Topics (203k) Articles (205k) Users (305) Discussions (237) Comments (383) Files (715) New article
Solving partial differential equations with the Fourier series by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16See: math.stackexchange.com/questions/579453/real-world-application-of-fourier-series/3729366#3729366 from heat equation solution with Fourier series.
Separation of variables of certain equations like the heat equation and wave equation are solved immediately by calculating the Fourier series of initial conditions!
This technique is crazy! It allows to both:You actually see discrete peaks at different minute counts on the other end.
- separate gaseous mixtures
- identify gaseous compounds
It is based on how much the gas interacts with the column.
Detection is usually done burning the sample to ionize it when it comes out, and then you measure the current produced.
Gas chromatography by Quick Biochemistry Basics (2019)
Source. Visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16Good ultra quick visual non-mathematical introduction to the Riemann hypothesis and analytic continuation.
Essentially, defining an holomorphic function on any open subset, no matter how small, also uniquely defines it everywhere.
This is basically why it makes sense to talk about analytic continuation at all.
One way to think about this is because the Taylor series matches the exact value of an holomorphic function no matter how large the difference from the starting point.
Therefore a holomorphic function basically only contains as much information as a countable sequence of numbers.
These people are serious.
Looking at most astronomical object through a telescope is boring because you only see a white ball or point every time. Such targets would likely only be interesting with spectroscopy analysis.
There are however some objects that you can see the structure of even with an amateur telescope, and that makes them very exciting.
Some good ones:
- The Moon, notably crater detail.
- Saturn. Clearly visible to the naked eye, but looks like a ball. But under an amateur telescope, you can clearly see that there is a disk. Clearly discerning that the disk is a ring, i.e. seeing the gap, is a bit harder though.
- Jupiter. Clearly visible to the naked eye, it is quite huge. The four Galilean moons, being Earth-sized, are incredibly clearly visible, tested on Celestron NexStar 4SE 25mm/9mm eyepiece. Colored gas clouds are hard though, you will likely just see it bright white. www.reddit.com/r/telescopes/comments/35xrbb/how_can_i_see_the_color_of_jupiter_with_my/
- a double star. As mentioned at www.relativelyinteresting.com/10-astronomical-targets-new-telescope/ Albireo are incredibly separated. Also it is is easy to find manually being in a major well known constellation. It is no wonder it is not quite even known if they are gravitationally bound or not!
- Andromeda Galaxy. This is when things start getting hard. You can see a faint cloud, but it is not super clear that it has a center.One important understanding is that it is not possible to see stars outside of the Milky Way by naked eye.It is at this point that you start to learn that pictures of faint objects require longer term exposure and averaging of the images taken. For this you need:Just looking through the scope to immediately see something is not enough.
- a digital camera attached to the scope
- a computerized scope that slowly moves to track the point of interest
- image processing software that does the averaging
Video "Andromeda Galaxy with only a Camera, Lens, & Tripod by Nebula Photos (2020)" gives a good notion of expectation adjustment.
Sometimes mathematicians go a little overboard with their naming.
Most of the videos are crap, but the following ones almost killed Ciro Santilli of laughter:
- Video "Bill Gates vs Steve Jobs by Epic Rap Battles of History (2012)"
- Video 1. "Rasputin vs Stalin by Epic Rap Battles of History (2013)"
Rasputin vs Stalin by Epic Rap Battles of History (2013)
Source. See also: cirosantilli.com/china-dictatorship/stalin.Monty Python has a few precursors to the "random famous people mixed together compting" format, although not in the rap fight format:
Communist Quiz by Monty Python (1970)
Source. Has a Wiki page: en.wikipedia.org/wiki/World_Forum/Communist_Quiz. Brings to mind: Video "Football, Football, Football by Mitchell and Webb". 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 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 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. 
- 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