Ciro Santilli @cirosantilli 40

Intuition, please? Example? mathoverflow.net/questions/278641/intuition-for-symplectic-groups The key motivation seems to be related to Hamiltonian mechanics. The two arguments of the bilinear form correspond to each set of variables in Hamiltonian mechanics: the generalized positions and generalized momentums, which appear in the same number each.

Seems to be set of matrices that preserve a skew-symmetric bilinear form, which is comparable to the orthogonal group, which preserves a symmetric bilinear form. More precisely, the orthogonal group has:and its generalization the indefinite orthogonal group has:where S is symmetric. So for the symplectic group we have matrices Y such as:where A is antisymmetric. This is explained at: www.ucl.ac.uk/~ucahad0/7302_handout_13.pdf They also explain there that unlike as in the analogous orthogonal group, that definition ends up excluding determinant -1 automatically.

Therefore, just like the special orthogonal group, the symplectic group is also a subgroup of the special linear group.

There is only a very fine difference between a very good film, and the best films of all time. Perhaps it is something to do on how epic the subject matter is? It is often very hard to tell, and switches between the categories are also possible.

Besides of course sexual selection, considering in this section only "formal learning" activities.

Consider e.g. the 2020 University of Oxford, where many many people are taking courses without any laboratory work (and also without much use at all) like literature and history, and they are paying about 9k pounds/year for it: how much it costs to study at the University of Oxford?.

Basically all of this could be done online from books.

Laboratories are impossible however, because expendables of every experiment you do cost from hundreds to thousands of dollars, not to mention crazy upfront equipment costs.

For this reason, the brick and mortar aspect universities should focus exclusively on laboratories, and ensuring that the students with the most relevant knowledge (which can be readily obtained online) get access to those laboratories. Students should of course fully master every aspect of theory pertinent to their experiments. principal investigators should hand pick whichever criteria they want to select their students, possibly based partly on exam as a service if they find it a useful metric.

Furthermore, the use of laboratories should put great focus on novel research. A lot of laboratory instruction could be done from video of an experiments. As much as possible, we should use laboratories for novel research. Related: Section "Videos of all key physics experiments".

As mentioned on the introduction, the main objective of the course is to try predict qualitative properties of materials, notably the existence of certain phase transitions, starting from first principle toy models.

Key phenomena covered include:

Low budget.

Good theory of Jesus.

List of similar feeling films: www.youtube.com/watch?v=zwYwFoanrNg 11 Underrated Hard Sci-fi Movies by Marvelous Videos (2021)

We are at a point in history where the electrical telegraph is well established.

But people don't want to press letters one by one on a switch. They want to talk!

The first phones appear to have used telegraph lines directly.

Also wired phones don't require modulation, which likely made their development much easier than wireless voice transmission. You just send the signal as a voltage differential directly obtained from the air pressure: how the telephone works.

Let's show that this definition is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.

Note that:and for that to be true for all possible $x$ and $y$ then we must have:i.e. the matrix inverse is equal to the transpose.

Conversely, if:is true, then

These matricese are called the orthogonal matrices.

TODO is there any more intuitive way to think about this?

Some cool ones: