Elements of a Lie algebra can (should!) be seen a continuous analogue to the generating set of a group in finite groups.
For continuous groups however, we can't have a finite generating set in the strict sense, as a finite set won't ever cover every possible point.
But the generator of a Lie algebra can be finite.
And just like in finite groups, where you can specify the full group by specifying only the relationships between generating elements, in the Lie algebra you can almost specify the full group by specifying the relationships between the elements of a generator of the Lie algebra.
The reason why the algebra works out well for continuous stuff is that by definition an algebra over a field is a vector space with some extra structure, and we know very well how to make infinitesimal elements in a vector space: just multiply its vectors by a constant that cana be arbitrarily small.
TODO concrete example, please...
Insanely active poster on Stack Overflow 4chan post (2023-07-03) Updated 2025-07-11 +Created 1970-01-01
archive.ph/Dd3aC web.archive.org/web/20230709141533/https://desuarchive.org/g/thread/94445084/#94448535 desuarchive.org/g/thread/94445084/#94448535
Most of the thread went into pro/anti gay trashtalk due to Ciro using Gay Putin at the time on his Stack Overflow profile as a useless way to protest the Russian invasion of Ukraine.
Some comments:
How does this guy manage to be so active on Stack Overflow? I feel like this disgusting avatar is on at least a quarter of all the active posts.
The answers are always pretty good though.
Obviously severe autism. Also racism homophobia Looks like everything is ok if it's Russia/Chinese...
The only new information:
Reminds me of Xah Lee.
Tested on Ubuntu 20.04:Add to your and then to use it on a shell e.g. with Python 3.9 create the environment with:and then use it with:Now you can use
mkdir -p ~/miniconda3
wget https://repo.anaconda.com/miniconda/Miniconda3-latest-Linux-x86_64.sh -O ~/miniconda3/miniconda.sh
bash ~/miniconda3/miniconda.sh -b -u -p ~/miniconda3
rm -rf ~/miniconda3/miniconda.sh.bashrc:PATH="$PATH:$HOME/miniconda3/bin"conda create -y -n mytest3.9 python=3.9eval "$(command conda 'shell.bash' 'hook' 2> /dev/null)"
conda activate mytest3.9python and pip normally from inside that mytest3.9 environment.At that time, the exact installer under
latest appears to have been: repo.anaconda.com/miniconda/Miniconda3-py311_23.11.0-2-Linux-x86_64.shIt basically came about because of the endless stream of useless software startups made since the 2000's by one or two people with no investments with the continued increase in computers and Internet speeds until the great wall was reached.
Deep tech means not one of those. More specifically, it means technologies that require significant investment in expensive materials and laboratory equipment to progress, such as molecular biology technologies and quantum computing.
And it basically comes down to technologies that wrestle with the fundamental laws of physics rather than software data wrangling.
Computers are of course limited by the laws of physics, but those are much hidden by several layers of indirection.
Full visibility, and full control, make computer tasks be tasks that eventually always work out more or less as expected.
The same does not hold true when real Physics is involved.
Physics is brutal.
To start with, you can't even see your system very clearly, and often doing so requires altering its behaviour.
For example, in molecular biology, most great discoveries are made after some new technique is made to be able to observe smaller things.
But you often have to kill your cells to make those observations, which makes it very hard to understand how they work dynamically.
What we would really want would be to track every single protein as it goes about inside the cell. But that is likely an impossible dream.
The same for the brain. If we had observations of every neuron, how long would it take to understand it? Not long, people are really good at reverse engineering things when there is enough information available to do so, see also science is the reverse engineering of nature.
Then, even when you start to see the system, you might have a very hard time controlling it, because it is so fragile. This is basically the case of quantum computing in 2020.
The next big things will come from deep tech. Failure is always a possibility, and you can't know before you try.
But that's also why its so fun to dare.
Stuff that Ciro Santilli considers "deep tech" as of 2020:
- brain-computer interface
- fusion power. The question there is, when is "deep", "too deep"?
A good definition is that the sparse matrix has non-zero entries proportional the number of rows. Therefore this is Big O notation less than something that has non zero entries. Of course, this only makes sense when generalizing to larger and larger matrices, otherwise we could take the constant of proportionality very high for one specific matrix.
Currently a redirect page on Wikipedia: en.wikipedia.org/?title=Department_of_Statistics,_University_of_Oxford&redirect=no Newbies!
Why is this not part of the Mathematical Institute of the University of Oxford? Who knows!
This is an important metric, because it takes some time for the quantum operations to propagate, and so the depth of a circuit gives you an idea of how long the coherence time a hardware needs to support a given circuit.
Bibliography:
The Dirac equation can be derived basically "directly" from the Representation theory of the Lorentz group for the spin half representation, this is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) 6.3 "Dirac Equation".
The Diract equation is the spacetime symmetry part of the quantum electrodynamics Lagrangian, i.e. is describes how spin half particles behave without interactions. The full quantum electrodynamics Lagrangian can then be reached by adding the internal symmetry.
As mentioned at spin comes naturally when adding relativity to quantum mechanics, this same method allows us to analogously derive the equations for other spin numbers.
Bibliography:
Like the rest of the Standard Model Lagrangian, this can be split into two parts:
- spacetime symmetry: reaches the derivation of the Dirac equation, but has no interactions
- add the internal symmetry to add interactions, which reaches the full equation
Deriving the qED Lagrangian by Dietterich Labs (2018)
Source. As mentioned at the start of the video, he starts with the Dirac equation Lagrangian derived in a previous video. It has nothing to do with electromagnetism specifically.
He notes that that Dirac Lagrangian, besides being globally Lorentz invariant, it also also has a global invariance.
However, it does not have a local invariance if the transformation depends on the point in spacetime.
He doesn't mention it, but I think this is highly desirable, because in general local symmetries of the Lagrangian imply conserved currents, and in this case we want conservation of charges.
To fix that, he adds an extra gauge field (a field of matrices) to the regular derivative, and the resulting derivative has a fancy name: the covariant derivative.
Then finally he notes that this gauge field he had to add has to transform exactly like the electromagnetic four-potential!
So he uses that as the gauge, and also adds in the Maxwell Lagrangian in the same go. It is kind of a guess, but it is a natural guess, and it turns out to be correct.
Applications:
- because it has an even number of nucleons it is transparent to NMR, and therefore is useful in solvents for NMR spectroscopy
Many devborads require a 5V power supply.
For battery power you need a transformer.
Raspberry Pi Battery Power by ExplainingComputers (2021)
Source. Lit: fish timber question answer.
The dialog is also known as allegory for an incredibly deep philosophical discussion between an idealized wise woodcutter and a fisherman, e.g. mentioned at: www2.kenyon.edu/Depts/Religion/Fac/Adler/Writings/Fisherman%20and%20Woodcutter.pdf
This song is just too slow for Ciro Santilli to make much out of it.
Bibliography:
Dialog between Fisherman and Woodcutter Chinese traditional painting by Xie Shichen
. Adds special relativity to the Schrödinger equation, and the following conclusions come basically as a direct consequence of this!
Experiments explained:
Experiments not explained: those that quantum electrodynamics explains like:See also: Dirac equation vs quantum electrodynamics.
- Lamb shift
- TODO: quantization of the electromagnetic field as photons?
The Dirac equation is a set of 4 partial differential equations on 4 complex valued wave functions. The full explicit form in Planck units is shown e.g. in Video 1. "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=1010:Then as done at physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600 from why are complex numbers used in the Schrodinger equation?, we could further split those equations up into a system of 8 equations on 8 real-valued functions.
Equation 1.
Expanded Dirac equation in Planck units
. PHYS 485 Lecture 14: The Dirac Equation by Roger Moore (2016)
Source. Predicts fine structure.
Bibliography:
Dirac equation for the electron and hydrogen Hamiltonian by Barton Zwiebach (2019)
Source. Uses perturbation theory to get to the relativistic corrections of fine structure! Part of MIT 8.06 Quantum Physics III, Spring 2018 by Barton ZwiebachHow To Solve The Dirac Equation For The Hydrogen Atom | Relativistic Quantum Mechanics by Dietterich Labs (2018)
Source. You gotta be born after the year 2000 to understand it.
This is becoming more and more popular as a group chat with channels and threads possibility as of 2020.
Very similar to Slack.
They force your username to have 4 random digits? www.reddit.com/r/discordapp/comments/43kjdl/whats_the_number_next_to_the_username/
Not possible to anonymously join just one server without creating a new account? What's the point of servers then! www.reddit.com/r/discordapp/comments/6gmjl7/changing_nick_before_joining_a_new_server/ Oh, also nicks don't hide your username from the server in any way, you can get the original username by just clicking on the person's username.
No proper threaded discussion without creating new channels? As of 2022 there is kind of a way, but it was a bit obtuse.
As of 2022 they also have a school hub: support.discord.com/hc/en-us/articles/4406046651927-Discord-Student-Hubs-FAQ which auto creates groups by university email access. Good idea, and shows popularity amongst that user group.
Servers don't have an ID to join them? www.reddit.com/r/discordapp/comments/b9zdt6/join_discord_server_from_id/
Can only make public servers if you have 1000 members?? support.discord.com/hc/en-us/articles/360023968311 Why so much bullshit?? www.reddit.com/r/discordapp/comments/6jouf8/how_do_i_make_my_server_public/
Output: another sequence of complex numbers such that:Intuitively, this means that we are braking up the complex signal into sinusoidal frequencies:and is the amplitude of each sine.
- : is kind of magic and ends up being a constant added to the signal because
- : sinusoidal that completes one cycle over the signal. The larger the , the larger the resolution of that sinusoidal. But it completes one cycle regardless.
- : sinusoidal that completes two cycles over the signal
- ...
- : sinusoidal that completes cycles over the signal
Motivation: similar to the Fourier transform:In particular, the discrete Fourier transform is used in signal processing after a analog-to-digital converter. Digital signal processing historically likely grew more and more over analog processing as digital processors got faster and faster as it gives more flexibility in algorithm design.
- compression: a sine would use N points in the time domain, but in the frequency domain just one, so we can throw the rest away. A sum of two sines, only two. So if your signal has periodicity, in general you can compress it with the transform
- noise removal: many systems add noise only at certain frequencies, which are hopefully different from the main frequencies of the actual signal. By doing the transform, we can remove those frequencies to attain a better signal-to-noise
Sample software implementations:
- numpy.fft, notably see the example: numpy/fft.py
DFT of with 25 points
. This is a simple example of a discrete Fourier transform for a real input signal. It illustrates how the DFT takes N complex numbers as input, and produces N complex numbers as output. It also illustrates how the discrete Fourier transform of a real signal is symmetric around the center point.Check which you you have:Tested on Ubuntu 23.10 I see:which means I have GNOME Display Manager.
systemctl status display-manager.service● gdm.service - GNOME Display Manager
Loaded: loaded (/lib/systemd/system/gdm.service; static)
Active: active (running) since Sun 2023-12-24 10:34:50 GMT; 23min ago
Process: 1827 ExecStartPre=/usr/share/gdm/generate-config (code=exited, status=0/SUCCESS)
Main PID: 1850 (gdm3)
Tasks: 4 (limit: 71817)
Memory: 6.8M
CPU: 119ms
CGroup: /system.slice/gdm.service
└─1850 /usr/sbin/gdm3 Unlisted articles are being shown, click here to show only listed articles.