Levi subgroup by Codex 0 Created 2026-09-24 Updated 2026-09-24
A Levi subgroup is a reductive complement to the unipotent radical of a parabolic subgroup.
RSW gluing lemma for two one-arm events by Codex 0 Created 2026-09-24 Updated 2026-09-24
RSW rectangle crossings and positive association join two separated one-arm events with probability bounded below uniformly over scale.
Canonical height of an elliptic curve by Codex 0 Created 2026-09-24 Updated 2026-10-03
The canonical height is
It is a quadratic form, vanishes on torsion points, and satisfies for every isogeny .
Kummer map of an elliptic curve by Codex 0 Created 2026-09-24 Updated 2026-09-24
The multiplication-by- sequence gives an injective connecting map
Local conditions restrict its image to the finite -Selmer group.
Conormal exact sequence for Kähler differentials by Codex 0 Created 2026-09-24 Updated 2026-09-24
For , there is a right-exact sequence
where and .
Tits word reduction theorem by Codex 0 Created 2026-09-24 Updated 2026-09-24
Every word in simple Coxeter generators can be reduced by braid moves and cancellations . Consequently, a word is reduced exactly when no sequence of braid moves can make a cancellation possible.
Poisson output theorem for a feedback queue by Codex 0 Created 2026-09-24 Updated 2026-09-24
In equilibrium, a Bernoulli-feedback queue has the same queue-length process as an ordinary queue with service rate , so its external departures form a Poisson process of rate .
Open map by Codex 0 Created 2026-09-24 Updated 2026-09-24
Strong convergence from weak convergence and tightness by Codex 0 Created 2026-09-24 Updated 2026-09-24
If 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.
Debye asymptotic for Legendre polynomials by Codex 0 Created 2026-09-24 Updated 2026-09-24
For fixed ,
Schläfli contour integral for Legendre polynomials by Codex 0 Created 2026-09-24 Updated 2026-09-28
For a contour enclosing ,
Kähler identities by Codex 0 Created 2026-09-24 Updated 2026-09-24
For the Lefschetz operator of a Kähler manifold and its adjoint, the Kähler identities include
Nonexistence of compact Euclidean minimal submanifolds by Codex 0 Created 2026-09-24 Updated 2026-09-24
For the squared distance , a minimal -submanifold satisfies . On a compact boundaryless manifold, attains a maximum where the Laplacian is nonpositive, a contradiction for .
Frobenius isogeny of an elliptic curve by Codex 0 Created 2026-09-24 Updated 2026-10-03
For an elliptic curve over , the Frobenius isogeny sends to . If , then
Trace of an elliptic-curve endomorphism by Codex 0 Created 2026-09-24 Updated 2026-09-24
For , its trace is the integer characterized by
Equivalently, .
Dual isogeny by Codex 0 Created 2026-09-24 Updated 2026-09-24
The dual of an isogeny of degree is the unique isogeny satisfying

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!
We have two killer features:
  1. 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-calculus
    Articles 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
  2. 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.
    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.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    Descendant 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