Existence and uniqueness of solutions of partial differential equations by
Ciro Santilli 40 Updated 2025-07-16
Unlike for ordinary differential equations which have the Picard–Lindelöf theorem, the existence and uniqueness of solution is not well solved for PDEs.
For example, Navier-Stokes existence and smoothness was one of the Millennium Prize Problems.
Often just called collimated light due to the collimator being the main procedure to obtain it.
However, you move very far away from the source, e.g. the Sun, you also get essentially parallel light.
Why do the electron and the proton have the same charge except for the opposite signs? by
Ciro Santilli 40 Updated 2025-07-16
Given the view of the Standard Model where the electron and quarks are just completely separate matter fields, there is at first sight no clear theoretical requirement for that.
As mentioned e.g. at QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga by Silvan Schweber (1994) chapter 1.6 "Hole theory", Dirac initially wanted to think of the holes in his hole theory as the protons, as a way to not have to postulate a new particle, the positron, and as a way to "explain" the proton in similar terms. Others however soon proposed arguments why the positron would need to have the same mass, and this idea had to be discarded.
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:
Deriving The Dirac equation by Andrew Dotson (2019)
Source. These videos can give some geometric insight and do have their value.
But they are sometimes too slow, there are never any mention of experiments, just "the truth".
The inaugural that predicted the Josephson effect.
Published on Physics Letters, then a new journal, before they split into Physics Letters A and Physics Letters B. True Genius: The Life and Science of John Bardeen mentions that this choice was made rather than the more prestigious Physical Review Letters because they were not yet so confident about the results.
A tiny idealized magnet! It is a very good model if you have a small strong magnet interacting with objects that are far away, notably other magnetic dipoles or a constant magnetic field.
The cool thing about this model is that we have simple explicit formulas for the magnetic field it produces, and for how this little magnet is affected by a magnetic field or by another magnetic dipole.
This is the perfect model for electron spin, but it can also be representative of macroscopic systems in the right circumstances.
Different macroscopic magnets can be approximated by a magnetic dipole when shrunk seen from far away
. Source. How to use an Oxford Nanopore MinION to extract DNA from river water and determine which bacteria live in it Conclusions by
Ciro Santilli 40 Updated 2025-07-16
- against all odds, the experiment worked and we got DNA out of the water, despite a bunch of non-bio newbs actively messing with random parts of the experiment
- PuntSeq and Biomakespace people, and all those tho do scientific outreach, are awesome!
- biology is hard
- creating insanely media rich articles like this is also hard, but the following helped enormously:
- Wikimedia Commons to store large media files out of Git
- Asciidoctor extensions to easily include those media files. The lessons learnt in this article were then an important motivation for Ciro's OurBigBook Markup, to which this article was later migrated.
- Nomacs to give Google Photos photos meaningful names and to edit people's faces out of pictures ;-)
- some scientific Wikipedia pages may or may not have been edited with better pictures during the course of writing this article
2023 Silk Road's Second-in-Command Gets 20 Years in Prison www.wired.com/story/silk-road-variety-jones-sentencing/
2016 "Exclusive: Our Thai prison interview with the alleged top advisor to Silk Road" arstechnica.com/tech-policy/2016/09/exclusive-our-thai-prison-interview-with-an-alleged-top-advisor-to-silk-road/
2015 The Variety Show On the trail of the man believed to be Variety Jones, one of the architects of the defunct drug marketplace Silk Road. www.vice.com/en/article/wnx5qn/the-variety-show
www.justice.gov/usao-sdny/file/797251/download some kind of case file of his trial.
Bibliography:
The curious thing about VJ is that he actually has some culture and says cool things, e.g.:
Chopsticks is a hand game typically played by two or more players. It's a game that involves using fingers to represent numbers, and it can be played with both strategy and skill. The objective is to eliminate all of your opponents' "fingers" (or hands) by touching them and using simple rules of movement and counting. ### Basic Rules: 1. **Starting Position**: Each player starts with one finger extended on each hand (usually two hands).
"Kaye and Laby" refers to the "Kaye and Laby: Tables of Physical and Chemical Constants," which is a reference book widely used in the fields of physics and chemistry. The book contains a comprehensive collection of tables that provide various physical and chemical constants, properties of materials, and other essential data that researchers and scientists often require. Originally compiled by Sir J. H. Kaye and Sir D. W.
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







