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.
The Purpose of Harvard is Not to Educate People by Sean Carroll (2008) Updated 2025-07-14 +Created 1970-01-01
Maybe they did try once though: Harvard Project Physics.
An important case is the discrete logarithm of the cyclic group in which the group is a cyclic group.
Ah, Ciro Santilli loved this one... games young Ciro Santilli played.
List of similar feeling films: www.youtube.com/watch?v=zwYwFoanrNg 11 Underrated Hard Sci-fi Movies by Marvelous Videos (2021)
Yet another awk-like domain-specific language to do things from the CLI in a ridiculously short humber of character? Oh yes.
Historian Alan B. Carr:
- www.youtube.com/@AlanBCarr. IMPORTANT NOTE: Although Alan B. Carr is a Los Alamos National Laboratory (LANL) employee, this page has absolutely no formal connection with LANL.
Man-made virus!
TODO: if we had cheap de novo DNA synthesis, how hard would it be to bootstrap a virus culture from that? github.com/cirosantilli/cirosantilli.github.io/issues/60
Is it easy to transfect a cell with the synthesized DNA, and get it to generate full infectious viral particles?
If so, then de novo DNA synthesis would be very similar to 3D printed guns: en.wikipedia.org/wiki/3D_printed_firearms.
It might already be possible to order dissimulated sequences online:
Bibliography:
- www.youtube.com/watch?v=j1PAxNKB_Zc Manifolds #6 - Tangent Space (Detail) by WHYB maths (2020). This is worth looking into.
- www.youtube.com/watch?v=oxB4aH8h5j4 actually gives a more concrete example. Basically, the vectors are defined by saying "we are doing the Directional derivative of any function along this direction".One thing to remember is that of course, the most convenient way to define a function and to specify a direction, is by using one of the coordinate charts.
- jakobschwichtenberg.com/lie-algebra-able-describe-group/ by Jakob Schwichtenberg
- math.stackexchange.com/questions/1388144/what-exactly-is-a-tangent-vector/2714944 What exactly is a tangent vector? on Stack Exchange
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.
Some cool ones:
- playinside.me
There are unlisted articles, also show them or only show them.