Basically a synonym of Lorentz covariance?
The elliptic curve group of all elliptic curve over the rational numbers is always a finitely generated group.
The number of points may be either finite or infinite. But when infinite, it is still a finitely generated group.
For this reason, the rank of an elliptic curve over the rational numbers is always defined.
TODO example.
The summary from www.geeksforgeeks.org/tree-traversals-inorder-preorder-and-postorder/ is a winner:
1
/ \
2 3
/ \
4 5- inorder DFS: 4 2 5 1 3
- preorder DFS: 1 2 4 5 3
- postorder DFS: 4 5 2 3 1
- breadth-first search: 1 2 3 4 5
In principle one could talk about tree traversal of unordered trees as a number of possible traversals without a fixed order. But we won't consider that under this section, only deterministic ordered tree traversals.
Some examples by Ciro Santilli follow.
Of the tutorial-subjectivity type:
- This edit perfectly summarizes how Ciro feels about Wikipedia (no particular hate towards that user, he was a teacher at the prestigious Pierre and Marie Curie University and actually as a wiki page about him):which removed the only diagram that was actually understandable to non-Mathematicians, which Ciro Santilli had created, and received many upvotes at: math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups/3732426#3732426. The removal does not generate any notifications to you unless you follow the page which would lead to infinite noise, and is extremely difficult to find out how to contact the other person. The removal justification is even somewhat ad hominem: how does he know Ciro Santilli is also not a professional Mathematician? :-) Maybe it is obvious because Ciro explains in a way that is understandable. Also removal makes no effort to contact original author. Of course, this is caused by the fact that there must also have been a bunch of useless edits not done by Ciro, and there is no reputation system to see if you should ignore a person or not immediately, so removal author has no patience anymore. This is what makes it impossible to contribute to Wikipedia: your stuff gets deleted at any time, and you don't know how to appeal it. Ciro is going to regret having written this rant after Daniel replies and shows the diagram is crap. But that would be better than not getting a reply and not learning that the diagram is crap.
rm a cryptic diagram (not understandable by a professional mathematician, without further explanations
- en.wikipedia.org/w/index.php?title=Finite_field&type=revision&diff=1044934168&oldid=1044905041 on finite fields with edit comment "Obviously: X ≡ α". Discussion at en.wikipedia.org/wiki/Talk:Finite_field#Concrete_simple_worked_out_example Some people simply don't know how to explain things to beginners, or don't think Wikipedia is where it should be done. One simply can't waste time fighting off those people, writing good tutorials is hard enough in itself without that fight.
- en.wikipedia.org/w/index.php?title=Discrete_Fourier_transform&diff=1193622235&oldid=1193529573 by user Bob K. removed Ciro Santilli's awesome simple image of the Discrete Fourier transform as seen at en.wikipedia.org/w/index.php?title=Discrete_Fourier_transform&oldid=1176616763:with message:
Hello. I am a retired electrical engineer, living near Washington,DC. Most of my contributions are in the area of DSP, where I have about 40 years of experience in applications on many different processors and architectures.
Thank you so much!!remove non-helpful image
Maybe it is a common thread that these old "experts" keep removing anything that is actually intelligible by beginners? Section "There is value in tutorials written by beginners"Also ranted at: x.com/cirosantilli/status/1808862417566290252
Figure 1. Ciro Santilli's awesome graph removed by Bob K. from the Discrete Fourier transform page.Source at: numpy/fft_plot.py. - when Ciro Santilli created Scott Hassan's page, he originally included mentions of his saucy divorce: en.wikipedia.org/w/index.php?title=Scott_Hassan&oldid=1091706391 These were reverted by Scott's puppets three times, and Ciro and two other editors fought back. Finally, Ciro understood that Hassan's puppets were likely right about the removal because you can't talk about private matters of someone who is low profile:even if it is published in well known and reliable publications like the bloody New York Times. In this case, it is clear that most people wanted to see this information summarized on Wikipedia since others fought back Hassan's puppet. This is therefore a failure of Wikipedia to show what the people actually want to read about.This case is similar to the PsiQuantum one. Something is extremely well known in an important niche, and many people want to read about it. But because the average person does not know about this important subject, and you are limited about what you can write about it or not, thus hurting the people who want to know about it.
Notability constraints, which are are way too strict:There are even a Wikis that were created to remove notability constraints: Wiki without notability requirements.
- even information about important companies can be disputed. E.g. once Ciro Santilli tried to create a page for PsiQuantum, a startup with $650m in funding, and there was a deletion proposal because it did not contain verifiable sources not linked directly to information provided by the company itself: en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/PsiQuantum Although this argument is correct, it is also true about 90% of everything that is on Wikipedia about any company. Where else can you get any information about a B2B company? Their clients are not going to say anything. Lawsuits and scandals are kind of the only possible source... In that case, the page was deleted with 2 votes against vs 3 votes for deletion.is very similar to Stack Exchange's own Stack Overflow content deletion issues. Ain't Nobody Got Time For That. "Ain't Nobody Got Time for That" actually has a Wiki page: en.wikipedia.org/wiki/Ain%27t_Nobody_Got_Time_for_That. That's notable. Unlike a $600M+ company of course.
should we delete this extremely likely useful/correct content or not according to this extremely complex system of guidelines"
In December 2023 the page was re-created, and seemed to stick: en.wikipedia.org/wiki/Talk:PsiQuantum#Secondary_sources It's just a random going back and forth. Author Ctjk has an interesting background:I am a legal official at a major government antitrust agency. The only plausible connection is we regulate tech firms
For these reasons reason why Ciro basically only contributes images to Wikipedia: because they are either all in or all out, and you can determine which one of them it is. And this allows images to be more attributable, so people can actually see that it was Ciro that created a given amazing image, thus overcoming Wikipedia's lack of reputation system a little bit as well.
Wikipedia is perfect for things like biographies, geography, or history, which have a much more defined and subjective expository order. But when it comes to "tutorials of how to actually do stuff", which is what mathematics and physics are basically about, Wikipedia has a very hard time to go beyond dry definitions which are only useful for people who already half know the stuff. But to learn from zero, newbies need tutorials with intuition and examples.
Bibliography:
- gwern.net/inclusionism from gwern.net:
Iron Law of Bureaucracy: the downwards deletionism spiral discourages contribution and is how Wikipedia will die.
- Quote "Golden wiki vs Deletionism on Wikipedia"
Quantum logic gates are needed for physical implementation by 
Ciro Santilli 35 Updated 2025-01-29 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-29 +Created 1970-01-01One direct practical reason is that we need to map the matrix to real quantum hardware somehow, and all quantum hardware designs so far and likely in the future are gate-based: you manipulate a small number of qubits at a time (2) and add more and more of such operations.
While there are "quantum compilers" to increase the portability of quantum programs, it is to be expected that programs manually crafted for a specific hardware will be more efficient just like in classic computers.
TODO: is there any clear reason why computers can't beat humans in approximating any unitary matrix with a gate set?
This is analogous to what classic circuit programmers will do, by using smaller logic gates to create complex circuits, rather than directly creating one huge truth table.
The most commonly considered quantum gates take 1, 2, or 3 qubits as input.
The gates themselves are just unitary matrices that operate on the input qubits and produce the same number of output qubits.
For example, the matrix for the CNOT gate, which takes 2 qubits as input is:
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0The final question is then: if I have a 2 qubit gate but an input with more qubits, say 3 qubits, then what does the 2 qubit gate (4x4 matrix) do for the final big 3 qubit matrix (8x8)? In order words, how do we scale quantum gates up to match the total number of qubits?
The intuitive answer is simple: we "just" extend the small matrix with a larger identity matrix so that the sum of the probabilities third bit is unaffected.
More precisely, we likely have to extend the matrix in a way such that the partial measurement of the original small gate qubits leaves all other qubits unaffected.
For example, if the circuit were made up of a CNOT gate operating on the first and second qubits as in:
0 ----+----- 0
|
1 ---CNOT--- 1
2 ---------- 2then we would just extend the 2x2 CNOT gate to:
TODO lazy to properly learn right now. Apparently you have to use the Kronecker product by the identity matrix. Also, zX-calculus appears to provide a powerful alternative method in some/all cases.
Ultimate explanation: math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups/3732426#3732426
Does not have to be isomorphic to a subgroup:
This is one of the reasons why the analogy between simple groups of finite groups and prime numbers is limited.
Compile MicroPython code for Micro Bit locally on Ubuntu 22.04 with your own firmware by Ciro Santilli 35 Updated 2025-01-29 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-29 +Created 1970-01-01TODO didn't manage from source Ubuntu 22.04, their setup bitrotted way too fast... it's shameful even. Until I gave up and went for the magic Docker of + github.com/bbcmicrobit/micropython, and it bloody worked:
git clone https://github.com/bbcmicrobit/micropython
cd micropython
git checkout 7fc33d13b31a915cbe90dc5d515c6337b5fa1660
docker pull ghcr.io/carlosperate/microbit-toolchain:latest
docker run -v $(pwd):/home --rm ghcr.io/carlosperate/microbit-toolchain:latest yt target bbc-microbit-classic-gcc-nosd@https://github.com/lancaster-university/yotta-target-bbc-microbit-classic-gcc-nosd
docker run -v $(pwd):/home --rm ghcr.io/carlosperate/microbit-toolchain:latest make all
# Build one.
tools/makecombinedhex.py build/firmware.hex examples/counter.py -o build/counter.hex
cp build/counter.hex "/media/$USER/MICROBIT/"
# Build all.
for f in examples/*; do b="$(basename "$f")"; echo $b; tools/makecombinedhex.py build/firmware.hex "$f" -o "build/${b%.py}.hex"; doneThe pre-Docker attempts:
sudo add-apt-repository -y ppa:team-gcc-arm-embedded
sudo apt update
sudo apt install gcc-arm-embedded
sudo apt install cmake ninja-build srecord libssl-dev
# Rust required for some Yotta component, OMG.
sudo snap install rustup
rustup default 1.64.0
python3 -m pip install yottaThe line:warns:and then the update/
sudo add-apt-repository -y ppa:team-gcc-arm-embeddedE: The repository 'https://ppa.launchpadcontent.net/team-gcc-arm-embedded/ppa/ubuntu jammy Release' does not have a Release file.
N: Updating from such a repository can't be done securely, and is therefore disabled by default.
N: See apt-secure(8) manpage for repository creation and user configuration details.sudo apt-get install gcc-arm-embedded fails, bibliography:Attempting to install Yotta:or:was failing with:Running:did not help. Bibliography:
sudo -H pip3 install yottapython3 -m pip install --user yottaException: Version mismatch: this is the 'cffi' package version 1.15.1, located in '/tmp/pip-build-env-dinhie_9/overlay/local/lib/python3.10/dist-packages/cffi/api.py'. When we import the top-level '_cffi_backend' extension module, we get version 1.15.0, located in '/usr/lib/python3/dist-packages/_cffi_backend.cpython-310-x86_64-linux-gnu.so'. The two versions should be equal; check your installation.python3 -m pip install --user cffi==1.15.1From a clean virtualenv, it appears to move further, and then fails at:So we install Rust and try again, OMG:which at the time of writing was
Building wheel for cmsis-pack-manager (pyproject.toml) ... error
error: [Errno 2] No such file or directory: 'cargo'sudo snap install rustup
rustup default stablerustc 1.64.0, and then OMG, it worked!! We have the yt command.However, it is still broken, e.g.:blows up:bibliography:
git clone https://github.com/lancaster-university/microbit-samples
cd microbit-samples
git checkout 285f9acfb54fce2381339164b6fe5c1a7ebd39d5
cp source/examples/invaders/* source
yt clean
yt buildannot import name 'soft_unicode' from 'markupsafe' Pinned article: ourbigbook/introduction-to-the-ourbigbook-project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivative - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. - Internal cross file references done right:
- Infinitely deep tables of contents:
Figure 6. Dynamic article tree with infinitely deep table of contents.Live URL: ourbigbook.com/cirosantilli/chordateDescendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact