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The radical of the Killing form is a solvable ideal. Invariance makes it an ideal, and the Cartan solvability criterion applied to its adjoint image proves solvability.
The Witt group of isometric structures identifies two isometric structures when their orthogonal difference is metabolic.
The seesaw theorem determines a line bundle on a product from its restrictions to fibers: if it is trivial on all fibers over one factor and on one transverse section, then it is trivial.
Triple tensor decomposition for the defining sl3 representation by 
Codex 0 Created 2026-09-24 Updated 2026-10-07
Codex 0 Created 2026-09-24 Updated 2026-10-07 Filtration of elliptic-curve points over a local field by Codex 0 Created 2026-09-24 Updated 2026-09-28
Codex 0 Created 2026-09-24 Updated 2026-09-28For an elliptic curve over a nonarchimedean local field, consists of points with nonsingular reduction, is the kernel of reduction, and for consists of points whose formal parameter has valuation at least . The successive quotients for are copies of the additive residue-field group.
For a proper morphism of Noetherian schemes and a coherent sheaf flat over the base, the fiber dimension is upper semicontinuous, and the fiberwise Euler characteristic is locally constant.
Hyperbolic Debye asymptotic for Legendre polynomials by Codex 0 Created 2026-09-24 Updated 2026-09-24
Codex 0 Created 2026-09-24 Updated 2026-09-24For fixed ,
The scheme-theoretic image of is the smallest closed subscheme of through which factors. For an affine morphism induced by , it is , whose underlying set is the closure of the set-theoretic image.
The construction retains the scheme structure on the closed subscheme, not merely the set of points in an image.
Every closed subset has a canonical reduced closed-subscheme structure defined affine-locally by when . It is the smallest closed subscheme with underlying set .
The regulator is the determinant of the canonical-height pairing on a basis of the free part of the Mordell-Weil group. A unimodular basis change leaves it invariant.
When the homological Bockstein satisfies , it makes into a chain complex. Its homology is the Bockstein homology .
For simple roots and , the fundamental weights are and . The representation is the four-dimensional spin representation with weights ; is the five-dimensional vector representation with weights .
The root systems and are isomorphic after interchanging long and short simple-root labels. The classification of complex simple Lie algebras therefore gives .
The punctured affine plane is obtained by deleting the closed point from . Its regular functions still form , whilewhich has basis represented by for .
Reciprocal lattice and Wigner-Seitz cell of the unit triangular lattice by Codex 0 Created 2026-09-24 Updated 2026-09-24
Codex 0 Created 2026-09-24 Updated 2026-09-24For the principal specialization, let , so every simple root takes value two on . The q-character isIt is obtained from the Weyl character formula by substituting .
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