Home Updated 2024-12-15 +Created 1970-01-01
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Check out: OurBigBook.com, the best way to publish your scientific knowledge. It's an open source note taking system that can publish from lightweight markup files in your computer both to a multi-user mind melding dynamic website, or as a static website. It's like Wikipedia + GitHub + Stack Overflow + Obsidian mashed up. Source code: github.com/ourbigbook/ourbigbook.
Sponsor me to work on this project: 100k USD = I quit me job and work on it one year full time. Status: ~144k / 200k USD reached: 1st year locked-in, 2nd year stretch goal open at 200k USD. 1M USD = I retire and do it forever. How to donate: Section "Sponsor Ciro Santilli's work on OurBigBook.com".
I reached 100k USD after a 1000 Monero donation, so I quit my job for 1 year starting 1st June 2024 to solve as many STEM courses as I can from a world leading university to try and kickstart The Higher Education Revolution. If I reach 200k USD, then I'll do it for two years instead. A second year greatly improve chances of success: year one I solve a bunch of courses, year two I come guns blazing with the content and expand further.
Mission: to live in a world where you can learn university-level mathematics, physics, chemistry, biology and engineering from perfect free open source books that anyone can write to get famous. More rationale: Section "OurBigBook.com"
Explaining things is my superpower, e.g. I was top user #39 on Stack Overflow in 2023[ref][ref] and I have a few 1k+ star educational GitHub repositories[ref][ref][ref][ref]. Now I want to bring that level of awesomeness to masters level Mathematics and Physics. But I can't do it alone! So I created OurBigBook.com to allow everyone to work together towards the perfect book of everything.
My life's goal is to bring hardcore university-level STEM open educational content to all ages. Sponsor me at github.com/sponsors/cirosantilli starting from 1$/month so I can work full time on it. Further information: Section "Sponsor Ciro Santilli's work on OurBigBook.com". Achieving what I call "free gifted education" is my Nirvana.
This website is written in OurBigBook Markup, and it is published on both cirosantilli.com (static website) and outbigbook.om/cirosantilli (multi-user OurBigBook Web instance). Its source code is located at: github.com/cirosantilli/cirosantilli.github.io and also at cirosantilli.com/_dir and it is licensed under CC BY-SA 4.0 unless otherwise noted.
To contact Ciro, see: Section "How to contact Ciro Santilli". He likes to talk with random people of the Internet.
https://raw.githubusercontent.com/cirosantilli/media/master/ID_photo_of_Ciro_Santilli_taken_in_2013.jpg https://raw.githubusercontent.com/cirosantilli/media/master/Ciro_Santilli's_learn_teach_apply_logo.png
Besides that, I'm also a freedom of speech slacktivist and recreational cyclist. I like Chinese traditional music and classic Brazilian pop. Opinions are my own, but they could be yours too. Tax the rich.
I offer:
My approach is to:
For minors, parents are welcome to join video calls, and all interactions with the student will be recorded and made available to parents.
I have a proven track of explaining complex concepts in an interesting and useful way. I work for the learner. Teaching statement at: Section "How to teach". Pricing to be discussed. Contact details at: Section "How to contact Ciro Santilli".
I am particularly excited about pointing people to the potential next big things, my top picks these days are:
I am also generally interested in:
Figure 1.
Ciro Santilli's amazing Stack Overflow profile
. Ciro contributes almost exclusively by answering question he Googles into out of his own need, and never by refreshing the newest question of big tags for low hanging fruit! More information at: Section "Ciro Santilli's Stack Overflow contributions".
Video 1.
Introduction to the OurBigBook Project
. Source.
Video 2.
OurBigBook Web topics demo
. Source. The OurBigBook topic feature allows users to "merge their minds" in a "sort by upvote"-stack overflow-like manner for each subject. This is the killer feature of OurBigBook Web. More information at: docs.ourbigbook.com/ourbigbook-web-topics.
Video 3.
OurBigBook dynamic article tree demo
. Source. The OurBigBook dynamic tree feature allows any of your headers to be the toplevel h1 header of a page, while still displaying its descendants. SEO loves this, and it also allows users to always get their content on the correct granularity. More information at: docs.ourbigbook.com/ourbigbook-web-dynamic-article-tree.
Video 4.
OurBigBook local editing and publishing demo
. Source. With OurBigBook you can store your content as plaintext files in a Lightweight markup, and then publish that to either OurBigBook.com to get awesome multi-user features, or as a static website where you are in full control. More information at: docs.ourbigbook.com/publish-your-content.
Video 5.
Top Down 2D continuous game with Urho3D C++ SDL and Box2D for Reinforcement learning by Ciro Santilli (2018)
Source. More information: Section "Ciro's 2D reinforcement learning games". This is Ciro's underwhelming stab at the fundamental question: Can AGI be trained in simulations?. This project could be taken much further.
Video 6.
Water Margin tribute to Chinese dissidents by Ciro Santilli (2022)
Source. Part of Ciro Santilli's campaign for freedom of speech in China, see also: cirosantilli.com/china-dictatorship/water-margin.
Figure 2.
Lenovo ThinkPad T430 running a BIOS hello world
. Source code: github.com/cirosantilli/x86-bare-metal-examples. Ciro's Linux Kernel Module Cheat is a closely related and much more important project that covers the Linux kernel and assembly language.
 -------------------------------------
|  Force of Will               3 U U  |
|  ---------------------------------  |
| |                  ////////////   | |
| |                ////() ()\////\  | |
| |               ///_\ (--) \///\  | |
| |        )      ////  \_____///\\ | |
| |       ) \      /   /   /    /   | |
| |    ) /   \     |   |  /   _/    | |
| |   ) \  (  (   /   / /   / \     | |
| |  / ) ( )  / (    )/(    )  \    | |
| |  \(_)/(_)/  /UUUU \  \\\/   |   | |
| .---------------------------------. |
| Interrupt                           |
| ,---------------------------------, |
| | You may pay 1 life and remove a | |
| | blue card in your hand from the | |
| | game instead of paying Force of | |
| | Will's casting cost.  Effects   | |
| | that prevent or redirect damage | |
| | cannot be used to counter this  | |
| | loss of life.                   | |
| | Counter target spell.           | |
| `---------------------------------` |
|                                     l
| Illus.  Terese Nelsen               |
 -------------------------------------
Code 1.
ASCII art of a Force of Will Magic: The Gathering card inscribed in the Bitcoin blockchain
.
Artist unknown, uploaded December 2014. Part of Section "Cool data embedded in the Bitcoin blockchain" where Ciro Santilli maintains a curated list of such interesting inscriptions.
This was a small project done by Ciro for artistic purposes that received some attention due to the incredible hype surrounding cryptocurrencies at the time. Ciro Santilli's views on cryptocurrencies are summarized at: Section "Are cryptocurrencies useful?".
Figure 3.
YellowRobot.jpg
. Source.
JPG image fully embedded in the Bitcoin blockchain depicting some kind of cut material art depicting a yellow robot, inscribed on January 29, 2017.
Ciro Santilli found this image and others during his research for Section "Cool data embedded in the Bitcoin blockchain" by searching for image fingerprints on every transaction payload of the blockchain with a script.
The image was uploaded by EMBII, co-creator of the AtomSea & EMBII upload mechanism, which was responsible for a large part of the image inscriptions in the Bitcoin blockchain.
The associated message reads:
Chiharu [EMBII's Japanese wife] and I found this little yellow robot while exploring Chicago. It will be covered by tar or eventually removed but this tribute will remain. N 41.880778 E -87.629210
This is one of Ciro Santilli's favorite AtomSea & EMBII uploads, as it perfectly encapsules the "medium as an art form" approach to blockchain art, where even non-novel works can be recontextualized into something interesting, here depicting an opposition between the ephemeral and the immutable.
At twitter.com/EMBII4U/status/1615389973343268871 EMBII announced that he would be giving off shares of that image on Sup!?, a Bitcoin-backed NFT system he was; making. In December 2023, he gave some shares of the robot to Ciro Santilli.
Figure 4.
2010 Wayback Machine archive of starwarsweb.net
.
This website was used as one of the CIA 2010 covert communication websites, a covert system the CIA used to communicate with its assets. More details at: Section "CIA 2010 covert communication websites".
Ciro Santilli had some naughty OSINT fun finding some of the websites of this defunct network in 2023 after he heard about the 2022 Reuters report on the matter, which for the first time gave away 7 concrete websites out of a claimed 885 total found. As of November 2023, Ciro had found about 350 of them.
Figure 5.
2010 Wayback Machine archive of noticiasmusica.net
.
This is another website that was used as one of the CIA 2010 covert communication websites. This website is written in Brazilian Portuguese, and therefore suggests that the CIA had assets in Brazil at the time, and thus was spying on a "fellow democracy".
Although Snowden's revelations made it extremely obvious to the world that the USA spies upon everyone outside of the Five Eyes, including fellow democracies, it is rare to have such a direct a concrete proof of it visible live right on the Wayback Machine. Other targeted democracies include France, Germany, Italy and Spain. More details at: USA spying on its own allies.
Video 7.
How I found a Star Wars website made by the CIA by Ciro Santilli
. Source. Slightly edited VOD of the talk Aratu Week 2024 Talk by Ciro Santilli: My Best Random Projects about the CIA 2010 covert communication websites.
Figure 6.
Diagram of the fundamental theorem on homomorphisms by Ciro Santilli (2020)
Shows the relationship between group homomorphisms and normal subgroups.
Figure 7.
Spacetime diagram illustrating how faster-than-light travel implies time travel by Ciro Santilli (2021)
Used in the Stack Exchange ansewr to Does faster than light travel imply travelling back in time?. One of Section "The best articles by Ciro Santilli".
Figure 8.
Average insertion time into heaps, binary search tree and hash maps of the C++ standard library by Ciro Santilli (2015)
Source. Used in the Stack Overflow answer to Heap vs Binary Search Tree (BST). One of Section "The best articles by Ciro Santilli".
Video 8.
Birch and Swinnerton-Dyer conjecture in two minutes by Ciro Santilli
. Source. Quick and direct explanation of the statement of the BSD conjecture for people who know basic university mathematics. This is one of the Millennium Prize Problems, and you will get a million dollars if you can solve it! This therefore falls in the Simple to state but hard to prove of Ciro Santilli's the beauty of mathematics aesthetics.
Figure 9.
Top view of an open Oxford Nanopore MinION
. Source. This is Ciro Santilli's hand on the Wikipedia article: en.wikipedia.org/wiki/Oxford_Nanopore_Technologies. He put it there after working a bit on Section "How to use an Oxford Nanopore MinION to extract DNA from river water and determine which bacteria live in it" :-) And he would love to document more experiments like that one Section "Videos of all key physics experiments", but opportunities are extremely rare.
A quick 2D continuous AI game prototype for reinforcement learning written in Matter.js, you can view it on a separate page at cirosantilli.com/_raw/js/matterjs/examples.html#top-down-asdw-fixed-viewport. This is a for-fun-only prototype for Ciro's 2D reinforcement learning games, C++ or maybe Python (for the deep learning ecosystem) seems inevitable for a serious version of such a project. But it is cute how much you can do with a few lines of Matter.js!
HTML snippet:
<iframe src="_raw/js/matterjs/examples.html#top-down-asdw-fixed-viewport" width="1000" height="850"></iframe>
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The best articles by Ciro Santilli Updated 2024-12-15 +Created 1970-01-01
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These are the best articles ever authored by Ciro Santilli, most of them in the format of Stack Overflow answers.
Ciro posts update about new articles on his Twitter accounts.
A chronological list of all articles is also kept at: Section "Updates".
Some random generally less technical in-tree essays will be present at: Section "Essays by Ciro Santilli".
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Fundamental theorem on homomorphisms Updated 2024-12-15 +Created 1970-01-01
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Figure 1. Source.
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Lie algebra Updated 2024-12-15 +Created 1970-01-01
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Like everything else in Lie groups, first start with the matrix as discussed at Section "Lie algebra of a matrix Lie group".
Intuitively, a Lie algebra is a simpler object than a Lie group. Without any extra structure, groups can be very complicated non-linear objects. But a Lie algebra is just an algebra over a field, and one with a restricted bilinear map called the Lie bracket, that has to also be alternating and satisfy the Jacobi identity.
Another important way to think about Lie algebras, is as infinitesimal generators.
Because of the Lie group-Lie algebra correspondence, we know that there is almost a bijection between each Lie group and the corresponding Lie algebra. So it makes sense to try and study the algebra instead of the group itself whenever possible, to try and get insight and proofs in that simpler framework. This is the key reason why people study Lie algebras. One is philosophically reminded of how normal subgroups are a simpler representation of group homomorphisms.
To make things even simpler, because all vector spaces of the same dimension on a given field are isomorphic, the only things we need to specify a Lie group through a Lie algebra are:
Note that the Lie bracket can look different under different basis of the Lie algebra however. This is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) page 71 for the Lorentz group.
As mentioned at Lie Groups, Physics, and Geometry by Robert Gilmore (2008) Chapter 4 "Lie Algebras", taking the Lie algebra around the identity is mostly a convention, we could treat any other point, and things are more or less equivalent.
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Normal subgroup Updated 2024-12-15 +Created 1970-01-01
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Only normal subgroups can be used to form quotient groups: their key definition is that they plus their cosets form a group.
One key intuition is that "a normal subgroup is the kernel" of a group homomorphism, and the normal subgroup plus cosets are isomorphic to the image of the isomorphism, which is what the fundamental theorem on homomorphisms says.
Therefore "there aren't that many group homomorphism", and a normal subgroup it is a concrete and natural way to uniquely represent that homomorphism.
The best way to think about the, is to always think first: what is the homomorphism? And then work out everything else from there.
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Semidirect product Updated 2024-12-15 +Created 1970-01-01
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As per en.wikipedia.org/w/index.php?title=Semidirect_product&oldid=1040813965#Properties, unlike the Direct product, the semidirect product of two goups is neither unique, nor does it always exist, and there is no known algorithmic way way to tell if one exists or not.
This is because reaching the "output" of the semidirect produt of two groups requires extra non-obvious information that might not exist. This is because the semi-direct product is based on the product of group subsets. So you start with two small and completely independent groups, and it is not obvious how to join them up, i.e. how to define the group operation of the product group that is compatible with that of the two smaller input groups. Contrast this with the Direct product, where the composition is simple: just use the group operation of each group on either side.
Product of group subsets
So in other words, it is not a function like the Direct product. The semidiret product is therefore more like a property of three groups.
The semidirect product is more general than the direct product of groups when thinking about the group extension problem, because with the direct product of groups, both subgroups of the larger group are necessarily also normal (trivial projection group homomorphism on either side), while for the semidirect product, only one of them does.
Conversely, en.wikipedia.org/w/index.php?title=Semidirect_product&oldid=1040813965 explains that if , and besides the implied requirement that N is normal, H is also normal, then .
Smallest example: where is a dihedral group and are cyclic groups. (the rotation) is a normal subgroup of , but (the flip) is not.
Note that with the Direct product instead we get and not , i.e. as per the direct product of two cyclic groups of coprime order is another cyclic group.
TODO:
  • why does one of the groups have to be normal in the definition?
  • what is the smallest example of a non-simple group that is neither a direct nor a semi-direct product of any two other groups?
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