious algorithmic relationship between $n$ and the actual group.

TODO initial motivation? Why did Mathieu care about k-transitive groups?

Their; k-transitive group properties seem to be the main characterization, according to Wikipedia:Looking at the classification of k-transitive groups we see that the Mathieu groups are the only families of 4 and 5 transitive groups other than symmetric groups and alternating groups. 3-transitive is not as nice, so let's just say it is the stabilizer of $M_{23}$ and be done with it.

Apparently only Mathieu group $M_{12}$ and Mathieu group $M_{24}$.

www.maths.qmul.ac.uk/~pjc/pps/pps9.pdf mentions:Hmm, is that 54, or more likely 5 and 4?

scite.ai/reports/4-homogeneous-groups-EAKY21 quotes link.springer.com/article/10.1007%2FBF01111290 which suggests that is is also another one of the Mathieu groups, math.stackexchange.com/questions/698327/classification-of-triply-transitive-finite-groups#comment7650505_3721840 and en.wikipedia.org/wiki/Mathieu_group_M12 mentions $M_{12}$.

The traffic is designed for cars, which makes many red stops for bicycles completely stupid.

In a bicycle you just have too much more control and awareness than in a car, so if the way is completely clear, you should be allowed to stop, look if the way is clear, and then run reds.

Of course, this does increase the chances of hitting pedestrians a little bit. But the risk change feels so little that it would be worth it. Studies quoted by Wikipedia corroborate. It just feels extremely unintuitive to make cyclists stop in certain places when the street is clear.

When it exists, which is not for all matrices, only invertible matrix, the inverse is denoted:

As usual, it is useful to think about how a bilinear form looks like in terms of vectors and matrices.

Unlike a linear form, which was a vector, because it has two inputs, the bilinear form is represented by a matrix $M$ which encodes the value for each possible pair of basis vectors.

In terms of that matrix, the form $B(x,y)$ is then given by:

The matrix ring of degree n $M_n$ is the set of all n-by-n square matrices together with the usual vector space and matrix multiplication operations.

This set forms a ring.

Related terminology:

Weight: heavy.

MIDI support is kind of secondary: www.youtube.com/watch?v=vnkJ0uYXMG8, e.g. how to export MIDI? discourse.ardour.org/t/export-an-entire-project-into-a-midi-file/88116

Ardour vmpk input: askubuntu.com/questions/709673/save-as-midi-when-playing-from-vmpk-qsynth/1298231#1298231

The following aspects of Maxwell's equations make no sense without special relativity:

When charged particle though experiment are seen from the point of view of special relativity, it becomes clear that magnetism is just a direct side effect of charges being viewed in special relativity. One is philosophically reminded of how spin is the consequence of quantum mechanics + special relativity.

Bibliography:

TODO what's the point of it.

Bibliography:

Name of the clade of archaea plus eukarya proposed at: www.frontiersin.org/articles/10.3389/fmicb.2015.00717/full. Much better term than prokaryote as that is not a clade. Let's hope it catches on!