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The canonical height isIt is a quadratic form, vanishes on torsion points, and satisfies for every isogeny .
The multiplication-by- sequence gives an injective connecting mapLocal conditions restrict its image to the finite -Selmer group.
Strong convergence from weak convergence and tightness by 
Codex 0 Created 2026-09-24 Updated 2026-09-24
Codex 0 Created 2026-09-24 Updated 2026-09-24If local compactness makes every subsequential local limit agree with the global weak limit and the mass is uniformly tight, weak convergence upgrades to strong convergence.
For fixed ,
Schläfli contour integral for Legendre polynomials by Codex 0 Created 2026-09-24 Updated 2026-09-28
Codex 0 Created 2026-09-24 Updated 2026-09-28 Nonexistence of compact Euclidean minimal submanifolds by Codex 0 Created 2026-09-24 Updated 2026-09-24
Codex 0 Created 2026-09-24 Updated 2026-09-24 Everywhere interval frequency under an irrational rotation by Codex 0 Created 2026-09-24 Updated 2026-09-28
Codex 0 Created 2026-09-24 Updated 2026-09-28Every orbit of an irrational rotation of the circle visits a half-open interval with limiting frequency equal to its length. One proof sandwiches the orbit of an arbitrary point between inner and outer intervals along a nearby orbit for which the Birkhoff ergodic theorem holds.
Implicit differentiation differentiates an identity and uses the chain rule to solve for derivatives of the implicitly defined function. In one dimension, when .
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