The set of all invertible matrices forms a group: the general linear group with matrix multiplication. Non-invertible matrices don't form a group due to the lack of inverse.
For this sub-case, we can define the Lie algebra of a Lie group as the set of all matrices such that for all :If we fix a given and vary , we obtain a subgroup of . This type of subgroup is known as a one parameter subgroup.
The immediate question is then if every element of can be reached in a unique way (i.e. is the exponential map a bijection). By looking at the matrix logarithm however we conclude that this is not the case for real matrices, but it is for complex matrices.
TODO example it can be seen that the Lie algebra is not closed matrix multiplication, even though the corresponding group is by definition. But it is closed under the Lie bracket operation.
Virgin hub 3.0. mega minimal manual: web.archive.org/web/20220930011319/https://www.virginmedia.com/content/dam/virginmedia/dotcom/documents/corporate/userg-guide_broadband_superhub3.pdf
Home 2023:
- cell phone wifi: 260 Mps download
The explosive moments presumably represent the intense rowing action of a dragon boat race competition.
TODO date composed.
Dragon boat performed by Yuanchun Yu (2021)
Source. Toy model of matter that exhibits phase transition in dimension 2 and greater. It does not provide numerically exact results by itself, but can serve as a tool to theorize existing and new phase transitions.
As mentioned at: stanford.edu/~jeffjar/statmech/intro4.html some systems which can be seen as modelled by it include:
- the spins direction (up or down) of atoms in a magnet, which can undergo phase transitions depending on temperature as that characterized by the Curie temperature and an externally applied magnetic fieldNeighboring spins like to align, which lowers the total system energy.
- the type of atom at a lattice point in a 2-metal alloy, e.g. Fe-C (e.g. steel). TODO: intuition for the neighbor interaction? What likes to be with what? And aren't different phases in different crystal structures?
Also has some funky relations to renormalization TODO.
Bibliography:
The Ising Model in Python by Mr. P Solver
. Source. The dude is crushing it on a Jupyter Notebook.
Liu Dehai peforming at the Musée Guimet in 2010
. Source. 
First proper nearest galaxy to the Milky Way. Everything in the middle in the Local group is either a satellite of the Milky Way or Andromeda.
As described on Wikipedia, the observational history of Andromeda is fascinating. Little by little, people noticed that it had a different nature to many other objects observed on the sky, and the hypothesis that there are other galaxies like ours grew in force.
Part of our fascination with Andromeda is due to how similar in size and shape and close it is to the Milky Way.
It is clearly the only thing so large and so close.
One can't help but wonder if there is some alien looking back at us when we are looking at them through our telescope.
Andromeda is also the furthest object from Earth that can be seen with the naked eye.[ref] Not surprising, as it literally shines with the strength of a trillion suns!
Andromeda Shun from Saint Seiya performing his Nebula Chain attack
. Source. The original Japanese music actually says "Nebula Chain" in English. The Andromeda Galaxy is shown on the back, the chain appears to go all the way to it and back towards the evil guys' head. Not very relativistic, but so be it.Whenever Ciro Santilli learns about molecular biology, he can't help but to feel that it feels like programming, and notably systems programming and computer hardware design.
In some sense, the comparison is obvious: DNA is clearly a programmable medium like any assembly language, but still, systems programming did give Ciro some further feelings.
- The most important analogy perhaps is observability, or more precisely the lack of it. For the computer, this is described at: The lower level you go into a computer, the harder it is to observe things.And then, when Ciro started learning a bit about biology techniques, he started to feel the exact same thing.For example when he played with E. Coli Whole Cell Model by Covert Lab, the main thing Ciro felt was: it is going to be hard to verify any of this data, because it is hard/impossible to know the concentration of each element in a cell as a function of time.More generally of course, this is exactly why making any biology discovery is so hard: we can't easily see what's going on inside the cell, and have to resort to indirect ways of doing so..This exact idea was highlighted by I should have loved biology by James Somers:
For a computer scientist, a biologist's methods can seem insane; the trouble comes from the fact that cells are too small, too numerous, too complex to analyze the way a programmer would, say in a step-by-step debugger.
And then just like in software, some of the methods biologists use to overcome the lack of visibility have direct software analogues:- add instrumentation to cells, e.g. GFP tagging comes to mind
- emulation, e.g. E. Coli Whole Cell Model by Covert Lab
- The boot process is another one. E.g. in x86 the way that you start in 16-bit mode, largely compatible into the 70's, then move to 32-bit and finally 64, does feel a lot the way a earlier stages of embryo development looks more and more like more ancient animals.
Ciro likes to think that maybe that is why a hardcore systems programmer like Bert Hubert got into molecular biology.
Some other people who mention similar things:
- I should have loved biology by James Somers highlights the computer abstraction layer analogy between the two:
But as of 2020, it is completely surrounded by extremely poor people, to the point that it makes you scared if you stand out in any way by showing any kind of middle/upper class wealth, or being a foreigner.
The market is basically a touristic spot that no person in Sao Paulo will ever go to (unless they are young, single, and can just walk in there by themselves) in the middle of this surreal environment.
In 2020 Ciro was there with his wife on a touristic visit. Living in Europe at the time, he felt even more privileged. So they went to a fruit stand, and the man started giving his wife amazing free samples of very exotic fruit, some of which Ciro had never tasted himself, without saying the price. It did feel like he was giving out too much for free. Then Ciro decided of course to buy some more fruits to pay for the show, which was a nice show. Then while buying, it came out a bit more expensive than would have been reasonable, but Ciro was too dazzled by the speed and noises, and he paid for it. Later on, he told his wife about it, and how he felt that they had added some ultra-expensive bulk fruits that were of a clearly lower level than the gold nuggets of the free samples (especially for Brazil's cost standards). The presenter was an extremely crafty con artist, and Ciro felt like they had specifically preyed on Ciro Santilli's self perceived compassionate personality, because it was apparent that those men were underprivileged and fighting for their living day by day with those over-expensive fruits. This was an extremely valuable lesson, Ciro was glad that it was learnt at a relatively low cost on that occasion.
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