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.
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
To test it, let's get two computers on the same local area network, e.g. connected to Wi-Fi on the same home modem router.
On computer B:
- find computer IP with the
ipCLI tool. Suppose it is 192.168.1.102 - then run Ciro's
ncHTTP test server
Output on terminal 1:TODO understand them all! Possibly correlate with Wireshark, or use
17:14:22.017001 IP ciro-p14s.55798 > 192.168.1.102.8000: Flags [S], seq 2563867413, win 64240, options [mss 1460,sackOK,TS val 303966323 ecr 0,nop,wscale 7], length 0
17:14:22.073957 IP 192.168.1.102.8000 > ciro-p14s.55798: Flags [S.], seq 1371418143, ack 2563867414, win 65160, options [mss 1460,sackOK,TS val 171832817 ecr 303966323,nop,wscale 7], length 0
17:14:22.074002 IP ciro-p14s.55798 > 192.168.1.102.8000: Flags [.], ack 1, win 502, options [nop,nop,TS val 303966380 ecr 171832817], length 0
17:14:22.074195 IP ciro-p14s.55798 > 192.168.1.102.8000: Flags [P.], seq 1:82, ack 1, win 502, options [nop,nop,TS val 303966380 ecr 171832817], length 81
17:14:22.076710 IP 192.168.1.102.8000 > ciro-p14s.55798: Flags [P.], seq 1:80, ack 1, win 510, options [nop,nop,TS val 171832821 ecr 303966380], length 79
17:14:22.076710 IP 192.168.1.102.8000 > ciro-p14s.55798: Flags [.], ack 82, win 510, options [nop,nop,TS val 171832821 ecr 303966380], length 0
17:14:22.076727 IP ciro-p14s.55798 > 192.168.1.102.8000: Flags [.], ack 80, win 502, options [nop,nop,TS val 303966383 ecr 171832821], length 0
17:14:22.077006 IP ciro-p14s.55798 > 192.168.1.102.8000: Flags [F.], seq 82, ack 80, win 502, options [nop,nop,TS val 303966383 ecr 171832821], length 0
17:14:22.077564 IP 192.168.1.102.8000 > ciro-p14s.55798: Flags [F.], seq 80, ack 82, win 510, options [nop,nop,TS val 171832821 ecr 303966380], length 0
17:14:22.077578 IP ciro-p14s.55798 > 192.168.1.102.8000: Flags [.], ack 81, win 502, options [nop,nop,TS val 303966384 ecr 171832821], length 0
17:14:22.079429 IP 192.168.1.102.8000 > ciro-p14s.55798: Flags [.], ack 83, win 510, options [nop,nop,TS val 171832824 ecr 303966383], length 0-A option to dump content.The 3D regular convex polyhedrons are super famous, have the name: Platonic solid, and have been known since antiquity. In particular, there are only 5 of them.
The counts per dimension are:
The cool thing is that the 3 that exist in 5+ dimensions are all of one of the three families:Then, the 2 3D missing ones have 4D analogues and the sixth one in 4D does not have a 3D analogue: the 24-cell. Yes, this is the kind of irregular stuff Ciro Santilli lives for.
How the hell are you supposed to develop an open source implementation of something that has a closed standard?
For a commented initial example, see: e. Coli K-12 MG1655 gene thrA.
But BioCyc is generally better otherwise.
Basically the opposite of security through obscurity, though slightly more focused on cryptography.
where:
- is the electromagnetic tensor
Note that this is the sum of the:Note that the relationship between and is not explicit. However, if we knew what type of particle we were talking about, e.g. electron, then the knowledge of psi would also give the charge distribution and therefore
- Dirac Lagrangian, which only describes the "inertia of bodies" part of the equation
- the electromagnetic interaction term , which describes term describes forces
As mentioned at the beginning of Quantum Field Theory lecture notes by David Tong (2007):
- by "Lagrangian" we mean Lagrangian density
- the generalized coordinates of the Lagrangian are fields
The hard part then is how to make any predictions from it:
- 2024 www.nature.com/articles/d41586-024-02935-z Fly-brain connectome helps to make predictions about neural activity. Summary of "Connectome-constrained networks predict neural activity across the fly visual system" by J. K. Lappalainen et. al.
2024: www.nature.com/articles/d41586-024-03190-y Largest brain map ever reveals fruit fly's neurons in exquisite detail
As of 2022, it had been almost fully decoded by post mortem connectome extraction with microtome!!! 135k neurons.
- 2021 www.nytimes.com/2021/10/26/science/drosophila-fly-brain-connectome.html Why Scientists Have Spent Years Mapping This Creature’s Brain by New York Times
That article mentions the humongous paper elifesciences.org/articles/66039 elifesciences.org/articles/66039 "A connectome of the Drosophila central complex reveals network motifs suitable for flexible navigation and context-dependent action selection" by a group from Janelia Research Campus. THe paper is so large that it makes eLife hang.
It can be seen as the limit case of an Einstein solid at high temperatures. At lower temperatures, the heat capacity depends on temperature.
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