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.
I caught and overcame a minor addiction to Cataclysm: Dark Days Ahead.
It does bring back the The Sims feeling from my teenage years, but with killer zombies added in.
I especially like going to sleep in that game, and how you need to setup a confy bed for it.
Some further comments at: Section "Cataclysm: Dark Days Ahead".
The way to quit is simple: delete your saves, then get annoyed with slowness of progressing back up, then use built-in debug/cheat menu to overcome that, then it's not fun anymore. This is a major advantage of single player open source games: addiction resistance!
Generalization of Laplace's equation where the value is not necessarily 0.
- twitter.com/saylor 2.6 M followers as of 2022. Because of Bitcoin? Or was he famous before that?
- www.michael.com/ yes, he is Michael.com lol. When did he get the domain, and at what price? He's mildlly obsessed with domain names it seems: www.domainsherpa.com/saylor/. He also bought
frank.com,mike.com.emma.com. This was between 1994 and 2000. He mentions that his email issaylor@strategy.com.strategy.comredirects tomicrostrategy.comas of 2022, so seems still valid.We actually got a backlink from tvt.phishlabs.com to this page in August 2022, a phising security provider, presumably for showing his email here. This reminds Ciro of the All GitHub Commit Emails event!A quick google for that address also leads to www.washingtonpost.com/wp-srv/liveonline/00/business/saylor0621.htm, a year 2000 profile page by the Washington Post, followed by a Q&A. In the Q&A, Michael himself gives his email in relation to Saylor Academy:so it was already beyond leaked. Ciro did hit that email address seeking funding for OurBigBook.com, but no reply unfortunately. Maybe you gotta be part of the Rockefeller family to get a reply from these people? Well, at least this led to Ciro learning about the Rockefeller Foundation.Kensington, Maryland: What has happened to your vision of an internet university? I believe it is an important initiative and it doesn't require $100 million to get started. We at the Rockefeller Foundation would like to become involved. How can we do this?Michael Saylor: The internet university is alive and well. My goal is to initially roll out a cyber-libary of digital video that can be used by any for-profit or non-profit organization in order to accelerate the free education initiative. Eventually, people may just go direct to the web site.I think the entire thing will unfold over about 10 years.Send me an email if you would like to get involved. (saylor@strategy.com)
Describes perfect lossless waves on the surface of a string, or on a water surface.
As mentioned at: math.stackexchange.com/questions/579453/real-world-application-of-fourier-series/3729366#3729366 from solving partial differential equations with the Fourier series citing courses.maths.ox.ac.uk/node/view_material/1720, analogously to the heat equation, the wave linear equation can be be solved nicely with separation of variables.
This section talks about solvers/simulators dedicated solving the wave equation. Of course, any serious solver will likely be able to solve a wider range of PDE, so this section contains mostly fun toys. For more serious stuff see: Section "PDE solver".
JavaScript toy solvers:
- jtiscione.github.io/webassembly-wave/index.html circular domain, create waves with mouse click
- dionyziz.com/graphics/wave-experiment/ with useless 3D WebGL visualization :-), waves with mouse click. Solving itself done on CPU, not GPU.
Start with: Section "String polarization".
Then go to: Section "Polarization of light".
2-dimensional heat equation example with Dirichlet boundary condition:
Elliptic partial differential equation by 
Ciro Santilli 35 Updated 2025-01-27 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-27 +Created 1970-01-01 Hyperbolic partial differential equation by Ciro Santilli 35 Updated 2025-01-27 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-27 +Created 1970-01-01This idea comes up particularly in the phase space coordinate of Hamiltonian mechanics.
Vs: image: the codomain is the set that the function might reach.
The image is the exact set that it actually reaches.
E.g. the function:could have:
- codomain
- image
Note that the definition of the codomain is somewhat arbitrary, e.g. could as well technically have codomain:even though it will obviously never reach any value in .
The exact image is in general therefore harder to characterize.
Most commonly, boundary conditions such as the Dirichlet boundary condition are taken to be fixed values in time.
But it also makes sense to think about cases where those values vary in time.
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