Sets are unordered, but we can use them to create ordered objects, which are of fundamental importance. Notably, they are used in the definition of functions.
The model is extremely simple, but has been proven to be able to solve all the problems that any reasonable computer model can solve, thus its adoption as the "default model".
The smallest known Turing machine that cannot be proven to halt or not as of 2019 is 7,918-states: www.scottaaronson.com/blog/?p=2725. Shtetl-Optimized by Scott Aaronson is just the best website.
A bunch of non-reasonable-looking computers have also been proven to be Turing complete for fun, e.g. Magic: The Gathering.
3 is not enough by Legendre's three-square theorem.
The subsets reachable with 2 and 3 squares are fully characterized by Legendre's three-square theorem.
In x86 systems, there may actually be 2 address translation steps:like this:
- first segmentation
- then paging
(logical) ------------------> (linear) ------------> (physical)
segmentation pagingThe major difference between paging and segmentation is that:
- paging splits RAM into equal sized chunks called pages
- segmentation splits memory into chunks of arbitrary sizes
This is the main advantage of paging, since equal sized chunks make things more manageable by reducing memory fragmentation problems. See also:
Paging came after segmentation historically, and largely replaced it for the implementation of virtual memory in modern OSs.
Paging has become so much more popular that support for segmentation was dropped in x86-64 in 64-bit mode, the main mode of operation for new software, where it only exists in compatibility mode, which emulates IA-32.
The side effects of ambitious goals are often the most valuable thing achieved by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16A quote by Ciro's Teacher R.:
Sometimes, even if our end goals are too far from reality, the side effects of trying to reach them can have meaningful impact.
If the goals are not ambitious enough, you risk not even having useful side effects so show in the end!
By doing the prerequisites of the impossible goal you desire, maybe the next generation will be able to achieve it.
This is basically why Ciro Santilli has contributed to Stack Overflow, which has happened while was doing his overly ambitious projects and notice that all kinds of basic pre-requisites were not well explained anywhere.
This is especially effective when you use backward design, because then you will go "down the dependency graph of prerequisites" and smoothen out any particularly inefficient points that you come across.
There are of course countless examples of such events:
- youtu.be/qrDZhAxpKrQ?t=174 Blitzscaling 11: Patrick Collison on Hiring at Stripe and the Role of a Product-Focused CEO by Greylock (2015)
The danger of this approach is of course spending too much time on stuff that will not be done enough times to be worth it, as highlighted by several xkcds:
xkcd 1319: Automation
. Source. Ciro Santilli has a bad memory for events that happened a medium time ago, for example in order of months/years. Especially if they are one-off things that have no relation to anything else.
For example, Ciro never remembers which places he travelled to just once, and who was in each trip! He has images of several places he travelled to in his head, and would recognize them, but he just doesn't know where they were!
Another example, Ciro was looking at the carpet at their house, and asked where it came from. His wife replied immeidately: from Bercy shopping quarter in Paris about 10 years ago, and you took it on your back for a long walk until we could find the bus back home because we were concerned it wouldn't fit in the train!
The same goes for scenes from movies and passages from music, which explains why Ciro's art consumption focuses on innovative discrete "what happened" and "general gist" ideas, rather than, analog details such as colors and shapes.
Going back even further in time, Ciro starts to forget the less close friends he had, because the events start to fade away.
Paradoxically however, Ciro believes that this bad memory is one of his greatest strengths and key defining characteristics, because it leads Ciro to want to write down every interesting thing he learns, which motivated OurBigBook.com and his Stack Overflow contributions and his related Ciro Santilli's documentation superpowers.
It also somewhat leads Ciro to like physics and mathematics, because in these fields you "can deduce everything" from very few base principles, so if you forget them, it does not matter that much as you can re-deduce stuff over and over. Which is somewhat where the high flying bird attitude comes from. It is hard to go deep when you have to re-prove everything every time. But the upside is that anything that sticks, does so because it has a broad net to stick to, and therefore allows Ciro to make unusual and unexpected connections that others might not.
Ciro believes that there are two types of people, and most notably software engineers, which are basically data wranglers: those with bad memory and those with good memory.
Those with bad memory, tend to focus on automating and improving their processes a lot. They take much longer to do one-off specific deep knowledge tasks however.
The downside of the good memory ones is that sooner or later they will find tasks that no matter how much memory they have, they cannot solve without automation, and they will fail at those.
This dichotomy also explains why Ciro sucks at code reviews, but is rather the person who runs the interesting patches by himself and finds some critical problems that the more theoretical code reviewers missed.
If Ciro had become a scientist, he would without doubt be an experimentalist, just like in this reality he is a GDB/runtime person rather than a "static source analysis" person. Those who have bad memory prefer to just run experiments over and over and observe system state at runtime.
Other effects of having a bad memory include:
- code duplication, or a constant fear of it at least, because Ciro forgets that some functionality exists already
- meeting aversion, because everything that is not recorded will fade away
- passion for backward design, because by the time a piece of knowledge learnt in school might be useful (and 99.99% won't), it will have been long forgotten
Related: jakobschwichtenberg.com/about/ from Jakob Schwichtenberg:
In some sense, one of the biggest benefits I have over other people in physics is that I'm certainly not the smartest guy! I usually can't grasp complex issues very easily. So I have to break down complex ideas into smaller chunks to understand it myself. This means, whenever I describe something to others, everyone understands, because it's broken down into such simple terms.
On C2 wiki, therefore it cannot be wrong wiki.c2.com/?QuasiGreatTeacher:
Only patches which were reviewed by at least one person with push permission will be listed here.
This may also include patches which were rejected in favor of another patch, but strongly influenced the merged patch.
For sizing see also: Ciro Santilli's body.
We select for the general Equation "Schrodinger equation":giving the full explicit partial differential equation:
- , the linear cartesian coordinate in the x direction
- , which analogous to the sum of kinetic and potential energy in classical mechanics
Equation 1.
Schrödinger equation for a one dimensional particle
. The corresponding time-independent Schrödinger equation for this equation is:
Equation 2.
time-independent Schrödinger equation for a one dimensional particle
. 
The color wheel in optics refers to a visual representation of colors arranged in a circular format, illustrating the relationship between different hues and their corresponding complementaries. Initially used in art and design, the color wheel has significant relevance in the field of optics as well.
In optics, a "domain wall" typically refers to a boundary between two different regions of a material where the optical properties change, often associated with phase transitions or other phenomena. This concept is similar to the idea of domain walls in magnetism, where they separate regions of differing magnetic orientations. In the context of nonlinear optics, domain walls can play a significant role in systems with spatial or temporal modulation of refractive index, leading to phenomena such as solitons or localized modes.
Live MOS (Metal-Oxide-Semiconductor) is a type of image sensor technology primarily used in digital cameras and camcorders. Developed by Panasonic, Live MOS sensors combine the advantages of both CCD (Charge-Coupled Device) and CMOS (Complementary Metal-Oxide-Semiconductor) technologies.
Pinned article: 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!
Intro to OurBigBook
. Source. 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/derivativeVideo 2. OurBigBook Web topics demo. Source. - 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 3. Visual Studio Code extension installation.
Figure 4. Visual Studio Code extension tree navigation.
Figure 5. Web editor. 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. 
- Infinitely deep tables of contents:
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